From e529c449b560893f75e3020dd6928e3f30dac2e1 Mon Sep 17 00:00:00 2001 From: nbright Date: Mon, 24 Aug 2026 12:53:50 +0900 Subject: [PATCH] Add the paper study material and set the bare-earth goal Six papers converted to Markdown with the local doc2md tool, figures extracted and annotated. The tool's venv had a CPU-only torch, so marker-pdf silently ran on CPU and stalled; swapping in 2.5.1+cu121 dropped a paper from "hung after six minutes" to three. Gemini then described all 86 figures in place, below each original caption. The PDFs themselves are gitignored - 88 MB of public arXiv downloads that convert_papers.sh regenerates. The .md and figures are tracked, because the annotations took a separate pass and do not reproduce byte-for-byte. sum-parts-explained.html gains two tabs: - PointVector. Why representing a scalar feature as a rotated 3D vector buys anisotropic aggregation without attention's cost, and why the paper predicts two independent angles rather than a rotation matrix whose nine elements are interdependent. - Bare Earth. Reframes the task as ground vs not-ground, and separates the five boundaries by their nature. Four of them are cuts; the slope boundary is the one that must NOT be cut, which is why "horizontal means ground" destroys road cut and fill. Notes that SUM Parts is flat Helsinki and cannot teach slopes at all, so that part needs a geometric filter rather than more training. NEXT.md carries the goal forward: separate bare earth from the rest as OBJ meshes, then reclassify the remainder. Removing the ground first is sound - it is 24-40% of the points, and without it the remaining objects fall apart into separate connected components instead of being joined through the floor. The gap that blocks step 4 is named: mesh_to_ply.py samples points without recording which face each came from, so there is no way back to the mesh yet. Co-Authored-By: Claude Opus 5 (1M context) --- .gitignore | 7 + NEXT.md | 159 +++ README.md | 2 + docs/papers/ANNOTATE_PROMPT.md | 177 ++++ docs/papers/md/PointNeXt_2206.04670.md | 515 ++++++++++ .../_page_15_Figure_4.jpeg | Bin 0 -> 30649 bytes .../_page_16_Figure_0.jpeg | Bin 0 -> 252804 bytes .../_page_17_Figure_0.jpeg | Bin 0 -> 258803 bytes .../_page_1_Figure_0.jpeg | Bin 0 -> 31384 bytes .../_page_3_Figure_0.jpeg | Bin 0 -> 71863 bytes docs/papers/md/PointNet++_1706.02413.md | 464 +++++++++ .../_page_11_Figure_2.jpeg | Bin 0 -> 31621 bytes .../_page_1_Picture_2.jpeg | Bin 0 -> 13620 bytes .../_page_2_Figure_0.jpeg | Bin 0 -> 66740 bytes .../_page_3_Figure_6.jpeg | Bin 0 -> 13667 bytes .../_page_5_Figure_12.jpeg | Bin 0 -> 14071 bytes .../_page_5_Figure_4.jpeg | Bin 0 -> 25433 bytes .../_page_5_Figure_5.jpeg | Bin 0 -> 20129 bytes .../_page_6_Figure_3.jpeg | Bin 0 -> 44523 bytes .../_page_6_Figure_9.jpeg | Bin 0 -> 10480 bytes .../_page_7_Figure_4.jpeg | Bin 0 -> 35623 bytes 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and every one is a public arXiv download. +# scripts/convert_papers.sh regenerates the markdown from them. +# The converted .md and extracted figures ARE tracked: they carry the Vision +# annotations, which took a GPU pass to produce and are not reproducible +# byte-for-byte. +docs/papers/*.pdf + # Source-tree backups left by the patch scripts *.orig *.bak diff --git a/NEXT.md b/NEXT.md new file mode 100644 index 0000000..d238b0a --- /dev/null +++ b/NEXT.md @@ -0,0 +1,159 @@ +# 다음 세션 시작 프롬프트 + +목표가 정해졌다. 이 문서를 읽고 아래 프롬프트로 이어간다. + +--- + +## 단기 목표 + +> **bare earth 모델과 나머지를 분리한다. OBJ(mesh) 형태로 분리한 뒤, +> 그 나머지 모델을 다시 분류한다. 지면이 없어지면 분류가 더 쉬울 것이다.** + +이 순서는 타당하다. 근거: + +- 지면이 전체 포인트의 24~40%를 차지한다 — 제거하면 남은 문제가 그만큼 작아진다 +- 지면을 걷어내면 건물·수목·차량이 **공간적으로 분리된 덩어리**가 된다. + 지면이 있으면 전부 하나로 이어져 있어 연결 성분 분석이 안 된다 +- 2단계 분류는 실패가 국소화된다. 1단계에서 지면을 놓쳐도 2단계가 무너지지 않는다 + +--- + +## 프롬프트 + +```` +docs/sum-parts-explained.html 의 "Bare Earth" 탭과 STATUS.md 를 먼저 읽어라. +그 다음 이어서 작업한다. + +## 목표 + +bare earth(지면)와 나머지를 분리해서 각각 OBJ 메시로 내보낸다. +그 다음 나머지를 건물/수목/차량으로 다시 분류한다. + +## 현재 가진 것 + +- PointVector 학습 완료 (3060, voxel_max 24000, 100 epoch) + SUM val 기준 지면 precision 95.24% / recall 66.10% +- 서산 명천 BlockYBA 타일 1장 변환·추론 완료 +- 클래스별 분리 산출물 D:\AI_Test\sum-part\{ground,building,tree,vehicle,unseen}\ + 단 포인트 클라우드(.ply)이고 메시가 아니다 + +## 해야 할 일 — 순서대로 + +### 1. 이진 학습이 실제로 이득인지 확인 + +가설: 학습부터 2클래스(지면/비지면)로 하면 경계 정확도가 오른다. +근거: 13클래스 51.83% → 4클래스 통합 72.14%. 합칠수록 올랐다. + +구현: 데이터로더에서 라벨 remap. + terrain(1) → 1, 나머지 → 2, unclassified(0) → 0(ignore) + num_classes: 3, 나머지 cfg 그대로 + +측정: 지면 precision / recall을 현재 95.24 / 66.10과 비교. +약 4시간. 3060으로 가능하다. + +### 2. CSF를 서산 타일에 단독 적용 + +Cloth Simulation Filter. 학습 불필요, 파라미터 몇 개. +비탈면(절토·성토)에 강한 것이 알려져 있고, SUM(헬싱키 평지)으로는 +비탈면을 배울 수 없으므로 이 부분은 기하 기법이 필요하다. + +파이썬 구현: CSF (pip install cloth-simulation-filter) 또는 PDAL filters.csf + +측정: 신경망 결과와 어디가 다른지. 특히 경사면에서. + +### 3. 조합 + +신경망(의미) → 나무·차량·건물 제거 → CSF(기하) → 비탈면 보존 +둘의 합의/불일치 지점을 본다. + +### 4. 포인트 → 메시 복원 ← 여기가 미구현이고 핵심 + +현재 변환기(scripts/mesh_to_ply.py)는 면적가중 샘플링만 하고 +**각 포인트가 어느 face에서 왔는지 기록하지 않는다.** +포인트 예측을 원본 메시로 되돌리려면 이 역매핑이 필요하다. + +해야 할 것: + a. mesh_to_ply.py에 face index 보존 추가 (PLY에 face_idx 속성) + b. 포인트 예측 → face 다수결 → face별 클래스 + c. 클래스별로 face를 분리해 OBJ 저장 + +원칙: 원본 vertex는 불가침. 분리는 재생성이 아니라 face 분할이다. +구멍 메우기는 새 vertex 덧대기만 허용된다. + +검증: 분할된 face 수의 합 == 원본 face 수. 누락도 중복도 없을 것. + +### 5. 나머지 재분류 + +지면 제거 후 남은 메시에서 건물/수목/차량 분리. +지면이 없으면 연결 성분(connected component)으로 물체가 자연히 나뉜다. +이 단계는 4번이 끝나야 의미가 있다. + +## 주의사항 + +- WSL2에서 VRAM 초과는 OOM을 안 낸다. 호스트 RAM으로 흘려서 + 25~100배 느리게 조용히 완주한다. peak VRAM과 전력(W)으로 판정해라. +- 파일을 만들었으면 scripts/verify_outputs.sh 로 읽히는지 확인해라. + "만들었다"고 보고하기 전에 파서로 검증한다. +- 포인트 클라우드는 PLY로 내라. 면 없는 OBJ는 뷰어가 아무것도 안 보여준다. +- 각 단계 게이트를 통과 못 하면 다음으로 넘어가지 마라. + +## 판정 기준 + +bare earth의 성패는 precision이다. 지면이라 부른 것에 구조물이 섞이면 +지형이 왜곡된다. 반대로 지면을 놓쳐 생긴 구멍은 보간으로 메운다. + +현재 기준선: precision 95.24% / recall 66.10% +```` + +--- + +## 참고 — 이 세션에서 확인한 것 + +### PointVector 핵심 + +특징값(스칼라)을 3D 벡터로 확장해서 **방향**을 얻는다. +회전각 α, β 두 개만 MLP로 예측한다 — 회전행렬을 직접 예측하면 +원소끼리 종속이라 최적화가 어렵기 때문이다. + +attention이나 dynamic conv 없이 이방성(anisotropy)을 얻으므로 +**PointNeXt 파라미터의 58%로 더 높은 정확도**가 나온다. + +### 왜 우리 과제에 맞나 + +지면과 벽의 구분은 본질적으로 방향 문제다. +지면은 이웃이 수평으로, 벽은 수직으로 퍼져 있다. +등방적 집계는 이 차이를 뭉갠다. + +측정치가 이를 뒷받침한다 — 비지면이 지면으로 샌 26,217 포인트 중 +**49.90%가 facade_surface**였다. 벽 하단, 지면과 만나는 경계다. + +### 경계 5종의 성질이 다르다 + +| 경계 | 성질 | 난이도 | +|---|---|---| +| 지면–벽 | 법선 급변 | 쉬움 | +| 지면–수목 | 수직 이격 | 중간 | +| 지면–자동차 | 얹힌 물체 | 중간 | +| 지면–지장물 | 얹힌 물체 | 중간 | +| **지면–비탈면** | **둘 다 지면** | **가장 어려움** | + +비탈면만 성질이 반대다 — 나머지는 잘라내야 하고 비탈면은 남겨야 한다. +"수평이면 지면"으로 가면 도로 절·성토를 통째로 날린다. + +### SUM Parts의 한계 + +헬싱키 평지 도시라 **비탈면을 배울 수 없다.** +재학습으로도 안 고쳐진다. 데이터에 없는 개념이다. +→ 비탈면은 CSF 같은 기하 기법으로 보완한다. + +--- + +## 문서 + +| 문서 | 내용 | +|---|---| +| [docs/sum-parts-explained.html](docs/sum-parts-explained.html) | 학습 정리 — 용어, 모델 계보, PointVector 원리, bare earth 전략 | +| [STATUS.md](STATUS.md) | 작업 현황, 실측 수치, 미해결 문제 | +| [docs/pipeline.html](docs/pipeline.html) | 6단계 공정 정의 | +| [docs/papers/md/](docs/papers/md/) | 논문 6편 마크다운 + 그림 86개 (Vision 주석 완료) | +| [SETUP.md](SETUP.md) · [TRAIN.md](TRAIN.md) | 새 머신 구축 | diff --git a/README.md b/README.md index a04de05..3663695 100644 --- a/README.md +++ b/README.md @@ -9,9 +9,11 @@ | 경로 | 내용 | |---|---| +| [NEXT.md](NEXT.md) | **다음 세션 시작점 — 목표와 프롬프트** | | [STATUS.md](STATUS.md) | **현재 상태·결과·미해결 문제 — 이어받을 때 여기부터** | | [SETUP.md](SETUP.md) | 1단계 — 환경 구축 (GPU 불필요) | | [TRAIN.md](TRAIN.md) | 2단계 — 학습 (GPU 필요) | +| [docs/sum-parts-explained.html](docs/sum-parts-explained.html) | 학습 정리 — 용어·모델 계보·PointVector 원리·bare earth 전략 | | [docs/pipeline.html](docs/pipeline.html) | 전체 6단계 공정 정의 (브라우저로 열 것) | | [docs/SUM-Parts-검토노트.md](docs/SUM-Parts-검토노트.md) | 트러블슈팅 16건, 데이터 스키마 실측, 라이선스 | | [scripts/](scripts/) | 환경 구축 · 학습 · 평가 · 변환 스크립트 | diff --git a/docs/papers/ANNOTATE_PROMPT.md b/docs/papers/ANNOTATE_PROMPT.md new file mode 100644 index 0000000..0954dc8 --- /dev/null +++ b/docs/papers/ANNOTATE_PROMPT.md @@ -0,0 +1,177 @@ +# Gemini 이미지 주석 작업 프롬프트 + +`docs/papers/md/`의 논문 마크다운에 들어 있는 그림 86개를 +Vision으로 분석해 각 그림 아래에 한국어 설명을 붙이는 작업. + +아래 블록을 Antigravity에 그대로 넣으면 된다. + +--- + +## 프롬프트 + +```` +# 작업: 논문 마크다운의 그림에 Vision 분석 주석 달기 + +## 대상 + +D:\MYCLAUDE_PROJECT\sum-parts-test\docs\papers\md\ 아래 .md 파일 6개. +각 파일 옆에 같은 이름의 _images\ 폴더가 있고, 거기에 그림 파일이 들어 있다. + +| 파일 | 그림 수 | +|---|---| +| PointNet_1612.00593.md | 24 | +| PointNet++_1706.02413.md | 10 | +| PointNeXt_2206.04670.md | 5 | +| PointVector_2205.10528.md | 8 | +| SUM_2021_dataset.md | 14 | +| SUM-Parts_2503.15300.md | 25 | +| **합계** | **86** | + +## 마크다운의 현재 구조 + +그림은 항상 이 형태로 들어 있다. 앞에 `` 앵커가 붙기도 한다. + +``` +![](PointVector_2205.10528_images/_page_0_Figure_11.jpeg) + +Figure 1. Illustrations of the core operations of the different methods. (a) The +features of each point are calculated separately... (논문 원본 캡션) +``` + +- 이미지 경로는 **.md 파일 기준 상대경로**다 +- 이미지 바로 아래 빈 줄, 그 다음에 **논문 원본 캡션**이 이미 있다 +- 캡션이 없는 그림도 있다 (표 이미지, 장식용 등) + +## 해야 할 일 + +각 이미지를 Vision으로 열어보고, **원본 캡션 아래에** 한국어 설명을 삽입한다. + +### 삽입 형식 + +원본 캡션 다음 빈 줄 뒤에 인용 블록으로 넣는다. + +``` +![](PointVector_2205.10528_images/_page_0_Figure_11.jpeg) + +Figure 1. Illustrations of the core operations of the different methods. (a) The +features of each point are calculated separately... + +> **[그림 해설]** 4개 패널을 가로로 배열한 비교 다이어그램. (a) attention 방식은 +> 고정 커널로 각 점을 따로 계산한 뒤 입력에서 만든 가중치로 이방성을 부여한다. +> (b) 변위 벡터로 커널 패턴에 가까운 점을 골라 집계한다. (c) 점마다 다른 동적 +> 커널을 적용한다. (d) 제안 방식은 특징에서 벡터 표현을 만들고, 벡터의 방향 +> 자체가 이방성을 만든다. 화살표 색으로 등방성(검정)과 이방성(색)을 구분한다. +``` + +**원본 캡션이 없는 그림**은 이미지 바로 아래에 넣는다. + +### 규칙 + +1. **`> **[그림 해설]**` 마커로 시작한다.** 이 마커로 나중에 검색·제거가 가능해야 한다. +2. **원본 텍스트는 절대 수정하지 않는다.** 캡션도, 본문도, 앵커 ``도 그대로 둔다. + 오직 삽입만 한다. +3. **이미 `[그림 해설]`이 붙은 그림은 건너뛴다.** 재실행해도 중복되지 않아야 한다. +4. 한국어로 쓴다. 기술 용어(attention, MLP, IoU, voxel 등)는 원어 그대로 둔다. +5. 인용 블록 전체를 `> `로 시작하는 여러 줄로 쓴다. + +## 무엇을 쓸 것인가 + +**"그림이 무엇을 보여주는지"를 쓴다. 캡션을 번역하지 않는다.** +원본 캡션은 바로 위에 이미 있으므로, 번역만 하면 아무 가치가 없다. + +그림 유형별로 다르게 접근한다. + +### 아키텍처 다이어그램 (가장 중요) + +- 블록이 몇 개이고 무엇이 무엇으로 흘러가는가 +- 화살표 방향, 분기와 합류 지점 +- 텐서 shape 표기가 있으면 그대로 옮긴다 (예: `N×3 → N×64 → 1024`) +- 색이나 선 스타일이 구분하는 것 +- **논문의 주장과 직결되는 지점을 짚는다** (예: "여기가 skip connection이고, 이게 + PointNet++ 대비 추가된 부분이다") + +### 성능 차트 (막대·꺾은선·산점도) + +- 축이 무엇인가 (단위 포함) +- 비교 대상이 몇 개이고 각각 무엇인가 +- **읽을 수 있는 수치는 옮긴다** — 나중에 검색된다 +- 추세와 교차점 + +### 정성 결과 (세그멘테이션 시각화 등) + +- 몇 개 열/행이고 각각 무엇인가 (입력 / GT / 각 방법의 결과) +- 색상 범례가 있으면 클래스-색 대응을 옮긴다 +- 방법 간 눈에 띄는 차이가 어디서 나타나는가 + +### 표를 이미지로 캡처한 것 + +- **표 내용을 마크다운 표로 복원한다.** 이게 가장 가치 있다. +- 행·열 헤더와 수치를 정확히 옮긴다 + +### 수식 이미지 + +- 수식을 LaTeX로 옮긴다 +- 각 기호가 무엇을 뜻하는지 본문에서 찾아 붙인다 + +## 하지 말 것 + +- **읽을 수 없는 것을 지어내지 않는다.** 흐리거나 잘려서 안 보이면 + "해상도가 낮아 세부 수치는 판독 불가" 라고 명시한다. +- 원본 캡션을 그대로 번역하지 않는다. +- 논문에 없는 해석이나 평가를 덧붙이지 않는다. + ("이 방법이 우수하다" 같은 것 — 그림에서 읽히는 사실만 쓴다) +- 그림과 무관한 배경 설명을 늘어놓지 않는다. + +## 분량 + +그림당 **3~8줄**. 아키텍처 다이어그램과 표 이미지는 더 길어져도 된다 +(표는 완전히 복원할 것). 장식용 그림은 1~2줄로 짧게. + +## 진행 방식 + +1. 파일 하나씩 처리한다. 한 파일을 끝내고 다음으로 간다. +2. 파일 안에서는 위에서 아래 순서로 그림을 처리한다. +3. 이미지를 열 때는 **.md 파일이 있는 디렉토리 기준**으로 상대경로를 해석한다. +4. 파일 하나가 끝나면 몇 개 그림에 주석을 달았는지 보고한다. + +## 검증 + +작업 후 이것들이 성립해야 한다. + +- `[그림 해설]` 개수 == 그 파일의 이미지 개수 +- 원본 텍스트 줄 수가 줄지 않았다 (삽입만 했으므로 늘어나야 정상) +- 마크다운이 깨지지 않았다 (표, 코드 블록, 수식이 그대로) +```` + +--- + +## 작업 후 확인 명령 + +Git Bash 또는 WSL에서: + +```bash +cd /d/MYCLAUDE_PROJECT/sum-parts-test/docs/papers/md + +# 그림 수 대 주석 수 대조 +for md in *.md; do + img=$(grep -c '!\[' "$md") + ann=$(grep -c '\[그림 해설\]' "$md") + printf '%-34s 그림 %2s 주석 %2s %s\n' "$md" "$img" "$ann" \ + "$([ "$img" -eq "$ann" ] && echo OK || echo MISMATCH)" +done +``` + +## 되돌리기 + +주석만 지우려면: + +```bash +# 백업 먼저 +cp -r md md.backup + +# [그림 해설] 인용 블록 제거 +sed -i '/^> \*\*\[그림 해설\]\*\*/,/^$/d' md/*.md +``` + +원본 PDF가 `docs/papers/`에 있으므로 최악의 경우 +`scripts/convert_papers.sh`로 재생성하면 된다. diff --git a/docs/papers/md/PointNeXt_2206.04670.md b/docs/papers/md/PointNeXt_2206.04670.md new file mode 100644 index 0000000..b432149 --- /dev/null +++ b/docs/papers/md/PointNeXt_2206.04670.md @@ -0,0 +1,515 @@ +## PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies + +Guocheng Qian1 , Yuchen Li1 , Houwen Peng2† , Jinjie Mai1 , Hasan Abed Al Kader Hammoud1 , Mohamed Elhoseiny1 , Bernard Ghanem1† 1King Abdullah University of Science and Technology (KAUST), 2Microsoft Research + +## Abstract + +PointNet++ is one of the most influential neural architectures for point cloud understanding. Although the accuracy of PointNet++ has been largely surpassed by recent networks such as PointMLP and Point Transformer, we find that a large portion of the performance gain is due to improved training strategies, *i.e.* data augmentation and optimization techniques, and increased model sizes rather than architectural innovations. Thus, the full potential of PointNet++ has yet to be explored. In this work, we revisit the classical PointNet++ through a systematic study of model training and scaling strategies, and offer two major contributions. First, we propose a set of improved training strategies that significantly improve PointNet++ performance. For example, we show that, without any change in architecture, the overall accuracy (OA) of PointNet++ on ScanObjectNN object classification can be raised from 77.9% to 86.1%, even outperforming state-of-theart PointMLP. Second, we introduce an inverted residual bottleneck design and separable MLPs into PointNet++ to enable efficient and effective model scaling and propose *PointNeXt*, the next version of PointNets. PointNeXt can be flexibly scaled up and outperforms state-of-the-art methods on both 3D classification and segmentation tasks. For classification, PointNeXt reaches an overall accuracy of 87.7% on ScanObjectNN, surpassing PointMLP by 2.3%, while being 10× faster in inference. For semantic segmentation, PointNeXt establishes a new state-of-theart performance with 74.9% mean IoU on S3DIS (6-fold cross-validation), being superior to the recent Point Transformer. The code and models are available at . + +## 1 Introduction + +Recent advances in 3D data acquisition have led to a surge in interest for point cloud understanding. With the rise of PointNet [\[29\]](#page-11-0) and PointNet++ [\[30\]](#page-11-1), processing point clouds in their unstructured format using deep CNNs become possible. Subsequent to "PointNets", many point-based networks are introduced with the majority focusing on developing new and sophisticated modules to extract local structures, *e.g*. the pseudo-grid convolution in KPConv [\[43\]](#page-12-0) and the self-attention layer in Point Transformer [\[56\]](#page-12-1). These newly proposed methods outperform PointNet++ by a large margin in a variety of tasks, leaving the impression that the PointNet++ architecture is too simple to learn complex point cloud representations. In this work, we revisit PointNet++, the classical and widely used network, and find that its full potential has yet to be explored, mainly due to two factors that were not present at the time of PointNet++: (1) superior training strategies and (2) effective model scaling strategies. + +Through a comprehensive empirical study on various benchmarks, *e.g*., ScanObjecNN [\[44\]](#page-12-2) for object classification and S3DIS [\[1\]](#page-10-0) for semantic segmentation, we discover that training strategies, *i.e*., data augmentation and optimization techniques, play an important role in the network's performance. In fact, a large part of the performance gain of state-of-the-art (SOTA) methods [\[46,](#page-12-3) [43,](#page-12-0) [56\]](#page-12-1) over Point-Net++ [\[30\]](#page-11-1) is due to improved training strategies that are, unfortunately, less publicized compared to + +Equal contribution. †Corresponding authors. + +![](PointNeXt_2206.04670_images/_page_1_Figure_0.jpeg) + +Figure 1: Effects of training strategies and model scaling on PointNet++ [\[30\]](#page-11-1). We show that improved training strategies (data augmentation and optimization techniques) and model scaling can significantly boost PointNet++ performance. The average overall accuracy and mIoU (6-fold cross-validation) are reported on ScanObjectNN [\[44\]](#page-12-2) and S3DIS [\[1\]](#page-10-0). + +> **[그림 해설]** 학습 전략 개선 및 모델 스케일링이 PointNet++의 성능 향상에 미치는 기여도 분석 차트. +> - **ScanObjectNN (상단 막대, Overall Accuracy %)**: +> - 원본 PointNet++(노란색): 77.9% +> - Data Augmentation(초록색): +5.8% $\to$ 83.7% +> - Optimization Techniques(보라색): +2.4% $\to$ 86.1% (기존 SOTA인 PointMLP 85.4% 추월) +> - Model Scaling (PointNeXt 구조, 분홍색): +1.6% $\to$ **87.7%** 달성. +> - **S3DIS (하단 막대, 6-fold mIoU %)**: +> - 원본 PointNet++(노란색): 54.5% +> - Data Augmentation(초록색): +12.3% $\to$ 66.8% +> - Optimization Techniques(보라색): +1.3% $\to$ 68.1% +> - Model Scaling(분홍색): +6.8% $\to$ **74.9%** 달성 (기존 SOTA인 Point Transformer 73.5% 추월). + +architectural changes. For example, randomly dropping colors during training can unexpectedly boost the testing performance of PointNet++ by 5.9% mean IoU (mIoU) on S3DIS [\[1\]](#page-10-0), as demonstrated in Tab. [5.](#page-7-0) In addition, adopting label smoothing [\[39\]](#page-11-2) can improve the overall accuracy (OA) on ScanObjectNN [\[44\]](#page-12-2) by 1.3%. These findings inspire us to revisit PointNet++ and equip it with new advanced training strategies that are widely used today. Surprisingly, as shown in Fig. [1,](#page-1-0) utilizing the improved training strategies alone improves the OA of PointNet++ by 8.2% on ScanObjectNN (from 77.9% to 86.1%), establishing a new SOTA without introducing any changes to the architecture (refer to Sec. [4.4.1](#page-7-1) for details). For the S3DIS segmentation benchmark, the mIoU evaluated in all areas by 6-fold cross-validation can increase by 13.6% (from 54.5% to 68.1%), outperforming many modern architectures that are subsequent to PointNet++, such as PointCNN [\[22\]](#page-11-3) and DeepGCN [\[21\]](#page-11-4). + +Moreover, we observe that the current prevailing models [\[20,](#page-11-5) [43,](#page-12-0) [56\]](#page-12-1) for point cloud analysis have employed many more parameters than the original PointNets [\[29,](#page-11-0) [30\]](#page-11-1). Effectively expanding PointNet++ from its original small scale to a larger scale is a topic worth studying because larger models are generally expected to enable richer representations and perform better [\[2,](#page-10-1) [19,](#page-10-2) [55\]](#page-12-4). However, we find that the naive way of using more building blocks or increasing the channel size in PointNet++ only leads to an overhead in latency and no significant improvement in accuracy (see Sec. [4.4.2\)](#page-8-0). For effective and efficient model scaling, we introduce residual connections [\[13\]](#page-10-3), an inverted bottleneck design [\[36\]](#page-11-6), and separable MLPs [\[32\]](#page-11-7) into PointNet++. The modernized architecture is named PointNeXt, the next version of PointNets. PointNeXt can be scaled up flexibly and outperforms SOTA on various benchmarks. As demonstrated in Fig. [1,](#page-1-0) PointNeXt improves the original PointNet++ by 20.4% mIoU (from 54.5% to 74.9%) on *S3DIS* [\[1\]](#page-10-0) 6-fold and achieves 9.8% OA gains on *ScanObjecNN* [\[44\]](#page-12-2), surpassing SOTA Point Transformer [\[56\]](#page-12-1) and PointMLP [\[28\]](#page-11-8). We summarize our contributions next: + +- We present the first systematic study of training strategies in the point cloud domain and show that *PointNet++ strikes back* (+8.2% OA on ScanObjectNN and +13.6% mIoU on S3DIS) by simply adopting *improved training strategies alone*. The improved training strategies are general and can be easily applied to improve other methods [\[29,](#page-11-0) [46,](#page-12-3) [28\]](#page-11-8). +- We propose PointNeXt, the next version of PointNets. PointNeXt is scalable and surpasses SOTA on all tasks studied, including object classification [\[44,](#page-12-2) [49\]](#page-12-5), semantic segmentation [\[1,](#page-10-0) [5\]](#page-10-4), and part segmentation [\[53\]](#page-12-6), while being faster than SOTA in inference. + +## 2 Preliminary: A Review of PointNet++ + +Our PointNeXt is built upon PointNet++ [\[30\]](#page-11-1), which uses a U-Net [\[35\]](#page-11-9) like architecture with an encoder and a decoder, as visualized in Figure [2.](#page-3-0) The encoder part hierarchically abstracts features of point clouds using a number of *set abstraction* (SA) blocks, while the decoder gradually interpolates the abstracted features by the same number of *feature propagation* blocks. The SA block consists of a *subsampling* layer to downsample the incoming points, a *grouping* layer to query neighbors for each point, a set of shared multilayer perceptrons (*MLPs*) to extract features, and a *reduction* layer to aggregate features within