Thank you very much for your encouraging comments and constructive suggestions. We have conducted additional experiments to test the proposed method and sincerely hope that we have addressed your concerns.

Re: Weaknesses

• Test on the SHREC'19 Dataset: Thanks. We have performed additional experiments on the SHREC'19 dataset, which includes 44 shapes and a total of 430 evaluation pairs. Following the suggested work, we report the mean geodesic errors in the subsequent table. It is noteworthy that, while our focus in this work is on optimizing a non-rigid deformation field with a particular emphasis to ensure physically plausible deformations for occluded parts, rather than on shape matching, our method still yields comparatively satisfactory results (albeit slightly inferior to those of specialized matching approaches). The results also demonstrate that our method can indeed provide an appropriate initialization which can then be enhanced through subsequent matching optimization processes.

• |Methods|SMM[1]|IFMatch[2]|Ours| |-|--|-|-| |Geodesic error|0.040|0.065|0.113|

• Test on the SHREC'16 Dataset: Thanks. We have conducted further experiments on partial shapes from the challenging SHREC’16 dataset, which contains 200 test shapes categorized into CUTS and HOLES types. The mean geodesic errors reported in the following table reveal that methods specifically tailored for shape matching, such as SMM, achieve higher accuracy. However, our method is capable of providing a suitable initialization for these challenging cases, even without pretraining.

• Methods	CUTS	HOLES
  SMM	0.076	0.159
  Ours	0.123	0.284

• Differences from and Discussions of the Suggested Methods: Thanks. There are two core contributions of our work: 1) The local and adaptive metric MCC effectively prevents physically implausible deformations, such as collapses and tearing, which are common in previous non-rigid registration methods. 2) We employ LLR to further ensure reasonable deformations of occluded parts, a problem that previous approaches have not adequately addressed. In contrast to our method, [1] primarily focuses on training a self-supervised network for multimodal shape matching. While it shows promising performance, it does not address the deformations of occluded parts. [2] utilizes an auto-decoder structure to implicitly align two volumes, which requires surface normals for training. Additionally, it employs a bi-directional Chamfer Distance for inferences, a metric that may be susceptible to occlusion. In comparison, our method stands out as an unsupervised, runtime-optimization approach, which ensures robust generalization without the need of labeled data for specific training. We have cited and added discussions of the suggested work in Sec. Related Work of the revised manuscript.

References

[1] Cao et al., Self-supervised learning for multimodal non-rigid 3d shape matching, CVPR 2023.

[2] Sundararaman et al., Implicit field supervision for robust non-rigid shape matching, ECCV 2022.

Dear Reviewer C5yq,

Thank you very much for reviewing our paper. Given that the end of the discussion period is approaching, we would like to ask if you have any further concerns or questions, particularly as a follow-up or update the score to our response?

We appreciate the discussion, your feedback, and your suggestions. Many thanks for your time and effort.

Best and sincere wishes,

The authors

Thank you very much for your encouraging comments and constructive suggestions. We have conducted additional experiments and give a detailed response in the following. Due to space limit, we first address the weakness concerns of our method, followed by responses to questions in the subsequent Part 2. We sincerely hope that we have addressed your concerns.

Re: Weaknesses

• Incorporation of Geometric Descriptor and how to Use it: Thanks. The core contribution of this work is the optimization of a non-rigid deformation field designed to align two shapes, especially under conditions of occlusion. This is somewhat different from geometric matching methods, which primarily focus on establishing correspondences between shapes. Deformation methods are often considered a subsequent step after the matching process to deal with large motions, as has been adapted in many previous approaches. However, our method is more effective for shapes with occlusion. We also demonstrate its versatility by integrating it with state-of-the-art correspondence methods.

  In our experiments, following [1], we employ a pre-trained descriptor for initial landmark point matching. Specifically, given the two matching sets $\mathcal{S}(\mathbf{X})$ and $\mathcal{S}(\mathbf{Y})$ with $|\mathcal{S}(\mathbf{X})|=|\mathcal{S}(\mathbf{Y})|<\min(|\mathbf{X}|,|\mathbf{Y}|)$, our optimization function is formally defined as $\mathbf{\Theta}^*=\mathop{\arg\min}\limits_{\mathbf{\Theta}}\mathcal{F}(\mathbf{\Theta})+\beta\mathcal{M}(\mathbf{\Theta})$
  where $\mathcal{F}(\mathbf{\Theta})$ and $\mathcal{M}(\mathbf{\Theta})$ denote the deformation loss and the pointwise matching loss, separately. We have added these explanations in Sec. I of the Appendix to make it more understandable.

