Occlusion-aware Non-Rigid Point Cloud Registration via Unsupervised Neural Deformation Correntropy
Authors: Mingyang Zhao
Gaofeng Meng
Dong-ming Yan
Venue: ICLR 2025 Poster (2025)
Keywords: Unsupervised Deformation
Neural Implicit Representations
Correntropy
Locally Linear Reconstruction
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: This paper aims to solve the occlusion challenge in non-rigid alignment of point clouds using neural deformation correntropy.

Abstract: Non-rigid alignment of point clouds is crucial for scene understanding, reconstruction, and various computer vision and robotics tasks. Recent advancements in implicit deformation networks for non-rigid registration have significantly reduced the reliance on large amounts of annotated training data. However, existing state-of-the-art methods still face challenges in handling occlusion scenarios. To address this issue, this paper introduces an innovative unsupervised method called Occlusion-Aware Registration (OAR) for non-rigidly aligning point clouds. The key innovation of our method lies in the utilization of the adaptive correntropy function as a localized similarity measure, enabling us to treat individual points distinctly. In contrast to previous approaches that solely minimize overall deviations between two shapes, we combine unsupervised implicit neural representations with the maximum correntropy criterion to optimize the deformation of unoccluded regions. This effectively avoids collapsed, tearing, and other physically implausible results. Moreover, we present a theoretical analysis and establish the relationship between the maximum correntropy criterion and the commonly used Chamfer distance, highlighting that the correntropy-induced metric can be served as a more universal measure for point cloud analysis. Additionally, we introduce
locally linear reconstruction to ensure that regions lacking correspondences between shapes still undergo physically natural deformations. Our method achieves superior or competitive performance compared to existing approaches, particularly when dealing with occluded geometries. We also demonstrate the versatility of our method in challenging tasks such as large deformations, shape interpolation, and shape completion under occlusion disturbances.

Published as a conference paper at ICLR 2025
OCCLUSION-AWARE NON-RIGID POINT CLOUD REG- ISTRATION VIA UNSUPERVISED NEURAL DEFORMA- TION CORRENTROPY
Mingyang Zhao', Gaofeng Meng1.2.3.- Dong-Ming Yan
1. CAIR. Hong Kong Institute of Science & Innovation, Chinese Academy of Sciences MAIS, Institute of Automation, Chinese Academy of Sciences
3. University of Chinese Academy of Sciences
cow http:/ikaiillim /OAReg
ABSTRACT
Non-rigid alignment of point clouds is crucial for scene understanding, recon- struction, and various computer vision and robotics tasks. Recent advancements in implicit deformation networks for non-rigid registration have significantly re- duced the reliance on large amounts of annotated training data. However, existing state-of-the-art methods still face challenges in handling occlusion scenarios. To address this issue, this paper introduces an innovative unsupervised method called Occlusion-A ware Registration (OAR) for non-rigidly aligning point clouds. The key innovation of our method lies in the utilization of the adaptive correntropy function as localized similarity measure, enabling us to treat individual points distinctly. In contrast to previous approaches that solely minimize overall devia- tions between two shapes, we combine unsupervised implicit neural representa- tions with the maximum correntropy criterion to optimize the deformation of un- occluded regions. This effectively avoids collapsed, tearing, and other physically implausible results. Moreover, we present a theoretical analysis and establish the relationship between the maximum correntropy criterion and the commonly used Chamfer distance, highlighting that the correntropy-induced metric can be served as a more universal measure for point cloud analysis. Additionally, we introduce locally linear reconstruction ensure that regions lacking correspondences be- tween shapes still undergo physically natural deformations. Our method achieves superior competitive performance compared existing approaches, particu- larly when dealing with occluded geometries. We also demonstrate the versatility of our method in challenging tasks such as large deformations, shape interpola- tion, and shape completion under occlusion disturbances.
INTRODUCTION
Non-rigid point cloud registration is a critical and challenging problem within the domains of com- puter vision, robotics, and medical imaging. Its objective is optimize a deformation field that enables precise alignment between geometric shapes. Due to its fundamental importance, this prob- lem has diverse applications, including reconstructions (Slavcheva et al., 2017; Bozic et al., 2021; Park et al., 2021), generation (Barber et al., 2007; Wang et al., 2008), and animations (Chen et al., 2021; Siarohin et al., 2023).
Thanks to recent advancements in neural representations for 3D modeling and rendering (Park et al., 2019; Mescheder et al., 2019; Sitzmann et al., 2020; Mildenhall et al., 2020), coordinate-based net- works (i.e., utilizing 3D spatial positions as inputs) have made significant progress in fully automated non-rigid point cloud registration (Li et al., 202 la; Li & Harada, 2022b; Prokudin et al., 2023). Despite their impressive results on benchmark tests, the systematic exploration of coordinate-basse based networks for non-rigid point cloud registration involving occlusion geometries is comparatively lim- ited and presents substantial challenges. Actually, the occlusion regime is quite relevant for practical
Corresponding authors.

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Published as a conference paper at ICLR 2025
Vioo
View
Source
Target
NSFP
NDP
DPF
Ours
Figure I: Non-rigid registration of point clouds under occlusion disturbances. The pre-operative liver (complete) and the intra-operative liver (occluded) serve as the source and target models, spectively. While competing approaches produce physically implausible results like collapses and tearing, method achieves successful registrations (top) while faithfully preserving the physical structure (bottom) such as the blood vessel present in the source liver.
applications. For instance, in Computer-Assisted Surgery (CAS) scenarios, accurate registration of complete pre-operative CT or MRI models with occluded intra-operative sean data crucial for surgical navigation, as the camera attached to the surgical instrument only provides limited view of the abdomen during laparoscopic surgeries (Robu et al., 2018). is surprising that current state- of-the-art algorithms struggle when dealing with scenarios where the source shape is complete and the target shape occluded (see Fig. 1).
We analyze and argue that the unsatisfactory results can be attributed to the choice of the dis- tance metric between two shapes. Previous approaches typically use the standard Chamfer Distance (CD) (Fan et al., 2017) the similarity evaluation protocol (Li et al., 2021a; Li & Harada, 2022b; Prokudin et al., 2023), which directly finds the nearest point in the other point cloud and minimizes holistic deviations between shapes. While this effective for registering two complete shapes, can lead to collapsed or physically implausible results when dealing with a complete source shape and an occluded target geometry, as shown Fig. 1.
To address the above issues, we present a novel and unsupervised non-rigid registration method for point clouds under occlusion. A key aspect of our method is that, instead of blindly optimizing the overall non-linear transformation, we introduce the use of the Maximum Correntropy Criterion (MCC) from statistical and information-theoree
learning (Erdogmus & Principe, 2002) effec- tively handle occlusion. Correntropy serves as adaptive and local similarity measure, allowing for differential treatment of individual points within the neural representations. Consequently, occluded points will make smaller contributions to the correntropy, enabling more precise and distinguished handling of occlusion. As demonstrated subsequent experiments, this correntropy -induuce metric effectively prevents collapses, tearing, and other physically implausible outcomes, in contrast to the commonly used CD metric.
Moreover, ensuring the physical plausibility of the deformation field for regions of the source geom- etry that lack correspondence the target shape presents another significant challenge. This problem considerably more complex than its rigid counterpart (Zhao et al., 2023), as inherently ill- posed and under-constrained. In non-rigid registration, there are numerous potential deformation so- lutions that can align with the partial deformations of the source surface. To mitigate this challenge, we draw inspiration from the concept of locally linear embedding in manifold learning (Roweis & Saul, 2000) and develop a robust geometric prior termed Locally Linear Reconstruction (LLR). The LLR approach formulates the problem as constrained non-linear least-squares optimization, in- corporating deformation regularity by linearly reconstructing the unknown out-of-part deformations using information from neighboring regions. As result, LLR significantly enhances the natural- ness of the deformation field and preserves higher-quality geometric details compared to previous approaches.
