Non-rigid Point Cloud Registration with Neural Deformation Pyramid
Authors: YANG LI
Tatsuya Harada
Venue: NeurIPS 2022 Accept (2022)
Keywords: Non-rigid registration
neural deformation field
motion decomposition
partial-to-partial point cloud registration
3D computer vision

Abstract: Non-rigid point cloud registration is a key component in many computer vision and computer graphics applications. The high complexity of the unknown non-rigid motion make this task a challenging problem. In this paper, we break down this problem via hierarchical motion decomposition. Our method called Neural Deformation Pyramid (NDP) represents non-rigid motion using a pyramid architecture. Each pyramid level, denoted by a Multi-Layer Perception (MLP), takes as input a sinusoidally encoded 3D point and outputs its motion increments from the previous level. The sinusoidal function starts with a low input frequency and gradually increases when the pyramid level goes down. This allows a multi-level rigid to nonrigid motion decomposition and also speeds up the solving by ×50 times compared to the existing MLP-based approach. Our method achieves advanced partial-to-partial non-rigid point cloud registration results on the 4DMatch/4DLoMatch
benchmark under both no-learned and supervised settings.

Non-rigid Point Cloud Registration with Neural Deformation Pyramid
Yang Li! liyangdmi. t. u- t-tokyo.... jc.
Tatsuya Harada 1.2 harada@mi t. t-u-tokyo jc.jp
The University of Tokyo
PRIKEN
Abstract
Non-rigid point cloud registration a key component many computer vision and computer graphics applications. The high complexity of the unknown non-rigid motion make this task challenging problem. this paper, we break down this problem via hierarchical motion decomposition. Our method called Neural Defor- mation Pyramid (NDP) represents non-rigid motion using a pyramid architecture. Each pyramid level, denoted by Multi-Layer Perception (MLP), takes input sinusoidally encoded 3D point and outputs its motion increments from the previous level. The sinusoidal function starts with low input frequency and gradually increases when the pyramid level goes down. This allows a multi-level rigid to non- rigid motion decomposition and also speeds up the solving by 50 times compared to the existing ML.P-assdd approach. Our method achieves advanced partial- to-partial non-rigid point cloud registration results on the 4DMatch/4DLoMin benchmark under both no-learned and supervised settings. Code is available at http://twbb.ooobbbbbb//eeooooaaa
Non-rigid point cloud registration a key component in many computer vision and computer graphics applications. The goal of non-rigid registration is to find the transformation that maps one point cloud another. With the availability of consumer range sensors that can measure time-varye time-varying surface points. non-rigid registration has been applied to dynamic shape reconstruction problems such as human performance capture, enabling a wide range of applications in XR and robotics.
Non-rigid registration a challenging problem. First, 3D sensor measurements often contain noise, outliers, and occlusions. Occlusions often lead disconnection of point cloud geometry. Point clouds may also have very low overlap ratios with each other due to the scene deformation and sensor's viewpoint change. The most challenging thing is, unlike rigid registration that only needs to determine the rotation and translation parameters, non-rigid registration needs to estimate the unknown movement of all points, making this problem especially complex solve.
In this paper, we alleviate this complexity through motion decomposition. We observe that natural non-rigid motion usually forms hierarchical structure: with higher hierarchies representing the global movements and lower hierarchies representing the local deformation. For instance, a walking person be roughly approximated by three levels: 1) global location and orientation change, 2) local articulated movements from arms, legs, etc, and 3) fine-grained cloth deformation caused by exterior forces. Each level represents motion at different scale and they have top-down dependencies.
Based on this observation, we propose a hierarchical motion representation called Neural Deformation Pyramid (NDP) for non-rigid registration. NDP has pyramid architecture. Each pyramid level contains a Multi-Layer Perception (MLP) that takes as input a sinusoidally encoded 3D point and outputs its motion increments from the previous level. We found that the frequency of the sinusoidal function controls MLP's s capacity of representing non-rigidity: low frequencies yield smooth signals
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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that are suitable for fitting relatively rigid motion; high freguencies produce more fluctuations that are capable of representing highly non-rigid motion. We start the sinusoidal function the first pyramid level with low frequency and gradually increase it when the pyramid level goes down. This allows a multi-level rigid to non-rigid motion decomposition and also achieves over 50 times faster solving than the existing MLP--aaed approach.
The paper is about developing an MLP-based hierarchical deformation model for non-rigid point cloud registration. The proposed method achieves state-of-the-at prrtial-toppattinn non-rigid regis- tration results on the challenging 4DMutch44DLMMath [19] benchmark under both no-learned and supervised settings. We also demonstrate the application for shape transfer.
