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Shape completion with azimuthal rotations using spherical gidding-based invariant and equivariant network

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Abstract

Point cloud completion aims to restore full shapes of objects from their partial views obtained by 3D optical scanners. In order to make point cloud completion become more robust to azimuthal rotations and more adaptive to real-world scenarios, we propose a novel network for simultaneous rotation invariant and equivariant completion with no need of data augmentation, while other existing approaches require separately trained models for different completion types. Our method includes several main steps: First, Density Compensation Mapping (DCM) as well as Aggregative Gaussian Gridding (AGG) modules are introduced to transfer partial point clouds to spherical signals and avoid unbalanced sampling. Second, an encoder based on group correlation is designed to extract rotation invariant global features and equivariant azimuthal features from spherical signals. Third, parallel groups of decoders are proposed to realize rotation invariant completion based on feature fusion. Finally, a feature remapping module as well as Pose Voting Alignment (PVA) algorithm are proposed to unify feature space and realize rotation equivariant completion. Based on these modules, we find that the application of group correlation can be extended to the domain of shape completion; equivariant and invariant completions can be unified in one pipeline, and our inherent rotation equivariant and invariant framework can achieve competitive performances when comparing with existing representative methods.

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Algorithm 1
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Data availability

The datasets used and analyzed during the current study are public and available at: https://shapenet.org/, https://cs.nyu.edu/~silberman/datasets/nyu_depth_v2.html and https://apolloscape.auto/.

Notes

  1. Detailed explanations of some major steps in Algorithm 1 are recorded in “Appendix 3”.

  2. Codes will be available at https://github.com/HangWu2020/RISC.

  3. The calculation methods of FLOPs are recorded in “Appendix 4”.

  4. https://github.com/Lyken17/pytorch-OpCounter.

  5. https://github.com/Microsoft/nni.

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Funding

This work is supported by National Natural Science Foundation of China (Grant No. 51975361).

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Authors and Affiliations

  1. School of Mechanical Engineering, Shanghai Jiao Tong University, 800 Dong Chuan Road, Shanghai, 200240, China

    Hang Wu, Yubin Miao & Ruochong Fu

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  1. Hang Wu
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Correspondence to Yubin Miao.

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Appendices

Supplementary

Please refer to Appendix for more discussions.

Appendix 1: Full expression of DCM and Inverse DCM

DCM changes the value of \(\theta\) in point x:

$$\begin{aligned} \mathcal {M}(\theta )= {\left\{ \begin{array}{ll} \frac{-B+A\theta _0+\sqrt{(B-A\theta _0)^2+2A\theta I_3/\pi }}{A} &{} \theta < L_1\\ 2\tan ^{-1}(e^{\frac{\theta I_3}{\pi }+\log (\tan 0.5\theta _0)-I_1}) &{} L_1 \leqslant \theta \leqslant L_2 \\ \frac{-A \theta _1 - B+\sqrt{(B+A\theta _1)^2+2AC(\theta )}}{-A} &{} \theta > L_2 \end{array}\right. } \end{aligned}$$
(22)

where \(A = \frac{-\cos \theta _0}{\sin \theta _0^2}\), \(B=\frac{1}{\sin \theta _0}\), \(L_1 = \frac{I_1\pi }{I_3}\) , \(L_2 = \frac{I_2\pi }{I_3}\), and:

$$\begin{aligned} \begin{aligned}&I_1 = -0.5A\theta _0^2 + B\theta _0\\&I_2 = I_1+\log (\tan 0.5\theta _1/\tan 0.5\theta _0)\\&I_3 = I_2-0.5A\pi ^2-\theta _1^2+A\theta _1\theta _0 + B\theta _0\\&C(\theta ) = I_2-\theta _1(A\theta _1+B)-\theta I_3/\pi +0.5A\theta _1^2\\ \end{aligned} \end{aligned}$$
(23)

In AGG, we need to transfer points back to their original shape using inverse DCM \(\mathcal {M}^{-1}\):

$$\begin{aligned} \mathcal {M}^{-1}(\theta )=\frac{1}{I_3\pi }{ {\left\{ \begin{array}{ll} 0.5A\theta ^{\prime 2} + (B-A\theta _0)\theta ^\prime &{} \theta ^{\prime }< \theta _0\\ -0.5A\theta ^{\prime 2} + (A\theta _1+B)\theta ^\prime + C &{} \theta > \theta _1\\ I_1+\log (\tan 0.5\theta /\tan 0.5\theta _0) &{} \mathrm{{otherwise}} \end{array}\right. } }\end{aligned}$$
(24)

where A, B, \(I_1\), \(I_2\), \(I_3\) are the same as (2) in Sect. 3.1, except for C:

$$\begin{aligned} C = I_2 - 0.5A{\theta _1}^2-B\theta _1 \end{aligned}$$
(25)

(24) illustrates that DCM and its inverse form are monotonicity continuous. In practice, an alternate is to record the corresponding relations between indices of points and grid cells.

