PaperlayerView exact revision on arXiv

Source-linked AI summary

Equivariant Point Network for 3D Point Cloud Analysis

Haiwei Chen, Shichen Liu, Weikai Chen, Hao Li

arXiv:2103.14147v2cs.CV

TL;DR

Point-cloud analysis must handle arbitrary 3D transformations, while invariant features can lose distinctions between internally symmetric instances. The paper introduces SE(3) separable convolution and attention for equivariant features, and reports consistent gains over strong baselines across varied tasks.

  • Problem

    Point clouds undergo arbitrary 3D translations and rotations, and invariant features can fail to distinguish instances with internal symmetries.

    Source: §1 · pp. 1–2

  • Method

    The framework factors SE(3) convolution into separable Euclidean and SO(3) operators and adds attention to harness equivariant features.

    Source: §1 · p. 2

  • Results

    The approach consistently outperforms strong baselines across varied tasks, and its equivariant framework significantly outperforms a non-equivariant framework on shape alignment.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The framework provides SE(3)-equivariant point-cloud features and uses attention to support rotation inference for shape alignment.

    Source: §1 · p. 2

    For circularly symmetric shapes whose prescribed canonical poses align with a symmetry axis, the attention layer may not predict a deterministic pose.

    Source: §4.1 · p. 7

Abstract

from arXiv · show

Features that are equivariant to a larger group of symmetries have been shown to be more discriminative and powerful in recent studies. However, higher-order equivariant features often come with an exponentially-growing computational cost. Furthermore, it remains relatively less explored how rotation-equivariant features can be leveraged to tackle 3D shape alignment tasks. While many past approaches have been based on either non-equivariant or invariant descriptors to align 3D shapes, we argue that such tasks may benefit greatly from an equivariant framework. In this paper, we propose an effective and practical SE(3) (3D translation and rotation) equivariant network for point cloud analysis that addresses both problems. First, we present SE(3) separable point convolution, a novel framework that breaks down the 6D convolution into two separable convolutional operators alternatively performed in the 3D Euclidean and SO(3) spaces. This significantly reduces the computational cost without compromising the performance. Second, we introduce an attention layer to effectively harness the expressiveness of the equivariant features. While jointly trained with the network, the attention layer implicitly derives the intrinsic local frame in the feature space and generates attention vectors that can be integrated into different alignment tasks. We evaluate our approach through extensive studies and visual interpretations. The empirical results demonstrate that our proposed model outperforms strong baselines in a variety of benchmarks

1. Introduction

The introduction motivates SE(3)-equivariant point-cloud features as a way to preserve transformation information, then proposes separable convolution and attention to address computational and feature-fusion challenges. It reports strong-baseline gains across varied tasks, including shape alignment.

1. Introduction

  • Point clouds undergo arbitrary 3D translations and rotations, while invariant features can fail to distinguish instances with internal symmetries.Equivariant features retain information about the input group transform through neural layers.Source: §1 · pp. 1–2
  • Matching equivariant features can require costly PnP optimization, while naive pooling may discard useful features and damage equivariant structure.These are the stated challenges motivating attention-based feature fusion.Source: §1 · p. 2
  • SE(3) separable convolution factors naive 6D convolution into operators in Euclidean and SO(3) spaces, significantly reducing computational cost.It lifts input points to a homogeneous space and uses finite rotation groups such as the icosahedral group.Source: §1 · p. 2
  • Group attentive pooling adaptively fuses rotation-equivariant features into invariant counterparts using learned attention weights and an implicit intrinsic local frame.The layer is trained jointly with the network.Source: §1 · p. 2
  • Across varied tasks, the approach consistently outperforms strong baselines, including a shape-alignment comparison with a non-equivariant framework.The introduction also notes ablation analysis and qualitative visualization of algorithmic components.Source: §1 · p. 2

2. Related Work

Related work spans point-cloud architectures and rotation-invariant descriptors, alongside rotation-equivariant methods for images, spherical signals, and voxels. The authors position their contribution as a unified hierarchical SE(3)-equivariant framework for point clouds within non-spectral group-equivariant approaches.

2. Related Work

  • Point-cloud methods include permutation-invariant functions [36], shift-equivariant hierarchies [38] [33], explicit kernels [22] [52], and image-projection or voxel representations [37].The passage points to [21] for a survey of point-cloud convolution.Source: §2 · p. 2
  • Rotation-invariant descriptors use local reference frames [17] [55] or point-pair folding [9], but invariant features may lose spatial information.The passage describes Perfect Match [17], a capsule network [55], and PPF-FoldNet [9].Source: §2 · pp. 2–3
  • 2D rotation-equivariant approaches use oriented filters [34], log-polar transforms [13], circular harmonics [50], filter rotation [31] [48], or group convolutions [5] [8].Source: §2 · p. 3
  • For 3D rotations, spectral methods use S2 or SO(3) correlations [12] and have been extended to voxel grids [49] [47] [12] [24], while point-cloud work remains sparse.Tensor field networks [44] achieve SE(3) equivariance on irregular point clouds using radial functions and spherical harmonics.Source: §2 · p. 3
  • The authors place their approach in non-spectral group-equivariant work [5] [7] and claim the first unified hierarchical SE(3)-equivariant framework for point-cloud convolution.The passage contrasts this scope with prior approaches focused only on rotational equivariance.Source: §2 · p. 3

3. Method

The method builds SE(3)-equivariant point-cloud features by approximating a 6D convolution with separable spatial and rotational convolutions, then uses attention to process equivariant features for shape matching and invariant pooling.