• Motivation for Utilizing Implicit Neural Fields (INFs): We appreciate the inquiry. There are several reasons for employing INFs in our work:

  (1). INFs leverage the universal approximation capability of neural networks to effectively describe deformation fields. This eliminates the need to manual and explicit definition of complex deformations, such as the traditional spline and kernel function methods [1].

  (2). The adoption of INFs enables automatic differentiation, which brings powerful optimization efficacy.

  (3). INFs also offer the advantage of runtime optimization, allowing for an unsupervised approach that ensures robust generalization.

  Yes, as LLR enables reasonable deformations even for unmatched parts, which in turn naturally ensures the optimization and smoothness of the deformation field for each point.

• Ablation Study of MCC and CD: Thanks for this suggestion. We have conducted additional ablation studies to demonstrate the superiority of the MCC metric against CD and added these experiments in the revised manuscript. Please refer to the first response to Reviewer FdAL for details.

• Further Comparison with the Suggested CD Variants: Thanks. We have conducted further experiments to evaluate MCC against the most recently or state-of-the-art Hyperbolic Chamfer Distance (HCD) in point cloud completion [2]. We use the author's source code to implement the HCD metric. Particularly, we adopt three distinct tests by adjusting the coefficient $\alpha$ within the HCD framework to achieve comprehensive results. The quantitative results reported in the following table demonstrate that our proposed method consistently achieves higher-quality deformations. Moreover, MCC shows potential for further exploration and application in general point analysis tasks, including point cloud completion.

• |Method |EPE↓|AccS↑| AccR↑|Outlier↓|EPE↓|AccS↑|AccR↑|Outlier↓| EPE↓|AccS↑|AccR↑|Outlier↓| |-|--|-|-|-|-|-|-|-|-|-|-|-| |HCD with $\alpha=1.0$ |28.946| 15.533| 33.077| 0.054|[29.164| 15.455| 32.369| 0.169| 29.384| 15.042| 33.103| 0.216| |HCD with $\alpha=0.5$|27.116| 12.727|33.527| 0. 000 |26.451| 11.944| 32.145|0.000|26.815| 12.826| 32.750| 0. 000| |HCD with $\alpha=0.3$|23.535|10.414| 33.338|0.000|23.074|9.730|32.881|0. 000 |24.291|10.872|33.744|0. 000 |MCC (Ours) |8.662|29.228|96.813|0.000|5.687|75.193|97.184|0.000|12.112|42.372|56.564|0.000

  We have added these experiments and analyses in Sec. H of the Appendix of the revised manuscript.

References

[1] Yang et al., Non-Rigid Point Cloud Registration with Neural Deformation Pyramid, NeurIPS, 2022.

[2] Myronenko et al., Point Set Registration: Coherent Point Drift, IEEE Transactions on Pattern Analysis and Machine Intelligence, 2010.

[3] F. Lin, et al., Hyperbolic Chamfer Distance for Point Cloud Completion, ICCV, 2023.

We give a detailed answer to other comments and sincerely hope that we have addressed your concerns.

Re: Questions

• Update $\mathcal{T}(\mathbf{Y})$: Thanks for this detailed comment. We have revised it in the update manuscript.

• Adding More Details of Lemma 1: Thanks. We can rewrite the correntropy as

Thank you very much for your encouraging comments and valuable suggestions. We have conducted additional experiments and give a detailed response in the following. We sincerely hope that we have addressed your concerns.

Re: Weaknesses

• Ablation Study of MCC and the Chamfer Distance: Thanks for this constructive suggestion. As suggested, we have conducted additional ablation experiments to replace MCC with CD on a series of models used in Fig. 5, testing them with progressively increasing levels of occlusion (0-5). The new added Fig. 9 in the Appendix demonstrates that MCC consistently achieves higher-quality deformation results, particularly with significantly high AccS and AccR metrics. Moreover, we provide a detailed comparison of the time consumption of both the Chamfer Distance and MCC in the subsequent table, measured in seconds. As observed, MCC also achieves significantly high efficiency with the computation time around 1 millisecond for a dataset comprising around $10^4$ points. We have added these time tests in Sec. G of the Appendix in the updated manuscript.

• |Cases |0|1|2|3|4|5|Ave. |-|-|-|-|-|-|-|-| |CD time $(\times 10^{-3}$s)|0.696|0.676|0.685|0.687|0.697|0.6900|0.689| |MCC time $(\times 10^{-3}$s)|1.332|1.327|1.466|1.323|1.285|1.316|1.342|

• Generalization Capability: Thanks. As an unsupervised and runtime optimization-based neural deformation pipeline, our method does not require data annotations. It reasons non-rigid deformations from a case-by-case geometric perspective, hence guaranteeing a high generalization to unknown categories. Moreover, indeed we have demonstrated the generalization capabilities of MCC across a variety of real-world point cloud scenarios, such as the human registration on occluded depth views (where different levels of occlusion are present), as detailed in Sec. 5.4, and shape completion applications (i.e., same-class level, not same-instance level), covered in Sect. 5.7. In these tests, the shapes are real data sets captured using scanners, thereby further validating the robust generalization of our method to real-world scenarios.