the neighbors. The combination of the grouping layer, MLPs, and the reduction layer is formulated as: + + +$$\mathbf{x}_{i}^{l+1} = \mathcal{R}_{j:(i,j)\in\mathcal{N}} \left\{ h_{\Theta} \left( \left[ \mathbf{x}_{j}^{l}; \mathbf{p}_{j}^{l} - \mathbf{p}_{i}^{l} \right] \right) \right\}, \tag{1}$$ + +where $\mathcal{R}$ is the reduction layer (e.g. max-pooling) that aggregates features for point i from its neighbors denoted as $\{j:(i,j)\in\mathcal{N}\}$ . $\mathbf{p}_i^l,\mathbf{x}_i^l,\mathbf{x}_j^l$ are the input coordinates, the input features, and the features of neighbor j in the $l^{th}$ layer of the network, respectively. $h_{\Theta}$ denotes the shared MLPs that take the concatenation of $\mathbf{x}_j^l$ and the relative coordinates $(\mathbf{p}_j^l - \mathbf{p}_i^l)$ as input. Note that, since PointNet++ with single-scale grouping that uses one SA block per stage is the default architecture used in the original paper [30], we refer to it as PointNet++ throughout and use it as our baseline. + +## 3 Methodology: From PointNet++ to PointNeXt + +In this section, we present how to modernize the classical architecture PointNet++ [30] into PointNeXt, the next version of PointNet++ with SOTA performance. Our exploration mainly focuses on two aspects: (1) training modernization to improve data augmentation and optimization techniques, and (2) architectural modernization to probe receptive field scaling and model scaling. Both aspects have important impact on the model's performance, but were under-explored by previous studies. + +#### 3.1 Training Modernization: PointNet++ Strikes Back + +We conduct a systematic study to quantify the effect of each data augmentation and optimization technique used by modern point cloud networks [46, 43, 56] and propose a set of improved training strategies. The potential of PointNet++ can be unveiled by adopting our proposed training strategies. + +#### 3.1.1 Data Augmentation + +Data augmentation is one of the most important strategies to boost the performance of a neural network; thus we start our modernization from there. The original PointNet++ used simple combinations of data augmentations from random rotation, scaling, translation, and jittering for various benchmarks [30]. Recent methods adopt stronger augmentations than those used in PointNet++. For example, KPConv [43] randomly drops colors during training, Point-BERT [54] uses a common point resampling strategy to randomly sample 1,024 points from the original point cloud for data scaling, while RandLA-Net [15] and Point Transformer [56] load the entire scene as input in segmentation tasks. In this paper, we quantify the effect of each data augmentation through an additive study. + +We start our study with PointNet++ [30] as the baseline, which is trained with the original data augmentations and optimization techniques. We remove each data augmentation to check whether it is necessary or not. We add back the useful augmentations but remove the unnecessary ones. We then systematically study all the data augmentations used in the representative works [46, 43, 32, 56, 28, 54], including data scaling such as point resampling [54] and loading the entire scene as input [15], random rotation, random scaling, translation to shift point clouds, jittering to add independent noise to each point, height appending [43] (*i.e.*, appending the measurement of each point along the gravity direction of objects as additional input features), color auto-contrast to automatically adjust color contrast [56], and color drop that randomly replaces colors with zero values. We verify the effectiveness of data augmentation incrementally and only keep the augmentations that give a better validation accuracy. At the end of this study, we provide a collection of data augmentations for each task that allow for the highest boost in the model's performance. Sec. 4.4.1 presents and analyzes in detail the uncovered findings. + +#### 3.1.2 Optimization Techniques + +Optimization techniques including loss functions, optimizers, learning rate schedulers, and hyperparameters are also vital to the performance of a neural network. PointNet++ uses the same optimization techniques throughout its experiments: CrossEntropy loss, Adam optimizer [16], exponential learning rate decay (Step Decay), and the same hyperparmeters. Owing to the development of machine learning theory, modern neural networks can be trained with theoretically better optimizers (e.g. AdamW [27] vs. Adam [16]) and more advanced loss functions (CrossEntropy with label smoothing [39]). Similarly to our study on data augmentations, we also quantify the effect of each modern optimization technique on PointNet++. We first perform a sequential hyperparameter search for the learning rate and weight decay. We then conduct an additive study on label smoothing, optimizer, and learning rate scheduler. We discover a set of improved optimization techniques that further + +![](PointNeXt_2206.04670_images/_page_3_Figure_0.jpeg) + +Figure 2: **PointNeXt architecture.** PointNeXt shares the same Set Abstraction and Feature Propagation blocks as PointNet++ [30], while adding an additional MLP layer at the beginning and scaling the architecture with the proposed Inverted Residual MLP (InvResMLP) blocks. + +> **[그림 해설]** PointNeXt의 전체 세그멘테이션 아키텍처 다이어그램 (빨간색 테두리가 현대화된 핵심 구성 요소). +> - **인코더 (U-Net 형태 백본)**: +> - 입력 포인트 $\to$ 초기 임베딩 **MLP [N, 32]** 추가. +> - 4단계 계층적 다운샘플링: Set Abstraction + **InvResMLP** 블록을 거치며 $[N/4, 64] \to [N/16, 128] \to [N/64, 256] \to [N/256, 512]$로 점진적 축소. +> - **핵심 블록 상세 (하단)**: +> - **Set Abstraction**: Subsample $\to$ Grouping $\to$ MLPs (64) $\to$ Reduction(Max-pool). +> - **InvResMLP (Inverted Residual MLP)**: Residual Connection(지름길 연결)을 포함하며, Grouping $\to$ MLP(256) $\to$ Reduction $\to$ MLP(1024, 4배 확장 역병목) $\to$ MLP(256) 구조로 파라미터 효율성과 특징 표현력을 극대화. +> - **Feature Propagation (디코더)**: +> - Interpolate $\to$ Skip Connection 결합 $\to$ MLPs(128)를 거쳐 $[N/64, 256] \to [N/16, 128] \to [N/4, 64] \to [N, 32]$ 순으로 원본 해상도를 복원한 후 최종 세그멘테이션 레이블을 출력. + +boost performance by a decent margin. In general, CrossEntropy with label smoothing, AdamW, and Cosine Decay can decently optimize models in various tasks. See Sec. 4.4.1 for detailed findings. + +#### **3.2** Architecture Modernization: Small Modifications → Big Improvements + +In this subsection, we modernize PointNet++ [30] into the proposed PointNeXt. The modernization consists of two aspects: (1) receptive field scaling and (2) model scaling. + +#### 3.2.1 Receptive Field Scaling + +The receptive field is a significant factor in the design space of a neural network [38, 7]. There are at least two ways to scale the receptive field in point cloud processing: (1) adopting a larger radius to query the neighborhood, and (2) adopting a hierarchical architecture. Since the hierarchical architecture has been adopted in the original PointNet++, we mainly study (1) in this subsection. Note that the radius of PointNet++ is set to an initial value r that doubles when the point cloud is downsampled. We study a different initial value in each benchmark and discover that the radius is dataset-specific and can have significant influence on performance. This is elaborated in Sec. 4.4.2. + +Furthermore, we find that the relative coordinates $\Delta_p = \mathbf{p}_j^l - \mathbf{p}_i^l$ in Eq. (1) make network optimization harder, leading to a decrease in performance. Thus, we propose relative position normalization ( $\Delta_p$ normalization) to divide relative position by the neighborhood query radius: + + +$$\mathbf{x}_{i}^{l+1} = \mathcal{R}_{j:(i,j)\in\mathcal{N}}\left\{h_{\Theta}\left(\left[\mathbf{x}_{j}^{l};(\mathbf{p}_{j}^{l} - \mathbf{p}_{i}^{l})/r^{l}\right]\right)\right\}. \tag{2}$$ + +Without normalization, values of relative positions $(\Delta_p = \mathbf{p}_j^l - \mathbf{p}_i^l)$ are considerably small (less than the radius), requiring the network to learn a larger weight to apply on $\Delta_p$ . This makes the optimization non-trivial, especially since weight decay is used to reduce the weights of the network and thus tends to ignore the effects of relative position. The proposed normalization alleviates this issue by rescaling and in the meantime reduces the variance of $\Delta_p$ among different stages. + +#### 3.2.2 Model Scaling + +PointNet++ is a relatively small network, where the encoder consists of only 2 stages in the classification architecture and 4 stages for segmentation. Each stage consists of only 1 SA block, and each block contains 3 layers of MLP. The model sizes of PointNet++ for both classification and segmentation are less than 2M, which is much smaller compared to modern networks that typically use more than 10M parameters [43, 28, 32]. Interestingly, we find that neither appending more SA blocks nor using more channels leads to a noticeable improvement in accuracy, while causing a significant drop in throughput (refer to Sec. 4.4.2), mainly due to vanishing gradient and overfitting. Therefore, in this subsection, we study how to scale up PointNet++ in an effective and efficient way. + +We propose an Inverted Residual MLP (InvResMLP) block to be appended after the first SA block, per stage, for effective and efficient model scaling. InvResMLP is built on the SA block and is + +illustrated at the bottom middle of Fig. [2.](#page-3-0) There are three differences between InvResMLP and SA. (1) A residual connection between the input and the output is added to alleviate the vanishing gradient problem [\[13\]](#page-10-3), especially when the network goes deeper. (2) Separable MLPs are introduced to reduce computation and reinforce pointwise feature extraction. While all 3 layers of MLPs in the original SA block are computed on the neighborhood features, InvResMLP separates the MLPs into a single layer computed on the neighborhood features (between the grouping and reduction layers) and two layers for point features (after reduction), as inspired by MobileNet [\[14\]](#page-10-8) and ASSANet [\[32\]](#page-11-7). (3) The inverted bottleneck design [\[36\]](#page-11-6) is leveraged to expand the output channels of the second MLP by 4 times to enrich feature extraction. Appending InvResMLP blocks is proven to significantly improve performance compared to the appending of the original SA blocks (see Sec. [4.4.2\)](#page-8-0). + +In addition to InvResMLP, we present three changes in the macro architecture. (1) We unify the design of PointNet++ encoder for classification and segmentation, *i.e*., scaling the number of SA blocks for classification from 2 to 4 while keeping the original number (4 blocks) for segmentation at each stage. (2) We utilize a symmetric decoder in which its channel size is changed to match the encoder. (3) We add a stem MLP, an additional MLP layer inserted at the beginning of the architecture, to map the input point cloud to a higher dimension. + +In summary, we present PointNeXt, the next version of PointNets [\[29,](#page-11-0) [52\]](#page-12-8), modified from PointNet++ by incorporating the proposed InvResMLP and the aforementioned macro-architectural changes. The architecture of PointNeXt is illustrated in Fig. [2.](#page-3-0) We denote the channel size of the stem MLP as C and the number of InvResMLP blocks as B. A larger C leads to an increase in the width of the network (*i.e*., width scaling), while a larger B leads to an increase in the depth of the network (*i.e*., depth scaling). Note that when B = 0, only one SA block and no InvResMLP blocks are used at each stage. The number of MLP layers in the SA block is set to 2, and a residual connection is added inside each SA block. When B 6= 0, InvResMLP blocks are appended after the original SA block. The number of MLP layers in the SA block in this case is set to 1 to save computation cost. The configuration of our PointNeXt family is summarized as follows: + +``` +• PointNeXt-S: C = 32, B = 0 +• PointNeXt-B: C = 32, B = (1, 2, 1, 1) + • PointNeXt-L: C = 32, B = (2, 4, 2, 2) + • PointNeXt-XL: C = 64, B = (3, 6, 3, 3) +``` + +## 4 Experiments + +We evaluate PointNeXt on five standard benchmarks: *S3DIS* [\[1\]](#page-10-0) and *ScanNet* [\[5\]](#page-10-4) for semantic segmentation, *ScanObjectNN* [\[44\]](#page-12-2) and *ModelNet40* [\[49\]](#page-12-5) for object classification, and *ShapeNetPart* [\[3\]](#page-10-9) for object part segmentation. + +Experimental Setups. We train PointNeXt using CrossEntropy loss with label smoothing [\[39\]](#page-11-2), AdamW optimizer [\[27\]](#page-11-10), an initial learning rate lr = 0.001, weight decay 104 , with Cosine Decay, and a batch size of 32, with a 32G V100 GPU, for all tasks, unless otherwise specified. The best model on the validation set is selected for testing. For S3DIS segmentation, point clouds are voxel downsampled with a voxel size of 0.04m following common practice [\[43,](#page-12-0) [32,](#page-11-7) [56\]](#page-12-1). PointNeXt is trained with an initial lr = 0.01, for 100 epochs (training set is repeated by 30 times), using a fixed number of points (24, 000) per batch with a batch size of 8 as input. During training, the input points are obtained by querying the nearest neighbors of a random point in each iteration. Following Point Transformer [\[56\]](#page-12-1), we evaluate PointNeXt using the entire voxel-downsampled scene as input. For ScanNet scene segmentation, we follow the Stratified Transformer [\[17\]](#page-10-10) and train PointNeXt with multi-step learning rate decay and decay at [70,90] epochs with a decay rate of 0.1 without label smoothing. The voxel size is set to 0.02m and input number of points in training is set to 64, 000. We train the model for 100 epochs (training set is repeated for 6 times) with a batch size of 2 per GPU with 8 GPUs. For ScanObjectNN classification, PointNeXt is trained with a weight decay of 0.05 for 250 epochs. Following Point-BERT [\[54\]](#page-12-7), the number of input points is set to 1, 024, where the points are randomly sampled during training and uniformly sampled during testing (denoted as point resampled augmentation). For ModelNet40 classification, PointNeXt is trained similarly as ScanObjectNN but for 600 epochs. For ShapeNetPart part segmentation, we train PointNeXt using a batch size of 8 per GPU with 4 GPUs, and Poly FocalLoss [\[18\]](#page-10-11) as criterion, for 400 epochs. Following PointNet++, 2,048 randomly sampled points with normals are used as input for training and testing. The details of data augmentations used in S3DIS, ScanNet, ScanObjectNN, ModelNet40 and ShapeNetPart are detailed in Sec. [4.4.1.](#page-7-1) + +Table 1: **3D semantic segmentation in S3DIS** (**evaluation by 6-Fold or in Area 5**) and **ScanNet V2.** For PointNeXt in S3DIS Area 5, the average results without voting in three random runs are reported. The improvements of PointNeXt over the original performance reported by PointNet++ [30] are highlighted in green color. PointNet++ (ours) denotes PointNet++ trained using our improved data augmentation and optmization techniques. Methods are in chronological order. + +| | S3DIS | 6-Fold | S3DIS | Area-5 | Scan | Net V2 | Doros | me FLO | Ps Throughput | +|------------------------|--------------|-------------|-------------------------|-----------------------|-------------|-------------|---------|-----------|------------------| +| Method | mIoU | OA | mIoU | OA | Val mIoU | Test mIoU | 1 ai ai | iiis. FLO | 1 8 Till oughput | +| | (%) | (%) | (%) | (%) | (%) | (%) | M | G | (ins./sec.) | +| PointNet [29] | 47.6 | 78.5 | 41.1 | - | - | - | 3.6 | 35.5 | 162 | +| PointCNN [22] | 65.4 | 88.1 | 57.3 | 85.9 | - | 45.8 | 0.6 | - | - | +| DGCNN [46] | 56.1 | 84.1 | 47.9 | 83.6 | - | - | 1.3 | - | 8 | +| DeepGCN [21] | 60.0 | 85.9 | 52.5 | - | - | - | 3.6 | - | 3 | +| KPConv [43] | 70.6 | - | 67.1 | - | 69.2 | 68.6 | 15.0 | - | 30 | +| RandLA-Net [15] | 70.0 | 88.0 | - | - | - | 64.5 | 1.3 | 5.8 | 159 | +| BAAF-Net [33] | 72.2 | 88.9 | 65.4 | 88.9 | - | - | 5.0 | - | 10 | +| Point Transformer [56] | 73.5 | 90.2 | 70.4 | 90.8 | 70.6 | - | 7.8 | 5.6 | 34 | +| CBL [41] | 73.1 | 89.6 | 69.4 | 90.6 | - | 70.5 | 18.6 | - | - | +| PointNet++ [30] | 54.5 | 81.0 | 53.5 | 83.0 | 53.5 | 55.7 | 1.0 | 7.2 | 186 | +| PointNet++ (ours) | 68.1(+13.6) | 87.6(+6.2) | $63.2\pm0.4(+9.7)$ | $87.5\pm0.2(+4.5)$ | 57.2(+3.7) | - | 1.0 | 7.2 | 186 | +| PointNeXt-S (ours) | 68.0(+13.5) | 87.4(+6.4) | 63.4±0.8(+9.9) | $87.9\pm0.3(+4.9)$ | 64.5(+11.0) | - | 0.8 | 3.6 | 227 | +| PointNeXt-B (ours) | 71.5(+17.0) | 88.8(+7.8) | $67.3\pm0.2(+13.8)$ | $89.4\pm0.1(+6.4)$ | 68.4(+14.9) | - | 3.8 | 8.9 | 158 | +| PointNeXt-L (ours) | 73.9(+19.4) | 89.8(+8.8) | 69.0±0.5(+15.5) | $90.0\pm0.1(+7.0)$ | 69.4(+15.9) | - | 7.1 | 15.2 | 115 | +| PointNeXt-XL (ours) | 74.9 (+20.4) | 90.3 (+9.3) | 70.5 ±0.3(+17.0) | $90.6 \pm 0.1 (+7.6)$ | 71.5(+18.0) | 71.2(+15.5) | 41.6 | 84.8 | 46 | + +For all experiments except ShapeNetPart segmentation, we do not conduct any voting $[23]^2$ , since it is more standard to compare the performance without using any ensemble methods as suggested by SimpleView [9]. However, we found that the performance in ShapeNetPart of nearly all models is quite close to each other, where it is hard to achieve state-of-the-art IoUs without voting. We also provide model parameters (Params.) and inference throughput (instances per second) for comparison. The throughput of all methods is measured using $128 \times 1024$ (batch size 128, number of points 1024) as input in ScanObjectNN and ModelNet40 and $64 \times 2048$ in ShapeNetPart. In S3DIS, $16 \times 15,000$ points are used to measure throughput following [32], since some methods [46, 20] could not process the whole scene due to memory constraints. The throughput of all methods is measured using an NVIDIA Tesla V100 32GB GPU and a 32 core Intel Xeon @ 2.80GHz CPU. + +#### 4.1 3D Semantic Segmentation in S3DIS and ScanNet + +S3DIS [1] (Stanford Large-Scale 3D Indoor Spaces) is a challenging benchmark composed of 6 large-scale indoor areas, 271 rooms, and 13 semantic categories in total. The standard 6-fold cross-validation results in S3DIS are reported in Tab. 1. Note that the official PointNet++ [30] did not conduct experiments in S3DIS. Here, we use the results reported by PointCNN [22] for comparison. Our PointNeXt-S, the smallest variant, outperforms PointNet++ by 13.5%, 6.4%, and 10.2% in terms of mean IoU (mIoU), overall accuracy (OA), and mean accuracy (mAcc), respectively, while being faster in terms of throughput. The increased speed is due to the reduced number of layers in the SA block for PointNeXt-S (see Sec. 3.2.2). With the proposed model scaling, the performance of PointNeXt can be gradually boosted. For example, PointNeXt-L outperforms SOTA Point Transformer [56] by 0.4% in mIoU while being 3× faster. Note that Point Transformer utilizes most of the improved training strategies of ours. PointNeXt-XL, the extra large variant, achieves mIoU/OA/mAcc of 74.9%/90.3%/83.0%, while running faster than Point Transformer. As a limitation, our PointNeXt-XL consists of more parameters and is more computationally expensive in terms of FLOPs, mainly due to channel expansion $(\times 4)$ in the inverted bottleneck and doubled initial channel size (C=64). We also provide the results of PointNeXt in S3DIS area 5 in the Tab. 1 with mean $\pm$ std in three random runs, where PointNeXt achieves similar improvements as the 6-fold experiments. + +ScanNet [5], another well-known large-scale segmentation dataset, contains 3D indoor scenes of various rooms with 20 semantic categories. We follow the public training, validation, and test splits, with 1201, 312 and 100 scans, respectively. For PointNet++, we use the results reported from the Stratified Transformer [17] for comparison. As shown in Tab. 1, we improve PointNet++ from 53.5% mIou to 57.2% mIoU in the validation set by adopting the improved training strategies (detailed in supplementary material). PointNeXt-S further gains +11.0 in val mIoU over the original PointNet++ mostly due to the use of a smaller radius $(0.1\text{m} \rightarrow 0.05\text{m})$ and relative position normalization. The performance in ScanNet improves steadily with the increase in model sizes. Our largest variant, + +<sup>2The voting strategy combines results by using randomly augmented points as input to enhance performance. + +Table 2: **3D object classification in ScanObjectNN and ModelNet40.** Averaged results in three random runs using 1024 points as input without normals and without voting are reported. + +| Method | ScanObjectNN
OA (%) | M (PB_T50_RS)
mAcc (%) | OA (%) | et40
mAcc (%) | Params.