We conduct extensive experiments to demonstrate the superiority and competitiveness of the pro- posed method baselines, including both traditional optimization- based and neural network-based approaches. This superiority is particularly pronounced in scenarios involving occlusion. Addition- ally, our method provides continuous deformation representations, enabling downstream applica-
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tions in shape interpolation and shape completion. To summarize, the main technical contributions of this work are threefold as follows:
We analyze the reasons why previous approaches fail to handle occlusion scenarios in non- rigid registration and introduce a novel unsupervised deformation framework to address this challenging problem.
We pioneer the combination of maximum correntropy criterion with implicit neural repre- sentations to ensure physically plausible deformations and reveal its relationship to Cham- fer distance.
We formulate the deformation of regions lacking correspondence local lincar recon- struction problem. This formulation can be solved through non-linear least-squares opti- mization and yields closed-form solution.
2 RELATED WORK
We review the work that is closely related to ours. Readers are referred to (Tam et al., 2012; Deng et al., 2022) for more comprehensive studies.
Non-rigid point cloud registration. In contrast shape matching, which aims to establish cor- rect correspondences, non-rigid ploud registration focuses on optimizing the deformation field. Amberg et al. (Amberg et al., 2007) extended the rigid Iterative Closest Point (ICP) algorithm (Besl & McKay, 1992) to non-rigid registration (NICP) by introducing stiffines parameters to control the deformation. Coherent Point Drift (CPD) represented point clouds using Gaussian mixture mod- els (GMMs) and formulated the registration problem probabilistic density estimation process, while (Jian Vemuri, 2010) directly minimized the Euclidean distance between two GMMs for registration. More recently. Hirose (Hirose, 2020; 2022) reformulated CPD in Bayesian setting, improving CPD scalability and robustness. Zhao et al. (2024) leveraged unsupervised cluster- ing analysis deal with non-rigid registration under large deformations. To enhance efficiency, deformation graph-basee based approaches (Sumner et al., 2007; Bozic et al., 2021; Zhao et al., 2022; Qin al., 2023) represent shapes using sparsely sub-sampled node graphs, where deformation is associated with each graph node and applied to nearby geometry.
With the advent of deep learning, neural network-based approaches have also been developed for non-rigid point cloud registration. However, many of these methods rely on neural networks to extract features for point correspondences and subsequently employ classical registration methods like NICP for deformation optimization. For instance, SyNoRiM (Huang et al., 2022) employed 3D CNN networks to learn non-orthogonal functional bases for shape matching without relying on the Laplace-Beltrang operators (Ovsjanikov et al., 2012). Lepard (Li & Harada, 2022a) disen- tangled point cloud representations into feature and position spaces and developed Transformer- based method for point-wiad wise matching. DFR (Sun et al., 2024) conducted registration by aligning the source mesh towards a target point cloud using correspondences induced by deep functional maps (Litany et al., 2017). Cao & Bernard (2023) presented a self-supervised network for multi- modal shape matching. While shows promising performance, it does not address the deformations of occluded parts. Sundararaman et al. (2022) utilized an auto-decoder structure to implicitly align two volumes, which requires surface normals for training. Additionally, it employs bi-directional Chamfer Distance for inferences, metric that may be susceptible to occlusion. Unlike these ap- proaches that primarily focus on shape matching and heavily depend on data annotations, our method unsupervised, enabling us achieve faithful registration that is more generalizable unknown categories.
Neural deformation representation. The representation of deformation fields a core aspect in non-rigid point cloud registration. Traditional methods often rely on manually defined deforma- tion functions, such as the thin-plate spline (Bookstein, 1989) and the radial basis function (Yuille & Grzywacz, 1989). However, the emergence of implicit neural representations has introduced new paradigm to describe deformations. Specifically, the coordinate- based Mulri-Layer Perception (MLP) architecture utilizes MLP to map the input coordinates to deformation fields, avoiding the need for explicit deformation definitions. For instance, NSFP (Li et al., 2021a) developed a coordinate- based MLP that implicitly regularized non-rigid deformations in scene flow estimation.
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This method directly minimizes the Chamfer distance runtime without relying extensive la- beled data to capture prior statistics. Building upon NSFP, NDP (Li & Harada, 2022b) extended the approach to hierarchical motion decomposition using a pyramid structure. This enables controlling of the coarse-to-
motion (from rigid to non-rigid) across low to high-frequency signals. More re- cently, DPF (Prokudin et al., 2023) introduced method for modeling non-rigid surfaces also based on the Chamfer distance, where the deformation field was further regularized by the well-established as-isometrie-ase asteriibl--p (AIAP) (Kilian et al., 2007) constraint. We also utilize a coordinate-bant based MLP to represent the continuous deformation field, which allows us to model and parameterize the deformations an unsupervised and flexible manner. However, unlike previous approaches that of- ten face challenges in occlusion scenarios, we thoroughly analyze and address these issues, leading to higher-quality non-rigid point cloud registration results.
3 PRELIMINARIES
Definition 1 (Santamaría et al. (2006); Liu et al. (2007)). (Cross) correntropy V is generalized local similarity measure between two arbitrary scalar random variables X and Y defined as
V(X,Y) = Exy Exx JJ ()xx)) (z, y)dxdy.
which is symmetric, positive, and bounded. Here k(;,)) .) is a kernel function that satisfies the Mercer theorem (Vapnik, 2013), E denotes the expectation operator, and pxy is the joint probability density function over the space Y).
2² In this work, we adopt the translation invariant Gaussian kernel k, exx( 20 to define the cor- rentropy. This choice is motivated by the symmetric and positive definite nature of the Gaussian kernel, as well as its property of approaching zero as points move away from the center. Moreover, the kernel bandwidth o allows flexible control over the decaying factor and enables accurate reg- istrations in the presence of various occlusion. We empirically investigate the influences of a on deformations Appendix D.
Definition 2 (Liu al. (2007)). Given the finite number of data points in the sample space, the correntropy-induced similarity metric (i.e., distance function) M between two vectors x (21,32, "2, .. ,n)) E Rn and y = (y1,y2,.". Yn) E R. is defined as
M( y) = (1)
1
Σ n j=1
1
(1)))=))))
1
In the machine learning community, for general optimization problems involving the unknown vari- able O minimizing M(x y) equivalent to maximizing the correntropy.
Lemma [Liu et al. (2007)) M(x, y) is equivalent to the & metric if a. y are close, behaves y get apart and approaches the to when they are far apart.
similarly to the & metric
Therefore, Eq. (2) can be utilized as novel cost function for adaptive training, referred to as the Maximum Correntropy Criterion (MCC). The pioneering work (He al., 2010) utilizes MCC for face recognition. We explore the combination of unsupervised implicit neural representations and the MCC-induced local metric to address occlusion geometries in non-rigid point cloud registration. We also demonstrate the superior performance of our method over previous competing approaches.
PROPOSED METHOD
Problem formulation. Let X = (x, E JR and Y (y, E R' denote two input point clouds, where X represents the target shape, possibly with occlusion, and Y is the source shape. Our objective is to solve the optimal non-linear map To parameterized by θ to minimize the shape deviation between the transformed source shape (Y) and X. Note that also aim to preserve the physical plausibility of the deformation Te regarding Y, ensuring that issues encountered in previous approaches, such collapsing or tearing of the geometric shape, avoided.
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4.1 NEURAL DEFORMATION CORRENTROPY
Motivation. Unlike previous approaches that directly minimize the overall difference between two shapes, leverage the advantage of local similarity induced by correntropy. This allows us to adaptively differentiate between various parts of a shape, ensuring that occluded regions have minimal influence on the deformation process.
Correntropy-indueee metric. We utilize the unsupervised neural presentations defined by a set of network parameters θ to implicitly parameterize the non-linear map T. Based on MCC, our goal R M 1== and Y = {v, E R 3 19=1
is to optimize the subsequent metric between X = {ox E R
min L()) B(M a ū.)) ((M((T(())
where y, = miny ETTYY I|; - (y 12.. ã. = ming, ex (T)() - x, x1112- As a result, Eq. (3) assigns adaptive importance during network training. i.e., giving significant importance to points with corresponding parts, while penalizing deformations in occluded regions. As demonstrated in subsequent experiments, method effectively avoids physically implausible limitations, leading to more accurate and reasonable deformations.