1 Related Work
Non-rigid point cloud registration. Non-rigid point cloud registration is about estimating the deformation field or assigning point-to-point mapping from one point cloud to another. The simplest way is to estimate the point-wise wise parameters, such affine transform, under motion smoothness regularization [22]. Optimal transport- registration method [11] finds displacement for each point using global bijective-matching constraint. Coherent Point Drift (CPD) [27, 14] constructs 3D displacement field using Gaussian Mixture Model (GMM) and encourages coherent motion of nearby points. Deformation graph [34] represents the scene using a sparsely sub-sampled graph from the surface and propagates deformation from node to surface via "skinning". The Non-rigid Iterative Closest Point (NICP) [16] achieve efficient registration by optimizing the alignment energy and deformation graph regularization cost [33], and has been adopted in many real-time 4D reconstruction systems [28]. Li al. [18] and Bozic et al. [4] learn dense feature alignment or correspondence re- weighting by differentiating through deformation graph- based non-rigid optimization. Learning- based scene flow estimation methods, e.g. FlowNet33 [8], use 3D siamese networks to regress the 3D displacement field between two point clouds and reach real-time inference but are not robust under large deformations and ambiguities. Lepard [19] use Transformer [37] to learn global point- to-point mapping and use it as landmark to guide global non-rigid registration. Functional map approaches [29] estimate the correspondence in the spectral domain between computed Laplacian- Beltrami basis functions, but usually assume connected surfaces thus not suitable for real-world partial point cloud data. To alleviate this problem, Synorim [15] uses 3D CNN networks to estimate the basis functions and derive scene flow from the functional mapping. While existing methods mainly represent non non-rigid motion or mapping at a single level, this paper propose a multi-level deformation model for non-rigid point cloud registration.
Pyramid for motion estimation. Bouguet et al. [3] show that the pyramid implementation of the Lucas-Kanade [5] method improves feature tracking. PWCNet [35] estimates optical flow pyramid of CNN feature maps. PointPWC-net [38] adopts similar idea for scene flow estimation from point cloud inputs. Pyramid-basee based camera 6-Dof pose estimation can be found in SLAM systems [9]. ZoomOut [23] adopts upsampling in the spectral domain for shape correspondence estimation. DynamicFusion [28] constructs tree of deformation graph for non-rigid tracking, and shows that it stabilizes tracking and reduces computation cost. DeepCap [12] combines an inner body pose with an outer cloth deformation for human performance capture. This paper is about using MLP to create a hierarchical deformation pyramid for non-rigid point cloud registration of general scenes.
Motion field with coOrdinate-MIPP Coordinate-MLl uses an MLP to map input coordinates to signal values. Coordinate-MI.P is continuous and memory-effiiiet.. It has shown promising results for representing ID sound wave [32], 2D images [32]. 3D shape [24]. and radiance field [25], etc. Tancik et al. [36] show that tuning the sinusoidal positional encoding of the input coordinate bias the network to fitting low- or high-frequency signals. Coordinate-ML. is also suitable for representing the deformation field. Close to our work, Li et al. [21] and Li et al. [17] use MLP to estimate scene flow, Nerfies [30] and SAPE [13] propose coarse-to- fine motion field MLP optimization technique by progressively expanding the frequency bandwidth of the input positional encoding. However, the aforementioned motion field MLPs are black box models that are designed to represent signals at a single scale, and usually need large network to fit complex motion, result their optimizations are usually cone-cons This paper decomposes the motion field using a sequence of smaller MLPs, achieving more interpretable and controllable motion representation and faster optimization.

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2 Non-rigid Point Cloud Registration Notation
Given source point cloud S {x,[x; 1,, R1) and target point cloud T (y,ly, E j= 1,..., n2), where 71, are the number of points, our goal is to recover the non-rigid warp function W: R + R that transforms points from to T. The simplest form the warp function is dense R3 vector field, which is also known scene flow. Scene flow is in theory sufficient to represent any continuous deformation, but in practice, it can not fit non- linear motions very well, such as 3D rotations. We therefore formulate the non-rigid warp function using the SF.(3) field.
Dense SE(3) warp field. Given a globally non-rigidly deforming point cloud, we consider each rigid body transform
individual point locally undergoes 3D rigid body movement.