M | FLOPs
G | Throughput (ins./sec.) | +|------------------------------|------------------------|---------------------------|----------------|------------------|--------------|------------|------------------------| +| PointNet [29] | 68.2 | 63.4 | 89.2 | 86.2 | 3.5 | 0.9 | 4212 | +| PointCNN [22] | 78.5 | 75.1 | 92.2 | 88.1 | 0.6 | - | 44 | +| DGCNN [46] | 78.1 | 73.6 | 92.9 | 90.2 | 1.8 | 4.8 | 402 | +| DeepGCN [20] | - | _ | 93.6 | 90.9 | 2.2 | 3.9 | 263 | +| KPConv [43] | - | _ | 92.9 | _ | 14.3 | - | - | +| ASSANet-L [32] | - | _ | 92.9 | _ | 118.4 | - | 153 | +| SimpleView [9] | 80.5±0.3 | _ | 93.0±0.4 | $90.5 \pm 0.8$ | 0.8 | - | - | +| MVTN [12] | 82.8 | _ | 93.5 | 92.2 | 3.5 | 1.8 | 236 | +| Point Cloud Transformer [11] | - | - | 93.2 | _ | 2.9 | 2.3 | - | +| CurveNet [50] | - | _ | 93.8 | _ | 2.0 | - | 22 | +| PointMLP [28] | 85.4±1.3 | $83.9 \pm 1.5$ | 94.1 | 91.3 | 13.2 | 31.3 | 191 | +| PointNet++ [30] | 77.9 | 75.4 | 91.9 | - | 1.5 | 1.7 | 1872 | +| PointNet++ (ours) | 86.1±0.7(+8.2) | $84.2 \pm 0.9 (+8.8)$ | 92.8±0.1(+0.9) | $89.9 \pm 0.8$ | 1.5 | 1.7 | 1872 | +| PointNeXt-S (ours) | 87.7±0.4(+9.8) | $85.8 \pm 0.6 (+10.4)$ | 93.2±0.1(+1.3) | $90.8 {\pm} 0.2$ | 1.4 | 1.6 | 2040 | + +PointNeXt-XL outperforms PointNet++ by 18.0% mIoU in validation and achieves 71.2% mIoU in testing, beating the recent methods Point Transformer [56] and CBL [41]. + +#### 4.2 3D Object Classification in ScanObjectNN and ModelNet40 + +ScanObjectNN [44] contains about 15,000 real scanned objects that are categorized into 15 classes with 2,902 unique object instances. Due to occlusions and noise, ScanObjectNN poses significant challenges to existing point cloud analysis methods. Following PointMLP [28], we experiment on PB\_T50\_RS, the hardest and most commonly used variant of ScanObjectNN. As reported in Tab. 2, the proposed PointNeXt-S surpasses existing methods by non-trivial margins in terms of both OA and mAcc, while using much fewer model parameters and running much faster. Built upon PointNet++ [30], PointNeXt achieves significant improvements over the originally reported performance of PointNet++, *i.e.* +9.8% OA and +10.4% mACC. This demonstrates the efficacy of the proposed training and model scaling strategies. PointNeXt also outperforms SOTA PointMLP [28] (*i.e.* +2.3% OA, +1.9% mACC), while running $10\times$ faster. This shows that PointNeXt is a simple, yet effective, and efficient baseline. Note that we did not experiment with upscaled variants of PointNeXt on this benchmark, since we found that the performance had saturated using PointNeXt-S mostly due to the limited scale of the dataset. + +ModelNet40 [49] was a commonly used 3D object classification dataset, which has 40 object categories, each of which contains 100 unique CAD models. However, recent works [12, 28, 34] show an increasing interest in the real-world scanned dataset ScanObejectNN compared to this synthesized dataset. Following this trend, we mainly benchmarked PointNeXt in ScanObjectNN. Here, we also provide our results in ModelNet40. Tab. 2 shows that advanced training strategies improve PointNet++ from 91.9% OA to 92.8% OA without any architecture change. PointNeXt-S (C=32) outperforms the original reported PointNet++ by 1.3% OA, while being faster. Note that PointNeXt-S with a larger width C=64 can achieve a higher overall accuracy (94.0%). + +## 4.3 3D Object Part Segmentation in ShapeNetPart + +ShapeNetPart [53] is a widely-used dataset for object-level part segmentation. It consists of 16,880 models from 16 different shape categories, 2-6 parts for each category, and 50 part labels in total. As shown in Tab. 3, our PointNeXt-S with default width (C=32) obtains a performance comparable + +Table 3: Part segmentation in ShapeNetPart. + + + +| Method | ins. mIoU | cls. mIoU | Params. | FLOPs | Throughput | +|------------------------|-----------------------|-----------------------|---------|-------|------------| +| PointNet [29] | 83.7 | 80.4 | 3.6 | 4.9 | 1184 | +| DGCNN [46] | 85.2 | 82.3 | 1.3 | 12.4 | 147 | +| KPConv [43] | 86.4 | 85.1 | - | - | 44 | +| CurveNet [50] | 86.8 | - | - | - | 97 | +| ASSANet-L [32] | 86.1 | - | - | - | 640 | +| Point Transformer [56] | 86.6 | 83.7 | 7.8 | - | 297 | +| PointMLP [28] | 86.1 | 84.6 | - | - | 270 | +| Stratifiedformer [17] | 86.6 | 85.1 | - | - | 398 | +| PointNet++ [30] | 85.1 | 81.9 | 1.0 | 4.9 | 708 | +| PointNeXt-S | $86.7 \pm 0.0 (+1.6)$ | $84.4 \pm 0.2 (+2.5)$ | 1.0 | 4.5 | 782 | +| PointNeXt-S (C=64) | $86.9\pm0.1(+1.8)$ | $84.8 \pm 0.5 (+2.9)$ | 3.7 | 17.8 | 331 | +| PointNeXt-S (C=160) | $87.0\pm0.1(+1.9)$ | $85.2 \pm 0.1 (+3.3)$ | 22.5 | 110.2 | 76 | + +to that of the SOTA CurveNet [50] and outperforms a large number of representative networks, such as KPConv [43] and ASSANet [32] in terms of both instance mean IoU (ins. mIoU) and throughput. Due to the small scale of ShapeNetPart, the model would overfit after being depth scaled. However, we find by increasing the width from 32 to 64 instead, PointNeXt can outperform CurveNet, while being over 4× faster. It is also worth highlighting that PointNeXt with an even larger width (C = 160) reaches 87.0% Ins. mIoU, whereas the performance of point-based methods has saturated below this value for years. We highlight that we used voting only in ShapeNetPart by averaging the results of 10 randomly scaled input point clouds, with scaling factors equal to [0.8,1.2]. Without voting, we notice a performance drop around 0.5 instance mIoU. + +#### 4.4 Ablation and Analysis + +Tab. 4 and Tab. 5 present additive studies for the proposed training and scaling strategies in ScanObjectNN [44] and S3DIS [1], respectively. We adopt the original PointNet++ as the baseline. In ScanObjectNN, PointNet++ was trained by [44] with CrossEntropy loss, Adam optimizer, a learning rate 1e-3, a weight decay of 1e-4, a step decay of 0.7 for every 20 epochs, and a batch size of 16, for 250 epochs, while using random rotation and jittering as data augmentations. The official PointNet++ did not conduct experiments in S3DIS dataset. We refer to the widely used reimplementation [52], where PointNet++ was trained with the same settings as ScanObjectNN except that only random rotation was used as augmentation. Note that for all experiments, we train all our models for 250 epochs in ScanObjectNN and for 100 epochs in S3DIS. + +#### 4.4.1 Training Strategies + +Data augmentation is the first aspect that we study to modernize PointNet++. We draw four conclusions based on observations in Tab. 4 and 5. (1) Data scaling improves performance for both classification and segmentation tasks. For example, point resampling is shown to boost the performance by 2.5% OA in ScanObjectNN. Taking the entire scene as input instead of using the block or sphere subsampled input as done in PointNet++ [30] and other previous works [43, 21, 32] improves the segmentation result by 1.1% mIoU. (2) Height appending improves performance, especially for object classification. Height appending makes the network aware of the actual size of the objects, thus leading to an increase in accuracy (+1.1% OA). (3) Color drop is a strong augmentation that significantly improves the performance of tasks where colors are available. Adopting color drop alone adds 5.9% mIoU in S3DIS area 5. We hypothesize that color drop forces the network to focus more on the geometric relationships between points, which in turn improves performance. (4) Larger models favor stronger data augmentation. Whereas random rotation drops the performance of PointNet++ by 0.3% mIoU in S3DIS ( $2^{nd}$ row in Tab. 5 data augmentation part), it is shown to be beneficial for larger-scale models (e.g. raises 1.5% mIoU on PointNeXt-B). Another example in ScanObjectNN shows that the removal of random jittering also adds 1.1% OA. In general, with the improved data augmentations, the OA of PointNet++ in ScanObjectNN and the mIoU in S3DIS area 5 are increased by 5.8% and 9.5%, respectively. + +Table 4: Additive study of sequentially applying train- Table 5: Additive study of sequentially applying and scaling strategies for classification on ScanOb- ing training and scaling strategies for segmenjectNN. We use light green, purple, yellow, and pink background colors to denote data augmentation, \top- ing/removing the strategy. timization techniques, receptive field scaling, and model scaling, respectively. + +| Improvements | OA (%) | Δ | +|-------------------------------------------|----------------|------| +| PointNet++ | 77.9 | - | +| + Point resampling | $81.4 \pm 0.6$ | +2.5 | +|
  • Jittering
| $82.5 \pm 0.4$ | +1.1 | +| + Height appending | $83.6 \pm 0.4$ | +1.1 | +| + Random scaling | $83.7 \pm 0.2$ | +0.1 | +| + Label Smoothing | $85.0 \pm 0.5$ | +1.3 | +| $+$ Adam $\rightarrow$ Adam $W$ | $85.6 \pm 0.1$ | +0.6 | +| $+$ Step Decay $\rightarrow$ Cosine Decay | $86.1 \pm 0.7$ | +0.5 | +| $+$ Radius $0.2 \rightarrow 0.15$ | $86.4 \pm 0.3$ | +0.3 | +| + Normalizing $\Delta_p$ (Eqn. (2)) | $86.7 \pm 0.3$ | +0.3 | +| + Scale up (PointNeXt-S) | $87.7 \pm 0.4$ | +1.0 | + +tation on S3DIS area 5. +/- denote adopt- + +| Improvements | mIoU (%) | Δ | +|-------------------------------------------|----------------|------| +| PointNet++ | 51.5 | - | +| + Entire scene as input | $52.6 \pm 0.5$ | +1.1 | +| - Rotation | $52.9 \pm 0.6$ | +0.3 | +| + Height appending | $53.4 \pm 0.4$ | +0.5 | +| + Color drop | $59.3 \pm 0.7$ | +5.9 | +| + Color auto-contrast | $61.0 \pm 0.4$ | +0.7 | +| $+ lr = 0.001 \rightarrow 0.01$ | $61.5 \pm 0.5$ | +0.5 | +| + Label Smoothing | $61.9 \pm 0.1$ | +0.4 | +| $+$ Adam $\rightarrow$ Adam $W$ | $62.5 \pm 0.6$ | +0.6 | +| $+$ Step Decay $\rightarrow$ Cosine Decay | $63.2 \pm 0.4$ | +0.7 | +| + Normalize $\Delta_p$ | $63.6 \pm 0.4$ | +0.4 | +| + Scale down (PointNeXt-S) | $63.4 \pm 0.8$ | -0.2 | +| + Scale up (PointNeXt-B) | $65.8 \pm 0.5$ | +2.4 | +| + Rotation | $67.3 \pm 0.2$ | +1.5 | +| + Scale up (PointNeXt-L) | $69.0 \pm 0.5$ | +1.7 | +| + Scale up (PointNeXt-XL) | $70.5 \pm 0.3$ | +1.5 | + +Optimization techniques involve loss functions, optimizers, learning rate schedulers, and hyperparameters. As shown in Tab. [4](#page-7-0) and [5,](#page-7-0) Label Smoothing, AdamW [\[27\]](#page-11-10) optimizer, and Cosine Decay consistently boost performance in both classification and segmentation tasks. This reveals that the more developed optimization methods such as label smoothing and AdamW are generally good for optimizing a neural network. Compared to Step Decay, Cosine Decay is also easier to tune (usually only the initial and minimum learning rates are required) and can achieve a performance similar to Step Decay. Regarding hyperparameters, using a learning rate greater than that used in PointNet++ improves the segmentation performance in S3DIS. + +In general, our training strategies consisted of stronger data augmentation and modern optimization techniques can increase the performance of PointNet++ from 77.9% to 86.1% OA in ScanObjectNN dataset, impressively surpassing SOTA PointMLP by 0.7%. The mIoUs in S3DIS area 5 and S3DIS 6-fold (illustrated in Fig. [1\)](#page-1-0) are boosted by 11.7 and 13.6 absolute percentage points, respectively. Our observations imply that *a significant portion of the performance gap between classical PointNet++ and SOTA is due to the training strategies.* + +Generalize to other networks. Although the training strategies are proposed for PointNet++ [\[30\]](#page-11-1), we find that they can be applied to other methods such as PointNet [\[29\]](#page-11-0), DGCNN [\[46\]](#page-12-3), and PointMLP [\[28\]](#page-11-8), and also improve their performance. Such generalizability is validated in ScanObjectNN [\[44\]](#page-12-2). As shown in Tab. [6,](#page-8-1) the OA of the representative methods can all be improved when equipped with our training strategies. + +Table 6: The generalizability of improved training strategies. OA on ScanObjectNN of networks trained with improved training strategies is reported. + +| Method | ours | ∆ | +|---------------|------------|------| +| PointNet [29] | 74.4 ± 0.9 | +6.2 | +| DGCNN [46] | 86.0 ± 0.5 | +7.9 | +| PointMLP [28] | 87.1 ± 0.7 | +1.7 | + +#### 4.4.2 Model Scaling + +Receptive field scaling includes both radius scaling and normalizing ∆p defined in Eqn. [\(2\)](#page-3-2), which are also validated in Tab. [4](#page-7-0) and [5.](#page-7-0) The radius is dataset specific, while down-scaling the radius from 0.2 to 0.15 improves 0.3% OA in ScanObjectNN, keeping the radius the same as 0.1 achieves the best performance in S3DIS. Regarding normalizing ∆p, it improves the performance in ScanObjectNN and S3DIS by 0.3 OA and 0.4 mIoU, respectively. Furthermore, in Tab. [7,](#page-8-2) we show that normalizing ∆p has a larger impact (2.3 mIoU in S3DIS dataset) on the bigger model PointNext-XL. + +Model scaling scales PointNet++ by the proposed InvResMLP and some macro-architectural changes (see Sec. [3.2.2\)](#page-3-1). In Tab. [4,](#page-7-0) we show that PointNeXt-S using the stem MLP, the symmetric decoder, and the residual connection in the SA block improves 1.0% OA in ScanObjectNN. Performance in the large-scale S3DIS dataset can be further unveiled (from 63.8% to 70.5% mIoU) by up-scaling PointNeXt-S using more blocks of the proposed InvResMLP, as demonstrated in Tab. [5.](#page-7-0) Furthermore, in Tab. [7,](#page-8-2) we ablate each component of the proposed InvResMLP + +block and different stage ratios in S3DIS area 5 using the best-performed model PointNeXt-XL as the baseline. As observed, each architectural change indeed contributes to increased performance. Among all changes, the residual connection is the most essential, without which the mIoU will drop from 70.5% to only 64.0%. The separable MLPs increase 3.9% mIoU while speeding up the network 3 times. Removing the inverted bottleneck from the baseline leads to a drop of 1.5% mIoU with less than a 1% gain in speed. Adding more blocks inside each stage after removing inverted bottleneck can improve its performance to 69.7 ± 0.3 but is still lower than the baseline. Tab. [7](#page-8-2) also shows the performance of naive width scaling that increases the width of PointNet++ from 32 to 256 to match the throughput of PointNeXt-XL, naive depth + +Table 7: Ablate architectural changes on S3DIS area 5. − denotes removing from baseline. TP denotes throughput. + +| Ablate | mIoU | ∆ | TP | +|-------------------------|------------|-------|----| +| baseline (PointNeXt-XL) | 70.5 ± 0.3 | - | 45 | +| − normalizing ∆p | 68.2 ± 0.7 | -2.3 | 45 | +| − residual connection | 64.0 ± 1.0 | -6.5 | 45 | +| − stem MLP | 70.1 ± 0.4 | -0.4 | 46 | +| − Separable MLPs | 66.6 ± 0.8 | -3.9 | 15 | +| − Inverted bottleneck | 69.0 ± 0.4 | -1.5 | 48 | +| − Inverted bottleneck | 69.7 ± 0.3 | -0.8 | 43 | +| stage ratio → (1:1:1:1) | 69.8 ± 0.6 | -0.7 | 52 | +| stage ratio → (2:1:1:1) | 69.4 ± 0.4 | -1.1 | 41 | +| stage ratio → (1:1:2:1) | 69.9 ± 0.6 | -0.6 | 47 | +| stage ratio → (1:1:1:2) | 69.5 ± 0.4 | -1.0 | 48 | +| stage ratio → (1:3:1:1) | 70.1 ± 0.4 | -0.4 | 39 | +| naive width scaling | 59.4 ± 0.1 | -11.1 | 43 | +| naive depth scaling | 63.4 ± 0.5 | -7.1 | 53 | +| naive compound scaling | 62.3 ± 1.2 | -8.2 | 24 | + +scaling to append more SA blocks in PointNet++ to obtain the same number of blocks of PointNext-XL whose B = (3, 6, 3, 3), and naive compound scaling to double the width of the naive depth scaled model to the same width as PointNeXt-XL (C = 64). Our proposed model scaling strategy achieves much higher performance than these naive scaling strategies, while being much faster. + +## 5 Related Work + +*Point-based methods* process point clouds directly using their unstructured format compared to voxel-based methods [\[10,](#page-10-15) [4\]](#page-10-16) and multi view-based methods [\[37,](#page-11-15) [12,](#page-10-13) [9\]](#page-10-12). PointNet [\[29\]](#page-11-0), the pioneering work of point-based methods, proposes to model the permutation invariance of points with shared MLPs by restricting feature extraction to be pointwise. PointNet++ [\[30\]](#page-11-1) is presented to improve PointNet by capturing local geometric structures. Currently, most point-based methods focus on the design of local modules. [\[46,](#page-12-3) [45,](#page-12-11) [31\]](#page-11-16) rely on graph neural networks. [\[51,](#page-12-12) [22,](#page-11-3) [43,](#page-12-0) [42\]](#page-12-13) project point clouds onto pseudo grids to allow for regular convolutions. [\[48,](#page-12-14) [23,](#page-11-13) [24\]](#page-11-17) adaptively aggregate neighborhood features through weights determined by the local structure. In addition, very recent methods leverage Transformer-like networks [\[56,](#page-12-1) [17\]](#page-10-10) to extract local information through self-attention. Our work does not follow this trend in local module design. In contrast, we shift our attention to another important but largely under-explored aspect, *i.e.*, the training and scaling strategies. + +*Training strategies* are studied recently in [\[2,](#page-10-1) [47,](#page-12-15) [26\]](#page-11-18) on image classification. In the point cloud domain, SimpleView [\[9\]](#page-10-12) is the first work to show that training strategies have a large impact on the performance of a neural network. However, SimpleView simply adopts the same training strategies as DGCNN [\[46\]](#page-12-3). On the contrary, we conducted a systematic study to quantify the effect of *each* data augmentation and optimization technique, and propose a set of improved training strategies that boost the performance of PointNet++ [\[30\]](#page-11-1) and other representative works [\[29,](#page-11-0) [46,](#page-12-3) [28\]](#page-11-8). + +*Model scaling* can significantly improve the performance of a network, as shown in pioneering works in various domains [\[40,](#page-12-16) [55,](#page-12-4) [21\]](#page-11-4). Compared to PointNet++ [\[30\]](#page-11-1) that uses parameters less than 2M, most current prevailing networks consist of parameters greater than 10 M, such as KPConv [\[43\]](#page-12-0) (15M) and PointMLP [\[28\]](#page-11-8) (13M). In our work, we explore model scaling strategies that can scale up PointNet++ in an effective and efficient manner. We offer practical suggestions on scaling technologies that improve performance, namely using residual connections and an inverted bottleneck design, while maintaining throughput by using separable MLPs. + +## 6 Conclusion and Discussion + +In this paper, we demonstrate that with improved training and scaling strategies, the performance of PointNet++ can be increased to exceed the current state of the art. More specifically, we quantify the effect of each data augmentation and optimization technique that are widely used today, and propose a set of improved training strategies. These strategies can be easily applied to boost the performance of PointNet++ and other representative works. We also introduce the Inverted Residual MLP block into PointNet++ to develop PointNeXt. We demonstrate that PointNeXt has superior performance and scalability over PointNet++ on various benchmarks while maintaining high throughput. This work aims to guide researchers toward paying more attention to the effects of training and scaling strategies and motivate future work in this direction. + +Limitation. Even though PointNeXt-XL is one of the largest models among all representative pointbased networks [\[30,](#page-11-1) [43,](#page-12-0) [15,](#page-10-5) [56\]](#page-12-1), its number of parameters (44M) is still below that of small networks in image classification such as Swin-S [\[25\]](#page-11-19) (50M), ConNeXt-S [\[26\]](#page-11-18) (50M), and ViT-B [\[8\]](#page-10-17) (87M), and is far from their large variants, including Swin-L (197M), ConvNeXt-XL (350M), and ViT-L (305M). In this work, we do not push the model size further, mainly due to the smaller-scale nature of point cloud datasets compared to their larger image counterparts, such as ImageNet [\[6\]](#page-10-18). Moreover, our work is limited to existing modules since the focus is not on introducing new architectural changes. + +Acknowledgement. The authors would like to thank the reviewers of NeurIPS'22 for their constructive suggestions. This work was supported by the King Abdullah University of Science and Technology (KAUST) Office of Sponsored Research through the Visual Computing Center (VCC) funding, as well as, the SDAIA-KAUST Center of Excellence in Data Science and Artificial Intelligence (SDAIA-KAUST AI). + +## References + +- [1] Iro Armeni, Ozan Sener, Amir R Zamir, Helen Jiang, Ioannis Brilakis, Martin Fischer, and Silvio Savarese. 3d semantic parsing of large-scale indoor spaces. In *Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)*, pages 1534–1543, 2016. +- [2] Irwan Bello, William Fedus, Xianzhi Du, Ekin Dogus Cubuk, Aravind Srinivas, Tsung-Yi Lin, Jonathon Shlens, and Barret Zoph. 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In *Proceedings of the IEEE/CVF International Conference on Computer Vision (ICCV)*, pages 16259–16268, 2021. + +# **PointNeXt: Revisiting PointNet++ with Improved Training and Scaling Strategies** + +## - Supplementary Material - + +In this appendix, we provide additional content to complement the main manuscript: + +- Appendix A: A detailed description of Tab. 7. +- Appendix B: Comparisons of training strategies for prior representative works and PointNeXt. +- Appendix C: Qualitative comparisons on S3DIS and ShapeNetPart. +- Appendix D: The architecture of PointNeXt for classification. +- Appendix E: Societal impact. + +## A Detailed Description for Manuscript Tab. 7 + +Naive width scaling increases the channel size of PointNet++ from 32 to 256 to match the throughput of the baseline model, PointNeXt-XL. Naive depth scaling refers to appending more SA blocks (B=(3,6,3,3)), the same as PointNext-XL) in PointNet++. Furthermore, naive compound scaling doubles the width of naive depth scaled model to the same as PointNeXt-XL (C=64). Compared to the PointNet++ trained with improved training strategies (63.2% mIoU, 186 ins./sec.), naive depth scaling (63.4% mIoU, 53 ins. / sec.) and naive width scaling (59.4% mIoU, 43 ins./sec.) only lead to a large overhead in throughput with insignificant improvement in accuracy. In contrast, our proposed model scaling strategy achieves much higher performance than the naive scaling strategies while being much faster. This can be observed by comparing PointNeXt-XL (70.5% mIoU, 45 ins./sec.) to the naive compound scaled PointNet++ (62.3% mIoU, 24 ins./sec.). + +#### **B** Training Strategies Comparison + +In this section, we summarize the training strategies used in representative point-based methods such as DGCNN [46], KPConv [43], PointMLP [28], Point Transformer [56], Stratified Transformer [17], PointNet++ [30], and our PointNeXt on S3DIS [1] in Tab. I, on ScanObjectNN [44] in Tab. II, on ScanNet [5] in Tab. III, and on ShapeNetPart [53] in Tab. IV, respectively. + +## C Qualitative Results + +We provide qualitative results of PointNeXt-XL for S3DIS (Fig. II) and PointNeXt-S (C=160) for ShapeNetPart (Fig. III). The qualitative results of PointNet++ trained with the original training strategies are also included in the figures for comparison. On both datasets, PointNeXt produces predictions closer to the ground truth compared to PointNet++. More specifically, on S3DIS shown in (Fig. II), PointNeXt is able to segment hard classes, including doors ( $1^{st}$ , $3^{rd}$ , and $4^{th}$ rows), clutter ( $1^{st}$ and $3^{rd}$ rows), chairs ( $2^{nd}$ row), and the board ( $4^{th}$ row), while PointNet++ fails to segment properly to some extent. On ShapeNetPart (Fig. III), PointNeXt precisely segments wings of an airplane ( $1^{st}$ row), microphone of an earphone( $2^{nd}$ row), body of a motorbike( $3^{rd}$ row), fin of a rocket( $4^{th}$ row), and bearing of a skateboard ( $5^{th}$ row). + +#### D Classification Architecture + +As illustrated in Fig. I, the classification architecture shares the same encoder as the segmentation one. The output features of the encoder are passed to a global pooling layer (*i.e.* global max-pooling) to acquire a global shape representation for classification. Note that the points are only downsampled by a factor of 2 in each stage, since the number of input points in classification tasks is usually small, *e.g.