Relationship to Chamfer distance. We further analyze the relationship between the maximum correntropy criterion and the Chamfer distance. Unlike CD that performs a holistic evaluation, MCC is local and adaptive similarity metric. Based Lemma we can draw the following conclusion.
Proposition 1. The Chamfer distance special case of Eq. (3), applicable only when both the source and target point clouds X and Y are complete and sufficiently close to each other. However, this equivalence does not hold when either of the inputs X and Y is occluded.
Therefore, the correntropy-induced metric can serve as a more universal measure, potentially appli- cable wider range of point cloud analysis tasks.
4.2 LOCALLY LINEAR RECONSTRUCTION
Neural deformation correntropy ensures physically plausible deformations for unoccluded parts through local measure. However, addressing the challenge of making reasonable deformation of parts in the source shape that lack correspondence in the target shape remains highly -oo-pdded. To tackle this, we propose incorporating a geometric prior by modeling the deformation field of occluded parts through Locally Linear Reconstruction (LLR),, which transforms the unconstrained problem into a constrained non-linear optimization. This allows us not only maintaining a smooth and coherent deformation field for unoccluded parts but also ensuring physically plausible results for parts that lack correspondence.
Inspired by the principle of locally linear embedding in nonlinear spectral dimensionality reduc- tion (Roweis Saul, 2000), we develop a novel method to preserve the linear pattern of the lo- cal geometric structure and reconstruct each point in the deformation space, especially for points that lack correspondences. by utilizing its neighboring structure. Concretely, for each source point y5 R in Y, we formulate the locally linear reconstruction process the subsequent optimization problem:
1 1 lyy Z, w G,wj, s.t. w. 1 1,
min L(w;) wa
(4)
2
where w, = [WJI. Wy2s'.. W)+]" E I is the set of unknown weight factors, Zj [zj1, Zyz. .... ,,4] E 13xk is the neneessst neighbors of y, (the investigation of the influences of k on deformations is presented in Appendix D), G, = (y, Z,) "(y,11 Z,) * the Gram matrix, 1 E I* is the k-dimensional vector of all ones, and represents the matrix transpose operator.
Eq. (4) is constrained non non-linear least-squares regression problem, utilizing the Lagrange multi-
plier, we obtain the closed-form solution of w for each y, (derivations are presented in Appendix B)
G 1
w =
(5)
1 G
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with the corresponding Lagrange multiplier
> G 1x
(6)
LLR regularization. Once obtaining the weight vectors (wj E [ for each yj E Y, we keep them fixed. Then, we utilize these weight vectors regularize the displacement field, aiming to promote its smoothness and ensure physical plausibility in the deformation space throughout the entire optimization process even without correspondences, i.e.,
R( - Σ (e) - To (Z,) wy
Geometric meaning of LLR. Unlike rigid registration, where each point shares the same spa- tial transformation (i.e, rotation and translation), non-rigid deformation requires solving for each point's individual displacement. In challenging occlusion scenarios, Eq. (7) demonstrates that the deformation of occluded parts can inferred from, or propagated through, the deformations of neighboring points via locally linear reconstruction. Fig. 2 presents schematic description of the geometric meaning.
4.3 UNSUPERVISED OPTIMIZAATION
Given the active research and powerful fitting abil- ity of implicit neural representations across various domains, we adopt a coordinate-bon based MLP with pe- riodic activations (Sitzmann et al., 2020) as de- formation network T. Our objective is to optimize a set of network parameters @ to define the deforma- tion space. Additionally, as a test-time optimization method, method is unsupervised, requiring no data annotations. The neural network is composed of three hidden layers, each consisting of 128 rons. We obtain the optimal neural network param- eters O' by optimizing the following cost function with automatic differentiation:
W
Deform
O
Figure 2: Geometric meaning of locally lin- ear reconstruction. Left: The initial source shape along with the reconstruction weight vector wy. Right: The deformed shape re- constructed using the same wy.
θ = arg min F(() an c,) a2
(8)
where a1, 02 E R+ are trade-off factors. Finally, the resulting deformed point cloud Te- TY- is obtained by applying the learned deformation network Te. to the source model
EXPERIMENTAL RESULTS
We conduct comprehensive experiments to validate the performance of our method and compare with state-of-the-art approaches in both occluded and complete settings. Additionally, showcase the versatility of our method by applying it to shape interpolation and shape completion tasks.
Implementation details. Our method is implemented using the PyTorch framework (Paszke al., 2019), and we employ the Adam optimizer (Kingma & Ba, 2014) with default settings 31 = 0.9, 82 = 0.999, and 6 10-8 to optimize the objective function defined in Eq. (8). The initial learning rate is set to Ir = 1e-4, 4, and the network is trained for 200 epochs with a batch size of 1. To dynam- ically update the learning rate, we utilize the ReduceLROnF llteau technique with a patience value of 1. During optimization, we fix across various scenarios. All experiments are conducted on a machine equipped with an NVIDIA A30 GPU and an Intel(R) Xeon(R) Gold 6248R CPU @ 3.00GHz. We utilize publicly available implementations of baseline approaches for evaluation, with their parameters either fine-tunel by ourselves suggested by the authors obtain their best results.
10a and 02 102, which yield appealing results

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Table I: Quantitative comparisons on the occluded Open-CAS CAS liver dataset. t means larger values are better while + means smaller values are better. Bond fonts indicate the top performer.
Metric
Liver
Laver
Laver
Mathod
EPF L
AndS 7
AadR
Chilier
EPF. J
Acc5 1
AccR: 7
Outlier J
EPF. 4
Acc8 1
AcoR
Outlier L
GBCPD (Harose, 2022)
11,849
18,770
66.263
0.000
9.109
48.871
75,534
0.000
16.240
31,770
53.162
0.000
AMM NKK Crao 2025)
28.505
21,549
36683
1.353
19.188
36,957
47,050
0.000
18.744
35,593
49.289
0.000
NSFP (Lú al. 2021a)
49.085
18.057
39.494
27.218
32,520
45.011
53,69
9,514
28.789
37,742
50.696
14.167
NDP (L. Harada, 2022b)
58.470
2,795
8.504
25,961
37,456
10.357
21,074
11.131
29.508
26.675
43,407
12.780
DPF (Prokadin all, 2023)
35.090
9.065
21.797
1.225
25,180
20.158
46 16.976
0,000
23,045
26.907
43.891
5.041
(Ouns)
29.228 29.
96.813
5,687
97.184
12.112
42.372
56.564
0.000
Evaluation metries. To quantitatively evaluate the results of non-rigid deformation, we adopt four metries described in (Li & Harada, 2022b). 1) End-Point Error (EPE): This metric calculates the average norm of the 3D deformation error vectors across all points; 2) 3D Accuracy Strict (AccS): AceS measures the percentage of points whose relative error is less than 2.5% or 2.5 cm. 3) 3D Accuracy Relaxed (AccR): AccR measures the percentage of points whose relative error is less than 5% or 5 cm; 4) Outlier Ratio: The outlier ratio calculates the percentage of points whose relative error is greater than 30%. Concrete definitions of these metrics are presented in Appendix A. Note that AceS and AceR are considered the most important metries as they directly assess the ratio of accurately registered points, providing a reliable assessment of the registration quality (Li & Harada, 2022b).
5.1 COMPARISONS
We first assess the proposed algorithm occlusion scenarios using the silico liver dataset from Open-CAS (ope). Open-CAS is an open collection of datasets specifically designed for validating computer-assisted surgery systems. The silico liver dataset consists of three pairs of liver models that have undergone deformation using a non-linear biomechanical model. Each pair, as illustrated in Fig. 1, consists of source complete liver model and target occluded liver model, along with its corresponding ground truth complete model. These pairs represent the pre-operative and intra- operative shapes, respectively. We adopt state-of-the-an approaches for comparison, including both traditional optimization-base based methods GBCPD (Hirose, 2022), AMM_NRR (Yao et al., 2023) as well as neural network-based competitors NSFP (Li al., 2021a), NDP (Li & Harada, 2022b), and DPF (Prokudin et al., 2023).