R 0 f)
SE(3) denotes rotation and translation in 3D, with R E SO(3) and E R33 We parameterize rotations with 3-dimensional axis-angle vector w E R We use the exponential map exp so(3) SO(3), @ e R convert from axis-angle to matrix rotation form, where the Sooerator creates a 3 x 3 skew-symmetric matrix from a 3-dimensional vector. The resulting 3D motion parameterization for point x; E R is therefore denoted by & (w,, t:) R. i.e. each point has 6 degrees of freedom (Dof). The warp function reads
W(x;, E,) = c +t,
(1)
Dense Sim(3) warp field. This paper mainly focuses on registering point clouds captured from the same scene with the same scale. For scale variant tasks such inter-shape registration (c.f. Sec. 4.3), we extend the above warp function by incorporating a scaling factor 8 E IR+, resulting 3D similarity transform (RR t)
1) (3). With the parameterization &, ($t, wa t.) E IR?, the warp function reads W(x;, Ei) = s,c x; + t,-
We shows ablation study of the dense warp field types including 3R SE(3), and Sim (3), and rotations representations including Axis-angle, Euler-angle, Quanterion, and 6D [40] in Sec. 4.2.
3 Neural Deformation Pyramid
As denoted Sec. 2, we aim to find the motion parameter &; for each 3D point x; in the source point cloud. However, due to the high complexity and non-convexity of the non-rigid registration, directly estimating & is usually difficult and sequence of sub-tra transformmations { estimate, and when combining them we can get the same registration effect as E
Therefore, we want to decompose & into .... <T" }, such that each uub-transformation is easier to
a
m
0
Figure 1: Neural Deformation Pyramid.

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Input
Level I
Level
Level 8
Figure 2: Hierarchical non-rigid registration of multiple seans of Dinosaur using LNDP. We show the outputs from the 1st, 4th, and 8th level of the deformation pyramid. The pink color denotes the target point cloud, the rests are the sources (the figures in this paper are best viewed on screen).
To this end, we introduce Neural Deformation Pyramid (NDP) which allows hierarchical motion decomposition using a pyramid architecture.
3.1 Hierarchical motion decomposition.
As shown Fig. 3, NDP is pyramid functions A [[T +x) 1,.,, m). where each pyramid level contains pair of continuous functions (T's, Os), and the total number of levels.
At level of the pyramid, F& is the positional encoding function that maps the input point from previous level x E R using the sinusoidal encoding
TA
R
k-1 +- [x(xx=>) (sin th x cos to
(2)
where ky a constant that controls the initial frequency the first level of the pyramid. The frequency of the sinusoidal function is enssential for motion decomposition: low frequencies yield smooth signals that are suitable for fitting relatively rigid motion; high frequencies produce more fluctuations that are capable of representing highly non-rigid motion. By gradually increasing frequency with pyramid level k; we can achieve a hierarchical rigid-to-nonrigid motion decomposition. Fig. 2 shows an example of hierarchical registration of of multiple scans of Dinosaur.
O₂ is an optimizable MLP network. It takes as input the encoded coordinate I'x the transformation increments at the current pyramid level. Formally, we have
(x (xx-1)
and estimates
O R + R..
Tx(x"=+) ~ O& (x(Txxx,
(E" a
where che E: R6 is the 6-Dof transformation parameter that is obtained from liner output head of the MLP. of E [0, 1] is a scalar that represents the network's confidence if the motion estimates are successful at the current level, it is obtained via Sigmoid output head of the MLP.
Given the D-Dof estimates & and confidence af we compose the transformed coordinate at the k-th level by
x af ((x-11, -,
where W is the warp function as defined in Eqn. 1. Note that af control the degree of deviation w.r.t to the previous more rigid pyramid level, thus regard it the level-wise deformability. To encourage srrigiddas-possibbee movement, we apply regularization terms on a which is shown in the next.
Merits of hierarchical motion decomposition. 1) provides a more controllable and interpretable coordinate MLP-based deformation representation. 2) it simplifies the task and speedup the optimiza- tion: we can use a small MLP for each level, because each level only needs to estimate the motion increments single frequency band, also found that the overall convergence of the pyramid is faster than using a single coordinate-Mil with the full frequency band, this significantly cut off the total optimization time compared to the existing MLP-based approaches, c.f. discussion in Sec. 4.2.
3.2 Cost function.
At level k we denote the transformed source point cloud as S (xxli 1,...,n) The target point cloud is T {yyli 1,.., n22.. The cost functions at level l are:

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Chamfer distance term. Chamfer distance finds the nearest point in the other point cloud, and sums the square of distance. Formally defined as
E
min p(x² yz) + Y
Σ N
min
p(x) - -y,) ES
(5)
S
T
X
where p(...) is the distance function, which we use L1 or L2 norm. We found that the robust L norm is more suitable for handling partial-to-partin registration, see ablation study in Section. 4.
Correspondence term. Given putative correspondence set M, we minimize
E
Σ (a.P)EM
p(x) -v.)