* 1024 or 2048 points. + +Table I: Training strategies used in different methods for S3DIS segmentation. + + + +| Method | DGCNN | KPConv | PointTransformer | PointNet++ | PointNeXt (Ours) | +|-------------------------------|--------------------|--------------------|------------------|--------------------|------------------| +| Epochs | 101 | 500 | 100 | 32 | 100 | +| Batch size | 12 | 10 | 16 | 16 | 8 | +| Optimizer | Adam | SGD | SGD | Adam | AdamW | +| LR | $1 \times 10^{-3}$ | $1 \times 10^{-2}$ | 0.5 | $1 \times 10^{-3}$ | 0.01 | +| LR decay | step | step | multi step | step | cosine | +| Weight decay | 0 | $10^{-3}$ | $10^{-4}$ | $10^{-4}$ | $10^{-4}$ | +| Label smoothing $\varepsilon$ | X | Х | × | × | 0.2 | +| Entire scene as input | X | Х | ✓ | × | ✓ | +| Random rotation | X | ✓ | X | ✓ | ✓ | +| Random scaling | X | [0.8, 1.2] | [0.9, 1.1] | X | [0.9, 1.1] | +| Random translation | X | X | X | X | X | +| Random jittering | X | 0.001 | X | X | ✓ | +| Height appending | X | ✓ | X | X | ✓ | +| Color drop | × | 0.2 | X | X | 0.2 | +| Color auto-contrast | X | × | ✓ | X | ✓ | +| Color jittering | X | X | ✓ | × | X | +| mIoU (%) | 56.1 | 70.6 | 73.5 | 54.5 | 74.9 | + +Table II: Training strategies used in different methods for ScanObecjectNN classification. + +| Method | DGCNN | PointMLP | PointNet++ | PointNeXt (Ours) | +|-------------------------------|--------------------|-----------|------------|--------------------| +| Epochs | 250 | 200 | 250 | 250 | +| Batch size | 32 | 32 | 16 | 32 | +| Optimizer | Adam | SGD | Adam | AdamW | +| LR | $1 \times 10^{-3}$ | 0.01 | $10^{-3}$ | $2 \times 10^{-3}$ | +| LR decay | step | cosine | step | cosine | +| Weight decay | $10^{-4}$ | $10^{-4}$ | $10^{-4}$ | 0.05 | +| Label smoothing $\varepsilon$ | 0.2 | 0.2 | × | 0.3 | +| Point resampling | Х | Х | × | ✓ | +| Random rotation | 1 | × | ✓ | ✓ | +| Random scaling | X | ✓ | X | ✓ | +| Random translation | X | ✓ | X | X | +| Random jittering | / | × | ✓ | X | +| Height appending | X | × | × | ✓ | +| OA (%) | 78.1 | 85.7 | 77.9 | 87.7 | + +## **E** Societal Impact + +We do not see an immediate negative societal impact from our work. We notice that the way we discover the improved training and scaling strategies may consume a little more computing resources and affect the environment. Nevertheless, the improved training and scaling strategies will make researchers pay more attention to aspects other than architectural changes, which in the long term makes research in computer vision more diverse and generally better. + +Table III: Training strategies used in different methods for ScanNet segmentation. + + + +| Method | KPConv | PointTransformer | Stratified Transformer | PointNet++ | PointNeXt (Ours) | +|-----------------------|--------------------|--------------------|-------------------------|--------------------|--------------------| +| Epochs | 500 | 100 | 100 | 200 | 100 | +| Batch size | 10 | 16 | 8 | 32 | 2 | +| Optimizer | SGD | SGD | AdamW | Adam | AdamW | +| LR | $1 \times 10^{-2}$ | $5 \times 10^{-1}$ | $6 \times 10^{-3}$ | $1 \times 10^{-3}$ | $1 \times 10^{-3}$ | +| LR decay | step | multi step | multi step with warm up | step | multi step | +| Weight decay | $10^{-3}$ | $10^{-4}$ | $5 \times 10^{-2}$ | $10^{-4}$ | $10^{-4}$ | +| Entire scene as input | X | 1 | / | × | / | +| Random rotation | 1 | X | ✓ | ✓ | ✓ | +| Random scaling | [0.9,1.1] | [0.9,1.1] | [0.8,1.2] | × | [0.8,1.2] | +| Random translation | X | X | × | × | Х | +| Random jittering | 0.001 | X | × | × | X | +| Height appending | / | X | × | × | ✓ | +| Color drop | X | X | 0.2 | × | 0.2 | +| Color auto-contrast | × | ✓ | × | × | ✓ | +| Color jittering | × | ✓ | × | / × | × | +| Test mIoU (%) | 68.6 | - | 73.7 | 55.7 | 71.2 | + +Table IV: Training strategies used in different methods for ShapeNetPart segmentation. + +| Method | DGCNN | KPConv | PointNet++ | PointNeXt (Ours) | +|-------------------------------|--------------------|--------------------|--------------------|------------------| +| Epochs | 201 | 500 | 201 | 300 | +| Batch size | 16 | 16 | 32 | 8 | +| Optimizer | Adam | SGD | Adam | AdamW | +| LR | $3 \times 10^{-3}$ | $1 \times 10^{-2}$ | $1 \times 10^{-3}$ | 0.001 | +| LR decay | step | step | step | multi step | +| Weight decay | 0.0 | $10^{-3}$ | 0.0 | $10^{-4}$ | +| Label smoothing $\varepsilon$ | X | X | × | × | +| Random rotation | Х | Х | × | ✓ | +| Random scaling | X | [0.9, 1.1] | X | [0.8, 1.2] | +| Random translation | X | × | X | X | +| Random jittering | X | 0.001 | ✓ | 0.001 | +| Normal Drop | X | X | X | ✓ | +| Height appending | X | ✓ | × | ✓ | +| mIoU (%) | 85.2 | 86.4 | 85.1 | 87.0 | + +![](PointNeXt_2206.04670_images/_page_15_Figure_4.jpeg) + +Figure I: **PointNeXt architecture for classification.** The classification architecture shares the same encoder as the segmentation architecture. + +> **[그림 해설]** PointNeXt의 3D 객체 분류(Classification) 아키텍처 다이어그램. +> - 입력 포인트 클라우드 $\to$ 초기 MLP $[N, 32]$. +> - 4단계 Set Abstraction + InvResMLP 인코더: $[N/2, 64] \to [N/4, 128] \to [N/8, 256] \to [N/16, 512]$로 계층적 특징 추출. +> - 최종단에 **Global Pooling**을 적용하여 전역 특징 벡터를 집계한 뒤 분류 점수를 도출 (세그멘테이션과 동일한 백본 인코더 설계 공유). + +![](PointNeXt_2206.04670_images/_page_16_Figure_0.jpeg) + +Figure II: Qualitative comparisons of PointNet++ ( $2^{nd}$ column), PointNeXt ( $3^{rd}$ column), and Ground Truth ( $4^{th}$ column) on S3DIS semantic segmentation. The input point cloud is visualized with original colors in the $1^{st}$ column. Differences between PointNet++ and PointNeXt are highlighted with red dash circles. Zoom-in for details. + +> **[그림 해설]** S3DIS 실내 씬 5개에 대한 시맨틱 세그멘테이션 정성적 결과 비교 (Input vs PointNet++ vs PointNeXt vs Ground Truth). +> - **색상 범례**: ceiling(초록), floor(파랑), wall(하늘), beam(노랑), column(자주), window(남색), door(황갈), table(보라), chair(빨강), sofa(연분홍), bookcase(청록), board(회색), clutter(검정). +> - 빨간 점선 원으로 표시된 확대 영역에서 PointNet++는 문(door), 의자(chair), 화이트보드(board) 경계를 벽면으로 오분류하거나 누락하는 반면, **PointNeXt**는 복잡한 가구의 세부 윤곽과 벽 부착물 경계를 Ground Truth와 거의 완벽히 일치하게 분할함. + +![](PointNeXt_2206.04670_images/_page_17_Figure_0.jpeg) + +Figure III: Qualitative comparisons of PointNet++ (left), PointNeXt (middle), and Ground Truth (right) on ShapeNetPart part segmentation. + +> **[그림 해설]** ShapeNetPart 3D 파트 분할 정성적 비교 (비행기, 헤드폰, 오토바이, 로켓, 스케이트보드 5개 객체). +> - **빨간 점선 원 표시 영역**: +> - 비행기 날개 밑 제트 엔진(노란색): PointNet++는 누락/과소예측하나 PointNeXt는 선명히 감지. +> - 헤드폰 이어패드와 헤드밴드 결합부: PointNeXt가 정밀한 경계 구분 성공. +> - 오토바이 핸들/미러 및 로켓 날개 핀, 스케이트보드 트럭(바퀴 축): PointNeXt가 미세한 구조적 파트를 GT와 일치하게 분할. \ No newline at end of file diff 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PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation + +Charles R. Qi\* Hao Su\* Kaichun Mo Leonidas J. Guibas Stanford University + +### Abstract + +*Point cloud is an important type of geometric data structure. Due to its irregular format, most researchers transform such data to regular 3D voxel grids or collections of images. This, however, renders data unnecessarily voluminous and causes issues. In this paper, we design a novel type of neural network that directly consumes point clouds, which well respects the permutation invariance of points in the input. Our network, named PointNet, provides a unified architecture for applications ranging from object classification, part segmentation, to scene semantic parsing. Though simple, PointNet is highly efficient and effective. Empirically, it shows strong performance on par or even better than state of the art. Theoretically, we provide analysis towards understanding of what the network has learnt and why the network is robust with respect to input perturbation and corruption.* + +## 1. Introduction + +In this paper we explore deep learning architectures capable of reasoning about 3D geometric data such as point clouds or meshes. Typical convolutional architectures require highly regular input data formats, like those of image grids or 3D voxels, in order to perform weight sharing and other kernel optimizations. Since point clouds or meshes are not in a regular format, most researchers typically transform such data to regular 3D voxel grids or collections of images (e.g, views) before feeding them to a deep net architecture. This data representation transformation, however, renders the resulting data unnecessarily voluminous — while also introducing quantization artifacts that can obscure natural invariances of the data. + +For this reason we focus on a different input representation for 3D geometry using simply point clouds – and name our resulting deep nets *PointNets*. Point clouds are simple and unified structures that avoid the combinatorial irregularities and complexities of meshes, and thus are easier to learn from. The PointNet, however, + +![](PointNet_1612.00593_images/_page_0_Figure_9.jpeg) + +Figure 1. Applications of PointNet. We propose a novel deep net architecture that consumes raw point cloud (set of points) without voxelization or rendering. It is a unified architecture that learns both global and local point features, providing a simple, efficient and effective approach for a number of 3D recognition tasks. + +> **[그림 해설]** PointNet이 다루는 세 가지 핵심 3D 태스크를 요약한 다이어그램. +> - **Classification(분류)**: 컵(mug), 테이블(table), 자동차(car) 등 단일 3D 포인트 클라우드 입력을 받아 객체의 전체 클래스 레이블을 판별. +> - **Part Segmentation(부품 분할)**: 램프, 비행기(동체·날개·엔진), 탁자(상판·다리) 등 단일 객체 내 각 부품 영역을 포인트 단위로 분할하여 색상별로 구분. +> - **Semantic Segmentation(시맨틱 분할)**: 실내 공간 전체 포인트 클라우드에서 바닥, 벽, 의자, 테이블 등 복합 환경의 각 구성 요소를 포인트별 시맨틱 클래스로 분류. + +still has to respect the fact that a point cloud is just a set of points and therefore invariant to permutations of its members, necessitating certain symmetrizations in the net computation. Further invariances to rigid motions also need to be considered. + +Our PointNet is a unified architecture that directly takes point clouds as input and outputs either class labels for the entire input or per point segment/part labels for each point of the input. The basic architecture of our network is surprisingly simple as in the initial stages each point is processed identically and independently. In the basic setting each point is represented by just its three coordinates (x, y, z). Additional dimensions may be added by computing normals and other local or global features. + +Key to our approach is the use of a single symmetric function, max pooling. Effectively the network learns a set of optimization functions/criteria that select interesting or informative points of the point cloud and encode the reason for their selection. The final fully connected layers of the network aggregate these learnt optimal values into the global descriptor for the entire shape as mentioned above (shape classification) or are used to predict per point labels (shape segmentation). + +Our input format is easy to apply rigid or affine transformations to, as each point transforms independently. Thus we can add a data-dependent spatial transformer network that attempts to canonicalize the data before the PointNet processes them, so as to further improve the results. + +\* indicates equal contributions. + +We provide both a theoretical analysis and an experimental evaluation of our approach. We show that our network can approximate any set function that is continuous. More interestingly, it turns out that our network learns to summarize an input point cloud by a sparse set of key points, which roughly corresponds to the skeleton of objects according to visualization. The theoretical analysis provides an understanding why our PointNet is highly robust to small perturbation of input points as well as to corruption through point insertion (outliers) or deletion (missing data). + +On a number of benchmark datasets ranging from shape classification, part segmentation to scene segmentation, we experimentally compare our PointNet with state-ofthe-art approaches based upon multi-view and volumetric representations. Under a unified architecture, not only is our PointNet much faster in speed, but it also exhibits strong performance on par or even better than state of the art. + +The key contributions of our work are as follows: + +- We design a novel deep net architecture suitable for consuming unordered point sets in 3D; +- We show how such a net can be trained to perform 3D shape classification, shape part segmentation and scene semantic parsing tasks; +- We provide thorough empirical and theoretical analysis on the stability and efficiency of our method; +- We illustrate the 3D features computed by the selected neurons in the net and develop intuitive explanations for its performance. + +The problem of processing unordered sets by neural nets is a very general and fundamental problem – we expect that our ideas can be transferred to other domains as well. + +### 2. Related Work + +Point Cloud Features Most existing features for point cloud are handcrafted towards specific tasks. Point features often encode certain statistical properties of points and are designed to be invariant to certain transformations, which are typically classified as intrinsic [\[2,](#page-8-0) [24,](#page-8-1) [3\]](#page-8-2) or extrinsic [\[20,](#page-8-3) [19,](#page-8-4) [14,](#page-8-5) [10,](#page-8-6) [5\]](#page-8-7). They can also be categorized as local features and global features. For a specific task, it is not trivial to find the optimal feature combination. + +Deep Learning on 3D Data 3D data has multiple popular representations, leading to various approaches for learning. *Volumetric CNNs:* [\[28,](#page-8-8) [17,](#page-8-9) [18\]](#page-8-10) are the pioneers applying 3D convolutional neural networks on voxelized shapes. However, volumetric representation is constrained by its resolution due to data sparsity and computation cost of 3D convolution. FPNN [\[13\]](#page-8-11) and Vote3D [\[26\]](#page-8-12) proposed special methods to deal with the sparsity problem; however, their operations are still on sparse volumes, it's challenging for them to process very large point clouds. *Multiview CNNs:* [\[23,](#page-8-13) [18\]](#page-8-10) have tried to render 3D point cloud or shapes into 2D images and then apply 2D conv nets to classify them. With well engineered image CNNs, this line of methods have achieved dominating performance on shape classification and retrieval tasks [\[21\]](#page-8-14). However, it's nontrivial to extend them to scene understanding or other 3D tasks such as point classification and shape completion. *Spectral CNNs:* Some latest works [\[4,](#page-8-15) [16\]](#page-8-16) use spectral CNNs on meshes. However, these methods are currently constrained on manifold meshes such as organic objects and it's not obvious how to extend them to non-isometric shapes such as furniture. *Feature-based DNNs:* [\[6,](#page-8-17) [8\]](#page-8-18) firstly convert the 3D data into a vector, by extracting traditional shape features and then use a fully connected net to classify the shape. We think they are constrained by the representation power of the features extracted. + +Deep Learning on Unordered Sets From a data structure point of view, a point cloud is an unordered set of vectors. While most works in deep learning focus on regular input representations like sequences (in speech and language processing), images and volumes (video or 3D data), not much work has been done in deep learning on point sets. + +One recent work from Oriol Vinyals et al [\[25\]](#page-8-19) looks into this problem. They use a read-process-write network with attention mechanism to consume unordered input sets and show that their network has the ability to sort numbers. However, since their work focuses on generic sets and NLP applications, there lacks the role of geometry in the sets. + +### 3. Problem Statement + +We design a deep learning framework that directly consumes unordered point sets as inputs. A point cloud is represented as a set of 3D points {Pi | i = 1, ..., n}, where each point Pi is a vector of its (x, y, z) coordinate plus extra feature channels such as color, normal etc. For simplicity and clarity, unless otherwise noted, we only use the (x, y, z) coordinate as our point's channels. + +For the object classification task, the input point cloud is either directly sampled from a shape or pre-segmented from a scene point cloud. Our proposed deep network outputs k scores for all the k candidate classes. For semantic segmentation, the input can be a single object for part region segmentation, or a sub-volume from a 3D scene for object region segmentation. Our model will output n × m scores for each of the n points and each of the m semantic subcategories. + +![](PointNet_1612.00593_images/_page_2_Figure_0.jpeg) + +> **[그림 해설]** 'Classification Network' 텍스트 배너. 아래 Figure 2 다이어그램의 상단 파란색 블록에 해당하는 객체 분류 네트워크 영역을 지칭한다. + +![](PointNet_1612.00593_images/_page_2_Figure_1.jpeg) + +Figure 2. **PointNet Architecture.** The classification network takes n points as input, applies input and feature transformations, and then aggregates point features by max pooling. The output is classification scores for k classes. The segmentation network is an extension to the classification net. It concatenates global and local features and outputs per point scores. "mlp" stands for multi-layer perceptron, numbers in bracket are layer sizes. Batchnorm is used for all layers with ReLU. Dropout layers are used for the last mlp in classification net. + +> **[그림 해설]** PointNet의 전체 신경망 구조도. +> - **Classification Network (상단 파란색 영역)**: +> - 입력: $n \times 3$ 점 좌표. +> - **Input Transform**: T-Net을 통해 $3 \times 3$ 변환 행렬을 예측하여 입력 좌표 공간을 정렬 $\to n \times 3$. +> - **Shared MLP (64, 64)**: 각 점을 독립적으로 $64$차원 공간으로 매핑 $\to n \times 64$. +> - **Feature Transform**: T-Net을 통해 $64 \times 64$ 특징 변환 행렬을 예측하고 직교 정규화($L_{reg}$)를 적용하여 특징 공간 정렬 $\to n \times 64$. +> - **Shared MLP (64, 128, 1024)**: 점별 특징을 1024차원으로 확장 $\to n \times 1024$. +> - **Max Pool**: 대칭 함수(Symmetric Function)로 점 순서 불변성을 확보하며 전체 $n$개 점의 최댓값을 집계 $\to 1024$차원 Global Feature 생성. +> - **MLP (512, 256, k)**: 완전연결 레이어와 Dropout을 거쳐 $k$개 클래스 분류 점수(output scores) 출력. +> - **Segmentation Network (하단 노란색 영역)**: +> - $n \times 64$의 로컬 점 특징과 1024차원의 글로벌 특징 벡터를 결합(concatenation)하여 $n \times 1088$ 크기의 통합 특징 구성. +> - **Shared MLP (512, 256, 128)** $\to n \times 128$. +> - **Shared MLP (128, m)** $\to n \times m$ 출력 점수를 계산하여 $n$개의 각 점마다 $m$개 시맨틱/부품 범주 점수를 도출. + +### 4. Deep Learning on Point Sets + +The architecture of our network (Sec 4.2) is inspired by the properties of point sets in $\mathbb{R}^n$ (Sec 4.1). + +#### **4.1. Properties of Point Sets in** $\mathbb{R}^n$ + +Our input is a subset of points from an Euclidean space. It has three main properties: + +- Unordered. Unlike pixel arrays in images or voxel arrays in volumetric grids, point cloud is a set of points without specific order. In other words, a network that consumes N 3D point sets needs to be invariant to N! permutations of the input set in data feeding order. +- Interaction among points. The points are from a space with a distance metric. It means that points are not isolated, and neighboring points form a meaningful subset. Therefore, the model needs to be able to capture local structures from nearby points, and the combinatorial interactions among local structures. +- Invariance under transformations. As a geometric object, the learned representation of the point set should be invariant to certain transformations. For example, rotating and translating points all together should not modify the global point cloud category nor the segmentation of the points. + +#### 4.2. PointNet Architecture + +Our full network architecture is visualized in Fig 2, where the classification network and the segmentation network share a great portion of structures. Please read the caption of Fig 2 for the pipeline. + +Our network has three key modules: the max pooling layer as a symmetric function to aggregate information from + +all the points, a local and global information combination structure, and two joint alignment networks that align both input points and point features. + +We will discuss our reason behind these design choices in separate paragraphs below. + +Symmetry Function for Unordered Input In order to make a model invariant to input permutation, three strategies exist: 1) sort input into a canonical order; 2) treat the input as a sequence to train an RNN, but augment the training data by all kinds of permutations; 3) use a simple symmetric function to aggregate the information from each point. Here, a symmetric function takes n vectors as input and outputs a new vector that is invariant to the input order. For example, + and \* operators are symmetric binary functions. + +While sorting sounds like a simple solution, in high dimensional space there in fact does not exist an ordering that is stable w.r.t. point perturbations in the general sense. This can be easily shown by contradiction. If such an ordering strategy exists, it defines a bijection map between a high-dimensional space and a 1d real line. It is not hard to see, to require an ordering to be stable w.r.t point perturbations is equivalent to requiring that this map preserves spatial proximity as the dimension reduces, a task that cannot be achieved in the general case. Therefore, sorting does not fully resolve the ordering issue, and it's hard for a network to learn a consistent mapping from input to output as the ordering issue persists. As shown in experiments (Fig 5), we find that applying a MLP directly on the sorted point set performs poorly, though slightly better than directly processing an unsorted input. + +The idea to use RNN considers the point set as a sequential signal and hopes that by training the RNN + +with randomly permuted sequences, the RNN will become invariant to input order. However in "OrderMatters" [25] the authors have shown that order does matter and cannot be totally omitted. While RNN has relatively good robustness to input ordering for sequences with small length (dozens), it's hard to scale to thousands of input elements, which is the common size for point sets. Empirically, we have also shown that model based on RNN does not perform as well as our proposed method (Fig 5). + +Our idea is to approximate a general function defined on a point set by applying a symmetric function on transformed elements in the set: + +$$f(\lbrace x_1, \dots, x_n \rbrace) \approx g(h(x_1), \dots, h(x_n)), \tag{1}$$ + +where +$$f: 2^{\mathbb{R}^N} \to \mathbb{R}$$ +, $h: \mathbb{R}^N \to \mathbb{R}^K$ and $g: \mathbb{R}^K \times \cdots \times \mathbb{R}^K \to \mathbb{R}$ is a symmetric function. + +Empirically, our basic module is very simple: we approximate h by a multi-layer perceptron network and g by a composition of a single variable function and a max pooling function. This is found to work well by experiments. Through a collection of h, we can learn a number of f's to capture different properties of the set. + +While our key module seems simple, it has interesting properties (see Sec 5.3) and can achieve strong performace (see Sec 5.1) in a few different applications. Due to the simplicity of our module, we are also able to provide theoretical analysis as in Sec 4.3. + +**Local and Global Information Aggregation** The output from the above section forms a vector $[f_1,\ldots,f_K]$ , which is a global signature of the input set. We can easily train a SVM or multi-layer perceptron classifier on the shape global features for classification. However, point segmentation requires a combination of local and global knowledge. We can achieve this by a simple yet highly effective manner. + +Our solution can be seen in Fig 2 (Segmentation Network). After computing the global point cloud feature vector, we feed it back to per point features by concatenating the global feature with each of the point features. Then we extract new per point features based on the combined point features - this time the per point feature is aware of both the local and global information. + +With this modification our network is able to predict per point quantities that rely on both local geometry and global semantics. For example we can accurately predict per-point normals (fig in supplementary), validating that the network is able to summarize information from the point's local neighborhood. In experiment session, we also show that our model can achieve state-of-the-art performance on shape part segmentation and scene segmentation. **Joint Alignment Network** The semantic labeling of a point cloud has to be invariant if the point cloud undergoes certain geometric transformations, such as rigid transformation. We therefore expect that the learnt representation by our point set is invariant to these transformations. + +A natural solution is to align all input set to a canonical space before feature extraction. Jaderberg et al. [9] introduces the idea of spatial transformer to align 2D images through sampling and interpolation, achieved by a specifically tailored layer implemented on GPU. + +Our input form of point clouds allows us to achieve this goal in a much simpler way compared with [9]. We do not need to invent any new layers and no alias is introduced as in the image case. We predict an affine transformation matrix by a mini-network (T-net in Fig 2) and directly apply this transformation to the coordinates of input points. The mininetwork itself resembles the big network and is composed by basic modules of point independent feature extraction, max pooling and fully connected layers. More details about the T-net are in the supplementary. + +This idea can be further extended to the alignment of feature space, as well. We can insert another alignment network on point features and predict a feature transformation matrix to align features from different input point clouds. However, transformation matrix in the feature space has much higher dimension than the spatial transform matrix, which greatly increases the difficulty of optimization. We therefore add a regularization term to our softmax training loss. We constrain the feature transformation matrix to be close to orthogonal matrix: + +$$L_{reg} = ||I - AA^T||_F^2, (2)$$ + +where A is the feature alignment matrix predicted by a mini-network. An orthogonal transformation will not lose information in the input, thus is desired. We find that by adding the regularization term, the optimization becomes more stable and our model achieves better performance. + +#### 4.3. Theoretical Analysis + +**Universal approximation** We first show the universal approximation ability of our neural network to continuous set functions. By the continuity of set functions, intuitively, a small perturbation to the input point set should not greatly change the function values, such as classification or segmentation scores. + +Formally, let $\mathcal{X} = \{S : S \subseteq [0,1]^m \text{ and } |S| = n\}, f : \mathcal{X} \to \mathbb{R}$ is a continuous set function on $\mathcal{X}$ w.r.t to Hausdorff distance $d_H(\cdot,\cdot)$ , i.e., $\forall \epsilon > 0, \exists \delta > 0$ , for any $S, S' \in \mathcal{X}$ , if $d_H(S,S') < \delta$ , then $|f(S) - f(S')| < \epsilon$ . Our theorem says that f can be arbitrarily approximated by our network given enough neurons at the max pooling layer, i.e., K in (1) is sufficiently large. + +![](PointNet_1612.00593_images/_page_4_Figure_0.jpeg) + +Figure 3. Qualitative results for part segmentation. We visualize the CAD part segmentation results across all 16 object categories. We show both results for partial simulated Kinect scans (left block) and complete ShapeNet CAD models (right block). + +> **[그림 해설]** 16개 카테고리에 대한 3D 파트 분할(Part Segmentation) 정성적 결과 시각화. +> - **좌측 (Partial Inputs, 불완전 입력)**: 가상 Kinect 스캔으로 생성된 한쪽 면만 스캔되고 결손이 있는 포인트 클라우드에 대한 결과(table, motorbike, car, airplane, mug, lamp, guitar, chair 8종). 결손 및 가림이 있는 상태에서도 바퀴, 손잡이, 날개 등의 부품이 정확한 색상으로 분할됨. +> - **우측 (Complete Inputs, 완전 입력)**: ShapeNet 3D CAD 모델의 완전한 포인트 클라우드에 대한 결과(bag, knife, cap, skateboard, pistol, rocket, earphone, laptop 8종). 