The quantitative comparison results presented in Tab. demonstrate that our method consistently outperforms competing approaches with higher registration accuracy across all livers, indicating its robustness and stability to align the geometric shapes under occlusion disturbances. Notably, our method achieves the highest AceS and AccR values in all tests, surpassing competitors by significant margin. Additionally, our method delivers the lowest EPE values among all approaches and maintains zero outlier ratio in all tests. These achievements highlight the success of our method in avoiding significant registration errors such as collapses and tearing, thereby ensuring a high level of accuracy. The qualitative comparison presented in Fig. 1 further illustrates the advantage of our method in preserving detailed physical structures such as the blood vessel of the source liver.
5.2 REGISTRATION UNDER DIFFERENT TYPES OF OCCLUSION
To further investigate the performance of our proposed method in handling occlusion, we create three occlusion datasets using three different shapes (cat, horse, and dog) from the TOSCA dataset (Bron- stein et al., 2008). Each dataset consists of five types of occlusion. such as the occlusion present in the body of the cat and the tail of the dog, as shown in Fig. 3. We exclude the outlier ratio from our analysis since the deformation solution is not unique in these cases. Additionally, we exclude AMM NRR from our analysis due to its significant deviations from the target poses, which render the error metrics unreliable. The average metrics of all tests are reported in Tab. 2. The results indicate that our method still achieves the highest registration accuracy across different occlusion scenarios, especially in terms of the AccR metric. Although our method is slightly slower than other neural networks- betsorks-based approaches, it is much faster than traditional optimization-bant GBCPD. Several qualitative comparison results are presented in Fig. 3. As observed, our method success-
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Table Quantitative comparisons on different types of occlusion. t means larger values are better while 4 means smaller values are better. Bond fonts indicate the top performer.
Metric
Cat
Horse
Dog
Method
EPF
AccS
4
AccST
AccR
Time (8)
GBCPD (Hirose, 2022)
4.118
17.123
41.104
3.273
38.353
52.435
4.359
54.472
77.375
185.621
NSFP (1.i et al., 2021a)
4.591
13.617
40.397
2.761
25.484
53.698
4,159
61.486
78.940
4,576
NDP (Li & Harada, 2022b)
7,280
13.401
32.340
4.147
30.922
48.070
11.141
14.710
42.522
5.858
DPF (Prokudin ol., 2023)
2.688
50.048
75.350
2,885
27.819
55.056
3.888
64,480
80.814
9,518
OAR (Ours)
2.559
50.569
77.171
2.249
40.046
65.317
2.467
76.986
89.779
13,646
Source
GBOPD
NSFP
DPF
Ours
Ground Truth
Figure 3: Qualitative comparison on different types of occlusion. We register the complete source cat and dog model to the target shape with body and tail occlusion, respectively.
fully retains the dog tail without any collapse. More comparison results are presented in Fig. 10 and Fig. 11 of the Appendix.
5.3 REGISTRATION WITH CORRESPONDENCE
In addition handling occlusion scenarios, we also demonstrate the capability of the proposed method for general large deformation registration tasks. To this end, we combine the pre-trained ge- ometric feature descriptor Lepard (I.i & Harada, 2022a) with our method for non-rigid registration on the 4DMatch and 4Dth animation benchmark datasets (Li & Harada, 2022a) (details are presented in Appendix I). These datasets are collected from the EeformingThinngoee
sequence (Li et al., 2021b), which contains challenging scenarios with partial overlap, occlusion, and large mo- tions. The 4DMatch dataset comprises point clouds with overlap ratios ranging from 45% to 92%, while the 4DLoMatch dataset exhibits lower overlap ratios of 13% - 45%. Following (Li & Harada, 2022b), we adopt the filtered dataset that has undergone the removal of near-rigid transformations for testing. Specifically, the 4DMatch dataset consists of 2,590 pairs of test, while the 4DtoMatch dataset contains 1,175 pairs.
Tab. summarizes the quantitative comparison results, with the exception of our method, where the remaining results are taken from (Li & Harada, 2022b). It is important to note that all supervised models re-trained on 4DMatch's 's training split prior to evaluation. Although our method unsu- pervised, it still consistently achieves superior or competitive results compared to the competitors, particularly in terms of the two most important metrics, AceS and AccR. Moreover, it is worth not- ing that our method also delivers competitive performance on the highly challenging 4DLoMatch dataset even without any fine-tuning. Qualitative results are presented in Fig. 13 of the Appendix.
5.4 HUMAN REGISTRATION ON OCCLUDED DEPTH VIEWS
We further evaluate the proposed method for registering human point clouds captured from RGB- cameras, which can used for non-rigid reconstruction. As shown Fig. 16, the sequences from (Innmann et al., 2016) exhibit occlusion disturbances due to human motion. The qualitative results presented in Fig. 17 evidence that our method still achieves high-quality deformations despite these challenges. In contrast DPF, which produces physically unreasonable deformations such as collapse and distortion of body parts, our method consistently preserves realistic deformations, particularly evident in the human hands and arms.
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Table 3: Quantitative comparisons on the 4DMatch and 4DLoMatch benchmark datasets. The best and second best performers of AceS and AccR are highlighted in bold and underlined, respectively.
Metric
4DMatch
4DLoMatch
Method
EPE
AccS f
AccR 1
Outlier
EPE
AccS
AccR 1 Outiier
Lepard (Li & Harada, 2022a)+SVD
0.137
6,91
24.50
43.43
0.160
5.27
19.77
44.16
Puin (Wu et al., 2019)
0.182
6.25
21.49
$2.07
0.279
1.69
8.15
55.70
FLOT (Pay et al., 2020)
0.133
7.66
27.15
40.49
0.210
2.73
13.08
42.51
GeomFmaps (Donati al., 2020)
0.152
12.34
32.56
37,90
0.148
1.85
6.51
64.63
SyNoRiM-pw (Huang et al., 2022)
0.099
22.91
49.86
26.01
0.170
10.55
30.17
31.12
LePard+NNICP (Li & Harada, 2022a)
0.097
51.93
65.32
23.02
0.283
16.80
26.39
52.99
Lepard+NNDP Harada, 2022b)
0.077
61.30
74.12
17.37
0.177
26.59
41.05
33.81
Lepard+OAR (Ours)
0.059
59.32
74.33
16.41
0.251
27.25
42.01
45.04
Source
AIAP
LLR
Target
AIAP
LLR
Target 2
Figure 4: Qualitative comparison of AIAP and LLR with increasing occlusion. As observed, LLR enables more natural registration results and maintains geometric details such the facial expres- sion more faithfully.
5.5 ROBUSTNESS TEST
We also assess the robustness our developed method against noise and outlier disturbances. To
Intensiry/Ratio
0.1
0.2
0.3
0.4
0.5
0..6
this end, we introduce Gaussian noise with in- creasing intensity and varying outlier ratios (%) to the point clouds. The inset table reports the statistical results, while Fig. 19 and Fig. 20 present the qualitative outcomes. As observed, our method demonstrates strong resilience against both noise and outliers, maintaining performance even with noise levels up to 0.6% and in the contamination of 2, 350 outliers.
Nuise Outlier
$8.712
89.101
89.678
85. 35.701
$4.395
81
88.017
86.171
83.999
5.6 ABLATION STUDY
The effect LLR. To demonstrate the effect of the proposed LLR for non-rigid point cloud reg- istration under occlusion disturbances, we construct a series of models with progressively increasing levels of occlusion. Fig. 4, we illustrate this by manually removing some parts of the lion model, starting from the tail and extending towards the body. The statistical results in Fig. 5 indicate that method equipped with the LLR constraint (i.e., MCC+LLR), which helps preserve the local neigh- boring geometry. consistently produces higher-quality registration results compared to the single MCC framework.