[M
where v) E .M are the indices of the matched points in S and T. To obtain correspondence, we leverage the learning-hased point cloud matching method Lepard [19], which predicts sparse
point-to-point matches. Lepard's prediction contains certain amount of outlier correspondences, which may lead to erroneous registration. To alleviate this, we design a Transformer-based outlier rejection method, which takes as input matched coordinates pairs (u, v) E 1 and estimates their outlier probabilities. The details can be seen in the supplemental material.
Deformability regularization term. Given the deformability score a in Eqn. 4, encourages zero predictions by minimizing the negative log- likeliiihoodd
E
- log(1 aa.)
S
X.ES
This is to encourage as-rigid-ass -eessible movement. We found this regularization help preserve the geometry of the point cloud given high frequency input signals.
Total cost function. The total cost function Ej combines the above terms with the weighting factors Aeds Acor, and Ayeg balance them:
F
Frotal
Add Fr
Acor Fb + Aveg Ei
8
In the case that we use the correspondence term Ecors we denote our method as LNDP where 'L' indicates the Learedd correspondences.
3.3 Non-rigid registration algorithm
Non-rigid registration using NDP is performed in a top-down way: the top-level MLP is firstly optimized, once it converges, proceeds to lower level, we repeat this till all MLPs are optimized. use gradient descent as the optimizer. Optimization of an MLP stops 1) the max jiterreerrrrrr 500 reached, 2) a given registration cost threshold 7 0.0001 is reached, or 3) the registration cost does not change for more than 15 iterations. Alg. shows the pseudocode the algorithm.
Algorithm Non-rigid registration using NDP
1: function REGISTRATION(S! T)
S denotes the raw source point cloud
2:
for is +- 1 to do
3:
θ₁ + Xavier Uniforr))
b MLP initialization
4:
for iter
max maa_jierr iler do
5:
0k)
6:
E +- ComputeeeegistzattooCCoo(((("""
7:
ConverenneCCutiiioaatttteeell/" iter) then
Early stop
8:
break
9:
O t GradiienDecenttStaar(1...
b Update MLP weights
10:
return TM

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Sinkhom
BCPD
Source & Target
NICP
Nerfies
NDP (Ours)
PointPWC
Synorim-pw
LNDP (Ours)
0.2
Quantitative non-rigid registration results for the Dragon. For each method, we show the point cloud alignment and enror map. Sinkhorn does not preserve the point cloud topology. CPD and NICP can easily fall into local minima. The scene flow produced by PointPWC and Synonym-pw distorts the geometry. Synonym-pw does not generalize well to uncommon shapes in the dataset such as this dragon.
Figure 3:
Source & Target
Synorim-pw
LNDP (Ours)
Figure 4: Quantitative non-rigid registration results for the Deer and Cow. Synorim-pu ppw runs on subsampled 8192 points. LNDP can directly warp all input points because it is continuous function. LNDP better preserves the geometry, especially for the non-overlapping regions.

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Experiments
4.1 Benchmarking partial-to-partial non-rigid point cloud registration
JDMatch/4DLoMate benchamrk. 4DMatch44DLoMach [19] is a benchmark for non-rigid point cloud registration. It is constructed using animation sequences from EnformingTTiinshhe [20]. This benchmark is extremely challenging due to the partial overlap, occlusions, and large motion present in the data. Point cloud pairs in this benchmark have wide range of overlap ratios: 45% 92% in 4DMatch and 15% 45% in 4DLoMatch. We found that the original benchmark contains a certain amount of examples that dominated by rigid movement. For fair evaluation, we remove data with near-rigid movements, please check the supplementary for details.
Table Quantitative non-rigid registration results on 4DMatch and 4DLoMatch.