칼날과 손잡이, 모자 챙과 본체, 노트북 모니터와 본체 등 복잡한 기하학적 세부 부품이 정밀하게 구분됨. + +**Theorem 1.** Suppose $f: \mathcal{X} \to \mathbb{R}$ is a continuous set function w.r.t Hausdorff distance $d_H(\cdot, \cdot)$ . $\forall \epsilon > 0$ , $\exists$ a continuous function h and a symmetric function $g(x_1, \ldots, x_n) = \gamma \circ MAX$ , such that for any $S \in \mathcal{X}$ , + +$$\left| f(S) - \gamma \left( \max_{x_i \in S} \{h(x_i)\} \right) \right| < \epsilon$$ + +where $x_1, \ldots, x_n$ is the full list of elements in S ordered arbitrarily, $\gamma$ is a continuous function, and MAX is a vector max operator that takes n vectors as input and returns a new vector of the element-wise maximum. + +The proof to this theorem can be found in our supplementary material. The key idea is that in the worst case the network can learn to convert a point cloud into a volumetric representation, by partitioning the space into equal-sized voxels. In practice, however, the network learns a much smarter strategy to probe the space, as we shall see in point function visualizations. + +**Bottleneck dimension and stability** Theoretically and experimentally we find that the expressiveness of our network is strongly affected by the dimension of the max pooling layer, i.e., K in (1). Here we provide an analysis, which also reveals properties related to the stability of our model. + +We define $\mathbf{u} = \max_{x_i \in S} \{h(x_i)\}$ to be the sub-network of f which maps a point set in $[0,1]^m$ to a K-dimensional vector. The following theorem tells us that small corruptions or extra noise points in the input set are not likely to change the output of our network: + +**Theorem 2.** Suppose $\mathbf{u}: \mathcal{X} \to \mathbb{R}^K$ such that $\mathbf{u} = \max_{x \in S} \{h(x_i)\}$ and $f = \gamma \circ \mathbf{u}$ . Then, + +(a) +$$\forall S, \exists C_S, \mathcal{N}_S \subseteq \mathcal{X}, f(T) = f(S) \text{ if } C_S \subseteq T \subseteq \mathcal{N}_S;$$ + +(b) +$$|\mathcal{C}_S| \leq K$$ + +| | input | #views | accuracy | accuracy | +|------------------|--------|--------|------------|----------| +| | | | avg. class | overall | +| SPH [11] | mesh | - | 68.2 | - | +| 3DShapeNets [28] | volume | 1 | 77.3 | 84.7 | +| VoxNet [17] | volume | 12 | 83.0 | 85.9 | +| Subvolume [18] | volume | 20 | 86.0 | 89.2 | +| LFD [28] | image | 10 | 75.5 | - | +| MVCNN [23] | image | 80 | 90.1 | - | +| Ours baseline | point | - | 72.6 | 77.4 | +| Ours PointNet | point | 1 | 86.2 | 89.2 | + +Table 1. Classification results on ModelNet40. Our net achieves state-of-the-art among deep nets on 3D input. + +We explain the implications of the theorem. (a) says that f(S) is unchanged up to the input corruption if all points in $\mathcal{C}_S$ are preserved; it is also unchanged with extra noise points up to $\mathcal{N}_S$ . (b) says that $\mathcal{C}_S$ only contains a bounded number of points, determined by K in (1). In other words, f(S) is in fact totally determined by a finite subset $\mathcal{C}_S \subseteq S$ of less or equal to K elements. We therefore call $\mathcal{C}_S$ the critical point set of S and K the bottleneck dimension of f. + +Combined with the continuity of h, this explains the robustness of our model w.r.t point perturbation, corruption and extra noise points. The robustness is gained in analogy to the sparsity principle in machine learning models. Intuitively, our network learns to summarize a shape by a sparse set of key points. In experiment section we see that the key points form the skeleton of an object. + +#### 5. Experiment + +Experiments are divided into four parts. First, we show PointNets can be applied to multiple 3D recognition tasks (Sec 5.1). Second, we provide detailed experiments to validate our network design (Sec 5.2). At last we visualize what the network learns (Sec 5.3) and analyze time and space complexity (Sec 5.4). + +#### 5.1. Applications + +In this section we show how our network can be trained to perform 3D object classification, object part segmentation and semantic scene segmentation 1. Even though we are working on a brand new data representation (point sets), we are able to achieve comparable or even better performance on benchmarks for several tasks. + +**3D Object Classification** Our network learns global point cloud feature that can be used for object classification. We evaluate our model on the ModelNet40 [28] shape classification benchmark. There are 12,311 CAD models from 40 man-made object categories, split into 9,843 for + +&lt;sup>1More application examples such as correspondence and point cloud based CAD model retrieval are included in supplementary material. + + + +| | mean | aero | bag | cap | car | chair | ear | guitar knife | | lamp | laptop | motor | | mug pistol | rocket | skate | table | +|----------|------|------|------|------|------|-------|-------|--------------|------|------|--------|-------|-----|------------|--------|-------|-------| +| | | | | | | | phone | | | | | | | | | board | | +| # shapes | | 2690 | 76 | 55 | 898 | 3758 | 69 | 787 | 392 | 1547 | 451 | 202 | 184 | 283 | 66 | 152 | 5271 | +| Wu [27] | - | 63.2 | - | - | - | 73.5 | - | - | - | 74.4 | - | - | - | - | - | - | 74.8 | +| Yi [29] | 81.4 | 81.0 | 78.4 | 77.7 | 75.7 | 87.6 | 61.9 | 92.0 | 85.4 | 82.5 | 95.7 | 70.6 | | 91.9 85.9 | 53.1 | 69.8 | 75.3 | +| 3DCNN | 79.4 | 75.1 | 72.8 | 73.3 | 70.0 | 87.2 | 63.5 | 88.4 | 79.6 | 74.4 | 93.9 | 58.7 | | 91.8 76.4 | 51.2 | 65.3 | 77.1 | +| Ours | 83.7 | 83.4 | 78.7 | 82.5 | 74.9 | 89.6 | 73.0 | 91.5 | 85.9 | 80.8 | 95.3 | 65.2 | | 93.0 81.2 | 57.9 | 72.8 | 80.6 | + +Table 2. Segmentation results on ShapeNet part dataset. Metric is mIoU(%) on points. We compare with two traditional methods [\[27\]](#page-8-22) and [\[29\]](#page-8-23) and a 3D fully convolutional network baseline proposed by us. Our PointNet method achieved the state-of-the-art in mIoU. + +training and 2,468 for testing. While previous methods focus on volumetric and mult-view image representations, we are the first to directly work on raw point cloud. + +We uniformly sample 1024 points on mesh faces according to face area and normalize them into a unit sphere. During training we augment the point cloud on-the-fly by randomly rotating the object along the up-axis and jitter the position of each points by a Gaussian noise with zero mean and 0.02 standard deviation. + +In Table [1,](#page-4-2) we compare our model with previous works as well as our baseline using MLP on traditional features extracted from point cloud (point density, D2, shape contour etc.). Our model achieved state-of-the-art performance among methods based on 3D input (volumetric and point cloud). With only fully connected layers and max pooling, our net gains a strong lead in inference speed and can be easily parallelized in CPU as well. There is still a small gap between our method and multi-view based method (MVCNN [\[23\]](#page-8-13)), which we think is due to the loss of fine geometry details that can be captured by rendered images. + +3D Object Part Segmentation Part segmentation is a challenging fine-grained 3D recognition task. Given a 3D scan or a mesh model, the task is to assign part category label (e.g. chair leg, cup handle) to each point or face. + +We evaluate on ShapeNet part data set from [\[29\]](#page-8-23), which contains 16,881 shapes from 16 categories, annotated with 50 parts in total. Most object categories are labeled with two to five parts. Ground truth annotations are labeled on sampled points on the shapes. + +We formulate part segmentation as a per-point classification problem. Evaluation metric is mIoU on points. For each shape S of category C, to calculate the shape's mIoU: For each part type in category C, compute IoU between groundtruth and prediction. If the union of groundtruth and prediction points is empty, then count part IoU as 1. Then we average IoUs for all part types in category C to get mIoU for that shape. To calculate mIoU for the category, we take average of mIoUs for all shapes in that category. + +In this section, we compare our segmentation version PointNet (a modified version of Fig [2,](#page-2-2) *Segmentation Network*) with two traditional methods [\[27\]](#page-8-22) and [\[29\]](#page-8-23) that both take advantage of point-wise geometry features and correspondences between shapes, as well as our own 3D CNN baseline. See supplementary for the detailed modifications and network architecture for the 3D CNN. + +In Table [2,](#page-5-0) we report per-category and mean IoU(%) scores. We observe a 2.3% mean IoU improvement and our net beats the baseline methods in most categories. + +We also perform experiments on simulated Kinect scans to test the robustness of these methods. For every CAD model in the ShapeNet part data set, we use Blensor Kinect Simulator [\[7\]](#page-8-24) to generate incomplete point clouds from six random viewpoints. We train our PointNet on the complete shapes and partial scans with the same network architecture and training setting. Results show that we lose only 5.3% mean IoU. In Fig [3,](#page-4-3) we present qualitative results on both complete and partial data. One can see that though partial data is fairly challenging, our predictions are reasonable. + +Semantic Segmentation in Scenes Our network on part segmentation can be easily extended to semantic scene segmentation, where point labels become semantic object classes instead of object part labels. + +We experiment on the Stanford 3D semantic parsing data set [\[1\]](#page-8-25). The dataset contains 3D scans from Matterport scanners in 6 areas including 271 rooms. Each point in the scan is annotated with one of the semantic labels from 13 categories (chair, table, floor, wall etc. plus clutter). + +To prepare training data, we firstly split points by room, and then sample rooms into blocks with area 1m by 1m. We train our segmentation version of PointNet to predict + + + +| | mean IoU | overall accuracy | +|---------------|----------|------------------| +| Ours baseline | 20.12 | 53.19 | +| Ours PointNet | 47.71 | 78.62 | + +Table 3. Results on semantic segmentation in scenes. Metric is average IoU over 13 classes (structural and furniture elements plus clutter) and classification accuracy calculated on points. + +| | table | chair | sofa | board | mean | +|-------------------|-------|-------|------|-------|-------| +| # instance | 455 | 1363 | 55 | 137 | | +| Armeni et al. [1] | 46.02 | 16.15 | 6.78 | 3.91 | 18.22 | +| Ours | 46.67 | 33.80 | 4.76 | 11.72 | 24.24 | + +Table 4. Results on 3D object detection in scenes. Metric is average precision with threshold IoU 0.5 computed in 3D volumes. + +![](PointNet_1612.00593_images/_page_6_Figure_0.jpeg) + +Figure 4. Qualitative results for semantic segmentation. Top row is input point cloud with color. Bottom row is output semantic segmentation result (on points) displayed in the same camera viewpoint as input. + +> **[그림 해설]** Stanford 3D Semantic Parsing 데이터셋 실내 환경 3개 씬(오피스 2곳, 회의실 1곳)에 대한 Semantic Segmentation 결과. +> - **상단 (Input)**: RGB 컬러 정보가 포함된 원본 실내 공간 3D 포인트 클라우드 입력. +> - **하단 (Output)**: PointNet이 예측한 포인트별 시맨틱 레이블. +> - 색상 대응: 바닥(파란색), 벽(하늘색/청록색), 천장(초록색), 테이블(보라색), 의자(빨간색), 보드(회색), 책장/도어(노란색/연두색) 등으로 객체 및 실내 구조물이 명확히 분리됨. + +per point class in each block. Each point is represented by a 9-dim vector of XYZ, RGB and normalized location as to the room (from 0 to 1). At training time, we randomly sample 4096 points in each block on-the-fly. At test time, we test on all the points. We follow the same protocol as [\[1\]](#page-8-25) to use k-fold strategy for train and test. + +We compare our method with a baseline using handcrafted point features. The baseline extracts the same 9 dim local features and three additional ones: local point density, local curvature and normal. We use standard MLP as the classifier. Results are shown in Table [3,](#page-5-1) where our PointNet method significantly outperforms the baseline method. In Fig [4,](#page-6-2) we show qualitative segmentation results. Our network is able to output smooth predictions and is robust to missing points and occlusions. + +Based on the semantic segmentation output from our network, we further build a 3D object detection system using connected component for object proposal (see supplementary for details). We compare with previous stateof-the-art method in Table [4.](#page-5-2) The previous method is based on a sliding shape method (with CRF post processing) with SVMs trained on local geometric features and global room context feature in voxel grids. Our method outperforms it by a large margin on the furniture categories reported. + +### 5.2. Architecture Design Analysis + +In this section we validate our design choices by control experiments. We also show the effects of our network's hyperparameters. + +### Comparison with Alternative Order-invariant Methods + +As mentioned in Sec [4.2,](#page-2-0) there are at least three options for consuming unordered set inputs. We use the ModelNet40 shape classification problem as a test bed for comparisons of those options, the following two control experiment will also use this task. + +The baselines (illustrated in Fig [5\)](#page-6-0) we compared with include multi-layer perceptron on unsorted and sorted + +| rnn
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symmetry function | | + +Figure 5. Three approaches to achieve order invariance. Multilayer perceptron (MLP) applied on points consists of 5 hidden layers with neuron sizes 64,64,64,128,1024, all points share a single copy of MLP. The MLP close to the output consists of two layers with sizes 512,256. + +points as n×3 arrays, RNN model that considers input point as a sequence, and a model based on symmetry functions. The symmetry operation we experimented include max pooling, average pooling and an attention based weighted sum. The attention method is similar to that in [\[25\]](#page-8-19), where a scalar score is predicted from each point feature, then the score is normalized across points by computing a softmax. The weighted sum is then computed on the normalized scores and the point features. As shown in Fig [5,](#page-6-0) maxpooling operation achieves the best performance by a large winning margin, which validates our choice. + +### Effectiveness of Input and Feature Transformations In + +Table [5](#page-6-3) we demonstrate the positive effects of our input and feature transformations (for alignment). It's interesting to see that the most basic architecture already achieves quite reasonable results. Using input transformation gives a 0.8% performance boost. The regularization loss is necessary for the higher dimension transform to work. By combining both transformations and the regularization term, we achieve the best performance. + +Robustness Test We show our PointNet, while simple and effective, is robust to various kinds of input corruptions. We use the same architecture as in Fig [5'](#page-6-0)s max pooling network. Input points are normalized into a unit sphere. Results are in Fig [6.](#page-7-2) + +As to missing points, when there are 50% points missing, the accuracy only drops by 2.4% and 3.8% w.r.t. furthest and random input sampling. Our net is also robust to outlier + +| Transform | accuracy | +|------------------------|----------| +| none | 87.1 | +| input (3x3) | 87.9 | +| feature (64x64) | 86.9 | +| feature (64x64) + reg. | 87.4 | +| both | 89.2 | + +Table 5. Effects of input feature transforms. Metric is overall classification accuracy on ModelNet40 test set. + +![](PointNet_1612.00593_images/_page_7_Figure_0.jpeg) + +Figure 6. PointNet robustness test. The metric is overall classification accuracy on ModelNet40 test set. Left: Delete points. Furthest means the original 1024 points are sampled with furthest sampling. Middle: Insertion. Outliers uniformly scattered in the unit sphere. Right: Perturbation. Add Gaussian noise to each point independently. + +> **[그림 해설]** 포인트 결손, 이상치, 노이즈에 대한 PointNet의 강건성(Robustness) 평가 그래프 (ModelNet40 테스트셋 기준). +> - **좌측 (Missing data ratio, 데이터 결손)**: 점을 무작위(Random, 빨간 사각) 또는 최원점 샘플링(Furthest, 파란 원)으로 삭제했을 때의 정확도 변화. 50%의 점이 누락되어도 정확도는 약 86% 수준을 유지하며, 75% 누락 시에도 74~81%로 유지되다가 90% 이상 누락 시 급격히 하락. +> - **중앙 (Outlier ratio, 이상치 비율)**: 공간 내 임의의 노이즈 점 추가 시 정확도 변화. $XYZ$ 좌표만 사용한 경우(파란 사각)와 밀도 정보($XYZ+\text{density}$, 갈색 다이아몬드)를 함께 사용한 경우 모두 이상치 비율 30%까지 70% 이상의 정확도를 견고하게 유지. +> - **우측 (Perturbation noise std, 점 섭동 노이즈)**: 점 좌표에 가우시안 노이즈(표준편차 $0 \sim 0.1$)를 가했을 때의 정확도 변화. 표준편차 0.05까지 약 80% 이상을 유지하다가 0.1에 도달하면 약 30%로 감소. + +points, if it has seen those during training. We evaluate two models: one trained on points with (x, y, z) coordinates; the other on (x, y, z) plus point density. The net has more than 80% accuracy even when 20% of the points are outliers. Fig [6](#page-7-2) right shows the net is robust to point perturbations. + +### 5.3. Visualizing PointNet + +In Fig [7,](#page-7-3) we visualize *critical point sets* CS and *upperbound shapes* NS (as discussed in Thm [2\)](#page-4-4) for some sample shapes S. The point sets between the two shapes will give exactly the same global shape feature f(S). + +We can see clearly from Fig [7](#page-7-3) that the *critical point sets* CS, those contributed to the max pooled feature, summarizes the skeleton of the shape. The *upper-bound shapes* NS illustrates the largest possible point cloud that give the same global shape feature f(S) as the input point cloud S. CS and NS reflect the robustness of PointNet, meaning that losing some non-critical points does not change the global shape signature f(S) at all. + +The NS is constructed by forwarding all the points in a edge-length-2 cube through the network and select points p whose point function values (h1(p), h2(p), · · · , hK(p)) are no larger than the global shape descriptor. + +![](PointNet_1612.00593_images/_page_7_Figure_7.jpeg) + +Figure 7. Critical points and upper bound shape. While critical points jointly determine the global shape feature for a given shape, any point cloud that falls between the critical points set and the upper bound shape gives exactly the same feature. We color-code all figures to show the depth information. + +> **[그림 해설]** PointNet이 학습한 임계 포인트 세트($\mathcal{C}_S$)와 상한 형상($\mathcal{N}_S$)의 시각화 (테이블, 권총, 램프, 스탠드 4종, 깊이에 따라 무지개색 코딩). +> - **1행 (Original Shape, $S$)**: 원본 입력 포인트 클라우드. +> - **2행 (Critical Point Sets, $\mathcal{C}_S$)**: Max Pooling 레이어의 1024개 글로벌 특징을 결정짓는 핵심 포인트들만 추출한 서브셋. 물체의 외곽 윤곽(스켈레톤) 형태를 띄며, 입력 포인트의 극히 일부만으로 구성됨. +> - **3행 (Upper-bound Shapes, $\mathcal{N}_S$)**: 글로벌 특징 벡터 $\mathbf{u}$의 출력을 변화시키지 않으면서 최대로 추가할 수 있는 포인트들의 점유 영역. 원본 형태를 둘러싼 두꺼운 볼륨 형태를 형성하여, 노이즈나 추가 점이 이 영역 내에 존재해도 네트워크 출력이 불변함을 입증. + +### 5.4. Time and Space Complexity Analysis + +Table [6](#page-7-4) summarizes space (number of parameters in the network) and time (floating-point operations/sample) complexity of our classification PointNet. We also compare PointNet to a representative set of volumetric and multiview based architectures in previous works. + +While MVCNN [\[23\]](#page-8-13) and Subvolume (3D CNN) [\[18\]](#page-8-10) achieve high performance, PointNet is orders more efficient in computational cost (measured in FLOPs/sample: *141x* and *8x* more efficient, respectively). Besides, PointNet is much more space efficient than MVCNN in terms of #param in the network (*17x* less parameters). Moreover, PointNet is much more scalable – it's space and time complexity is O(N) – *linear* in the number of input points. However, since convolution dominates computing time, multi-view method's time complexity grows *squarely* on image resolution and volumetric convolution based method grows *cubically* with the volume size. + +Empirically, PointNet is able to process more than one million points per second for point cloud classification (around 1K objects/second) or semantic segmentation (around 2 rooms/second) with a 1080X GPU on Tensor-Flow, showing great potential for real-time applications. + +| | #params | FLOPs/sample | +|--------------------|---------|--------------| +| PointNet (vanilla) | 0.8M | 148M | +| PointNet | 3.5M | 440M | +| Subvolume [18] | 16.6M | 3633M | +| MVCNN [23] | 60.0M | 62057M | + +Table 6. Time and space complexity of deep architectures for 3D data classification. PointNet (vanilla) is the classification PointNet without input and feature transformations. FLOP stands for floating-point operation. The "M" stands for million. Subvolume and MVCNN used pooling on input data from multiple rotations or views, without which they have much inferior performance. + +### 6. Conclusion + +In this work, we propose a novel deep neural network *PointNet* that directly consumes point cloud. Our network provides a unified approach to a number of 3D recognition tasks including object classification, part segmentation and semantic segmentation, while obtaining on par or better results than state of the arts on standard benchmarks. We also provide theoretical analysis and visualizations towards understanding of our network. + +Acknowledgement. The authors gratefully acknowledge the support of a Samsung GRO grant, ONR MURI N00014- 13-1-0341 grant, NSF grant IIS-1528025, a Google Focused Research Award, a gift from the Adobe corporation and hardware donations by NVIDIA. + +### References + +- [1] I. Armeni, O. Sener, A. R. Zamir, H. Jiang, I. Brilakis, M. Fischer, and S. 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Sun, M. Ovsjanikov, and L. Guibas. A concise and provably informative multi-scale signature based on heat diffusion. In *Computer graphics forum*, volume 28, pages 1383–1392. Wiley Online Library, 2009. [2](#page-1-0) +- [25] O. Vinyals, S. Bengio, and M. Kudlur. Order matters: Sequence to sequence for sets. *arXiv preprint arXiv:1511.06391*, 2015. [2,](#page-1-0) [4,](#page-3-2) [7](#page-6-4) +- [26] D. Z. Wang and I. Posner. Voting for voting in online point cloud object detection. *Proceedings of the Robotics: Science and Systems, Rome, Italy*, 1317, 2015. [2](#page-1-0) +- [27] Z. Wu, R. Shou, Y. Wang, and X. Liu. Interactive shape cosegmentation via label propagation. *Computers & Graphics*, 38:248–254, 2014. [6,](#page-5-3) [10](#page-9-0) +- [28] Z. Wu, S. Song, A. Khosla, F. Yu, L. Zhang, X. Tang, and J. Xiao. 3d shapenets: A deep representation for volumetric shapes. In *Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition*, pages 1912–1920, 2015. [2,](#page-1-0) [5,](#page-4-5) [11](#page-10-0) +- [29] L. Yi, V. G. Kim, D. Ceylan, I.-C. Shen, M. Yan, H. Su, C. Lu, Q. Huang, A. Sheffer, and L. Guibas. A scalable active framework for region annotation in 3d shape collections. *SIGGRAPH Asia*, 2016. [6,](#page-5-3) [10,](#page-9-0) [18](#page-17-0) + +### Supplementary + +### A. Overview + +This document provides additional quantitative results, technical details and more qualitative test examples to the main paper. + +In Sec [B](#page-9-1) we extend the robustness test to compare PointNet with VoxNet on incomplete input. In Sec [C](#page-9-2) we provide more details on neural network architectures, training parameters and in Sec [D](#page-10-1) we describe our detection pipeline in scenes. Then Sec [E](#page-10-2) illustrates more applications of PointNet, while Sec [F](#page-11-0) shows more analysis experiments. Sec [G](#page-12-1) provides a proof for our theory on PointNet. At last, we show more visualization results in Sec [H.](#page-14-1) + +## B. Comparison between PointNet and VoxNet (Sec 5.2) + +We extend the experiments in Sec 5.2 Robustness Test to compare PointNet and VoxNet [\[17\]](#page-8-9) (a representative architecture for volumetric representation) on robustness to missing data in the input point cloud. Both networks are trained on the same train test split with 1024 number of points as input. For VoxNet we voxelize the point cloud to 32 × 32 × 32 occupancy grids and augment the training data by random rotation around up-axis and jittering. + +At test time, input points are randomly dropped out by a certain ratio. As VoxNet is sensitive to rotations, its prediction uses average scores from 12 viewpoints of a point cloud. As shown in Fig [8,](#page-9-3) we see that our PointNet is much more robust to missing points. VoxNet's accuracy dramatically drops when half of the input points are missing, from 86.3% to 46.0% with a 40.3% difference, while our PointNet only has a 3.7% performance drop. This can be explained by the theoretical analysis and explanation of our PointNet – it is learning to use a collection of *critical points* to summarize the shape, thus it is very robust to missing data. + +## C. Network Architecture and Training Details (Sec 5.1) + +PointNet Classification Network As the basic architecture is already illustrated in the main paper, here we provides more details on the joint alignment/transformation network and training parameters. + +The first transformation network is a mini-PointNet that takes raw point cloud as input and regresses to a 3 × 3 matrix. It's composed of a shared MLP(64, 128, 1024) network (with layer output sizes 64, 128, 1024) on each point, a max pooling across points and two fully connected layers with output sizes 512, 256. The output matrix is initialized as an identity matrix. All layers, except the last one, include ReLU and batch normalization. The second + +![](PointNet_1612.00593_images/_page_9_Figure_10.jpeg) + +Figure 8. PointNet v.s. VoxNet [\[17\]](#page-8-9) on incomplete input data. Metric is overall classification accurcacy on ModelNet40 test set. Note that VoxNet is using 12 viewpoints averaging while PointNet is using only one view of the point cloud. Evidently PointNet presents much stronger robustness to missing points. + +> **[그림 해설]** 데이터 결손율(Missing Data Ratio, 0~1.0)에 따른 PointNet(파란 원)과 3D 복셀 기반 VoxNet(빨간 사각)의 분류 정확도(Accuracy %) 비교 그래프. +> - 데이터 결손이 0일 때 두 모델 모두 약 87%의 정확도로 시작함. +> - 데이터 결손율이 50%(0.5)일 때 PointNet은 약 84%의 높은 정확도를 유지하는 반면, VoxNet은 46% 수준으로 급락함. +> - 결손율 75%(0.75)에서도 PointNet은 약 74%를 기록하나 VoxNet은 18%로 추락하여, 불완전/부분 스캔 데이터에 대한 PointNet의 압도적인 강건성을 실증함. + +transformation network has the same architecture as the first one except that the output is a 64 × 64 matrix. The matrix is also initialized as an identity. A regularization loss (with weight 0.001) is added to the softmax classification loss to make the matrix close to orthogonal. + +We use dropout with keep ratio 0.7 on the last fully connected layer, whose output dimension 256, before class score prediction. The decay rate for batch normalization starts with 0.5 and is gradually increased to 0.99. We use adam optimizer with initial learning rate 0.001, momentum 0.9 and batch size 32. The learning rate is divided by 2 every 20 epochs. Training on ModelNet takes 3-6 hours to converge with TensorFlow and a GTX1080 GPU. + +PointNet Segmentation Network The segmentation network is an extension to the classification PointNet. Local point features (the output after the second transformation network) and global feature (output of the max pooling) are concatenated for each point. No dropout is used for segmentation network. Training parameters are the same as the classification network. + +As to the task of shape part segmentation, we made a few modifications to the basic segmentation network architecture (Fig 2 in main paper) in order to achieve best performance, as illustrated in Fig [9.](#page-10-3) We add a one-hot vector indicating the class of the input and concatenate it with the max pooling layer's output. We also increase neurons in some layers and add skip links to collect local point features in different layers and concatenate them to form point feature input to the segmentation network. + +While [\[27\]](#page-8-22) and [\[29\]](#page-8-23) deal with each object category independently, due to the lack of training data for some categories (the total number of shapes for all the categories in the data set are shown in the first line), we train our PointNet across categories (but with one-hot vector input to indicate category). To allow fair comparison, when testing + +![](PointNet_1612.00593_images/_page_10_Figure_0.jpeg) + +Figure 9. **Network architecture for part segmentation.