ALAP VS. LLR. We also compare the performance of the commonly used AIAP regular- ization with the proposed LLR in handling challenging occlusion scenarios. As observed in Fig. 5, MCC+AIAP delivers similar results to MCC, while MCC+LLR consistently outperforms MCC+AIAP across different levels of occlusion. Moreover, benefiting from the reconstruction scheme and maintaining the local structure, LLR excels preserving essential geometric details,
VCCAMAA
MCC
Occkssien
Oockssin
Figure 5: Ablation study of the LLR effect and the comparison between AIAP and LLR manners.

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1-1
t= 0.3
t= 0.5
t= 0.7
t 0.9
t= 1.0
Figure Application of the proposed non-rigid registration method shape interpolation.
Source
Shape 1
Result 1
Shape 2
Result 2
Shape 3
Result 3
Figure 7: Application of the proposed non-rigid registration method to mesh hole filling.
such as the intricate facial features the lion Fig. 4, after non-rigid deformation. This stands in contrast to AIAP under occlusion disturbances, which struggles to maintain such details.
5.7 APPLICATIONS
Shape interpolation. Since method successfully avoids physically implausible scenarios, such as collapses and tearing, and employs an unsupervised neural implicit representation, inherently allows for continuous deformation. In Fig. 6, illustrate the shape interpolation effect using our non-rigid deformation under occlusion, where we vary the timestamp / within the range of 10.1,1.0) For instance, the interpolation shape with
0.5 is represented by t=005 Y + v(Y) Y + 0.5 * v(Y) with y denoting the learned displacement field. As observed, our method not only achieves accurate registration
1.0 but also maintains substantially smooth and physically feasible interpolation configurations throughout the entire deformation process.
Shape completion. We also apply our method to address problem of shape completion. In Fig. 7, we utilize the source mesh model (Yang et al., 2014) the standard template, while the other shapes exhibit various types of holes caused by occlusion, serving as the targets. It is evident that by deforming the template model to align with the target shapes and leveraging the highly accurate outcomes, our method offers a feasible solution to the ill-posed mesh hole filling task without requiring any additional annotations and constraints. More hole filling results reported in Fig. 14 of the Appendix.
CONCLUSIONS
We introduced novel unsupervised solution address the challenging issue of occlusion distur- bances in non-rigid point cloud registration, a problem that has not received adequate attention in prior approaches. We provide comprehensive analysis elucidate the reasons why previous meth- ods struggle with occlusion scenarios. The first contribution of our method is maximizing the local similarity between source and target geometries, accomplished through the utilization of the adap- tive maximization correntropy critcrion. We also establish a connection between the correntropy- induced metric and the commonly used Chamfer distance, highlighting that this correntropy-indus induced metric can serve as more general measure. Additionally, we develop a locally linear reconstruction paradigm to ensure that deformations in the occluded regions are physically plausible while preserv- ing intricate geometric details of deformed shapes, in contrast to the commonly used AIAP metric. Our method achieves higher-quality non-rigid registration results across a range of datasets, while effectively mitigating the issues of collapses and tearing that encountered in previous approaches. Moreover, our method ensures continuous deformation, allowing for promising applications in shape interpolation and shape completion tasks.
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ACKNOWLEDGEMENTS
We sincerely thank the anonymous reviewers for their helpful comments which improved the qual- ity of the paper. This work partially funded by the National Natural Science Foundation of China (12494553, 12494550, 62172415, and 62376267), the Strategic Priority Research Program of the Chinese Academy of Sciences (XDB0640200 and XDB0640000), the Beijing Natural Science Foun- dation (Z240002), and the innoHK project.
REPRODUCIBILITY STATEMENT
The main contribution of this work is the development of an unsupervised neural deformation method that combines correntropy -inuuue metric and locally linear reconstruction with implicit neural representations ensure physically reasonable deformations of occluded point clouds. To reproduce the experimental results, we elaborate on the data pre- pre-preesssigg metric definitions, and hyper-parameter settings in Appendix A and Appendix D. To facilitate reproducibility, we also pub- licly release our code.
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APPENDIX
In this appendix, provide additional content support our paper. Concretely, we present the data pre-processing step of normalization and the definition of used evaluation metrics in Appendix A, followed by the theoretical derivation of the closed-form weight solution in Appendix B. Appendix C presents more details and analyses of Lemma 1. Appendix D investigates the influences of the num- ber of nearest neighbors k for each y) E Y and the kernel bandwidth o² on registration under occlusion disturbances. We present further ablation studies to investigate the regularization effect of implicit neural networks and activation functions Appendix E. The differences and connections of MCC with robust functions are presented in Appendix F. Additional ablation studies of MCC and Chamfer distance as well as its variants are reported Appendix G and Appendix H, separately. Appendix gives the details of how to integrate the geometric descriptors with our proposed opti- mization framework. Then, we discuss the limitations and future work of the proposed algorithm in Appendix J. Finally, in Appendix K, we present more quantitative and qualitative results. These results include the registration on various types of occlusion from the TOSCA dataset, liver dataset, 4DMatch and 4DL Motchh datasets, shape completion for the challenging mesh hole filling task, human RGB-D views, failure cases, as well as robustness test.
A DATA PRE-PROCESSING AND THE METRIC DEFINITION
Given pair of input point clouds X = (2; E R and Y = {uy E R represent the target and source point clouds, separately, we first normalize them to follow a normal distribution, i.e.,
J==1 where X and Y
N(X) = (X IMXx)/oxx
N(Y) (Y INNY)/OY.
(10)
where μsa E R represents the mean point in the set of and o, IR+ is the farthest Euclidean distance from H the elements in 1M RM and iN E RN the vectors of all ones. However, when evaluating the registration results, we still use the original inputs by first performing denormalization.
Specifically, we compute the deviations between the deformed shape Ý = (X)) + 1x/x and the complete shape X of X in terms of the metrics EnE, AccS, AccR, and Outlier Ratio by
EPE Ý X)
V Tr X) -X),,
N I(lū, -2y122 < 0.025) x 100%,
AccS(Y. x)
N
j=1
N
(1I)
AccR(Y, C.R(Y.X) Σ
I( E1112 < 0.05) 100%,
j=l
Outlier Y. x
-1112 > 0.3) 100%,
where Tr and T are the trace operator and transpose operator of a matrix, separately. is the indicator function and ||]|a represents the Euclidean distance.
B DERIVATION OF THE CLOSED-FORM SOLUTION IN EQ. (5) OF THE PAPER
To make the proposed method self-contained, we present the derivation of the closed-form solution in Eq. (5) as follows. According to the Lagrange multiplier, we have
L(w,, 21) = 1 G,,,, - \m 1 1).
(12)
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By equating the partial derivatives of L(w,, >ÿ) to zero, obtain
ac
G - 1,
Jwj
i G. xyزı - A,
(13)
0c
w 1,
1=0 1=0 1/111 1+
01
Then, we have
A 1 G 1
(14)
Thus,
Gj'L
w 1 G,
(15)
This completes the derivation of Eq. (5) and Eq. (6).
C
MORE DETAILS AND ANALYSES OF LEMMA 1
We present more details of Lemma
We can rewrite the correntropy of Gaussian kernel
Y) - 1/(V2To)
Σ ((-1))) n=0
E ((x- (
(16)
which involves all the even moments of the random variable X Y. When two points are in close proximity, the second-order moment tends to be dominant, causing correntropy to approach correlation, as the higher-order moments decay more rapidly. As the points become more distant, as shown in (Liu al., 2007), correntropy transitions from behaving like é norm to resembling lo norm. This transition further illustrates the inherent robustness of the MCC metric.