4DMatch
4DLoMatch
Method
EPEL
AceS
AccR
Outlier]
EPED
AccSt
AccRt
Outlier]
Time
ICP [2)
0.296
2.96
12.06
71.50
0.565
0.14
0.74
90.87
0.10
ZoomOut [23]
0.598
1.82
4.23
89.27
0.663
0.22
0.81
90.39
151.10
CPD [27
0.274
1.57
7.30
74.52
0.463
0.07
0.48
84,49
4.52
No-Leamed
BCPD [14]
0.291
5.13
12.35
73.74
0.492
0.20
0.86
86.88
5.81
Sinkhorn [11]
0.308
2.76
8.13
79.86
0.505
0.20
0.81
89.47
3.76
NICP (28)
0.325
6.44
12.10
80.04
0.517
0.18
0.73
92.37
4.80
NSFP [17]
0,265
8.66
18.65
64.96
0.495
0.38
1.56
84.77
39.54 1
Nerfies [30]
0,280
12.65
25.41
58.91
0.498
1.05
3.01
82.21
115.94
NDP (Ours)
0,195
18.69
35.64
45.04
0.467
0.79
3.05
80.47
2.31
Lepard [19]+SVD
0.137
6.91
24.50
43.43
0.160
5.27
19.77
44.16
1
PoinPWC [38]
0.182
6.25
21.49
52.07
0.279
1.69
8.15
55.70
0.06 :
Supervised
FLOT [31]
0.133
7.66
27.15
40.49
0.210
2.73
13.08
42.51
0.07 1
GcomIFmaps [6]
0.152
12.34
32.56
37.90
0.148
1.85
6.51
64.63
135.18
Synorim-pw [15]
0.099
22,91
49.86
26.01
0.170
10.55
30.17
31.12
0.41
Lepard+NICP [19]
0.097
51.93
65.32
23.02
0.283
16.80
26.39
52.99
3.93 1
LNDP (Ours)
0.075
62.85
75.26
16.78
0.169
28.65
43.37
32.14
2.39 1
rigid registration methods.
with GPU accerleration (NVIDIA A100).
The metrics for evaluating non-rigid registration quality are 1) End-Point Error (EPF), i.e., the average norm of the 3D warp vectors over all points, 2) 3D Accuracy Striet (AceS), the percentage of points whose relative error < 2.5% or 2.5 cm, 3) 3D Accuracy Relaxed (AccR), the percentage of points whose relative error < 5% 5 em, and 4) Outlier Ratio, the percentage of points whose relative error > 30%. We consider AccS and AccR as the most important metrics as they exactly measure the ratio of accurately registered points.
Metrics.
Baselines. We benchmark non-rigid registration using a number of baselines under both the no- learned and supervised settings. We use the underlined names for brevity:
Point-to-point Iterative Closest Point (ICP) [2] implemented in Open3D [39], Coherent Point Drift (27) and its Bayesian formulation BCPD [14), ZoomOut [23], Sinkhorn optimal transport method implemented in Geomloss [11] and Keops [10], point-to-point Non- rigid ICP (NICP) [28], coordinate-MLA. based approaches including Neural Scene Flow Prior (NSFP) [17] and Nerfies [30].
No-Learned.
Supervised. Procrustes approach using Lepard [19]'s feature matching followed by SVD solver [1] (V)),, Deep Geometric Maps (GeomFmaps) [7], Synorim [15] the pair-wise setting (Synorim-pw), scene flow estimation methods including FLOT [31] and PointPWC [38], feature matching enhanced NICP as [19] NIIP)) supervised models are re-trained on 4DMatch's 's training split before evaluation.
7

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Benchmarking results. Tab. shows the quantitative non-rigid registration results. The coordinate- MLP-based methods, including NSFP, Nerfies, and our NDP get clearly better results than other no-learned baselines. This indicates the advantages of using the continuous coordinate-MILP to represent motion. A major drawback of Nerfies and NSFP that their optimization is usually time-consuming. With hierarchical motion decomposition, our NDP runs around 50 times faster than Nerfies and still be equally or more accurate. Note that on 4DMatch, NDP even outperforms the supervised methods FLOT and PointPWC on the AccS/AccR metrics by significant margin. On 4DLoMatch, due to the small point cloud overlap (15% - 45%), none of the no-learned methods produce reasonable result. LNDP obtains significantly better non-rigid registration results than other supervised baselines. Fig. 2, 3, and shows the qualitative non-rigid registration results.
AccR (NDP)
70%
Acc5 (NDP)
25%
AccR (ICP)
20%
Acc5 (ICP)
40%
15%
30%
AccR (LNDP)
Acc5 (LNDP)
20%
AccR (Lepard
5%
(Lepard+SVD)
lter. number (NDP)
Iter. number (LNDP)
100
100
0
0
2
5
Pyramid Level
Pyramid Level
Figure 5: Point cloud registration accuracies (top) and gradient descent iterations (bottom) at each pyramid level the 4DMatch benchmark. Left: NDP, right: LNDP.
Table 2: Ablation study of loss functions.
4DMatch AccS
4DLoMatch
Method
EPF.
AccR
Outlier]
EPF]
AccSt
AccRt
Outlier]
(L2
Chamfer distance term )
0.205
14.34
29.79
48.4
0.473
3.03
NDP (L,g Chamfer distance
0.195
18.69
35.64
45.04
0.467
0.79
3.05
80.47
LNDP (L2 Correspondence term
0.078
61.27
74.10
17.50
0.177
26.59
41.50
33.81
LNDP (L1 Correspondence term
0.075
62.85
75.26
16.78
0.169
28.65
43.37
32.14
Table
Ablation study of warp field type and rotation representation in LNDP.