** T1 and T2 are alignment/transformation networks for input points and features. FC is fully connected layer operating on each point. MLP is multi-layer perceptron on each point. One-hot is a vector of size 16 indicating category of the input shape. + +> **[그림 해설]** ShapeNet Part Segmentation을 위한 심층 PointNet 세그멘테이션 네트워크 구조 다이어그램. +> - 입력 $n \times 3$ $\to$ T1(Spatial Transform) $\to n \times 3 \to$ FC(64) $\to n \times 64 \to$ FC(128) $\to n \times 128 \to$ FC(128) $\to n \times 128 \to$ T2(Feature Transform) $\to n \times 128 \to$ FC(512) $\to n \times 512 \to$ FC(2048) $\to n \times 2048 \to$ Max Pooling $\to 2048$차원 Global Feature. +> - 다중 계층 특징 결합: 각 단계의 점별 특징($64 + 128 + 128 + 128 + 512 = 960$), 2048차원 글로벌 특징, 그리고 객체 카테고리를 나타내는 one-hot 벡터를 모두 결합하여 $n \times 3024$ 통합 특징 벡터 구성. +> - 최종 MLP(256, 256, 128)을 거쳐 $n \times 50$ 파트 예측 점수(part scores) 출력. + +![](PointNet_1612.00593_images/_page_10_Figure_2.jpeg) + +Figure 10. **Baseline 3D CNN segmentation network.** The network is fully convolutional and predicts part scores for each voxel. + +> **[그림 해설]** 파트 분할 성능 비교를 위한 3D 복셀 기반 CNN(Voxel-CNN) 베이스라인 아키텍처 다이어그램. +> - $32 \times 32 \times 32$ 크기의 3D 복셀 그리드 입력을 받음. +> - 인코더: 32개 필터(커널 5, stride 1) 컨볼루션 4회 $\to$ 32개 필터(커널 3, stride 1) 컨볼루션 1회를 거쳐 형상 특징 추출. +> - 디코더: 1개 크기의 특징을 스킵 연결(Skip Connection) 및 64 필터 $\to$ 64 필터 $\to$ 50 필터(커널 1, stride 1) 컨볼루션을 거쳐 $32 \times 32 \times 32$ 복셀 공간에 대해 각 복셀의 파트 카테고리를 예측(in-category prediction). + +these two models, we only predict part labels for the given specific object category. + +As to semantic segmentation task, we used the architecture as in Fig 2 in the main paper. + +It takes around six to twelve hours to train the model on ShapeNet part dataset and around half a day to train on the Stanford semantic parsing dataset. + +Baseline 3D CNN Segmentation Network In ShapeNet part segmentation experiment, we compare our proposed segmentation version PointNet to two traditional methods as well as a 3D volumetric CNN network baseline. In Fig 10, we show the baseline 3D volumetric CNN network we use. We generalize the well-known 3D CNN architectures, such as VoxNet [17] and 3DShapeNets [28] to a fully convolutional 3D CNN segmentation network. + +For a given point cloud, we first convert it to the volumetric representation as a occupancy grid with resolution $32 \times 32 \times 32$ . Then, five 3D convolution operations each with 32 output channels and stride of 1 are sequentially applied to extract features. The receptive field is 19 for each voxel. Finally, a sequence of 3D convolutional layers with kernel size $1 \times 1 \times 1$ is appended to the computed feature map to predict segmentation label for each voxel. ReLU and + +batch normalization are used for all layers except the last one. The network is trained across categories, however, in order to compare with other baseline methods where object category is given, we only consider output scores in the given object category. + +### D. Details on Detection Pipeline (Sec 5.1) + +We build a simple 3D object detection system based on the semantic segmentation results and our object classification PointNet. + +We use connected component with segmentation scores to get object proposals in scenes. Starting from a random point in the scene, we find its predicted label and use BFS to search nearby points with the same label, with a search radius of 0.2 meter. If the resulted cluster has more than 200 points (assuming a 4096 point sample in a 1m by 1m area), the cluster's bounding box is marked as one object proposal. For each proposed object, it's detection score is computed as the average point score for that category. Before evaluation, proposals with extremely small areas/volumes are pruned. For tables, chairs and sofas, the bounding boxes are extended to the floor in case the legs are separated with the seat/surface. + +We observe that in some rooms such as auditoriums lots of objects (e.g. chairs) are close to each other, where connected component would fail to correctly segment out individual ones. Therefore we leverage our classification network and uses sliding shape method to alleviate the problem for the chair class. We train a binary classification network for each category and use the classifier for sliding window detection. The resulted boxes are pruned by non-maximum suppression. The proposed boxes from connected component and sliding shapes are combined for final evaluation. + +In Fig 11, we show the precision-recall curves for object detection. We trained six models, where each one of them is trained on five areas and tested on the left area. At test phase, each model is tested on the area it has never seen. The test results for all six areas are aggregated for the PR curve generation. + +#### E. More Applications (Sec 5.1) + +Model Retrieval from Point Cloud Our PointNet learns a global shape signature for every given input point cloud. We expect geometrically similar shapes have similar global signature. In this section, we test our conjecture on the shape retrieval application. To be more specific, for every given query shape from ModelNet test split, we compute its global signature (output of the layer before the score prediction layer) given by our classification PointNet and retrieve similar shapes in the train split by nearest neighbor search. Results are shown in Fig 12. + +![](PointNet_1612.00593_images/_page_11_Figure_0.jpeg) + +Figure 11. Precision-recall curves for object detection in 3D point cloud. We evaluated on all six areas for four categories: table, chair, sofa and board. IoU threshold is 0.5 in volume. + +> **[그림 해설]** 3D 씬 객체 검출(Object Detection)에서 주요 4개 카테고리에 대한 Precision-Recall(정밀도-재현율) PR 곡선. +> - **table (상단 좌측)**: Recall 0.5 부근까지 Precision 0.7~0.8 이상을 유지하다가 점진적으로 하강 (최종 Recall 약 0.68). +> - **chair (상단 우측)**: Precision이 0.8에서 출발하여 Recall 0.65 부근(Precision 약 0.2)까지 완만하게 선형 하강. +> - **sofa (하단 좌측)**: 초기 급격한 하강 후 낮은 Precision 영역에서 Recall 0.24 부근까지 형성. +> - **board (하단 우측)**: 초기 Precision 약 0.8까지 상승 후 Recall 0.2 부근에서 급격히 감소. + +![](PointNet_1612.00593_images/_page_11_Picture_2.jpeg) + +Figure 12. Model retrieval from point cloud. For every given point cloud, we retrieve the top-5 similar shapes from the ModelNet test split. From top to bottom rows, we show examples of chair, plant, nightstand and bathtub queries. Retrieved results that are in wrong category are marked by red boxes. + +> **[그림 해설]** 불완전 쿼리 포인트 클라우드에 대한 PointNet 글로벌 특징 기반 Top-5 CAD 모델 검색(3D Shape Retrieval) 결과. +> - 4개 쿼리(좌측: 불완전 스캔 의자, 식물, 캐비닛/수납장, 세면대)에 대해 가장 유사한 CAD 모델 5개를 우측에 순서대로 배열. +> - 1~3행의 의자, 식물, 가구는 모두 올바른 카테고리의 유사 형상이 완벽히 검색됨. +> - 4행(세면대 쿼리)의 경우 1번째, 3번째, 5번째는 세면대(sink, 빨간 사각 박스)로 올바르게 검색되었으나, 2번째와 4번째는 외형이 유사한 욕조(bathtub)가 검색된 검색 실패/혼동 사례를 표시. + +Shape Correspondence In this section, we show that point features learnt by PointNet can be potentially used to compute shape correspondences. Given two shapes, we compute the correspondence between their *critical point sets* CS's by matching the pairs of points that activate the same dimensions in the global features. Fig [13](#page-11-3) and Fig [14](#page-11-4) show the detected shape correspondence between two similar chairs and tables. + +### F. More Architecture Analysis (Sec 5.2) + +Effects of Bottleneck Dimension and Number of Input Points Here we show our model's performance change with regard to the size of the first max layer output as well as the number of input points. In Fig [15](#page-11-5) we see that performance grows as we increase the number of points however it saturates at around 1K points. The max layer size plays an important role, increasing the layer size from + +![](PointNet_1612.00593_images/_page_11_Figure_7.jpeg) + +Figure 13. Shape correspondence between two chairs. For the clarity of the visualization, we only show 20 randomly picked correspondence pairs. + +> **[그림 해설]** 서로 다른 두 개의 의자 포인트 클라우드(빨간색, 파란색) 간의 기하학적 부품 대응점(Shape Correspondence) 시각화. +> - 무작위로 선택된 20개 대응 쌍을 색상 직선으로 연결. +> - 등받이 상단, 좌판 모서리, 의자 다리 끝단 등 구조적으로 일치하는 부품 위치끼리 평행하게 연결되어 PointNet 특징이 의미론적 형상 대응을 정확히 학습했음을 입증. + +![](PointNet_1612.00593_images/_page_11_Figure_9.jpeg) + +Figure 14. Shape correspondence between two tables. For the clarity of the visualization, we only show 20 randomly picked correspondence pairs. + +> **[그림 해설]** 형태가 서로 다른 두 개의 테이블 포인트 클라우드(빨간색 직사각형 테이블, 파란색 원형/타원형 테이블) 간 형상 대응 시각화. +> - 무작위 20개 포인트 쌍이 상판의 둘레, 모서리, 테이블 다리 하단부 등 대응하는 구조적 위치로 정확히 매핑되어 연결선을 형성. + +64 to 1024 results in a 2−4% performance gain. It indicates that we need enough point feature functions to cover the 3D space in order to discriminate different shapes. + +It's worth notice that even with 64 points as input (obtained from furthest point sampling on meshes), our network can achieve decent performance. + +![](PointNet_1612.00593_images/_page_11_Figure_13.jpeg) + +Figure 15. Effects of bottleneck size and number of input points. The metric is overall classification accuracy on Model-Net40 test set. + +> **[그림 해설]** 병목 차원(Bottleneck size, 64~1024) 및 입력 포인트 수(#points: 64, 128, 512, 1024, 2048)에 따른 ModelNet40 테스트 분류 정확도(Accuracy %) 변화 그래프. +> - X축: Bottleneck size (0, 200, 400, 600, 800, 1000). Y축: Accuracy % (81% ~ 88%). +> - 입력 포인트 수가 64개(하늘색 사각)일 때는 정확도 82~84.5% 수준이나, 1024개(초록 삼각) 및 2048개(주황 원)로 증가하면 87% 이상으로 향상됨. +> - 병목 크기 256~512 이상에서 성능이 포화(87.3% 수준)에 도달하며, 1024 차원에서 최대 성능을 기록함. + +MNIST Digit Classification While we focus on 3D point cloud learning, a sanity check experiment is to apply our network on a 2D point clouds - pixel sets. + +To convert an MNIST image into a 2D point set we threshold pixel values and add the pixel (represented as a point with (x, y) coordinate in the image) with values larger than 128 to the set. We use a set size of 256. If there are more than 256 pixels int he set, we randomly sub-sample it; if there are less, we pad the set with the one of the pixels in the set (due to our max operation, which point to use for the padding will not affect outcome). + +As seen in Table [7,](#page-12-2) we compare with a few baselines including multi-layer perceptron that considers input image as an ordered vector, a RNN that consider input as sequence from pixel (0,0) to pixel (27,27), and a vanilla version CNN. While the best performing model on MNIST is still well engineered CNNs (achieving less than 0.3% error rate), it's interesting to see that our PointNet model can achieve reasonable performance by considering image as a 2D point set. + +| | input | error (%) | +|-----------------------------|-----------|-----------| +| Multi-layer perceptron [22] | vector | 1.60 | +| LeNet5 [12] | image | 0.80 | +| Ours PointNet | point set | 0.78 | + +Table 7. MNIST classification results. We compare with vanilla versions of other deep architectures to show that our network based on point sets input is achieving reasonable performance on this traditional task. + +Normal Estimation In segmentation version of PointNet, local point features and global feature are concatenated in order to provide context to local points. However, it's unclear whether the context is learnt through this concatenation. In this experiment, we validate our design by showing that our segmentation network can be trained to predict point normals, a local geometric property that is determined by a point's neighborhood. + +We train a modified version of our segmentation Point-Net in a supervised manner to regress to the groundtruth point normals. We just change the last layer of our segmentation PointNet to predict normal vector for each point. We use absolute value of cosine distance as loss. + +Fig. [16](#page-12-3) compares our PointNet normal prediction results (the left columns) to the ground-truth normals computed from the mesh (the right columns). We observe a reasonable normal reconstruction. Our predictions are more smooth and continuous than the ground-truth which includes flipped normal directions in some region. + +Segmentation Robustness As discussed in Sec 5.2 and Sec [B,](#page-9-1) our PointNet is less sensitive to data corruption and missing points for classification tasks since the global shape feature is extracted from a collection of *critical points* from the given input point cloud. In this section, we show that the robustness holds for segmentation tasks too. The per-point part labels are predicted based on the combination of perpoint features and the learnt global shape feature. In Fig [17,](#page-13-0) + +![](PointNet_1612.00593_images/_page_12_Figure_8.jpeg) + +Figure 16. PointNet normal reconstrution results. In this figure, we show the reconstructed normals for all the points in some sample point clouds and the ground-truth normals computed on the mesh. + +> **[그림 해설]** PointNet을 이용한 포인트별 표면 법선 벡터(Normal Vector) 재구성 결과 비교. +> - 3개 객체 샘플(의자 등받이/좌판, 비행기, 변기)에 대해 PointNet이 예측한 법선 벡터(Prediction, 좌측)와 3D 메시에서 계산된 실제 정답 법선 벡터(Ground-truth, 우측)를 파란색 선분으로 시각화. +> - 각 점의 국소 기하 구조(평면, 곡면, 경계선)에 수직인 법선 방향이 GT와 거의 완벽하게 일치하게 재구성됨. + +we illustrate the segmentation results for the given input point clouds S (the left-most column), the *critical point sets* CS (the middle column) and the *upper-bound shapes* NS. + +Network Generalizability to Unseen Shape Categories In Fig [18,](#page-13-1) we visualize the *critical point sets* and the *upperbound shapes* for new shapes from unseen categories (face, house, rabbit, teapot) that are not present in ModelNet or ShapeNet. It shows that the learnt per-point functions are generalizable. However, since we train mostly on manmade objects with lots of planar structures, the reconstructed upper-bound shape in novel categories also contain more planar surfaces. + +### G. Proof of Theorem (Sec 4.3) + +Let +$$\mathcal{X} = \{S : S \subseteq [0, 1] \text{ and } |S| = n\}.$$ + +f : X → R is a continuous function on X w.r.t to Hausdorff distance dH(·, ·) if the following condition is satisfied: + +$$\forall \epsilon > 0, \exists \delta > 0$$ +, for any $S, S' \in \mathcal{X}$ , if $d_H(S, S') < \delta$ , then $|f(S) - f(S')| < \epsilon$ . + +We show that f can be approximated arbitrarily by composing a symmetric function and a continuous function. + +![](PointNet_1612.00593_images/_page_13_Figure_0.jpeg) + +Input Point Cloud Critical Point Sets Upper-bound Shapes + +Figure 17. The consistency of segmentation results. We illustrate the segmentation results for some sample given point clouds S, their *critical point sets* $\mathcal{C}_S$ and *upper-bound shapes* $\mathcal{N}_S$ . We observe that the shape family between the $\mathcal{C}_S$ and $\mathcal{N}_S$ share a consistent segmentation results. + +> **[그림 해설]** 형상 집합군에서의 파트 세그멘테이션 일관성 시각화 (탁자, 머그컵, 자동차 3개 예시). +> - **1열 (Input Point Cloud, $S$)**: 원본 포인트 클라우드와 예측된 파트 분할 결과 (색상별 부품 분할). +> - **2열 (Critical Point Sets, $\mathcal{C}_S$)**: 해당 형상의 임계 포인트들만 남긴 서브셋에서도 원본과 동일한 부품 경계와 분할 결과가 유지됨. +> - **3열 (Upper-bound Shapes, $\mathcal{N}_S$)**: 포인트가 두껍게 확장된 상한 형상에서도 동일한 파트 분할 레이블이 완벽하게 일관성을 유지하며 보존됨을 확인. + +![](PointNet_1612.00593_images/_page_13_Figure_3.jpeg) + +Figure 18. The critical point sets and the upper-bound shapes for unseen objects. We visualize the *critical point sets* and the *upper-bound shapes* for teapot, bunny, hand and human body, which are not in the ModelNet or ShapeNet shape repository to test the generalizability of the learnt per-point functions of our PointNet on other unseen objects. The images are color-coded to reflect the depth information. + +> **[그림 해설]** 훈련 세트(ModelNet/ShapeNet)에 포함되지 않은 새로운 미학습 객체 4종(주전자 Teapot, 토끼 Stanford Bunny, 손 Hand, 인체 Human body)에 대한 일반화 성능 평가. +> - **1행 (Original Shape)**: 원본 객체 포인트 클라우드 (깊이 정보에 따른 무지개색 코딩). +> - **2행 (Critical Point Sets)**: 임계 포인트 세트가 주전자 주구/손잡이, 토끼 귀/발, 손가락 끝/관절, 인체 사지 등 핵심 외곽 스켈레톤 구조를 정확히 포착. +> - **3행 (Upper-bound Shapes)**: 점유 상한 형상 또한 각 객체의 전체 볼륨 윤곽을 충실하게 감싸며 모델의 뛰어난 일반화 능력을 입증. + +**Theorem 1.** Suppose $f: \mathcal{X} \to \mathbb{R}$ is a continuous set function w.r.t Hausdorff distance $d_H(\cdot,\cdot)$ . $\forall \epsilon > 0$ , $\exists$ a continuous function h and a symmetric function $g(x_1, \dots, x_n) = \gamma \circ MAX$ , where $\gamma$ is a continuous function, MAX is a vector max operator that takes n vectors as input and returns a new vector of the element-wise maximum, such that for any $S \in \mathcal{X}$ , + +$$|f(S) - \gamma(MAX(h(x_1), \dots, h(x_n)))| < \epsilon$$ + +where $x_1, \ldots, x_n$ are the elements of S extracted in certain + +order, + +*Proof.* By the continuity of f, we take $\delta_{\epsilon}$ so that $|f(S) - f(S')| < \epsilon$ for any $S, S' \in \mathcal{X}$ if $d_H(S, S') < \delta_{\epsilon}$ . + +Define $K = \lceil 1/\delta_{\epsilon} \rceil$ , which split [0,1] into K intervals evenly and define an auxiliary function that maps a point to the left end of the interval it lies in: + +$$\sigma(x) = \frac{\lfloor Kx \rfloor}{K}$$ + +Let $\tilde{S} = {\sigma(x) : x \in S}$ , then + +$$|f(S) - f(\tilde{S})| < \epsilon$$ + +because $d_H(S, \tilde{S}) < 1/K \le \delta_{\epsilon}$ . + +Let $h_k(x) = e^{-d(x, \lfloor \frac{k-1}{K}, \frac{k}{K} \rfloor)}$ be a soft indicator function where d(x, I) is the point to set (interval) distance. Let $\mathbf{h}(x) = [h_1(x); \dots; h_K(x)]$ , then $\mathbf{h} : \mathbb{R} \to \mathbb{R}^K$ . + +Let $v_j(x_1, \ldots, x_n) = \max\{\hat{h}_j(x_1), \ldots, \hat{h}_j(x_n)\}$ , indicating the occupancy of the j-th interval by points in S. Let $\mathbf{v} = [v_1; \ldots; v_K]$ , then $\mathbf{v} : \underbrace{\mathbb{R} \times \ldots \times \mathbb{R}}_n \to \{0, 1\}^K$ + +is a symmetric function, indicating the occupancy of each interval by points in S. + +Define $\tau:\{0,1\}^K\to\mathcal{X}$ as $\tau(v)=\{\frac{k-1}{K}:v_k\geq 1\}$ , which maps the occupancy vector to a set which contains the left end of each occupied interval. It is easy to show: + +$$\tau(\mathbf{v}(x_1,\ldots,x_n)) \equiv \tilde{S}$$ + +where $x_1, \ldots, x_n$ are the elements of S extracted in certain order + +Let $\gamma:\mathbb{R}^K\to\mathbb{R}$ be a continuous function such that $\gamma(\mathbf{v})=f(\tau(\mathbf{v}))$ for $v\in\{0,1\}^K$ . Then, + +$$|\gamma(\mathbf{v}(x_1,\ldots,x_n)) - f(S)|$$ + +=|f(\tau(\mathbf{v}(x\_1,\ldots,x\_n))) - f(S)| < \epsilon + +Note that $\gamma(\mathbf{v}(x_1,\ldots,x_n))$ can be rewritten as follows: + +$$\gamma(\mathbf{v}(x_1,\ldots,x_n)) = \gamma(\mathbf{MAX}(\mathbf{h}(x_1),\ldots,\mathbf{h}(x_n)))$$ +$$= (\gamma \circ \mathbf{MAX})(\mathbf{h}(x_1),\ldots,\mathbf{h}(x_n))$$ + +Obviously $\gamma \circ MAX$ is a symmetric function. $\square$ + +Next we give the proof of Theorem 2. We define $\mathbf{u} = \underset{x_i \in S}{\operatorname{MAX}} \{h(x_i)\}$ to be the sub-network of f which maps a point set in $[0,1]^m$ to a K-dimensional vector. The following theorem tells us that small corruptions or extra noise points in the input set is not likely to change the output of our network: + +**Theorem 2.** Suppose $\mathbf{u}: \mathcal{X} \to \mathbb{R}^K$ such that $\mathbf{u} = \max_{x \in S} \{h(x_i)\}$ and $f = \gamma \circ \mathbf{u}$ . Then, + +*(a)* ∀S, ∃ CS, NS ⊆ X *,* f(T) = f(S) *if* CS ⊆ T ⊆ NS*;* + +*(b)* |CS| ≤ K + +*Proof.* Obviously, ∀S ∈ X , f(S) is determined by u(S). So we only need to prove that ∀S, ∃ CS, NS ⊆ X , f(T) = f(S)if CS ⊆ T ⊆ NS. + +For the jth dimension as the output of u, there exists at least one xj ∈ X such that hj (xj ) = uj , where hj is the jth dimension of the output vector from h. Take CS as the union of all xj for j = 1, . . . , K. Then, CS satisfies the above condition. + +Adding any additional points x such that h(x) ≤ u(S) at all dimensions to CS does not change u, hence f. Therefore, TS can be obtained adding the union of all such points to NS. + +![](PointNet_1612.00593_images/_page_14_Picture_5.jpeg) + +Figure 19. Point function visualization. For each per-point function h, we calculate the values h(p) for all the points p in a cube of diameter two located at the origin, which spatially covers the unit sphere to which our input shapes are normalized when training our PointNet. In this figure, we visualize all the points p that give h(p) > 0.5 with function values color-coded by the brightness of the voxel. We randomly pick 15 point functions and visualize the activation regions for them. + +> **[그림 해설]** PointNet 내부의 per-point 함수 $h$ 중 무작위로 선택된 15개 뉴런의 3D 공간 활성화 영역($h(p) > 0.5$) 시각화. +> - 원점을 중심으로 한 3D 큐브 공간 내에서 각 뉴런이 활성화되는 3D 볼륨(회색 음영 영역)을 3행 5열 격자로 배열. +> - 특정 뉴런은 구의 상단 반구, 모서리 단면, 쐐기형 슬라이스, 중심부 국소 구체 등 다양한 형태의 기하학적 3D 공간 필터(공간 분할 함수) 역할을 수행함을 확인. + +### H. More Visualizations + +Classification Visualization We use t-SNE[\[15\]](#page-8-28) to embed point cloud global signature (1024-dim) from our classification PointNet into a 2D space. Fig [20](#page-15-0) shows the embedding space of ModelNet 40 test split shapes. Similar shapes are clustered together according to their semantic categories. + +Segmentation Visualization We present more segmentation results on both complete CAD models and simulated Kinect partial scans. We also visualize failure cases with error analysis. Fig [21](#page-16-0) and Fig [22](#page-16-1) show more segmentation results generated on complete CAD models and their simulated Kinect scans. Fig [23](#page-17-1) illustrates some failure cases. Please read the caption for the error analysis. + +Scene Semantic Parsing Visualization We give a visualization of semantic parsing in Fig [24](#page-18-0) where we show input point cloud, prediction and ground truth for both semantic segmentation and object detection for two office rooms and one conference room. The area and the rooms are unseen in the training set. + +Point Function Visualization Our classification Point-Net computes K (we take K = 1024 in this visualization) dimension point features for each point and aggregates all the per-point local features via a max pooling layer into a single K-dim vector, which forms the global shape descriptor. + +To gain more insights on what the learnt per-point functions h's detect, we visualize the points pi's that give high per-point function value f(pi) in Fig [19.](#page-14-2) This visualization clearly shows that different point functions learn to detect for points in different regions with various shapes scattered in the whole space. + +![](PointNet_1612.00593_images/_page_15_Figure_0.jpeg) + +Figure 20. 