3. Method

  • SPConv approximates a 6D SE(3) convolution with consecutive spatial and rotational convolutions, preserving equivariance at substantially lower computational cost.The spatial kernel varies by translation in the rotated reference frame, while the group kernel varies by rotation.Source: §3 · p. 4, Figure 2 · p. 4

3.1. SE(3) point convolution

  • SE(3) point convolution aggregates local point correlations under rotation, using explicit parameterized kernels or an implicit formulation that scales better for global filtering.The explicit form uses kernel points with associated weight matrices and linear or Gaussian correlations; the network uses it in convolutional layers and the implicit form in its final global-filtering layer.Source: §3.1 · pp. 4–5

3.2. SE(3) group convolution

  • SE(3) group convolution computes SO(3) correlations over a discrete rotation group using rotation-indexed kernels; the implementation can use the icosahedron group and precomputed permutation indices.The kernel aggregates information from neighboring rotation signals.Source: §3.2 · p. 5
  • The naive convolution costs O(KpKgCN), while SPConv costs O((Kp + Kg)CN), reducing complexity especially when Kp and Kg are large.Here Kp and Kg count point- and group-convolution kernels, C is the product of input and output channels, and N combines points and rotation-group elements.Source: §3.2 · p. 5

3.3. Shape Matching with Attention mechanism

  • Rotation-based attention assigns weights to equivariant features and supports SO(3) detection, where group elements act as rotation anchors with classification and rotational-offset regression.The detection objective combines anchor classification with regression for the selected anchor.Source: §3.3 · p. 5
  • Group attentive pooling weights rotation-equivariant features to produce an invariant representation, with a temperature controlling response sharpness and optional canonical-axis supervision.Its loss combines the task-specific loss with an optional cross-entropy term when ground-truth canonical pose is available.Source: §3.3 · p. 5

3.4. Implementation Details

  • The network uses five hierarchical layers with two SPConv blocks per layer, then applies GA pooling; classification adds fully connected and softmax layers, while metric learning uses L2 normalization.The first SPConv block in each layer is strided by a factor of two.Source: §3.4 · p. 6

4. Experiments

Across rotated ModelNet40 and 3DMatch, the experiments report strong results for the proposed model, including improved rotation prediction, classification and retrieval, and shape alignment. Ablations and attention-confidence analysis examine how pooling, group convolution, and object symmetry relate to those results.

4.1. Experiments on Rotated ModelNet40

  • For pose estimation, equivariant models outperform KPConv, with Ours-60 reporting the lowest errors and equivariant-model maximum errors kept within 9 degrees.The comparison uses angular errors; Figure 3 plots error-percentile curves across models with different SO(3) group sizes.Source: Table 2 · p. 6, Figure 3 · p. 6, Table 1 · p. 6
  • The model achieves the best classification and retrieval performance when ModelNet40 objects are randomly rotated.The experiments use random rotations and 1,024 sampled points per object.Source: §4.1 · pp. 6–7
  • Ablations show that supervised attentive pooling improves classification accuracy over max and mean pooling at the same parameter count, while unsupervised attentive pooling does not beat max pooling.Using only point convolution also reduces performance, supporting the effectiveness of group convolution.Source: §4.1 · p. 7, Table 3 · p. 7
  • Attention confidence is ambiguous for circularly symmetric categories such as cones and flower pots, but high for categories with distinctive canonical orientations.The authors report that GA pooling helps classification for rotationally asymmetric objects; for symmetric objects it can reduce to average pooling.Source: §4.1 · p. 7, Figure 4 · p. 7

4.2. Shape Alignment on 3DMatch

  • The 3DMatch evaluation uses 433 overlapping fragments from eight indoor scenes, with RGB-D training data from 62 indoor scenes and keypoint-correspondence average recall.Test fragments contain 150,000 to 600,000 points, and the reported evaluation omits RANSAC.Source: §4.2 · p. 7
  • On 3DMatch, the model outperforms all baselines in average recall without precomputing invariant representations or local reference frames, using sparse 1024-point sets and no point normals.It also exceeds, the state-of-the-art method discussed, without normal information.Source: Table 4 · p. 8, §4.2 · p. 8

5. Conclusions and Discussions

The framework combines an efficient way to compute SE(3)-equivariant point-cloud features with attention that preserves their expressiveness and supports alignment and pooling. The authors report performance gains across challenging tasks.

5. Conclusions and Discussions

  • SE(3) separable convolution factorizes naive SE(3) convolution into two concatenated operators in separate subspaces.This formulation is presented as an efficient way to compute equivariant features for point-cloud analysis.Source: §5 · p. 8
  • An attention mechanism maintains the expressiveness of SE(3)-equivariant features and supports 3D alignment and pooling into invariant features.The invariant counterparts are described as more ready to use.Source: §5 · p. 8
  • The framework is reported to produce performance gains across a variety of challenging tasks and is among the earliest investigations of SE(3)-equivariant point-cloud features.The authors note opportunities for more efficient methods and broader applications.Source: §5 · p. 8
Loading 2103.14147v2…