• Place Table 4 in the Main Text: Thank you for this careful comment. We have placed Table 4 in the main text to enhance clarity in the updated manuscript.

• Mesh hole filling with a mean shape: Thanks. We have conducted additional experiments to apply our proposed method to the challenging task of mesh hole filling, particularly in cases where the source or template models also contain holes. The source models are selected from Fig. 7 of the main paper and Fig. 14 of the appendix, with the middle model in each case serving as the mean shape.

  As presented in the added Item 5 of Sec. K and Fig. 15 of the updated Appendix, our method consistently achieves high-quality deformation results, as evidenced by the 'Result 1' and 'Result 2' deformed shapes. It is observed that, due to the superior non-rigid deformation capabilities of our method, the majority of the holes in the original shapes have been effectively filled. Moreover, we present the boolean shapes, which are obtained from the union operation between the deformed surfaces and their corresponding target models (that is, Shape 1 $\cup$ Result 1 and Shape 2 $\cup$ Result 2). This boolean operation not only further completes the holes but also demonstrates the seamless integration of the deformed shapes with their target models.

Re: Questions

• Explanation of the Timestamp $t$: Thanks. The timestamp $t\in[0, 1]$ represents the deformation process from the source shape to the target one. For instance, the interpolation shape with $t=0.5$ is represented by $\mathbf{Y}_{t=0.5}=\mathbf{Y}+t*\nu(\mathbf{Y})=\mathbf{Y}+0.5*\nu(\mathbf{Y})$ with $\nu$ denoting the learned displacement field. We have added these explanations in the revised manuscript to make it more understandable.

Dear Reviewer FdAL,

Thank you very much for reviewing our paper. Given that the end of the discussion period is approaching, we would like to ask if you have any further concerns or questions, particularly as a follow-up or update the score to our response?

We appreciate the discussion, your feedback, and your suggestions. Many thanks for your time and effort.

Best and sincere wishes,

The authors

Thank you very much for your insightful comments and valuable suggestions. Due to space limit, we first give a detailed answer to the two key concerns regarding regularization and the relationship to robust functions and present additional answers in the subsequent Part 2. By these clarifications we sincerely hope that we have addressed your concerns.

Re: Weaknesses

• Analysis of the Regularization: Thanks. Indeed, we have reported an ablation study in the paper (Sec. 5.6) to reveal that regularizing the deformation field of occluded regions is mainly from LLR rather than the network. This is because a network alone is insufficient to handle occlusions effectively, often resulting in significant deviations (Fig. 5). As suggested, we added additional ablation studies on the OpenCAS dataset to investigate the role of activation functions (i.e., ReLU and SIREN) with the LLR for regularization. The results presented in the following table reveal that the regularization of deformations in occluded areas is primarily attributed to LLR rather than the activation function, as ReLU+CD and ReLU+MCC still generate significant deformation errors, especially for both the EPE and Outlier metrics.

• |Method |EPE↓|AccS↑| AccR↑|Outlier↓|EPE↓|AccS↑|AccR↑|Outlier↓| EPE↓|AccS↑|AccR↑|Outlier↓| |-|--|-|-|-|-|-|-|-|-|-|-|-| |ReLU+CD |47.994|9.106|20.466|17.957|39.171|9.684|23.321|8.533| 27.494|28.766|44.443|11.272| |ReLU+MCC|31.582| 28.446| 40.614| 4.378|12.012| 47.258| 69.975| 0.000|17.245| 32.841| 51.744| 1.326| |ReLU+MCC+LLR|14.690|34.144| 57.860|0.000|6.984|66.543|89.956|0.000|14.306| 36.670| 52.982|0.000

• Moreover, to investigate the intrinsic regularization capabilities of activation functions, we have conducted an additional experiment by applying recursive deformations to complete shapes without occlusion. The EPE metric detailed in the subsequent table demonstrates that ReLU and SIREN exhibit comparable performance on complete shapes. This finding further substantiates our conclusion that while activation functions do possess a regularization effect, they may fall short when addressing the complexities of challenging occlusions (i.e., the motivation and core contributions of this work). In such cases, LLR proves to be particularly beneficial. We have added these additional ablation studies in Sec. E of the Appendix of the revised manuscript.

• |Method |Liver 1->Liver 2|Liver 2->Liver 3|Liver 3->Liver 1 |-|-|-|-| |ReLU+CD |4.685|5.550|6.787| |SIREN+CD|3.912| 5.063| 6.183|

• Differences and Connections with Robust Functions: Thanks for this insight. We have conducted an in-depth analysis comparing correntropy with robust functions.