2D embedding of learnt shape global features. We use t-SNE technique to visualize the learnt global shape features for the shapes in ModelNet40 test split. + +> **[그림 해설]** ModelNet40 테스트셋 전체 형상들에 대해 PointNet이 학습한 1024차원 글로벌 형상 특징 벡터의 2D t-SNE 임베딩 시각화. +> - 비행기, 의자, 테이블, 자동차, 병, 화분 등 동일 범주 및 유사 기하 구조를 가진 3D 객체들이 2차원 공간 상에서 밀집된 클러스터를 형성하고 상호 분리됨을 보여줌. + +![](PointNet_1612.00593_images/_page_16_Figure_0.jpeg) + +Figure 21. PointNet segmentation results on complete CAD models. + +> **[그림 해설]** 완전한 ShapeNet 3D CAD 모델에 대한 파트 분할 결과 (총 16개 카테고리, 카테고리당 3개씩 총 48개 객체 시각화). +> - 좌측: airplane, bag, cap, car, chair, earphone, guitar, knife. +> - 우측: rocket, pistol, table, skateboard, motorbike, mug, laptop, lamp. +> - 세부 부품(비행기 엔진, 가방 손잡이, 자동차 바퀴/유리, 의자 등받이/다리, 헤드폰 밴드/패드, 기타 넥/바디 등)이 선명한 고유 색상으로 정확하게 분할됨. + +![](PointNet_1612.00593_images/_page_16_Figure_2.jpeg) + +Figure 22. PointNet segmentation results on simulated Kinect scans. + +> **[그림 해설]** 시뮬레이션된 Kinect 스캔(단면 결손 및 가림이 존재하는 부분 스캔 데이터)에 대한 파트 분할 결과 (16개 카테고리별 각 3개씩 시각화). +> - 심한 점 결손과 가림 현상 속에서도 비행기 날개, 자동차 프레임, 의자 좌판, 모자 챙 등 부품 영역을 견고하게 인식하여 분할함. + +![](PointNet_1612.00593_images/_page_17_Figure_0.jpeg) + +Figure 23. PointNet segmentation failure cases. In this figure, we summarize six types of common errors in our segmentation application. The prediction and the ground-truth segmentations are given in the first and second columns, while the difference maps are computed and shown in the third columns. The red dots correspond to the wrongly labeled points in the given point clouds. (a) illustrates the most common failure cases: the points on the boundary are wrongly labeled. In the examples, the label predictions for the points near the intersections between the table/chair legs and the tops are not accurate. However, most segmentation algorithms suffer from this error. (b) shows the errors on exotic shapes. For examples, the chandelier and the airplane shown in the figure are very rare in the data set. (c) shows that small parts can be overwritten by nearby large parts. For example, the jet engines for airplanes (yellow in the figure) are mistakenly classified as body (green) or the plane wing (purple). (d) shows the error caused by the inherent ambiguity of shape parts. For example, the two bottoms of the two tables in the figure are classified as table legs and table bases (category *other* in [\[29\]](#page-8-23)), while ground-truth segmentation is the opposite. (e) illustrates the error introduced by the incompleteness of the partial scans. For the two caps in the figure, almost half of the point clouds are missing. (f) shows the failure cases when some object categories have too less training data to cover enough variety. There are only 54 bags and 39 caps in the whole dataset for the two categories shown here. + +> **[그림 해설]** PointNet 세그멘테이션의 6가지 주요 실패/오류 유형 분석 ((a)~(f) 각 유형별로 1열: Prediction, 2열: Ground-truth, 3열: 빨간 점으로 표시된 Difference map). +> - **(a) Boundary Error (경계 오류)**: 탁자/의자 상판과 다리 결합 부위 등 경계면 포인트의 오분류 (가장 흔한 오류). +> - **(b) Exotic Shapes (특이 형상)**: 샹들리에나 특이한 형태의 비행기 등 훈련 세트에 드문 희귀 형상에서의 오류. +> - **(c) Overwriting Small Parts (소형 부품 덮어쓰기)**: 제트 엔진(노란색) 같은 작은 부품이 인접한 큰 부품(동체/날개) 레이블로 덮어쓰여지는 현상. +> - **(d) Inherent Part Ambiguity (부품 정의의 모호성)**: 테이블 하단 지지 구조를 다리(leg)로 보는지 받침대(base)로 보는지에 대한 GT와의 정의 불일치. +> - **(e) Incomplete Scans (불완전 스캔)**: 모자 포인트 클라우드의 절반 가까이가 누락되어 형상 전체 맥락 파악 실패. +> - **(f) Lack of Training Data (데이터 부족)**: 가방(54개), 모자(39개) 등 훈련 데이터 수가 극히 적은 클래스에서 다양성 부족으로 인한 실패. + +![](PointNet_1612.00593_images/_page_18_Figure_0.jpeg) + +Figure 24. Examples of semantic segmentation and object detection. First row is input point cloud, where walls and ceiling are hided for clarity. Second and third rows are prediction and ground-truth of semantic segmentation on points, where points belonging to different semantic regions are colored differently (chairs in red, tables in purple, sofa in orange, board in gray, bookcase in green, floors in blue, windows in violet, beam in yellow, column in magenta, doors in khaki and clutters in black). The last two rows are object detection with bounding boxes, where predicted boxes are from connected components based on semantic segmentation prediction. + +> **[그림 해설]** 실내 대규모 3D 씬(3개 방/오피스)에 대한 시맨틱 분할 및 3D 바운딩 박스 객체 검출(Object Detection) 결과 (5개 행으로 구성). +> - **1행 (Input Point Cloud)**: 천장과 벽을 일부 제거하여 내부를 가시화한 실내 포인트 클라우드 입력. +> - **2행 (pred - Semantic Segmentation)**: PointNet의 포인트별 시맨틱 클래스 예측 결과 (의자: 빨강, 테이블: 보라, 책장: 초록, 바닥: 파랑, 빔: 노랑, 소파: 주황 등). +> - **3행 (GT - Semantic Segmentation)**: 실제 정답 시맨틱 분할. +> - **4행 (pred - Object Detection)**: 시맨틱 분할의 연결 요소(Connected Components)를 기반으로 추출된 3D 바운딩 박스 객체 검출 결과. +> - **5행 (GT - Object Detection)**: 실제 정답 3D 바운딩 박스. 예측 박스가 정답 박스의 위치와 크기를 거의 정확히 추정함. \ No newline at end of file diff --git a/docs/papers/md/PointNet_1612.00593_images/_page_0_Figure_9.jpeg b/docs/papers/md/PointNet_1612.00593_images/_page_0_Figure_9.jpeg new file mode 100644 index 0000000000000000000000000000000000000000..f0bb14dd03b8424ec53630ffe13d73d492e8dcf2 GIT binary 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on Point Sets in a Metric Space + +Charles R. Qi Li Yi Hao Su Leonidas J. Guibas Stanford University + +# Abstract + +Few prior works study deep learning on point sets. PointNet [\[20\]](#page-8-0) is a pioneer in this direction. However, by design PointNet does not capture local structures induced by the metric space points live in, limiting its ability to recognize fine-grained patterns and generalizability to complex scenes. In this work, we introduce a hierarchical neural network that applies PointNet recursively on a nested partitioning of the input point set. By exploiting metric space distances, our network is able to learn local features with increasing contextual scales. With further observation that point sets are usually sampled with varying densities, which results in greatly decreased performance for networks trained on uniform densities, we propose novel set learning layers to adaptively combine features from multiple scales. Experiments show that our network called PointNet++ is able to learn deep point set features efficiently and robustly. In particular, results significantly better than state-of-the-art have been obtained on challenging benchmarks of 3D point clouds. + +# 1 Introduction + +We are interested in analyzing geometric point sets which are collections of points in a Euclidean space. A particularly important type of geometric point set is point cloud captured by 3D scanners, e.g., from appropriately equipped autonomous vehicles. As a set, such data has to be invariant to permutations of its members. In addition, the distance metric defines local neighborhoods that may exhibit different properties. For example, the density and other attributes of points may not be uniform across different locations — in 3D scanning the density variability can come from perspective effects, radial density variations, motion, etc. + +Few prior works study deep learning on point sets. PointNet [\[20\]](#page-8-0) is a pioneering effort that directly processes point sets. The basic idea of PointNet is to learn a spatial encoding of each point and then aggregate all individual point features to a global point cloud signature. By its design, PointNet does not capture local structure induced by the metric. However, exploiting local structure has proven to be important for the success of convolutional architectures. A CNN takes data defined on regular grids as the input and is able to progressively capture features at increasingly larger scales along a multi-resolution hierarchy. At lower levels neurons have smaller receptive fields whereas at higher levels they have larger receptive fields. The ability to abstract local patterns along the hierarchy allows better generalizability to unseen cases. + +We introduce a hierarchical neural network, named as PointNet++, to process a set of points sampled in a metric space in a hierarchical fashion. The general idea of PointNet++ is simple. We first partition the set of points into overlapping local regions by the distance metric of the underlying space. Similar to CNNs, we extract local features capturing fine geometric structures from small neighborhoods; such local features are further grouped into larger units and processed to produce higher level features. This process is repeated until we obtain the features of the whole point set. + +The design of PointNet++ has to address two issues: how to generate the partitioning of the point set, and how to abstract sets of points or local features through a local feature learner. The two issues + +are correlated because the partitioning of the point set has to produce common structures across partitions, so that weights of local feature learners can be shared, as in the convolutional setting. We choose our local feature learner to be PointNet. As demonstrated in that work, PointNet is an effective architecture to process an unordered set of points for semantic feature extraction. In addition, this architecture is robust to input data corruption. As a basic building block, PointNet abstracts sets of local points or features into higher level representations. In this view, PointNet++ applies PointNet recursively on a nested partitioning of the input set. + +One issue that still remains is how to generate overlapping partitioning of a point set. Each partition is defined as a neighborhood ball in the underlying Euclidean space, whose parameters include centroid location and scale. To evenly cover the whole set, the centroids are selected among input point set by a farthest point sampling (FPS) algorithm. Compared with volumetric CNNs that scan the space with fixed strides, our local receptive fields are dependent on both the input data and the metric, and thus more efficient and effective. + +![](PointNet++_1706.02413_images/_page_1_Picture_2.jpeg) + +Figure 1: Visualization of a scan captured from a Structure Sensor (left: RGB; right: point cloud). + +> **[그림 해설]** 실제 센서 측정 시 발생하는 포인트 클라우드의 불균일 밀도(Non-uniform density) 문제 시각화. +> - **좌측 (RGB 이미지)**: 실제 실내 환경 사진 (소파, 테이블, 의자). +> - **우측 (포인트 클라우드)**: 시점 및 거리에 따라 색상 코딩된 포인트 집합. 센서에 가까운 전경(파란색 영역)은 점 밀도가 매우 조밀한 반면, 거리가 먼 후경(연두색/흰색 영역)은 점이 희소하고 차폐(Occlusion)로 인한 결손이 크게 발생함을 보여줌. + +Deciding the appropriate scale of local neighborhood balls, however, is a more challenging yet intriguing problem, due to the entanglement of feature scale and non-uniformity of input point set. We assume that the input point set may have variable density at different areas, which is quite common in real data such as Structure Sensor scanning [18] (see Fig. 1). Our input point set is thus very different from CNN inputs which can be viewed as data defined on regular grids with uniform constant density. In CNNs, the counterpart to local partition scale is the size of kernels. [25] shows that using smaller kernels helps to improve the ability of CNNs. Our experiments on point set data, however, give counter evidence to this rule. Small neighborhood may consist of too few points due to sampling deficiency, which might be insufficient to allow PointNets to capture patterns robustly. + +A significant contribution of our paper is that PointNet++ leverages neighborhoods at multiple scales to achieve both robustness and detail capture. Assisted with random input dropout during training, the network learns to adaptively weight patterns detected at different scales and combine multi-scale features according to the input data. Experiments show that our PointNet++ is able to process point sets efficiently and robustly. In particular, results that are significantly better than state-of-the-art have been obtained on challenging benchmarks of 3D point clouds. + +#### 2 Problem Statement + +Suppose that $\mathcal{X}=(M,d)$ is a discrete metric space whose metric is inherited from a Euclidean space $\mathbb{R}^n$ , where $M\subseteq\mathbb{R}^n$ is the set of points and d is the distance metric. In addition, the density of M in the ambient Euclidean space may not be uniform everywhere. We are interested in learning set functions f that take such $\mathcal{X}$ as the input (along with additional features for each point) and produce information of semantic interest regrading $\mathcal{X}$ . In practice, such f can be classification function that assigns a label to $\mathcal{X}$ or a segmentation function that assigns a per point label to each member of M. + +#### 3 Method + +Our work can be viewed as an extension of PointNet [20] with added hierarchical structure. We first review PointNet (Sec. 3.1) and then introduce a basic extension of PointNet with hierarchical structure (Sec. 3.2). Finally, we propose our PointNet++ that is able to robustly learn features even in non-uniformly sampled point sets (Sec. 3.3). + +### 3.1 Review of PointNet [20]: A Universal Continuous Set Function Approximator + +Given an unordered point set $\{x_1, x_2, ..., x_n\}$ with $x_i \in \mathbb{R}^d$ , one can define a set function $f : \mathcal{X} \to \mathbb{R}$ that maps a set of points to a vector: + + +$$f(x_1, x_2, ..., x_n) = \gamma \left( \max_{i=1,...,n} \{ h(x_i) \} \right)$$ + (1) + +![](PointNet++_1706.02413_images/_page_2_Figure_0.jpeg) + +Figure 2: Illustration of our hierarchical feature learning architecture and its application for set segmentation and classification using points in 2D Euclidean space as an example. Single scale point grouping is visualized here. For details on density adaptive grouping, see Fig. 3 + +> **[그림 해설]** PointNet++의 계층적 신경망 구조도 (Set Abstraction 인코더와 태스크별 헤드). +> - **Hierarchical feature learning (좌측 회색 인코더 영역)**: +> - 입력: $(N, d+C)$ 포인트 ($d$차원 좌표 + $C$차원 특징). +> - **Set Abstraction (SA) 모듈**: Farthest Point Sampling(FPS)으로 중심점 선정 $ o$ 반경 기반 Grouping으로 국소 이웃 군집화 $ o$ 국소 PointNet을 적용해 특징 요약. +> - 점 수를 점진적으로 축소하며 다중 스케일 국소 기하 구조를 계층적으로 학습: $(N, d+C) \to (N_1, d+C_1) \to (N_2, d+C_2)$. +> - **Segmentation (우측 상단 디코더 영역)**: +> - Feature Propagation(FP) 계층: 역거리 가중치 기반 $k$-NN 보간(Interpolate)과 스킵 연결(Skip link concatenation)을 통해 축소된 해상도를 원래 포인트 개수 $N$으로 점진적 복원. +> - Unit PointNet(1x1 Conv)을 거쳐 최종 $(N, k)$ 점별 분류 점수(per-point scores) 도출. +> - **Classification (우측 하단 분류 헤드)**: +> - 마지막 SA 계층의 포인트들을 전체 집계하여 $(1, C_4)$ 글로벌 특징 벡터 생성 $ o$ Fully Connected 레이어 $ o (k)$개 클래스 점수 출력. + +where $\gamma$ and h are usually multi-layer perceptron (MLP) networks. + +The set function f in Eq. 1 is invariant to input point permutations and can arbitrarily approximate any continuous set function [20]. Note that the response of h can be interpreted as the spatial encoding of a point (see [20] for details). + +PointNet achieved impressive performance on a few benchmarks. However, it lacks the ability to capture local context at different scales. We will introduce a hierarchical feature learning framework in the next section to resolve the limitation. + +### 3.2 Hierarchical Point Set Feature Learning + +While PointNet uses a single max pooling operation to aggregate the whole point set, our new architecture builds a hierarchical grouping of points and progressively abstract larger and larger local regions along the hierarchy. + +Our hierarchical structure is composed by a number of *set abstraction* levels (Fig. 2). At each level, a set of points is processed and abstracted to produce a new set with fewer elements. The set abstraction level is made of three key layers: *Sampling layer*, *Grouping layer* and *PointNet layer*. The *Sampling layer* selects a set of points from input points, which defines the centroids of local regions. *Grouping layer* then constructs local region sets by finding "neighboring" points around the centroids. *PointNet layer* uses a mini-PointNet to encode local region patterns into feature vectors. + +A set abstraction level takes an $N \times (d+C)$ matrix as input that is from N points with d-dim coordinates and C-dim point feature. It outputs an $N' \times (d+C')$ matrix of N' subsampled points with d-dim coordinates and new C'-dim feature vectors summarizing local context. We introduce the layers of a set abstraction level in the following paragraphs. + +**Sampling layer.** Given input points $\{x_1, x_2, ..., x_n\}$ , we use iterative farthest point sampling (FPS) to choose a subset of points $\{x_{i_1}, x_{i_2}, ..., x_{i_m}\}$ , such that $x_{i_j}$ is the most distant point (in metric distance) from the set $\{x_{i_1}, x_{i_2}, ..., x_{i_{j-1}}\}$ with regard to the rest points. Compared with random sampling, it has better coverage of the entire point set given the same number of centroids. In contrast to CNNs that scan the vector space agnostic of data distribution, our sampling strategy generates receptive fields in a data dependent manner. + +**Grouping layer.** The input to this layer is a point set of size $N \times (d+C)$ and the coordinates of a set of centroids of size $N' \times d$ . The output are groups of point sets of size $N' \times K \times (d+C)$ , where each group corresponds to a local region and K is the number of points in the neighborhood of centroid points. Note that K varies across groups but the succeeding *PointNet layer* is able to convert flexible number of points into a fixed length local region feature vector. + +In convolutional neural networks, a local region of a pixel consists of pixels with array indices within certain Manhattan distance (kernel size) of the pixel. In a point set sampled from a metric space, the neighborhood of a point is defined by metric distance. + +Ball query finds all points that are within a radius to the query point (an upper limit of K is set in implementation). An alternative range query is K nearest neighbor (kNN) search which finds a fixed + +number of neighboring points. Compared with kNN, ball query's local neighborhood guarantees a fixed region scale thus making local region feature more generalizable across space, which is preferred for tasks requiring local pattern recognition (e.g. semantic point labeling). + +**PointNet layer.** In this layer, the input are N' local regions of points with data size $N' \times K \times (d+C)$ . Each local region in the output is abstracted by its centroid and local feature that encodes the centroid's neighborhood. Output data size is $N' \times (d+C')$ . + +The coordinates of points in a local region are firstly translated into a local frame relative to the centroid point: $x_i^{(j)} = x_i^{(j)} - \hat{x}^{(j)}$ for i=1,2,...,K and j=1,2,...,d where $\hat{x}$ is the coordinate of the centroid. We use PointNet [20] as described in Sec. 3.1 as the basic building block for local pattern learning. By using relative coordinates together with point features we can capture point-to-point relations in the local region. + +# 3.3 Robust Feature Learning under Non-Uniform Sampling Density + +As discussed earlier, it is common that a point set comes with non-uniform density in different areas. Such non-uniformity introduces a significant challenge for point set feature learning. Features learned in dense data may not generalize to sparsely sampled regions. Consequently, models trained for sparse point cloud may not recognize fine-grained local structures. + +Ideally, we want to inspect as closely as possible into a point set to capture finest details in densely sampled regions. However, such close inspect is prohibited at low density areas because local patterns may be corrupted by the sampling deficiency. In this case, we should look for larger scale patterns in greater vicinity. To achieve this goal we propose density adaptive PointNet layers (Fig. 3) that learn to + +![](PointNet++_1706.02413_images/_page_3_Figure_6.jpeg) + +Figure 3: (a) Multi-scale grouping (MSG); (b) Multi-resolution grouping (MRG). + +> **[그림 해설]** 밀도 불균일성에 대응하는 밀도 적응형 계층(Density Adaptive Layers) 2종 다이어그램. +> - **(a) Multi-scale grouping (MSG)**: 단일 중심점에 대해 크기가 다른 여러 동심 반경 구(sphere)를 정의하고, 각 반경별 국소 영역에서 PointNet 특징을 따로 추출한 뒤 하나로 결합(concat). 다양한 스케일의 기하 정보를 동시에 포착. +> - **(b) Multi-resolution grouping (MRG)**: 연산량을 줄이기 위해 이전 계층(하위 해상도)에서 이미 요약된 특징 벡터 집합과, 원본 국소 영역 포인트들로부터 직접 추출한 특징을 결합(concat). 저밀도 영역에서는 큰 영역 특징을, 고밀도 영역에서는 세밀한 국소 특징을 적응적으로 활용. + +combine features from regions of different scales when the input sampling density changes. We call our hierarchical network with density adaptive PointNet layers as *PointNet++*. + +Previously in Sec. 3.2, each abstraction level contains grouping and feature extraction of a single scale. In PointNet++, each abstraction level extracts multiple scales of local patterns and combine them intelligently according to local point densities. In terms of grouping local regions and combining features from different scales, we propose two types of density adaptive layers as listed below. + +**Multi-scale grouping (MSG).** As shown in Fig. 3 (a), a simple but effective way to capture multi-scale patterns is to apply grouping layers with different scales followed by according PointNets to extract features of each scale. Features at different scales are concatenated to form a multi-scale feature. + +We train the network to learn an optimized strategy to combine the multi-scale features. This is done by randomly dropping out input points with a randomized probability for each instance, which we call random input dropout. Specifically, for each training point set, we choose a dropout ratio $\theta$ uniformly sampled from [0,p] where $p \leq 1$ . For each point, we randomly drop a point with probability $\theta$ . In practice we set p=0.95 to avoid generating empty point sets. In doing so we present the network with training sets of various sparsity (induced by $\theta$ ) and varying uniformity (induced by randomness in dropout). During test, we keep all available points. + +**Multi-resolution grouping (MRG).** The MSG approach above is computationally expensive since it runs local PointNet at large scale neighborhoods for every centroid point. In particular, since the number of centroid points is usually quite large at the lowest level, the time cost is significant. + +Here we propose an alternative approach that avoids such expensive computation but still preserves the ability to adaptively aggregate information according to the distributional properties of points. In Fig. 3 (b), features of a region at some level $L_i$ is a concatenation of two vectors. One vector (left in figure) is obtained by summarizing the features at each subregion from the lower level $L_{i-1}$ using the set abstraction level. The other vector (right) is the feature that is obtained by directly processing all raw points in the local region using a single PointNet. + +When the density of a local region is low, the first vector may be less reliable than the second vector, since the subregion in computing the first vector contains even sparser points and suffers more from sampling deficiency. In such a case, the second vector should be weighted higher. On the other hand, + +when the density of a local region is high, the first vector provides information of finer details since it possesses the ability to inspect at higher resolutions recursively in lower levels. + +Compared with MSG, this method is computationally more efficient since we avoids the feature extraction in large scale neighborhoods at lowest levels. + +### 3.4 Point Feature Propagation for Set Segmentation + +In set abstraction layer, the original point set is subsampled. However in set segmentation task such as semantic point labeling, we want to obtain point features for *all* the original points. One solution is to always sample all points as centroids in all set abstraction levels, which however results in high computation cost. Another way is to propagate features from subsampled points to the original points. + +We adopt a hierarchical propagation strategy with distance based interpolation and across level skip links (as shown in Fig. [2\)](#page-2-1). In a *feature propagation* level, we propagate point features from Nl × (d + C) points to Nl−1 points where Nl−1 and Nl (with Nl ≤ Nl−1) are point set size of input and output of set abstraction level l. We achieve feature propagation by interpolating feature values f of Nl points at coordinates of the Nl−1 points. Among the many choices for interpolation, we use inverse distance weighted average based on k nearest neighbors (as in Eq. [2,](#page-4-0) in default we use p = 2, k = 3). The interpolated features on Nl−1 points are then concatenated with skip linked point features from the set abstraction level. Then the concatenated features are passed through a "unit pointnet", which is similar to one-by-one convolution in CNNs. A few shared fully connected and ReLU layers are applied to update each point's feature vector. The process is repeated until we have propagated features to the original set of points. + + +$$f^{(j)}(x) = \frac{\sum_{i=1}^{k} w_i(x) f_i^{(j)}}{\sum_{i=1}^{k} w_i(x)} \quad \text{where} \quad w_i(x) = \frac{1}{d(x, x_i)^p}, \ j = 1, ..., C$$ + (2) + +# 4 Experiments + +Datasets We evaluate on four datasets ranging from 2D objects (MNIST [\[11\]](#page-8-3)), 3D objects (Model-Net40 [\[31\]](#page-9-0) rigid object, SHREC15 [\[12\]](#page-8-4) non-rigid object) to real 3D scenes (ScanNet [\[5\]](#page-8-5)). Object classification is evaluated by accuracy. Semantic scene labeling is evaluated by average voxel classification accuracy following [\[5\]](#page-8-5). We list below the experiment setting for each dataset: + +- MNIST: Images of handwritten digits with 60k training and 10k testing samples. +- ModelNet40: CAD models of 40 categories (mostly man-made). We use the official split with 9,843 shapes for training and 2,468 for testing. +- SHREC15: 1200 shapes from 50 categories. Each category contains 24 shapes which are mostly organic ones with various poses such as horses, cats, etc. We use five fold cross validation to acquire classification accuracy on this dataset. +- ScanNet: 1513 scanned and reconstructed indoor scenes. We follow the experiment setting in [\[5\]](#page-8-5) and use 1201 scenes for training, 312 scenes for test. + +### 4.1 Point Set Classification in Euclidean Metric Space + +We evaluate our network on classifying point clouds sampled from both 2D (MNIST) and 3D (ModleNet40) Euclidean spaces. MNIST images are converted to 2D point clouds of digit pixel locations. 3D point clouds are sampled from mesh surfaces from ModelNet40 shapes. In default we use 512 points for MNIST and 1024 points for ModelNet40. In last row (ours normal) in Table [2,](#page-5-0) we use face normals as additional point features, where we also use more points (N = 5000) to further boost performance. All point sets are normalized to be zero mean and within a unit ball. We use a three-level hierarchical network with three fully connected layers [1](#page-4-1) + +Results. In Table [1](#page-5-1) and Table [2,](#page-5-0) we compare our method with a representative set of previous state of the arts. Note that PointNet (vanilla) in Table [2](#page-5-0) is the the version in [\[20\]](#page-8-0) that does not use transformation networks, which is equivalent to our hierarchical net with only one level. + +Firstly, our hierarchical learning architecture achieves significantly better performance than the non-hierarchical PointNet [\[20\]](#page-8-0). In MNIST, we see a relative 60.8% and 34.6% error rate reduction + +1 See supplementary for more details on network architecture and experiment preparation. + +| Method | Error rate (%) | +|-----------------------------|----------------| +| Multi-layer perceptron [24] | 1.60 | +| LeNet5 [11] | 0.80 | +| Network in Network [13] | 0.47 | +| PointNet (vanilla) [20] | 1.30 | +| PointNet [20] | 0.78 | +| Ours | 0.51 | + + + +| Table 1: | MNIST digi | t classification. | +|----------|------------|-------------------| + +| Input | Accuracy (%) | +|-------|------------------------| +| vox | 89.2 | +| img | 90.1 | +| pc | 87.2 | +| pc | 89.2 | +| рс | 90.7 | +| pc | 91.9 | +| | vox
img
pc