D INVESTIGATION OF l AND d² ON DEFORMATION
We adopt the occluded liver dataset (ope) and vary the number of neighbors E [2,30] with Ak 2 and o² E [0.2, 2] with Ao" 0.2 to investigate their influences non-rigid registration under occlusion disturbances. As shown in Fig. 8, our method consistently achieves significantly lower outlier ratios across different values of and o². With increasing values of k. method not only demonstrates higher AceS and AccR results but also with slower EPE. Nevertheless, larger values of & typically result in higher computational complexity due to the matrix inverse operation in Eq. (5) the paper. Therefore, we suggest setting A 30 for general use. Regarding o², we suggest using o² 1.0 for general applications its variation has a minimal impact on the results.
EPE
Metric result
Acc5
Metric result
60
AccR
63
EPE
Outlier
AccS
50
50
AcCR Outlier
40
1.0
0.2
0.0
0.8
1.2
Figure 8: Influences of and o² on non-rigid registration under occlusion disturbance scenarios.
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Table Ablation study of the regularization effect of the activation functions on the occluded Open- CAS liver dataset. means larger values are better while + means smaller values are better.
Metric
Liver 1
Liver 2
Liver
Method
EPH L
AceS 7
AccR
Outlier
HPE
AccS f
AccR
Outlier
BPS:
AccS 7
Aock 1
Outlier
ReLU4CD
47,994
9.106
20.466
17.957
39.171
9.684
23.321
8,533
27.494
28.766
44,443
11.272
ReLU4MOC
31,582
28. 28.446
40.614
4,378
120012
47.258
69,975
0,000
17.245
32,841
51,744
1.326
RELUAMCCHLLR
14,690
34.144
57.860
0.000
6,944
66.543
89.956
0.000
14.306
36,670
52.922
0.000
Table 5: Ablation study of the intrinsic regularization effect of the activation functions on the com- plete Open-CAS liver dataset. better.
means larger values are better while means smaller values are
Method
Liver 1 Liveiver 2
Liver 2->Liver 3
Liver 3-veiver
ReLU+CD
4.685
5,550
6.787
SIREN+CD
3.912
5.063
6.183
E ANALYSIS OF THE REGULARIZATION
Fig. 5 reveals 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. We conduct 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 Tab. 4 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.
Moreover, to investigate the intrinsic regularization capabilities of activation functions, conduct an additional experiment by applying recursive deformations to complete shapes without occlusion. The EPE metrie detailed in Tab. 5 demonstrates that ReLU and SIREN exhibit comparable perfor- mance on complete shapes. This finding further substantiates our conclusion that while activation functions do possess 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 particularly beneficial.
F DIFFERENCES AND CONNECTIONS WITH ROBUST FUNCTIONS
Moreover, we conduct an in-depth analysis comparing correntropy with robust functions.
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 certain type of non Gaussian noise), which is related to the Renyi's quadratic entropy. Besides, MCC is a local measure that provides a probabilistic meaning of maximizing the error probability density the origin according the information potential.
Connections. We derive the relationship between MCC and robust functions. By setting p(e) (1- exp (-e2/203)) / V2To, can prove that p(e) is influence function satisfying all the criteria of a robust function, i.e.
p(e) 0,)(0) 0, = -(-e))
(17)
p p(e:) (((e) ( for lell > leyl.
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MCCHIL
Occkeion bruel
Seclux learl
Occknion
Oceanion
Figure Ablation study of the MCC and the CD metries.
Moreover,
N
min
(()) e max Σ exp (-c:/20²) /
(18)
with the corresponding weight function of p(e) as wle) = exp (-02/20²) //TT0. For compari- son, the Bi-square's weight function is
WPF (c) (1- (L-(e)1)) le le 2 h²
(19)
O,
where the tuning threshold. Therefore, the nonzero part of WBr(e) equivalent (with constant) the square of the first-order Taylor expansion of w(e). MCC is analogous to robust function but with the specific influence function p(e). However, unlike the Huber or Bi-square function, correntropy does not need predefined threshold such as h and the kernel size entirely governs the properties. Moreover, our derivation between correntropy and robust functions offers practical way to select an appropriate threshold for robust functions or determining the kernel size for correntropy.
G ABLATION STUDY OF MCC AND THE CHAMFER DISTANCE
Additionally, we conduct additional ablation experiments to replace MCC with CD series of models used in Fig. 5, testing them with progressively increasing levels of occlusion (0-5). The quantitative comparison results in Fig. demonstrates that MCC consistently achieves higher-quality deformation results, particularly with significantly high AceS and AccR metrics. Moreover, we provide detailed comparison of the time consumption of both the Chamfer Distance and MCC in Tab. 6, measured in seconds. As observed, MCC also achieves significantly high efficiency with the computation time around millisccond for a dataset comprising around 10* points.
Table 6: Comparison of time for MCC and CD metrics in seconds (s).
Cases
0
2
Ave.
CD (x10 8s)
0.696
0.676
0.685
0.687
0.697
0.690
0.689
MCC (x10~*3) $s)
1.332
1.327
1.466
1.323
1.285
1.316
1.342
H FURTHER COMPARISON WITH OTHER CD VARIANTS
To further demonstrate the advantages and versatility of the MCC metric, we conduct additional ex- periments to evaluate MCC against the most recent or state-of-the-art Hyperbolic Chamfer Distance (HCD) point cloud completion (Lin et al., 2023). We use the author's 's source code implement the HCD metric. Particularly, we adopt three distinct tests by adjusting the coefficient within the HCD framework to achieve comprehensive results. The quantitative results reported in Tab. demonstrate that proposed method consistently achieves higher-quality deformations. More- over, MCC shows potential for further exploration and application general point analysis tasks, including point cloud completion.
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Table 7: Quantitative comparisons between the MCC and the HCD metries on the occluded Open- CAS liver dataset. t means larger values are better while means smaller values are better. Bond fonts indicate the top performer.
Metric
Liver 1
Liver
Liver
Method
EPE 1
AceS
AccR1
Outlier
EPE
AccS 1
AccR
Outlier
EPE
Ace5 1
AccR 1
Outlier J
HCD with or 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 a 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 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
I HOw TO INCORPORATE THE GEOMETRIC DESCRIPTOR
Given the two matching sets S(X) and S(Y) extracted by the pre-trained geometric descriptor Lepard, with |S(X)] = [S( < min | Y)). our optimization function is formally defined as
O arg min F(O) + B.(());
(20)
where F(O) and M()) denote the deformation loss and the pointwise matching loss, separately.
J LIMITATIONS AND FUTURE WORK
While the proposed registration method demonstrates impressive performance on various occlu- sion disturbances, it is worth noting that the problem itself is highly challenging and severely ill- posed. When the occlusion region is significantly large, the method may yield unsatisfactory results (Fig. 18). However, integrating shape analysis theory, such as the geometric symmetry of models, could potentially mitigate this limitation.
Besides, the computational complexity of the locally linear reconstruction step is relatively high due to the matrix inverse operation. One possible approach is to explore low-rank approximations of the Gram matrix to further speed up the deformation process.
Another future work is to further speed up the computation of the MCC metric. As demonstrated in (Li et al., 2023), distance transform is utilized expedite the computation of the Chamfer dis- tance, and in principle, this manner can be extended to the MCC-induced metric as our proposed method also incorporates pairwise computation. However, the authors noted, we must carefully consider the trade-off between discretization error, grid resolution, memory consumption, and esti- mation accuracy.
K MORE QUANTITATIVE AND QUALITATIVE RESULTS
We provide additional results in the following to show the superiority of the proposed method. Specifically,
1. Fig. 10 to Fig. 11 showcase the test results on the occluded TOSCA dataset (Bronstein et al., 2008), where various occlusion scenarios are present.
2. Fig. 12 demonstrates the registration results of our proposed method by inversely deforming the occluded geometry to the complete one. It can be seen that our method achieves highly accurate registrations, preserves the original topology of the organs (including their distinct parts), and avoids the generation of physically infeasible parts.
3. Fig. 13 presents qualitative results on the 4DMatch or 4DLoMatch datasets (Li & Harada, 2022a), where our method performs impressively despite the existence of challenging sce- narios such as large deformation, low overlap, and occlusion.
4. Fig. 14 showcases additional shape completion results by deforming the source model (Yang al., 2014) to the shapes with occluded holes. Thanks the highly ac-
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Published as a conference paper at ICLR 2025
curate registration results, method provides a viable solution to this challenging mesh hole filling problem.
5. Fig. 15 further reports the mesh hole filling results where the source or template models also contain holes. The source models are selected from Fig. 7 and Fig. 14, with the middle model each case serving as the mean shape. 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 Result and Shape 2 U 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.
6. Fig. 16 to Fig. 18 report the registration results of our method on real-world RGB-D point clouds under occlusion disturbance.
7. Fig. 19 and Fig. 20 showcase qualitative registration results under significant disturbances caused by noise and outliers, respectively. As demonstrated, our method continues to achieve highly accurate deformations, attributable to the adaptive correntropy function.
Source
GBCPD
NSFP
NDP
Target
DPF
Ours
Ground Truth
Figure 10: Qualitative comparisons on the occluded cat dataset, where the body and tail of the cat are occluded.
Source
GBCPD
NSFP
NDP
Target
DPF
Ours
Ground Truth
Figure 11: Qualitative comparisons on the occluded dog dataset, where the left front leg and the tail of the dog occluded.
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Input
Result
Input 2
Result 2
Input
Result 3
Figure 12: Application the proposed method to register occluded livers to the complete ones. As observed, our method not only delivers highly-accurate registration results but also preserves the original topology of the organs, including their distinct parts, while avoiding the generation of additional physically infeasible parts.
Input
Deformation
Input
Deformation
Figure 13: Non-rigid registration on point clouds from the challenging 4DMatch or 4DLoMatch datasets, which exhibit significant deformations. The blue and gray point clouds represent the source and target models, separately, while the yellow color indicates our registrations. Our method still yields satisfactory results even in the presence of substantial deformations.
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Source
Shape
Result
Shape 2
Result
Source
Shape
Result
Shape 2
Result
Figure 14: Application of the proposed method to mesh hole filling by deforming the source model the target shape.
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Source
Shape
Result
Boolean
Shape
Result 2
Boolean 2
Source
Shape
Result
Boolcan
Shape
Result
Boolean 2
Figure 15: Application of the proposed non-rigid registration method to mesh hole filling. This figure illustrates the process of deforming source model containing holes align with other models that also have holes. "Result 1" and "Result 2" represent the deformed shapes by our non-rigid registration method, which consistently delivers high-quality deformations regardless "Boolean 1" and "Boolean 2" are obtained by taking the union operation of the deformed surfaces with their respective targets (i.e., Shape I U Result and Shape 2 U Result 2), highlighting the seamless integration of the deformed shapes with their target models.
the holes.
155
Figure 16: RGB sequence images with occlusion used in generating real RGB-D point clouds.
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Sourcs
Target
Inputs
OPF
Ours
Ourputs
Figure 17: Non-rigid registration on RGB-D point clouds with occlusion indicated by the red box. As observed, DPF suffers from physically implausible issues such as collapse pull of the body parts as shown in the green box, instead, our method not only delivers successful registration, but also preserves the physical rationality of the human body, such as the hands and arms in the blue box.
Source
Target
Deformed
Figure 18: Failure cases due to significantly large occlusion (i.e., the occlusion of both human arms and body). Nonetheless, our method preserves high-quality deformation in the overlapping regions and still strives deliver physically plausible deformation for occluded areas to the best extent possible.
Figure 19: Non-rigid point cloud registration under heavy noise disturbances (intensity=0.6%))
Our method still delivers satisfactory registration thanks to the utilized adaptive correntropy func- tion.
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Figure 20: Non-rigid point cloud registration under heavy outlier disturbances (#out)ers=2,3500,
Our method exhibits substantial robustness against outliers, attributed to the introduction of the adaptive correntropy function.
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Reviews:
[{"review_id": "nHAsJ5crOb", "reviewer": "Reviewer_Jzhm", "summary": "Authors proposed a new approach to non-rigid point alignment using implicit deformation networks. Solving non-rigid point alignment using implicit deformation is attractive as it is self-supervised can generalize to well to unknown categories. There is specific focus is on how best to handle occlusion using such networks. The authors propose a maximum correntropy criterion. They argue that it effectively avoids collapsed, tearing, and other physically implausible results – leading to substantially better results than current state of the art.\n\nThe authors also demonstrate impressive results on a variety of benchmarks including 4DMatch and OpenCAS. One concern, however, with the approach is that it is essentially a well documented loss function, and well known regularizer applied to the problem of non-rigid point alignment. There is no real explicit handling of occlusion, it is instead handled implicitly through the loss function and regularizer. Also, there needs to be further ablation  into how the network architecture itself regularizes the solution. This seems to have been discarded as a minor detail in Sec 4.3. As it stands, I am on the fence with the paper. The results are good, but the motivation, ablation and new ideas are lacking. \n\nOne major concern I have is the proposed objective seems quite similar to classical robust error functions used for decades within robotics and computer vision literature. It would be good if the authors could relate their loss function to these other methods. My initial feeling is that they would almost get identical results, but happy to hear back from the authors about why I am wrong.\n\nMost famously the Huber loss function, \n\n- P. J. Huber, “Robust Estimation of a Location Parameter,” The Annals of Mathematical Statistics, vol. 35, no. 1, pp. 73–101, 1964.\n\nHuber loss functions and their variants have been used for decades in both 3D point and image alignment problems to deal with occlusion and outliers. A great example can be found in:-\n\n- A. W. Fitzgibbon, “Robust registration of 2D and 3D point sets,” Image and Vision Computing, vol. 21, no. 13-14, pp. 1145–1153, 2003.    \n\nSubsequent works have evaluated broad families of robust error functions. \n\nOne notable example includes:-\n\n- B. Theobald, I. Matthews, and S. Baker. “Evaluating error functions for robust active appearance models,” in IEEE International Conference on Automatic Face and Gesture Recognition, 2006.", "questions": "- In Definition 1, why are the authors referring to bandwidth? It seems the statement is quite generic (covering all kernels that satisfy Mercer’s theorem). Kernels like Gaussian have bandwidth, but that seems much more restrictive. \n- Lemma 1 seems wrong. The authors have not defined the kernel, and just said that it needs to satisfy Mercer’s theorem. It is trivial to show that if k(x,y) = ||x – y||, then it would not be L2 close, L1 as they get apart, and L0 far apart. It seems this statement should be dependent upon the type of kernel used? For example, for the specific case of a Gaussian kernel I would believe such a statement. \n- Eq. 2 seems to have a mistake. They use k(x – y), but I suspect it should be k(x,y)?", "limitations": "", "rating": 5, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The results are impressive compared to current state of the art (especially in the presence of significant occlusion), and the authors do a good job of applying their approach across a number of application domains.\n- I especially like use of LLR in the method, using classical work from Roweis and Saul (’00). The Application of the LLR regularization loss is critical to the success of the propose approach. However, it does lead to questions of where does the regularization of the method is stemming (implicitly from the network, all solely through the LLR loss). \n- Authors do a good job on the ablation of their method, especially with respect to the number of neighbors and bandwidth (see Appendix C).\n- The evaluation and visualization of results are compelling.", "weaknesses": "- In general I do have some concerns over novelty. At the end of the day the authors are proposing an existing loss, and regularizer to a new problem. The novelty is really in how well it works for the problem of non-rigid point alignment. \n- In Sec 4.3, the authors use a SIREN style implicit neural function (INF) using sinusoid activations. This departs quite heavily from the NSFP work of Li et al. which use ReLU activations. One motivation in NSFP for ReLU is that it leverages the inherent spectral bias of ReLU to find low-frequency (i.e. smooth) solutions. By using SIREN style INFs the authors would kill this property. It seems this would be an additional motivation for the LLR regularization. Some sort of ablation on the role of activation with the LLR would be useful, as they seem to both be regularizing the solution.   \n- There seems to be a number of mistakes in the Lemma and Definitions, they also seem quite trivial and do not add much to the paper. \n- The authors report timing information for their method in Table 2. NSFP is currently considered quite slow for an INR method. Approaches like Li et al. “Fast Neural Scene Flow” (ICCV’23) show a 30 times speedup with almost no loss in performance. It would be useful for the authors to discuss this, and talk about how feasible such a strategy would be to their method.\n- See my other concerns in the summary concerning the relation of the author's loss function with classical robust error functions (e.g. Huber loss).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "QFIFBAzZnN", "reviewer": "Reviewer_FdAL", "summary": "This paper proposed a novel method for non-rigid registration between two point clouds under partial occlusion. The core idea of the proposed method is to use the Maximum Correntropy Criterion (MCC) to measure the similarity between two point clouds. By the combination of MCC and implicit neural representations, efficient unsupervised point cloud registration algorithm can be developed. Furthermore, a local linear reconstruction (LLR) formulation has been utilized to regularize the deformation of occluded region of point cloud. Experiment on the Open-CAS liver and several self-built dataset shows that the proposed method achieves impressive result under occlusion and large deformation.", "questions": "1. In Fig.6 shape interpolation experiment, what the timestamp t means? is it some mixture ratio between two shapes? it would be good to briefly describe the settings for shape interpolation", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. Although the MCC is an existing metric, but its usage in the framework of implicit neural deformation optimization is simple, elegant and effective\n2. The experiment result is impressive, the proposed method consistently outperforms competing approaches by large margin on Open-CAS liver dataset\n3. The proposed LLR has demonstrated its effectiveness in ablation study, especially in high level of occlusion\n4. The presentation is clear and easy to follow, and the result is reproducible with source code", "weaknesses": "1. There is no ablation study on the use of MCC and Chamfer distance for registration. As Proposition 1 says that the Chamfer distance is a special case of MCC induced metric, it would be good to see how the generalization of the MCC induced metric works on real-world point cloud registration, for example, with different level of occlusion, deformation, or even registration in same-class level, not same-instance level. It would also be good to see the comparison on computational efficiency\n2. Table 4 is in appendix, not in the main text\n3. The shape completion task in section 5.7 requires a complete source mesh model as template, which may not be available in practical scenarios. It would be good to see whether a 'mean' shape mesh model could serve as the template and achieve good mesh hole filling result.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Wc9imwC5S8", "reviewer": "Reviewer_C5yq", "summary": "The paper proposes an unsupervised method for deformable point cloud registration, aiming to handle the case where parts of the point cloud are missing due to occlusions. Prior works have mainly used the Chamfer Distance to measure the similarity between the registered point clouds, which does not account for occlusion. In contrast, the proposed method utilizes the correntropy-based metric. This metric includes a decaying kernel function that reduces the influence of the occluded region on the deformation result and enables the registration of point clouds with occlusion. To accommodate the deformation of the occluded parts, the authors include a linear reconstruction regularization term to preserve the original point local structure in the deformed point cloud. The point deformation field is parametrized as a coordinate-based neural network, where the network's parameters are optimized subject to the correntropy and regularization term. The method is applied to several point cloud benchmarks with occlusion, as well as for shape interpolation and completion applications, and demonstrates favorable performance.", "questions": "Please address the concerns raised in the \"weakness\" section.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "The paper is well-written and easy to understand. The method is technically sound, can be applied to various applications, and improves over the compared alternatives. That said, important evaluations and comparisons are missing.", "weaknesses": "The method is evaluated quantitatively on the Open-CAS, 4DMatch, and 4DLoMatch, and several shapes from the TOSCA dataset. Still, a major dataset in the non-rigid deformation literature is SHREC’19 [1]. The shape completion application hints that the method can handle human body non-rigid matching and an evaluation on this dataset can further demonstrate the method's utility. Additionally, the evaluation on partial animal shapes should be done with the common SHREC'16 benchmark [2] instead of several selected shapes from TOSCA.\n\nComparison:\nThere are prior works that use an implicit field [3] or self-supervision [4] for non-rigid partial shape registration. Such works should be discussed and compared.\n\nI think the paper passes the acceptance threshold, though adding the missing evaluations and comparisons can strengthen the submission much further. \n\n[1] Melzi, S., Marin, R., Rodol`a E., Castellani U., Ren J., Poulenard A., Wonka P., Ovsjanikov M. SHREC 2019: Matching humans with different connectivity. In Eurographics Workshop on 3D Object Retrieval. vol. 7 (2019)\n[2] Cosmo L., Rodol`a E., Bronstein M M., Torsello A., Cremers D., Sahillioglu Y. SHREC '16: Partial matching of deformable shapes. In Proc. 3DOR, 2(9):12 (2016).\n[3] Cao D, Bernard F. Self-supervised learning for multimodal non-rigid 3d shape matching. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition (2023).\n[4] Sundararaman, R., Pai, G., Ovsjanikov, M. Implicit field supervision for robust non-rigid shape matching. In the European Conference on Computer Vision (2022).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "HkKAwyPe16", "reviewer": "Reviewer_Rxao", "summary": "In this paper, the authors propose an occlusion-aware non-rigid point cloud registration method based on cross correntropy between deformed source shape $T(Y)$ and occluded target shape $X$.  The paper argues that the main failure reason of previous non-rigid point cloud registration methods is that most of them are based on the standard Chamfer Distance (CD), which can lead to collapsed or physically implausible results. To overcome the limitation, the paper demonstrates theoretical analysis between maximum correntropy criterion (MCC) and CD and concludes that the standard CD is a special case of MCC and MCC is more robust in occluded regions. Furthermore, the paper proposes a regularization based on locally linear reconstruction (LLR) inspired by local linear embedding (LLE) to regularize the deformation smoothness of unmatched points and experimentally demonstrate the superior performance of LLR compared to as-isometric-as-possible (AIAP). In the experiment part, the paper conducts extensive experiments under different settings (e.g. medical data, human, animal, different level of occlusions) and demonstrates the superior performance of the proposed method compared to prior works, including both axiomatic and learning-based methods.", "questions": "1. In Eq.3, the definition of $\\tilde{y}_{i}$ should be  $T(Y)$ rather than $Y$?\n2. More details of Lemma 1 should be provided, since it is the crucial difference between MCC and standard CD.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 2, "contribution": 2, "strengths": "1. The paper conducts exhaustive experiments in different scenarios ranging from medical data to animal and humans and demonstrates superior performance of the proposed method compared to both axiomatic and learning-based method. \n2. The idea of using maximum correntropy criterion (MCC) and local linear reconstruction (LLR) for non-rigid point cloud registration is novel and technically sound. In the ablation study, the paper also demonstrates the superior performance of LLR in comparison to as-isometric-as-possible (AIAP) regularization, which is commonly used in previous methods.", "weaknesses": "1. In the experiment part, the paper only demonstrates rather synthetic point cloud datasets in the context of occlusion ratio as well as point cloud sampling density. When it comes to more challenging dataset like 4DMatch and 4DLoMatch with less overlap and large motions. The proposed method needs to use the pre-trained geometric feature descriptor Lepard. Meanwhile, it would be better to shortly describe how to incorporate the pre-trained feature descriptor into the proposed method. \n2. The motivation of using neural implicit representation is not well illustrated. To my understanding, the proposed regularization can also be directly used to optimize the deformation field defined on each point in the point cloud. \n3. It would be better to conduct an ablation study, which replaces MCC with standard Chamfer Distance to better demonstrate the effectiveness of MCC.\n4. In the literature of point cloud completion, some works also propose variants of CD to address the problem of occlusion, it would be better to also compare MCC with their proposed variants, e.g.\nT. Wu, et al.: Density-aware Chamfer Distance as a Comprehensive Metric for Point Cloud Completion (NeurIPS 2021)\nF. Lin, et al.: Hyperbolic chamfer distance for point cloud completion (ICCV 2023)", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)