Appendix 2: Rotation equivariance of group correlation

We hereby illustrate that module \(E_1\) in Encoder is equivariant to azimuthal rotation on partial point cloud \(X_p\). Given the definition that:

$$\begin{aligned} H^1_\mathrm{{r}}(R) = \phi \star \mathcal {F}(X_p; G) \end{aligned}$$
(26)

where \(\mathcal {F}=\mathrm{{AGG}} \circ \mathrm{{DCM}}\). When \(X_p\) is rotated by \(\mathrm{{Rot}}_z\), without loss of generality, we illustrate the one-dimensional group correlation case (i.e., \(K=1\)) as:

$$\begin{aligned} \begin{aligned} \left[ \psi \star \mathcal {F}(\mathrm{{Rot}}_zX_p; G) \right] (R)&= \left[ \psi \star [L_{\mathrm{{Rot}}_z} \mathcal {F}] \right] (R) \\&= \int \limits _{{\textbf {G}}} \psi (R^{-1}g)\mathcal {F}(\mathrm{{Rot}}_z^{-1}g)\mathrm{{d}}g \\&= [L_{\mathrm{{Rot}}_z}[\psi \star \mathcal {F}]](R) \\&= [L_{\mathrm{{Rot}}_z}H^1_r](R) \end{aligned} \end{aligned}$$
(27)

A vital prerequisite of (27) is \(\mathcal {F}\) should be azimuthal rotation equivariant on \(S^2\) sphere, this is intuitive because the rotations of points induced by DCM are about the axes in \(x-y\) plane, which are all orthogonal to z-axis. In addition, the derivation process from step 2 to step 3 in (27) is a conclusion of [49], so we do not replicate the full steps here.

Appendix 3: More explanation of PVC

In this section, we provide more explanations of some steps in Algorithm 1.

Step 3 This step simply executes cyclic shift on \(H_\mathrm{{que}}\) by i bits.

Step 7 Z is a set of L2 distances \(z_i\), and V is a set of vote candidates \(v_i\).

Step 11 C is a changeable set of vote cluster centers. \(\mathrm{{max}}(\mathrm{{dist}}(V, C))\) is the unidirectional Hausdorff distance from V to C, while \(\mathrm{{argmax}}(\mathrm{{dist}}(V, C))\) is the index of \(v_i\) that is furthest to C. We set the threshold distance r as 6.

Step 18 Function \(\mathrm{{knn}}(V, C)\) means cluster all the elements in set V according to predefined centers in set C, it returns several clusters. Function \(\mathrm{{sort}}()\) sorts the clusters from the largest to the smallest.

Step 20 Function \(\mathrm{{mean}}(G)\) not only computes the mean of elements in G, but also transfers the mean of votes to angle.

Appendix 4: Computation of FLOPs

In Table 4, we compare FLOPs of different methods. For the neural network in feature remapping module, we double check the calculation accuracy using thopFootnote 4 and nni,Footnote 5 and report the larger value. For CD calculation of two point clouds \(X\in \mathbb {R}^{n\times 3}\) and \(Y\in \mathbb {R}^{n\times 3}\), it requires to compute L2 distance between two 3D points for \(n^2\) times, and each computation requires 8 FLOPs (i.e., 2 plusses, 3 minuses, and 3 multiplications).

Note that we ignore some lightweight operations here, including rotations of point clouds, L2 distance of feature vectors, dynamic vote clustering. However, all these operations are much lighter than neural network and monodirectional CD calculation and will not effect the overall comparison between different pose alignment methods.

Appendix 5: More completion results

Figure 13 illustrates more completion examples, including some asymmetric shapes. We also compare our network performances with existing methods.

Fig. 13
figure 13

Visualization of rotation equivariant completion results from different input poses

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Wu, H., Miao, Y. & Fu, R. Shape completion with azimuthal rotations using spherical gidding-based invariant and equivariant network. Neural Comput & Applic 36, 13269–13292 (2024). https://doi.org/10.1007/s00521-024-09712-z

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