4DMatch
4DLoMatch
Warp field
Rotation format
EPE4
AccS
AccR
Outliers]
EPEL
AccSt
AccRt
Outliers]
Iter.)
Rd
0.084
58.03
70.91
19.15
0.197
21.46
34,30
40.00
882
6D [40]
0.080
56.15
72.04
18.71
0.172
26.95
43.09
32.14
1077
SE(3)
Quaternion
0.080
55.47
71.87
18.73
0.168
26.79
43.29
31.33
1010
Euler angle
0.075
62.72
75.21
16.80
0.169
28.89
43.68
32,17
799
Axis-angle (Default)
0.075
62.85
75.26
16.78
0.169
28.65
43.37
32.14
785
Sim(3)
Axis-angle
0.079
59.33
72.35
19.86
0.173
24.74
42.56
33.01
1388
4.2 Ablation study
Pyramid level. We test pyramid levels with the initial frequency parameters in Eqn. set to kn = 88. Fig. 5 shows that, 1) the first level only slightly surpasses the rigid registration baseline, 2) the registration accuracies gradually increase with the pyramid level, i.e., the capacity of representing non-rigid motion gradually grows, and 3) lower levels tend to need more iterations to converge.

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Why is NDP faster than Nerfies? Summing up the iterations from all levels (c.f. Fig. 5), NDP needs total of 738 gradient decent iterations to converge. The coarse coarse-to-fine optimization in Nerfies uses a single MLP with the full frequency band as input. As comparison, Nerfies needs 3792 iterations for convergence. This indicates that our motion decomposition method speeds up the overall convergence. In addition, to represent complex motions, Nerfies needs large MLP of (witdth, depth) = (128, 7). As a comparison, for each pyramid level we use a small MLP (witdth, depth) = (128, 3), because each level only needs to estimate the motion increments at single frequency band. Note that NDP only queries a single level MLP at an iteration (c.f. Algorihtm. 1). As a result, the overall computation overhead of NDP is much smaller than Nerfies.
L1 norm vs L2 norm for partial registration. We show in Table. that the L1 norm more suitable for partial-ton sortalltt--aaaa to-partial registration. because it is more robust to large errors. For example, the chamber distance term based on the sum of the squared distance from each point the other model. Minimizing this loss would attempt to reduce the distance for every point, including those that do not correspond any point on the other point cloud due the partial overlap. Compared to the I.2 norm, the L1 norm allow for large distances on some points. Similarly, the Lj norm is also more tolerant to outliers in the correspondence term.
Motion field type. Tab. 3 shows that, 1) SE(3) warp field gets better results than Sim(3) field and R$ vector field, and 2) (3) field requires the most iterations to converge, possibly because it needs to estimate the extra scale factor.
Rotation representation. Tab. shows that Axis-angle and Fuler angle get similar results and better and converge faster than Quanterion and 6D.
4.3 Scale variant registration with Sim(3) warp field
In Fig. 6, we provide examples of the "shape transfer" application of our non-rigid registration method. To handle scale change between different shapes, we employ the dense Sim warp field (c.f. Sec. 2). The dense formulation allows NDP to reflect different scale changes at different regions of the body, e.g. character A's stomach is expanding while the legs are shrinking. The original mesh vertices of the shapes the chamfer distance cost as in Eqn. 5, therefore we use uniformly sub-sampled point clouds from the mesh's surface input points. Since NDP is continuous function, the deformed mesh can be obtained by querying NDP using the original mesh's 's vertices. We use the registration parameters (m, ko) (9, -)), this allows the transformed shapes roughly match the target shapes while retaining the geometrical details of the source shapes.
unevenly distributed, which is not suitable for computing
A + B
C
E F
C+D
E
B-A
B
D D
D
F-E
Figure 6: Shape transfer using NDP. The input shapes A, B, C. and D "Alien Soldier". "Ortiz". "Doozy", and "Jackie" from Mixamo (htts://www.miii wwwp:wwwwwaao.oo..... com/); E and F are "Racoon" and "Bear" from Deforming Things4D [20]. The arrows indicate the directions of transfer.

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5 Conclusion
We show that non-rigid point cloud registration can be decomposed into hierarchical motion estimation schema by stacking coordinate networks with growing input frequency. Our method demonstrates superior non-rigid registration results on the 4DMatch parttal-to-partial non-rigid registration benchmark under both no-learned and supervised settings. Our method runs over 50 times faster than the existing coordinate-netue networes--ttooos-beeeeee based approach.