pc | + +Table 2: ModelNet40 shape classification. + +![](PointNet++_1706.02413_images/_page_5_Figure_4.jpeg) + +> **[그림 해설]** 무작위 점 탈락(Random Point Dropout)을 적용한 의자 포인트 클라우드 시각화. 1024개에서 512, 256, 128개로 점 수가 급감함에 따라 형상의 디테일이 점차 희소해지는 과정을 보여준다. + +![](PointNet++_1706.02413_images/_page_5_Figure_5.jpeg) + +Figure 4: Left: Point cloud with random point dropout. Right: Curve showing advantage of our density adaptive strategy in dealing with non-uniform density. DP means random input dropout during training; otherwise training is on uniformly dense points. See Sec.3.3 for details. + +> **[그림 해설]** 테스트 시 점 개수 감소(1000 $\to$ 128개)에 따른 ModelNet40 분류 정확도(Accuracy %) 평가 곡선. +> - DP(Random Point Dropout) 없이 학습한 모델: PointNet vanilla(파란색)와 Ours SSG(주황색)는 점 수가 줄어들면 500개 미만에서 성능이 75% 이하로 급격히 추락함. +> - DP 적용 모델: PointNet vanilla(DP, 초록) 및 Ours SSG+DP(노랑)는 점 감소에 대해 상대적으로 안정적임. +> - 밀도 적응형 모델: **Ours MSG+DP(빨간색)** 및 **Ours MRG+DP(청록색)**는 1000개 포인트에서 90% 이상의 최고 성능을 기록하며, 점 개수가 256개 이하로 떨어져도 88~89% 이상의 높은 정확도를 견고하게 유지하여 밀도 적응 전략의 우수성을 입증. + +from PointNet (vanilla) and PointNet to our method. In ModelNet40 classification, we also see that using same input data size (1024 points) and features (coordinates only), ours is remarkably stronger than PointNet. Secondly, we observe that point set based method can even achieve better or similar performance as mature image CNNs. In MNIST, our method (based on 2D point set) is achieving an accuracy close to the Network in Network CNN. In ModelNet40, ours with normal information significantly outperforms previous state-of-the-art method MVCNN [26]. + +**Robustness to Sampling Density Variation.** Sensor data directly captured from real world usually suffers from severe irregular sampling issues (Fig. 1). Our approach selects point neighborhood of multiple scales and learns to balance the descriptiveness and robustness by properly weighting them. + +We randomly drop points (see Fig. 4 left) during test time to validate our network's robustness to non-uniform and sparse data. In Fig. 4 right, we see MSG+DP (multi-scale grouping with random input dropout during training) and MRG+DP (multi-resolution grouping with random input dropout during training) are very robust to sampling density variation. MSG+DP performance drops by less than 1% from 1024 to 256 test points. Moreover, it achieves the best performance on almost all sampling densities compared with alternatives. PointNet vanilla [20] is fairly robust under density variation due to its focus on global abstraction rather than fine details. However loss of details also makes it less powerful compared to our approach. SSG (ablated PointNet++ with single scale grouping in each level) fails to generalize to sparse sampling density while SSG+DP amends the problem by randomly dropping out points in training time. + +## 4.2 Point Set Segmentation for Semantic Scene Labeling + +To validate that our approach is suitable for large scale point cloud analysis, we also evaluate on semantic scene labeling task. The goal is to predict semantic object label for points in indoor scans. [5] provides a baseline using fully convolutional neural network on voxelized scans. They purely rely on scanning geometry instead of RGB information and report the accuracy on a per-voxel basis. To make a fair comparison, + +![](PointNet++_1706.02413_images/_page_5_Figure_12.jpeg) + +Figure 5: Scannet labeling accuracy. + +> **[그림 해설]** ScanNet 3D 시맨틱 세그멘테이션 복셀 레이블링 정확도(Accuracy) 비교 막대 그래프. +> - **ScanNet (균일 샘플링, 파란색)**: 3DCNN(0.730), PointNet(0.739), Ours SSG(0.833), Ours MSG+DP(0.845), Ours MRG+DP(0.834). +> - **ScanNet non-uniform (불균일 밀도, 노란색)**: PointNet(0.680, 0.059 하락), Ours SSG(0.727, 0.106 하락), **Ours MSG+DP(0.804, 0.041 하락으로 불균일 환경에서 최고 성능 및 최고 강건성 달성)**, Ours MRG+DP(0.762). + +we remove RGB information in all our experiments and convert point cloud label prediction into voxel labeling following [5]. We also compare with [20]. The accuracy is reported on a per-voxel basis in Fig. 5 (blue bar). + +Our approach outperforms all the baseline methods by a large margin. In comparison with [5], which learns on voxelized scans, we directly learn on point clouds to avoid additional quantization error, + +and conduct data dependent sampling to allow more effective learning. Compared with [20], our approach introduces hierarchical feature learning and captures geometry features at different scales. This is very important for understanding scenes at multiple levels and labeling objects with various sizes. We visualize example scene labeling results in Fig. 6. + +Robustness to Sampling Density Variation To test how our trained model performs on scans with non-uniform sampling density, we synthesize virtual scans of Scannet scenes similar to that in Fig. 1 and evaluate our network on this data. We refer readers to supplementary material for how we generate the virtual scans. We evaluate our framework in three settings (SSG, MSG+DP, MRG+DP) and compare with a baseline approach [20]. + +Performance comparison is shown in Fig. 5 (yellow bar). We see that SSG performance greatly falls due to the sampling density shift from uniform point cloud to virtually scanned scenes. MRG network, on the other hand, is more robust to the sampling density shift since it is able to automatically switch to features depicting coarser granularity when the sampling is sparse. Even though there is a domain + +![](PointNet++_1706.02413_images/_page_6_Figure_3.jpeg) + +• Wall • Floor • Chair • Desk • Bed • Door • Table Figure 6: Scannet labeling results. [20] captures the overall layout of the room correctly but fails to discover the furniture. Our approach, in contrast, is much better at segmenting objects besides the room layout. + +> **[그림 해설]** ScanNet 실내 씬 2개에 대한 시맨틱 분할 정성적 결과 비교 (PointNet vs Ours vs Ground Truth). +> - **색상 범례**: Wall(빨강), Floor(노랑), Chair(초록), Desk(파랑), Bed(보라), Door(하늘), Table(분홍). +> - PointNet(좌측)은 방의 기본 외곽(벽, 바닥)만 대략적으로 구분하고 실내 가구 객체들을 놓치거나 뭉개는 반면, **Ours(PointNet++, 중앙)**는 책상(파랑), 침대(보라), 의자(초록) 등 세부 가구들의 경계를 Ground Truth(우측)와 거의 일치하게 정확히 분할해냄. + +gap between training data (uniform points with random dropout) and scanned data with non-uniform density, our MSG network is only slightly affected and achieves the best accuracy among methods in comparison. These prove the effectiveness of our density adaptive layer design. + +#### 4.3 Point Set Classification in Non-Euclidean Metric Space + +In this section, we show generalizability of our approach to non-Euclidean space. In non-rigid shape classification (Fig. 7), a good classifier should be able to classify (a) and (c) in Fig. 7 correctly as the same category even given their difference in pose, which requires knowledge of intrinsic structure. Shapes in SHREC15 are 2D surfaces embedded in 3D space. Geodesic distances along the surfaces naturally induce a metric space. We show through experiments that adopting PointNet++ in this metric space is an effective way to capture intrinsic structure of the underlying point set. + +For each shape in [12], we firstly construct the metric space induced by pairwise geodesic distances. We follow [23] to obtain an embedding metric that mimics geodesic distance. Next we extract intrinsic point features in this metric space including WKS [1], HKS [27] and multi-scale Gaussian curvature [16]. We use these features as input and then sample and group points according to the underlying metric space. In this way, our network learns to capture multi-scale intrinsic structure that is not influenced by the specific pose of a shape. Alternative design choices include using XYZ coordinates as points feature or use Euclidean space $\mathbb{R}^3$ as the underlying metric space. We show below these are not optimal choices. + +![](PointNet++_1706.02413_images/_page_6_Figure_9.jpeg) + +(a) Horse (b) Cat (c) Horse Figure 7: An example of non-rigid shape classification. + +> **[그림 해설]** 비강체(Non-rigid) 3D 형상 분류 예시 (SHREC15 데이터셋). +> - (a) 뒷다리로 일어선 말(Horse), (b) 뒷다리로 일어선 고양이(Cat), (c) 네 발로 선 말(Horse). +> - 유클리드 좌표($XYZ$)만 사용하면 포즈가 유사한 (a)와 (b)를 같은 클래스로 오인하기 쉬우나, 표면 측지선 거리(Geodesic metric) 기반 내재적 특징(Intrinsic features)을 활용함으로써 포즈 변화가 극심한 (a)와 (c)를 동일한 말(Horse) 카테고리로 올바르게 분류 가능함을 설명. + +**Results.** We compare our methods with previous state-of-the-art method [14] in Table 3. [14] extracts geodesic moments as shape features and use a stacked sparse autoencoder to digest these features to predict shape category. Our approach using non-Euclidean metric space and intrinsic features achieves the best performance in all settings and outperforms [14] by a large margin. + +Comparing the first and second setting of our approach, we see intrinsic features are very important for non-rigid shape classification. XYZ feature fails to reveal intrinsic structures and is greatly influenced by pose variation. Comparing the second and third setting of our approach, we see using geodesic neighborhood is beneficial compared with Euclidean neighborhood. Euclidean neighborhood might include points far away on surfaces and this neighborhood could change dramatically when shape affords non-rigid deformation. This introduces difficulty for effective weight sharing since the local structure could become combinatorially complicated. Geodesic neighborhood on surfaces, on the other hand, gets rid of this issue and improves the learning effectiveness. + +| | Metric space | Input feature | Accuracy (%) | +|-------------|-----------------------------------------|-------------------------------------------------|--------------------------------| +| DeepGM [14] | - | Intrinsic features | 93.03 | +| Ours | Euclidean
Euclidean
Non-Euclidean | XYZ
Intrinsic features
Intrinsic features | 60.18
94.49
96.09 | + +Table 3: SHREC15 Non-rigid shape classification. + +#### 4.4 Feature Visualization. + +In Fig. 8 we visualize what has been learned by the first level kernels of our hierarchical network. We created a voxel grid in space and aggregate local point sets that activate certain neurons the most in grid cells (highest 100 examples are used). Grid cells with high votes are kept and converted back to 3D point clouds, which represents the pattern that neuron recognizes. Since the model is trained on ModelNet40 which is mostly consisted of furniture, we see structures of planes, double planes, lines, corners etc. in the visualization. + +![](PointNet++_1706.02413_images/_page_7_Figure_4.jpeg) + +### 5 Related Work + +The idea of hierarchical feature learning has been very successful. Among all the learning models, convolutional neural network [10, 25, 8] is one of the most prominent ones. However, convolution does not apply to unordered point sets with distance metrics, which is the focus of our work. + +Figure 8: 3D point cloud patterns learned from the first layer kernels. The model is trained for ModelNet40 shape classification (20 out of the 128 kernels are randomly selected). Color indicates point depth (red is near, blue is far). + +> **[그림 해설]** PointNet++ 1단계 계층의 커널들이 학습한 국소 3D 포인트 클라우드 기하학적 패턴 시각화 (무작위 20개 커널, 깊이에 따라 빨강(근거리)~파랑(원거리) 코딩). +> - 평면 조각, 원통/기둥, 모서리(Edge), 구면 곡면, 이중 원반(Disk), 쐐기형 코너 등 3D 공간을 구성하는 다양한 기본 기하 요소(Geometric primitives)를 1차 레이어에서 감지하도록 성공적으로 학습됨을 확인. + +A few very recent works [20, 28] have studied how to apply deep learning to unordered sets. They ignore the underlying distance metric even if the point set does possess one. As a result, they are unable to capture local context of points and are sensitive to global set translation and normalization. In this work, we target at points sampled from a metric space and tackle these issues by explicitly considering the underlying distance metric in our design. + +Point sampled from a metric space are usually noisy and with non-uniform sampling density. This affects effective point feature extraction and causes difficulty for learning. One of the key issue is to select proper scale for point feature design. Previously several approaches have been developed regarding this [19, 17, 2, 6, 7, 30] either in geometry processing community or photogrammetry and remote sensing community. In contrast to all these works, our approach learns to extract point features and balance multiple feature scales in an end-to-end fashion. + +In 3D metric space, other than point set, there are several popular representations for deep learning, including volumetric grids [21, 22, 29], and geometric graphs [3, 15, 33]. However, in none of these works, the problem of non-uniform sampling density has been explicitly considered. + +#### 6 Conclusion + +In this work, we propose PointNet++, a powerful neural network architecture for processing point sets sampled in a metric space. PointNet++ recursively functions on a nested partitioning of the input point set, and is effective in learning hierarchical features with respect to the distance metric. To handle the non uniform point sampling issue, we propose two novel set abstraction layers that intelligently aggregate multi-scale information according to local point densities. These contributions enable us to achieve state-of-the-art performance on challenging benchmarks of 3D point clouds. + +In the future, it's worthwhile thinking how to accelerate inference speed of our proposed network especially for MSG and MRG layers by sharing more computation in each local regions. 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Syncspeccnn: Synchronized spectral cnn for 3d shape segmentation. *arXiv preprint arXiv:1612.00606*, 2016. + +# Supplementary + +# A Overview + +This supplementary material provides more details on experiments in the main paper and includes more experiments to validate and analyze our proposed method. + +In Sec [B](#page-10-0) we provide specific network architectures used for experiments in the main paper and also describe details in data preparation and training. In Sec [C](#page-12-0) we show more experimental results including benchmark performance on part segmentation and analysis on neighborhood query, sensitivity to sampling randomness and time space complexity. + +# B Details in Experiments + +Architecture protocol. We use following notations to describe our network architecture. + +SA(K,r,[l1, ..., ld]) is a set abstraction (SA) level with K local regions of ball radius r using PointNet of d fully connected layers with width li (i = 1, ..., d). SA([l1, ...ld]) is a global set abstraction level that converts set to a single vector. In multi-scale setting (as in MSG), we use SA(K, [r (1), ..., r(m) ], [[l (1) 1 , ..., l(1) d ],...,[l (m) 1 , ..., l(m) d ]]) to represent MSG with m scales. + +FC(l,dp) represents a fully connected layer with width l and dropout ratio dp. FP(l1, ..., ld) is a feature propagation (FP) level with d fully connected layers. It is used for updating features concatenated from interpolation and skip link. All fully connected layers are followed by batch normalization and ReLU except for the last score prediction layer. + +# B.1 Network Architectures + +For all classification experiments we use the following architecture (Ours SSG) with different K (number of categories): + +``` +SA(512, 0.2, [64, 64, 128]) → SA(128, 0.4, [128, 128, 256]) → SA([256, 512, 1024]) → +F C(512, 0.5) → F C(256, 0.5) → F C(K) +``` + +The multi-scale grouping (MSG) network (PointNet++) architecture is as follows: + +``` +SA(512, [0.1, 0.2, 0.4], [[32, 32, 64], [64, 64, 128], [64, 96, 128]]) → +SA(128, [0.2, 0.4, 0.8], [[64, 64, 128], [128, 128, 256], [128, 128, 256]]) → +SA([256, 512, 1024]) → F C(512, 0.5) → F C(256, 0.5) → F C(K) +``` + +The cross level multi-resolution grouping (MRG) network's architecture uses three branches: + +``` +Branch 1: SA(512, 0.2, [64, 64, 128]) → SA(64, 0.4, [128, 128, 256]) +``` + +Branch 2: SA(512, 0.4, [64, 128, 256]) using r = 0.4 regions of original points + +Branch 3: SA(64, 128, 256, 512) using all original points. + +Branch 4: SA(256, 512, 1024). + +Branch 1 and branch 2 are concatenated and fed to branch 4. Output of branch 3 and branch4 are then concatenated and fed to F C(512, 0.5) → F C(256, 0.5) → F C(K) for classification. + +Network for semantic scene labeling (last two fully connected layers in FP are followed by dropout layers with drop ratio 0.5): + +``` +SA(1024, 0.1, [32, 32, 64]) → SA(256, 0.2, [64, 64, 128]) → +SA(64, 0.4, [128, 128, 256]) → SA(16, 0.8, [256, 256, 512]) → +F P(256, 256) → F P(256, 256) → F P(256, 128) → F P(128, 128, 128, 128, K) +``` + +Network for semantic and part segmentation (last two fully connected layers in FP are followed by dropout layers with drop ratio 0.5): + +``` +SA(512, 0.2, [64, 64, 128]) → SA(128, 0.4, [128, 128, 256]) → SA([256, 512, 1024]) → +F P(256, 256) → F P(256, 128) → F P(128, 128, 128, 128, K) +``` + +# B.2 Virtual Scan Generation + +In this section, we describe how we generate labeled virtual scan with non-uniform sampling density from ScanNet scenes. For each scene in ScanNet, we set camera location 1.5m above the centroid of the floor plane and rotate the camera orientation in the horizontal plane evenly in 8 directions. In each direction, we use a image plane with size 100px by 75px and cast rays from camera through each pixel to the scene. This gives a way to select visible points in the scene. We could then generate 8 virtual scans for each test scene similar and an example is shown in Fig. [9.](#page-11-0) Notice point samples are denser in regions closer to the camera. + +![](PointNet++_1706.02413_images/_page_11_Figure_2.jpeg) + +Figure 9: Virtual scan generated from ScanNet + +> **[그림 해설]** ScanNet 데이터로부터 합성한 가상 스캔(Virtual scan) 포인트 클라우드 비교. +> - **(a) ScanNet labeled scene (좌측)**: 완전한 3D 메시에서 균일하게 샘플링된 고밀도 실내 씬 포인트 클라우드. +> - **(b) ScanNet non-uniform (우측)**: 단일 가상 센서 시점에서 레이 캐스팅(Ray casting)을 시뮬레이션하여 거리에 따른 점 밀도 감소와 시점 가림(Occlusion)에 의한 포인트 결손을 사실적으로 반영한 불균일 스캔 데이터. + +## B.3 MNIST and ModelNet40 Experiment Details + +For MNIST images, we firstly normalize all pixel intensities to range [0, 1] and then select all pixels with intensities larger than 0.5 as valid digit pixels. Then we convert digit pixels in an image into a 2D point cloud with coordinates within [−1, 1], where the image center is the origin point. Augmented points are created to add the point set up to a fixed cardinality (512 in our case). We jitter the initial point cloud (with random translation of Gaussian distribution N (0, 0.01) and clipped to 0.03) to generate the augmented points. For ModelNet40, we uniformly sample N points from CAD models surfaces based on face area. + +For all experiments, we use Adam [\[9\]](#page-8-26) optimizer with learning rate 0.001 for training. For data augmentation, we randomly scale object, perturb the object location as well as point sample locations. We also follow [\[21\]](#page-8-8) to randomly rotate objects for ModelNet40 data augmentation. We use Tensor-Flow and GTX 1080, Titan X for training. All layers are implemented in CUDA to run GPU. It takes around 20 hours to train our model to convergence. + +### B.4 ScanNet Experiment Details + +To generate training data from ScanNet scenes, we sample 1.5m by 1.5m by 3m cubes from the initial scene and then keep the cubes where ≥ 2% of the voxels are occupied and ≥ 70% of the surface voxels have valid annotations (this is the same set up in [\[5\]](#page-8-5)). We sample such training cubes on the fly and random rotate it along the up-right axis. Augmented points are added to the point set to make a fixed cardinality (8192 in our case). During test time, we similarly split the test scene into smaller cubes and get label prediction for every point in the cubes first, then merge label prediction in all the cubes from a same scene. If a point get different labels from different cubes, we will just conduct a majority voting to get the final point label prediction. + +### B.5 SHREC15 Experiment Details + +We randomly sample 1024 points on each shape both for training and testing. To generate the input intrinsic features, we to extract 100 dimensional WKS, HKS and multiscale Gaussian curvature respectively, leading to a 300 dimensional feature vector for each point. Then we conduct PCA to reduce the feature dimension to 64. We use a 8 dimensional embedding following [\[23\]](#page-8-10) to mimic the geodesic distance, which is used to describe our non-Euclidean metric space while choosing the point neighborhood. + +# C More Experiments + +In this section we provide more experiment results to validate and analyze our proposed network architecture. + +## C.1 Semantic Part Segmentation + +Following the setting in [\[32\]](#page-9-4), we evaluate our approach on part segmentation task assuming category label for each shape is already known. Taken shapes represented by point clouds as input, the task is to predict a part label for each point. The dataset contains 16,881 shapes from 16 classes, annotated with 50 parts in total. We use the official train test split following [\[4\]](#page-8-27). + +We equip each point with its normal direction to better depict the underlying shape. This way we could get rid of hand-crafted geometric features as is used in [\[32,](#page-9-4) [33\]](#page-9-3). We compare our framework with traditional learning based techniques [\[32\]](#page-9-4), as well as state-of-the-art deep learning approaches [\[20,](#page-8-0) [33\]](#page-9-3) in Table [4.](#page-12-1) Point intersection over union (IoU) is used as the evaluation metric, averaged across all part classes. Cross-entropy loss is minimized during training. On average, our approach achieves the best performance. In comparison with [\[20\]](#page-8-0), our approach performs better on most of the categories, which proves the importance of hierarchical feature learning for detailed semantic understanding. Notice our approach could be viewed as implicitly building proximity graphs at different scales and operating on these graphs, thus is related to graph CNN approaches such as [\[33\]](#page-9-3). Thanks to the flexibility of our multi-scale neighborhood selection as well as the power of set operation units, we could achieve better performance compared with [\[33\]](#page-9-3). Notice our set operation unit is much simpler compared with graph convolution kernels, and we do not need to conduct expensive eigen decomposition as opposed to [\[33\]](#page-9-3). These make our approach more suitable for large scale point cloud analysis. + +| | mean | aero | bag | cap | car | chair ear | phone | | guitar knife | lamp | laptop motor | | mug | | pistol rocket skate | board | table | +|--------------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|--------------|---------------------|--------------|--------------| +| Yi [32]
PN [20] | 81.4
83.7 | 81.0
83.4 | 78.4
78.7 | 77.7
82.5 | 75.7
74.9 | 87.6
89.6 | 61.9
73.0 | 92.0
91.5 | 85.4
85.9 | 82.5
80.8 | 95.7
95.3 | 70.6
65.2 | 91.9
93.0 | 85.9
81.2 | 53.1
57.9 | 69.8
72.8 | 75.3
80.6 | +| SSCNN [33] | 84.7 | 81.6 | 81.7 | 81.9 | 75.2 | 90.2 | 74.9 | 93.0 | 86.1 | 84.7 | 95.6 | 66.7 | 92.7 | 81.6 | 60.6 | 82.9 | 82.1 | +| Ours | 85.1 | 82.4 | 79.0 | 87.7 | 77.3 | 90.8 | 71.8 | 91.0 | 85.9 | 83.7 | 95.3 | 71.6 | 94.1 | 81.3 | 58.7 | 76.4 | 82.6 | + +Table 4: Segmentation results on ShapeNet part dataset. + +# C.2 Neighborhood Query: kNN v.s. Ball Query. + +Here we compare two options to select a local neighborhood. We used radius based ball query in our main paper. Here we also experiment with kNN based neighborhood search and also play with different search radius and k. In this experiment all training and testing are on ModelNet40 shapes with uniform sampling density. 1024 points are used. As seen in Table [5,](#page-12-2) radius based ball query is slightly better than kNN based method. However, we speculate in very non-uniform point set, kNN based query will results in worse generalization ability. Also we observe that a slightly large radius is helpful for performance probably because it captures richer local patterns. + +| kNN (k=16) | kNN (k=64) | radius (r=0.1) | radius (r=0.2) | +|------------|------------|----------------|----------------| +| 89.3 | 90.3 | 89.1 | 90.7 | + +Table 5: Effects of neighborhood choices. Evaluation metric is classification accuracy (%) on ModelNet 40 test set. + +### C.3 Effect of Randomness in Farthest Point Sampling. + +For the *Sampling layer* in our set abstraction level, we use farthest point sampling (FPS) for point set sub sampling. However FPS algorithm is random and the subsampling depends on which point is selected first. Here we evaluate the sensitivity of our model to this randomness. In Table [6,](#page-13-0) we test our model trained on ModelNet40 for feature stability and classification stability. + +To evaluate feature stability we extract global features of all test samples for 10 times with different random seed. Then we compute mean features for each shape across the 10 sampling. Then we compute standard deviation of the norms of feature's difference from the mean feature. At last we average all std. in all feature dimensions as reported in the table. Since features are normalized into 0 to 1 before processing, the 0.021 difference means a 2.1% deviation of feature norm. + +For classification, we observe only a 0.17% standard deviation in test accuracy on all ModelNet40 test shapes, which is robust to sampling randomness. + + + +| Feature difference std. | Accuracy std. | +|-------------------------|---------------| +| 0.021 | 0.0017 | + +Table 6: Effects of randomness in FPS (using ModelNet40). + +# C.4 Time and Space Complexity. + +Table [7](#page-13-1) summarizes comparisons of time and space cost between a few point set based deep learning method. We record forward time with a batch size 8 using TensorFlow 1.1 with a single GTX 1080. The first batch is neglected since there is some preparation for GPU. While PointNet (vanilla) [\[20\]](#page-8-0) has the best time efficiency, our model without density adaptive layers achieved smallest model size with fair speed. + +It's worth noting that ours MSG, while it has good performance in non-uniformly sampled data, it's 2x expensive than SSG version due the multi-scale region feature extraction. 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