  1) Differences. Unlike common robust functions, correntropy is initially proposed in information-theoretic learning to handle nonzero mean and non-Gaussian noise (e.g., the occlusion part can be seen as a certain type of non-Gaussian noise), which is related to the Renyi’s quadratic entropy. Besides, Maximum correntropy criterion (MCC) is a local measure that provides a probabilistic meaning of maximizing the error probability density at the origin according to the information potential.

  2) Connections. As suggested, we derive the relationship between MCC and robust functions. By setting $\rho(e)=\left(1-\exp \left(-e^2 / 2 \sigma^2\right)\right) / \sqrt{2 \pi} \sigma$, we can prove that $\rho(e)$ is an influence function satisfying all the criteria of a robust function, i.e., 1) $\rho(e) \geq 0$; 2) $\rho(0)=0$; 3) $\rho(e)=\rho(-e)$; 4) $\rho\left(e_i\right) \geq \rho\left(e_j\right)$ for $\left|e_i\right|>\left|e_j\right|$. Moreover, $\min_{\theta}\sum_{i=1}^{N}\rho\left(e_{i}\right)\\Leftrightarrow \max_{\theta} \sum_{i=1}^{N} \exp\left(-e_{i}^{2} / 2 \sigma^{2}\right)/ \sqrt{2 \pi} \sigma$ with the corresponding weight function of $\rho(e)$ as $w(e)=\exp \left(-e^2 / 2 \sigma^2\right) / \sqrt{2 \pi} \sigma^3$. For comparison, the Bi-square's weight function is $w_{\mathrm{Bi}}(e)= \left[1-(e / h)^2\right]^2 \quad \text{if} \quad |e| \leq h \\ \quad \text{else} \quad 0$, where $h$ is the tuning threshold. Therefore, the nonzero part of $w_{\mathrm{Bi}}(e)$ is equivalent to (with a constant) the square of the first-order Taylor expansion of $w(e)$. MCC is analogous to a robust function but with the specific influence function $\rho(e)$. However, unlike the Huber or Bi-square function, correntropy does not need a predefined threshold such as $h$ and the kernel size entirely governs the properties. Moreover, our derivation between correntropy and robust functions offers a practical way to select an appropriate threshold for robust functions or determining the kernel size for correntropy. We added these analyses in Sec. F of the Appendix of the updated manuscript to relate MCC with robust functions.

We give a detailed answer to other comments and sincerely hope that we have addressed your concerns.

Re: Weaknesses

• Our Motivation and Contributions: In this work, we develop a systematic unsupervised and runtime-optimization non-rigid registration method to address the challenging issue of occlusions, a problem that has not received adequate attention in prior approaches. Our method features two key contributions: 1) A correntropy-induced local metric that adaptively distinguishes between points, thereby preventing the collapse, tearing, and other physically implausible outcomes encountered in prior methods. Moreover, given the widespread adoption of the Chamfer distance, we believe that the MCC metric can serve as a valuable enhancement across various domains such as for the point cloud anlaysis field. 2) We pioneer the use of local linear reconstruction for non-rigid registration to ensure physically reasonable deformations for occluded parts. Our method demonstrates superior performance over competitors and offers a valuable algorithm for managing occlusions effectively.

• Discussion of the Suggested Work: Thanks for the valuable suggestion. A distance transform is utilized to expedite the computation of the Chamfer distance in [1], and in principle, this manner can be extended to the MCC-induced metric as our proposed method also incorporates pairwise computation. However, as the authors noted, we must carefully consider the trade-off between discretization error, grid resolution, memory consumption, and estimation accuracy. We have cited and added these discussions of the recommended work in the revision.

Re: Questions

• Bandwith in Definition 1: Yes, Definition 1 is applicable to general Mercer kernels. We use bandwith here as our analysis is centered around the Gaussian kernel. We have updated the manuscript for general Mercer kernels and emphasized the use of Gaussian kernel.

• Kernel in Lemma 1: The kernel used in Lemma 1 is also the Gaussian kernel. Moreover, as Gaussian kernel is a translation-invariant kernel, we use $k(x-y)$.

References

[1] Li et al., Fast Neural Scene Flow, ICCV23.

Dear Reviewer Jzhm,

Thank you very much for reviewing our paper. We have carefully and diligently updated the manuscript to reflect the changes mentioned above. We hope our responses have addressed your concerns.

Since it is approaching the end of the discussion period, should you have any further questions, we're happy to discuss and clarify.

We appreciate the discussion, your feedback, and your suggestions. Many thanks for your time and effort.

Best and sincere wishes,

The authors