Limitations. 1) NDP uses input coordinates that are defined over the 3D Fuclidean space, therefore it can not handle topological changes very well, extending NDP the manifold surface space would be interesting. 2) Common to the unsupervised approach, NDP does not handle non-isometric cases. failure case could be seen in Figure. 6: i.c. A's chest is mapped B's belly. 3) Though NDP runs about 50 times faster than Nerfies, the current implementation still does not run at a real-time rate, further speedup could leverage the tiny CUDA neural network framework [26]. 4) Finally, non-rigid registration in the low-overlap cases, such as examples in 4DLoMatch, is still challenging, our method cannot solve all the cases.
Broader Impact. Our paper presents non-rigid registration, which is needed for a variety of applications ranging from XR robotics. In the former, a precise modeling of dynamic and deformable objects of major importance to provide an immersive experience to the user. In the later, localizing and understanding dynamic objects such as humans and animals in the envoriment using 3D sensor is essential for safe and intellegent robot operation. On the other hand, as a low-level building block, our work has no direct negative outcome, other than what could arise from the aforementioned applications.
Acknowledgments and Disclosure of F Funding
This work was partially supported by JST AIP Acceleration Research JPMJCR20U3, Moonshot R&D Grant Number JPMJPS2011, CREST Grant Number JPMJCR2015, JSPS KAKENHI Grant Number JP19H01 11101115 15 and Basic Research Grant (Super AT) of Institute for Al and Beyond of the University of Tokyo.
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12


Reviews:
[{"review_id": "vXqyWKZso6A", "reviewer": "Reviewer_AmBj", "summary": "The paper proposes a neural model for non-rigid point cloud registration. The key idea is to decompose the desired non-rigid deformation into a hierarchy of rigid/similarity deformations, and train a neural model to predict the deformation at each level. The proposed method performs well in the presented evaluation.\n\nThe main contribution of this paper is the idea of hierarchical deformation and the proposed neural model to predict it.", "questions": "See the weaknesses above.", "limitations": "It is adequate.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The idea of hierarchical neural deformation is interesting.\n- The method performs well in the evaluation.\n\nWeaknesses:\n- It is unclear why the proposed loss function is suitable for partial-to-partial registration. In particular, the chamber distance is based on sum of squared distance from each point to the other model. Minimizing this loss would attempt to reduce the distance for *every* point, including those that do not correspond to any point on the other model due to the partial overlap. In fact, existing work for robust non-rigid registration typically identifies such L2 distance as a source of poor performance for partial-to-partial registration and replaces it with other robust norms (e.g., L1) to allow for large distances on some points. It is surprising that using such a term in the training loss would work well for partial-to-partial registration in the current paper (for the unsupervised case at least).", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "tw3X98-zvZ", "reviewer": "Reviewer_iTyU", "summary": "The paper proposes a non-rigid registration algorithm that also works in the partial-to-partial point clouds setting. The idea is to regularize the deformation in three ways: 1) by dividing the problem into pyramidal (hierarchical) sub-problems; 2) by considering first low-frequency deformations, and scaling to higher frequencies later; 3) by defining for each point a rigid (or affine) deformation (i.e., rotation, translation, and optional scaling). The method is tested on 4DMatch and 4DLoMatch, showing compelling results and outperforming several SOTA methods.\n", "questions": "- How much the method is sensitive to the introduction of noise, clutter, and point-cloud density?\n- How does the method computation timing scale at a different number of points?", "limitations": "Both societal impact and limitations are fairly discussed.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "== STRENGHTS ==\n- The method seems novel in the composition of its elements. I like the proposed regularizations on the different aspects of the problem. The approach seems relatively straightforward, while probably to be implementable by scratch, further details are required (e.g., a deeper explanation of the architecture implementation, fewer pointers to different papers for details)\n- The shown results are convincing on a significant variety of shapes. The attached video gives a nice insight into the low-to-high frequencies deformation.\n\n== WEAKNESSES ==\n- The method is tested on various shapes, but the quantitative results are only on near-isometric deformations of synthetic data. The only experiment between significantly different shapes (Figure 6) is not convincing in terms of structural preservation of the shapes (the left-most example shows that A's chest is mapped to B's belly).\n- No analysis of noise or clutter is provided. This makes unclear the applicability of the method in real-case scenarios. Also, it mentions the possibility of including the scale factor, but no quantitative results are provided in this case. No analysis of different amounts of point clouds partiality or computational scaling at the different number of points.\n\nMinor fixes:\n- line 125: Fig.3 -> Fig. 1\n- line 137: 'registration of of multiple scans'\n- line 168: 'this regularization help to preserve' -> helps\n- [14] is not introduced in the previous works; [a] would be a better alternative than [6] since it overcome some LBO limitations in the point coluds setting\n\n[a]: Correspondence learning via linearly-invariant embedding, Marin et al., 2020\n\n== AFTER REBUTTAL ==\n\nAfter the rebuttal and the discussion phase, I am prone to keep my initial rating and vote for acceptance. For the final version, I suggest including in the main manuscript the feedback about the method limitations.", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "Zyhj4IcUCl_", "reviewer": "Reviewer_hpiz", "summary": "This paper presents Neural Deformation Pyramid (NDP), a learning-based method for non-rigid point cloud registration.\nGiven a source point cloud and a destination point cloud, NDP learns a dense SE(3) warp field (or dense Sim(3) warp field in the case of scaling difference), that is, to learn a rigid transformation for each point in the source cloud, so that the source cloud is best aligned with the destination cloud.\n\nThe proposed NDP seeks this warp filed in a hierarchical manner. Specifically, the warp field is decomposed into multiple layers of learnable MLP's, where at each layer the input 3D coordinates are passed through sinusoidal encoding with increasing frequencies. In this way, the NDP decomposes the non-rigid transformation into rigid transformations that go from a global scale to more local refinements. I found this design to be very intuitive.\n\nExperiments compare NDP with a pretty large set of both non-learned methods and learned methods, and demonstrates the superior performance of NDP in terms of accuracy and efficiency.", "questions": "Is NDP (instead of LNDP) basically solving an optimization problem without any supervision?\nThe Chamfer distance and regularization term do not seem to require any groundtruth data.\nIf so, do you find the initialization of the network weights affect the performance of the optimization problem? (in terms of quality of registration)", "limitations": "Limitations included.", "rating": 7, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n+ The intuition of hierarchical motion decomposition is intuitive and well motivated in the introduction.\n+ The design of the network and its description is succinct and to the point.\n+ Experiments are comprehensive, and support the claimed benefits of the algorithm.\n+ Limitations are honestly acknowledged.\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ["I don’t know"]}, "extra_text": {}}, {"review_id": "Vuutvcw8Xmp", "reviewer": "Reviewer_xwkn", "summary": "This paper proposes a novel method - Neural Deformation Pyramid (NDP) for non-rigid point cloud registration. Non-rigid point cloud registration remains a challenging problem as the input 3D data often contains noise, outliers, and occlusions. The proposed method NDP deals with the above complexity by decomposing motion hierarchically. The NDP method uses coordinate-MLP as the basic structure, similar to existing works Nerfies [30] and SAPE [13]. The NDP outperforms existing methods on the 4DMatch/4DLoMatch [19] dataset and is faster than existing MLP-based methods.\n", "questions": "- The authors claim that the proposed method is faster than the existing method; however, there is no detailed data about the runtime speed of the proposed method and existing methods. The author should report the speed in detail in their paper.\n- In line 53, what does 'via skinning’ mean here?\n- There are not enough implementation details to reproduce the proposed method, such as the parameters of the MLP structures.\n- In 3.3 (line 174), the authors propose the non-rigid registration algorithm. However, there is not enough justification for the algorithm. Such as why the authors optimize different levels in order instead of updating all levels simultaneously. Also, there is no justification why the max_iter is set to 500 and the theta to 15 iterations. \n", "limitations": "The authors have adequately addressed the limitations and potential negative societal impact.\n", "rating": 4, "confidence": 3, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "## Strengths\n\n- The proposed method NDP is well-motivated, and the connection to existing methods is clearly stated.\n- The proposed method achieves better performance on both no-learned and supervised settings quantitatively and qualitatively than previous methods and is faster than the existing method Nerfies [30]. \n- The authors compared their proposed methods with a variety of existing methods.\n- This paper is well written and easy to follow.\n\n\n## Weaknesses\n- One of the paper's main contributions is introducing a pyramid architecture to the non-rigid motion registration task. However, there is a range of methods that have already discussed pyramid structures, as justified in Related Work (line 68 to 76). Therefore, there is a lack of technical contribution in this paper. Similarly, the proposed framework is based on coordinate-MLP structure, similar to existing methods Nerfies [30] and SAPE [13]. \n- The authors only conducted experiments on a single benchmark 4DMatch/4DLoMatch [19]. The authors should consider conducting experiments on other benchmarks, such as the 3DMatch/3DLoMatch used in previous research [19].\n- In line 25, the authors claim that the proposed method alleviates the complexity of non-rigid registration task (data noise, outliers, and occlusions). However, there are no quantitative experimental results regarding the above complexities. The authors should consider conducting experiments with the proposed method to data noise, outliers, and occlusions.\n", "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": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept