Implicit Neural Surface Deformation with Explicit Velocity Fields
Authors: Lu Sang
Zehranaz Canfes
Dongliang Cao
Florian Bernard
Daniel Cremers
Venue: ICLR 2025 Poster (2025)
Keywords: Implicit Shape Representation
Neural Implicit Representation
Shape Deformation
Neural Surfaces
Primary area: applications to computer vision, audio, language, and other modalities
TL;DR: We propose a novel method that can simultaneously predicts time-varying neural implicit surfaces and deformations between pairs of point clouds, and reconstruct physically plausible intermediate shapes without supervision.

Abstract: In this work, we introduce the first unsupervised method that simultaneously predicts time-varying neural implicit surfaces and deformations between pairs of point clouds. We propose to model the point movement using an explicit velocity field and directly deform a time-varying implicit field using the modified level-set equation. This equation utilizes an iso-surface evolution with Eikonal constraints in a compact formulation, ensuring the integrity of the signed distance field. By applying a smooth, volume-preserving constraint to the velocity field, our method successfully recovers physically plausible intermediate shapes. Our method is able to handle both rigid and non-rigid deformations without any intermediate shape supervision. Our experimental results demonstrate that our method significantly outperforms existing works, delivering superior results in both quality and efficiency.

Published as a conference paper at ICLR 2025
IMPLICIT NEURAL SURFACE DEFORMATION WITH EXPLICIT VELOCITY FIELDS
Lu Sang" 12, Zehranaz Canfes', Dongliang Cao Florian Bernard", Daniel Cremers 1.2 'Technical University of Munich, MMunich Center of Machine Learning (lu. sang, zehranaz. canfes, cremers } utmmc... de University of Bonn {dcao, fbj@uni-bont bon. de
Figure Given two point clouds Po and Pi, our method predicts time-varying neural implicit surface that represents smooth and physically plausible deformation from Po to Pi. To ensure physical plausibility, we utilize a velocity network that leverages smoothness and divergence-fret constraints.
ABSTRACT
In this work, we introduce the first unsupervised method that simultaneously predicts time-varying neural implicit surfaces and deformations between pairs point clouds. We propose to model the point movement using an explicit velocity field and directly deform vary-ng implicit field using the modified level-set equation. This equation utilizes an iso-surface evolution with Eikonal constraints in compact formulation, ensuring the integrity of the signed distance field. By applying a smooth, volume-preservin method successfully recovers physically plausible intermediate shapes. Our method is able to handle both rigid and non-rigid deformations without any intermediate shape supervision. Our experimental results demonstrate that our method significantly outperforms existing works, delivering superior results in both quality and efficiency!
constraint to the velocity field,
INTRODUCTION
Representing surfaces using implicit methods, such as signed distance fields, offers significant advan- tages over explicit methods in some applications. For example, it allows flexible topological changes and is more memory-effiiient compared to storing an explicit representation of high-resolution surface. Additionally, implicit representations allow for differentiable operations, the respective surfaces are encoded in smooth fields, which in turn enhances variety of downstream tasks, such as radiance field rendering by Yariv et al. (2021); Wang et al. (2023). Embedding signed distance field within a neural network to represent a single surface demonstrated many successful outcomes, such work from Sitzmann al. 2020): Gropp al. (2020); Mescheder et al. (2018). However, using implicit representations to model surface deformation or a dynamic surface evolution, especially with
the code is available:
https://gittubb.com
www.a//memeeie..momo 1mp11 implich surf-iteoormatioon

---

Published as a conference paper at ICLR 2025
physically plausible deformations, still remains challenging. The challenges stem mainly from two inherent characteristics with implicit methods: (i) implicit representations do not store explicit surface point locations, which makes it hard to directly manipulate surfaces during deformation. (ii) the lack of traceable neighboring information in implicit fields prevents the use of efficient physical constraints, for example, as-rigid-as-postie
regularisation, proposed by |Sorkine & Alexa (2007).
which is crucial in many mesh-based methods such as the work from Alexa et al. (2023); Eisenberger et al. (2021) p) Cao et al. (2024a). In this paper, we aim to tackle these core problems of implicit surface representations. To this end, we introduce method that simultaneously recovers implicit neural representations of two given point cloud inputs, together with time-varying intermediate shapes between them. Most notably, our approach distinguishes itself from previous deformation methods based on implicit representations by recovering physically plausible intermediate shapes - without supervision from ground truth intermediate shapes. To achieve this goal, we model the deformation of surface points by training a velocity network that utilizes smoothness and divergence-free constraints, thereby ensuring natural and physically plausible deformations. Our approach circumvents the need for mesh rendering during training, facilitating an end-to-ens and fully differentiable training process. Our method supports both intrinsic and extrinsic deformations of the given point clouds, enhancing its versatility and application scope. In summary, we claim the following contributions:
We propose a novel end-to-end framework that recovers the underlying surfaces of given point clouds together with physically plausible intermediate shapes.
Our method directly deforms the implicit field by the explicit velocity field based on the level-set equation to avoid explicit mesh rendering.
We propose to use a modified level-set equation that combines Eikonal constraint and thereby enables a compact joint optimization while preventing degenerated signed distance fields.
We validate our method on different datasets and demonstrate that our methods give rise to high-quality interpolations for challenging inputs, both quantitatively and qualitatively.
2 RELATED WORKS
Surface representation methods We roughly divide shape representation into explicit and implicit approaches. While explicit representations, such as polygon meshes, store mesh properties, e.g. vertices, edges, and faces explicitly, implicit methods encode the surface information into function fields, such signed distance fields (SDF). With explicit methods, it is relatively straightforward to edit the shapes, since shape properties can directly be manipulated. However, there are some drawbacks to explicit surface representations. For instance, meshes can only have a fixed topology. It is not trivial to adapt vertices and the configuration of their connections (such as edges). Implicit methods, the contrary, allow arbitrary topological changes since no explicit surface and structural information are stored. Additionally, neural implicit representations enable arbitrary resolutions during inference, without memory increase during storage.
Mesh-based deformation Mesh-based shape deformation is a well-studied problem in computer graphics. The most common strategy is to directly deform vertices based on some local deformation measurements (e.g. as-rigid-as--ossiblle (ARAP) proposed by (Sorkine& & Alexa (2007). PriMo Botsch al. (2006), etc.). Another direction is deform intrinsic quantities like dihedral angles such as
work from Alexa et al. (2023): Back et al. (2015) before reconstructing 3D shapes. Meanwhile, other
works formulate shape detormation as a time-dependent velocity field Charpiat et al. (2007); Eckstein
et al. (2007) and incorporate specific constraints (e.g. volume preservation used by Eisenberger al. (2018); Eisenberger & Cremers (2020). Despite the great success achieved by based shape deformation methods, they rely on the local neighborhood information obtained from edges (or triangles) during deformation. Therefore, shapes have a fixed topology during deformation, which limits the applications for shapes with inconsistent topology or partiality. In contrast, method directly works on an implicit surface, and thus has no constraint on the shape topology and is applicable for partial shapes based on spatial smoothness regularisation. In the experiment part, we demonstrate that our method is capable of deforming shapes with significant variations of shape resolution and partiality.
Implicit-field based deformation Deforming implicit fields presents a significant challenge, as all information is encoded within a function field, preventing direct operations on the object. Previous
2

---

Published as a conference paper at ICLR 2025
works that studied this topic are typically for physical simulation, such as the work of Osher & Paragios (2003); Museth et al. (2002); Jones et al. (2006). and these works use classical discretely
stored implicit fields. More recently, the widespread adoption of neural networks to represent
implicit fields in shape modeling, work From/Yang et al. (202T); Sitzmann et al.] (2020) Ma et al.
implicit fields. Several works, e.g.. Peng ct al. (2021) utilizing neural networks for shape deformation have concentrated exclusively on
(2020) inspired works to study shape deformation
human body movements ,Chen et al. (2021) >; Deng al. (2020) Božič et al. (2021) requiring prior information such as skinning details or intermediate point clouds. Others. like Cao et al. (2024b) have explored deforming implicit fields through generative or diffusion models, but these still necessitate intermediate point clouds for supervision. Some works have trained on datasets of shapes from specific object categories, Deng et al. (202T): ); Genova et al. (2019) Iglesias et al. (2017): Hao et al. (2020) aiming to deform from one category to another, rather than recovering plausible intermediate shapes. Some works address this problem by defining a latent space and getting a deformed shape via latent space interpolations Liu et al. (2022). Yang et al. (202l) introduced pioneering work that enables direct editing of implicit fields using user-defined handle points, ensuring the deformation remains consistent with the original object. Following this, |Mehta et al. (2022) proposed using the level-set equation to deform the implicit network, providing theoretical insights into implicit field deformation. Building on this foundation, Novello al. (2023) extended these concepts, applying them to 3D shapes, primarily focusing on smoothing surface deformations. In terms of directly deforming the implicit field using velocity field, their work models the velocity field through linear interpolation between two pre-trained implicit networks representing the target shapes. Therefore, these works are limited to work on pre-defined velocity field and fail to predict reasonable intermediate shapes when the deformation is not a linear translation.
In this paper, we adopt neural implicit surface representations and tackle the challenging problem of directly deforming the implicit field while recovering physically plausible intermediate shapes without any rendering or intermediate ground truth supervision. We achieve this by modeling deformations using velocity field and directly deforming the implicit neural network using modified level-set-equee -etussiiss Moreover, our training is end-to-end without any pre-trained SDF network needed.
METHOD
Given two 3D point clouds Po = {x{]+ and PP = {x]] J., we aim to reconstruct a time- varying implicit representation of the inputs to- gether with natural and physically plausible in- termediate surfaces. To this end, we adopt Lagrangian perspective from fluid mechanics to track the trajectory of individual points through velocity field V Ω + R², for Ω C 3 be- ing the point domain, and directly deform the time-varying implicit field f Ω I -> R, where = [0, 1] is the time interval. The pri- mary challenges are (i) modeling deformations that are realistic according to physical laws, i.e. modeling physically plausible movements, and (ii) deforming the implicit surfaces without relying on explicit mesh rendering or intermediate shape supervision. To tackle these issues, we employ smoothness constraint and a divergence-frr free constraint on the velocity field to ensure realistic movement, Additionally, we utilize a modified level-set equation to directly deform the time-varying implicit surfaces without rendering meshes explicitly.
SDF
Iaplict
Velociy
Valocity-Ni Net
adkitive accumbation
2 pesiieea enceding
MLP
Figure 2: Pipeline of our method: given two point cloud Po and Pl, we train a time-varying Implicit- Net to predict SDF in different time steps and Velocity-Ner to predict the velocity of the point at each time step. We directly deform the implicit field using MLSE loss.
3.1 TIME- VARYING IMPLICIT FIELDS
The time- varying implicit field f(...) takes point location x E Q and a time E I as input, where I = [0, 1] is the time interval and Q is the surface domain. Its purpose is to encode the evolution of a surface, that is, the shape S, at time /. is represented by the zero-level-set of the implicit function f(, i... i.e.
S, = (x E Ω f(x t) = =0}
(1)
3

---

Published as a conference paper at ICLR 2025
Signed diseaddeittncce fields (SDFs) have many outstanding properties for representing surfaces: for C²- smooth surfaces, the gradient of an SDF coincides with the surface normal direction n on the zero-crossing contour (0S2), and the curvature coincides with the divergence, that is
Vf(x,)) () nx, t) IV 0)l x(x,t) = V nxx t), for x E an.
(2)
The vary-ng signed distance function should also satisfy the Eikonal equation, since at every time t, f still is a signed distance field, as proposed by Bothe et al. (2024), i.c. at any time E I,
V t)| =1.
3.2 VELOCITY FIELDS
Inspired by the Lagrangian representation in fluid mechanics, which tracks surface points by modeling the particle trajectory Ø Ω x I Ω, we track the point by estimating its velocity and integrating the velocity field to form the point trajectory. The trajectory specifically consists the position of particle time t. Assuming the points are moved by external velocity field V Ω IR3 where V E V, and V is a Hilbert space of a smooth and compactly supported vector field on 52. The velocity of the moving particle satisfies the following ordinary differential (ODE) equation with the initial condition: the trajectory derivative w.r.t. time is the velocity and the initial point location is given by x
{
(x) do(x.)) for t EII, 6(x,0)= x
(4)
Note that in our setting, we also enforce the ending point of particle trajectory by 6(x, 1) = x² where x is the point in the target point cloud. To control the smoothness of the movement and ensure propose to constrain the velocity field by two aspects: spatial smoothness of the velocity fields, and physical constraints.
physical plausibility,
Velocity fields that generate diffeomorphisms As the particle path Ω x t - Ω represents trajectory from one point cloud to another, we would like to recover a smooth transformation between two point clouds, which is consistent in both directions. Thus, seek velocity fields that generate diffeomorphisms when integrated using Eq. (4 i.e. & (-,t) 6(,1-1)
((x, t) = x² + L v(o(x, t))dr
Inspired by Dupuis al. (1998a), this can be achieved by regularizing on the space V through the differentiable operator L. -aA 7I such that
|Vlly = |++Vl = -// [+)(()) dx,
(6)
where I is the identity matrix. A more detailed explanation is provided in the Appendix A
Divergence-free velocity fields To model physically plausible deformations, we consider the basic conservation laws. One direct conservation law move points on the surface along a trajectory, we assume that no particles are moved across the surface boundary at any time. Then, the total mass inside the surface stays the same, which directly follows from the divergence theorem Kreyszig et al. (2011). i.e.
borrow is volume conservation. Since
Vxx) = 0.
similar idea has also been explored in previous work from Eisenberger al. (2018); Cosmo et al. (2020) for the case of explicit polygon meshes.
Velocity-Nee integration Our smooth velocity field is approximated by an ML.P Velocity-Net V. To track the point trajectory, we follow the forward Fuler method for integrating ODEs, i.e. we take certain discrete time steps and integrate the velocity step by step using step length 8T = 1/T. Then, the discrete trajectory of points is formed by
Φ(x, t 5t) oxx t) + V( t))
(8)
The relation between Po and P is then given o(x, 0) x", e Po and d(x. =x1, x PP.
4

---

Published as a conference paper at ICLR 2025
3.3 DIRECT IMPLICIT FIELD DEFORMATION
In the previous sections, we introduced the time-varying implicit fields and velocity fields that represent the shapes and the deformation of points, respectively. In this section, discuss how to directly deform the implicit field using the external velocity. We borrow the idea from fluid dynamics and treat every point as fluid particle. Since points stay on the surface of any intermediate shape (i.e. f(o(x, t), t) = O for any E I and x E 022. it implies there is no in- or outflow at the surface boundary an
d dt / ((o(x,t), = =.
Together with the initial condition, that is, the surface deforms from the underlying surface of point cloud Po. Eq. (9) implies that
vf= =0 { a V1-V-V=-- 1(x,0)= 10
in QxI,
(10)
where (x) = 0 for x E Po. The linear transport in Eq. 10 is called the Level-Set Equation (LSE). However. the function f is signed distance function in our scenario, which means the Eikonal equation in Eq. (3) should also hold to prevent degenerated level-set functions. Previous work from Sussman et al. (1994); (Sethian (199)); [Susmman Fatemi] (T999) proposed to solve by
introducing reinitialization equation at pseudo time т, c.g. solving
0J + sgm (f) - 1), fir= = f
(11)
However, this requires solving an additional partial differential equation (PDE), and requires that the signed distance field time 0 is well initialized. To avoid pre-training neural network to fit the starting mesh and solve the problem more compactly, we follow the idea of Bothe et al. (2024)
and Fricke et al. 2022). combining Eq. (3) and
Vf
V
V (
I)I = (VVV
=0,
(12)
dt
|V
|
with Eq. (10 to get our Modified Level-Set Equation (MLSE) that reads
{
0.S +v Vf = - (x/(((x () 2xI, f(x, 0) = fo,
(13)
where R(x,() = - πVР TVVT
V
Vf | Our MLSE in Eq. (13) preserves the norm of the gradient
at the zero-crossing contour. We also adapt the original proposed level-set equation in BBothe al. (2024): Fricke et al. (2022) by adding a weight A₁. We find that helps to achieve better implicit surfaces while still preserving the desired properties. Compared to the original level-set equation Eq. (10) and discretely enforced Eikonal constraint Eq. (3) on each time step, the modified level-set equation is more compact and solves single partial differentiable equation (PDE) in an integrated way.
Our MLSE is the bridge between the velocity field and the implicit field. Eq. allows us deform the implicit field without rendering explicit meshes. Moreover, every component of the formulation differentiable, thus it enables end-to-end training. Our method jointly recovers the implicit surface for both given point clouds and intermediate shapes without the need for pre-training SDF neural implicit networks for given point clouds or meshes.
3.4 LOSS
We set up the training loss as follows. Velocity-Net loss L, contains smoothness (Eq. 6 and divergence-free (Eq. terms. Implicit-Net loss L, contains MLSE (Eq. (13) term
/
Lg
|LV|| dx + Adiv
V Vldx
Jn
(14)
T, 10r +V. V Af
5

---

Published as a conference paper at ICLR 2025
The divergence-fred free weight Adiv can be set to 0 to disable volume preservation. We show examples and the influence of the divergence- free term in the experiment section (Sec. A.
Finally, to fit the network to the given point clouds, we propose the matching loss
Lm - [](x, 0)|dx + / [f(x, 1)fdx + Jpg ||,(x, 1) x x'I Po
/
(15)
The last term is used to indicate double integral over Q and Z, which is needed as o(x, 1) = x2 + So (7)dx) We use the forward Euler step, as described in Sec. 3.2 training. Moreover, the last term also implies that one-to-one correspondence is needed for the given two point clouds. However, thanks to the spatial smoothness of the velocity field, we only need a small part of the given correspondence to achieve satisfactory results, thus Pi C Po is the set of points for which correspondence information is available. We will discuss the number of correspondences that are needed in Sec. Our total loss term is defined as
Sec. to integrate during
L = XyI. + Aa + Am Lem.
(16)
where Ay, Ae, and Am are weights to balance the joint training of velocity and implicit function term.
EXPERIMENTS
Neural network architectures and implementation To ensure smooth and diffeomorphic transfor- mations between shapes, as discussed in Sec. 3 use the following architectures for the two neural networks: (i) Velocity-Net consists of 8-layer MLP with 256 nodes per layer; (ii) Implicit-Ne f also consists of 8-layer MLP with 512 nodes per layer. We use Softplus Zhao et al. (2018) as the activation function. To handle gigeffreyeency information and maintain a diffeomorphism, we incorporate Lipschitz continuous positional encoding Yang al. (202I).
Datasets We evaluated our methods using several datasets: Faust Bogo et al. (2014), SMAL Zuffi et al. (017), SHREC16 Cosmo al. (2016) and DeformingThinng4ll SMAL provide shapes with different categories and movements. Cross-category deformations involve non-rigid transformations between distinct objects, often with significant topological changes. Move- ment deformation involves changes in gestures within a single object, adhering to physical laws such as rigidity or conformality. DeformingThingsse provides continuous ground-truth displacements of the source mesh vertices in each frame. We used this data set to evaluate our interpolated meshes.
al. (2021) Faust and
Training strategy To generate training data, we sample 20,000 points on the surface of each mesh to create point clouds with partial correspondences. Each point cloud maintains ground-truth correspondences between 5% to 20% of its points We jointly estimate the velocity and the time- varying signed distance field. To ensure the good initialized deformation of the implicit surfaces, we train 2,000 warm-up epochs only for velocity fields. Then we gradually increase the loss term L weight xy for Implicit-Net. We implement our code using Jax [Bradbury et al, al.((2211 (2018) to enable fast higher-order derivative computations. We train for a total of 10,000 epochs with batch size 4,000. The run time is approximately 20 minutes on a GeForee GTX TITAN X GPU with CUDA for each pair.
4.1 SHAPE DEFORMATION
We test our method on various deformation scenarios and compare it with other methods: LipMLP from Liu et al. (2022) uses MLP layers that satisfy Lipschitz continuity ensure smooth transitions between source and target shapes. NFGP from Yang et al. (2021) deforms shapes based on source mesh and user-defined handle points. They estimate the implicit neural surface of the source mesh and then compute the deformed surface to match the target handle points. That means the method requires separate training for each time step. NISE fits two neural networks to the source and target meshes and trains implicit field with time dimension to estimate intermediate deformations. We use ground truth meshes to train both NISE and NFGP. NISE takes about 1.5 hours, and NEGP takes around 15 hours for each deformation step (over 75 hours for five steps). Our method requires only 1/5 of the time compared to NISE, excluding the pre-training time for SDF networks of the source and target meshes, which takes around another 20 minutes.

---

Published as a conference paper at ICLR 2025
Extrinsic (pose) deformation Extrinsic deformation refers only pose changes. The transforma- tions occurring within the same object category and no changes to the object type. In this context, we incorporate a divergence- volume does not change during the deformation. Fig. 3 illustrates the results on the Faust dataset, benchmarked against other methods. While other methods fail produce physically plausible inter- mediate shapes, both NFGP Nang et al. (2021) and our method successfully recover reasonable shapes. However, NFGP requires user-defined handle points for each step and must be trained incrementally, preventing it from generating smooth deformed implicit neural surface.
constraint (c.f. Eq. ( with Adiv 0) ensure that only the object
Instrinsic (non-rigid) deformation Non-rigid deformation typically involves different objects for the source and target point clouds. In this case, we disable the divergence-free constraint by setting Adi O. Fig. A shows an example that includes different categories and poses deformation (intrinsic and extrinsic) in the source and target point clouds. Since NFGP Yang et al. (2021) cannot handle non-rigid deformation, we only provide qualitative visualization results compared with the other two methods. While all methods can recover the source and target meshes, the comparison methods fail to generate reasonable intermediate meshes.
Quantitative evaluation To quantitatively evaluate the interpolated meshes, we use the fox and bear animation from the Deforming TrinninggggigDSDD Li et al. (2021) dataset. These examples contain relatively small deformations per frame, making them suitable as ground truth baseline. Each sequence contains 55 meshes. For each dataset, we select 5 key meshes and estimate the deformations between them. We calculate the Chamfer Distance (CD) and Hausdorff Distance (HD) of the recovered meshes compared to the ground truth. We compare our method with LipMLP Liu et al. (2022) and NISE Novello et al. (2023). Fig. 5 shows the average error table over the 55 interpolated meshes (left) and the error plot for each mesh in the bear dataset (right). While all methods accurately recover the mesh at input time steps. LipMLP and NISE exhibit increasing errors at intermediate steps. Our method maintains consistently low error on the intermediate meshes. Visualization results and error plots for the fox dataset are provided in Appendix A
4.2 INCOMPLETE AND SPARSE INPUT
Implicit methods showcase remarkable flexibility in representing shapes with varying typologies. In this section, we show some challenging cases that our method can still tackle.
Different sparsity inputs Most existing approaches require fitting two separate networks to esti- mate the Signed Distance Fields (SDF) for the initial and final shapes, as highlighted in previous works Novello al. (203);; Yang et al. (02T)); Liu et al. (002)). Consequently, the quality of the results heavily dependent on the characteristics such as the sparsity of the source and target point clouds. If the fitting process for one shape is unsuccessful, these methods fail estimate intermediate shapes. Our approach overcomes these challenges through the Velocity-Net, which enables tracking the dense initial point cloud Po to the sparse final point cloud Pi, and recovering the underlying shapes without compromising result quality. Fig. iillusttates the varying sparsity levels of the input data. The source point cloud Po contains 20,000, points while the target point cloud P, only has 2,000 points. While the source point cloud is dense enough train a neural network for fitting an SDF; the target point cloud is much sparser than the source. Although one could employ densification
LOMEL
NICPP
Figure 3: Experiment on extrinsic deformation. LipMLP Liu et al. 2022) and NISE Novello et al. (2023) fail to estimate the physically plausible intermediate shapes. NFGP Yang et al. (2021) can recover reasonable meshes but it is trained separately for each time step. Our method can recover realistic intermediate shapes in model.
7

---

Published as a conference paper at ICLR 2025
NISE
Pi
Po
Figure 4: Experiment involves both extrinsic and intrinsic deformation. While LipMLP Liu et al. (2022) and NISE Novello et al. (2023)) fail to create reasonable middle-step meshes, our method generates appropriate transition meshes from two given point clouds.
datasets
-10-
method
metric
fox
bear
LipMLPICD)
P(HD)
LipMLP
CD4
1.745
2.649
NISFACT)
HD
2.456
1.401
NISEIDD
CD4
0.178
0.366
Our
NISE
HD
0.262
0.560
OurxTII)
2
CD1
0.108
0.260
Ours
HD
0.114
0.265
(a) Average error evaluated interme- diate meshes. LipMLP[Liu and NISE Novello al. (2023) can fit well for the given meshes but produce big errors the intermediate meshes. Our proposed method maintains small error even on the middle shapes.
(2022)
10
20
30
40
50
(b) Error plot of each intermediate mesh of bear dataset.
Figure 5: Quantitative evaluation of the deformed shapes. Chamfer Distance (CD) scaled by 10 and Hausdorff Distance (HD) scaled by 102 for the 55 intermediate shapes.
strategies Sang et al. (2023): Zhao et al. (2022) or utilize priors Ma et al. (2022) before fitting the networks, our method successfully reconstructs both the final and intermediate shapes without any additional modifications.
Incomplete inputs Additionally, our time-varying implicit network is ca- pable of completing the shape even when both input sets are incomplete.
We study the situation that the given point clouds are incomplete in differ- ent areas. We sample point clouds from the incomplete shapes So and S to create point clouds with holes Po and Pı input data. Results in Fig. 6 demonstrate that our method can com- plete the shape. This experiment re- sult implies that we do not need consistent topology from the inputs, i.e. we can handle inputs with different topology features. Note that this is typically challenging for mesh-based methods, both with and without ground truth correspondences. Meshes have a fixed topology and deforming a mesh to another one with substantially varying topology is challenging endeavour, Moreover, mesh-based methods struggle with completing the mesh without giving complete shape as prior. We provide an analysis and comparison with mesh-based methods in Appendix A
Onn
Figure 7: Compared to previous methods, method can still recover both the target shape and the intermediate shapes even one input point cloud is excessively sparse.
4.3 ABLATIONS
Modified level-set equation In this section, we demonstrate that the modified level-set equa- tion Eq. (13 leads more stable results compared the original level-set equation. Fig. 8 shows the
8

---

Published as a conference paper at ICLR 2025
& U
6.
Figure 6: We sample point cloud Po and P1 from ground truth meshes with holes So and Sj. respectively. The input point clouds are incomplete. The proposed method can still recover neural implicit surfaces with complete shapes together with the intermediate steps.
OLSE
MLSE
Po
10 (first row) and modified level set equation (MLSE) Eq. (13) (second row). The OLSE does not coincide with the eikonal constraint Eq. 3 while MLSE implies it. The reconstruction results show the MLSE produces more stable and non-degenerate signed distance field.
Figure 8: Ablation study for original level-set equation (OLSE) Eq.
qualitative results of the two different level-set equations. The Original Level-Set Equation (OLSE) the formulation Eq. (1O) plus Eq. (3) at every discrete time step in training. The Modified Level-Set Equation (MLSE) embeds the Eikonal constraint compactly. Fig. shows that MLSE prevents the degenerated meshes.
Volume preservation effect We explore the influence of our divergence-free regularizer Eq. proposed in Sec. 3.2 We show that the divergence-free regularizer the velocity field indeed preserves the total volume of the recovered shape. Fig. Da shows that if the divergence-free constraint enforced, the network produces surface that preserves the volume the source point cloud. generates meshes that adopt the appearance features of the target point cloud but do not expand the volume to the target point cloud. The whole visualization provided in Appendix A
dataset Metric 1%
5%
10%
20%
CDJ 0. 1.119|0.100 0. 0.
fox
HD4 0.148|0.14] 0.114|0..166
CDJ
0.257|0.255 0.251 |0.250
bear
Adio >
Autc
HDJ 0.3 0.308 00.291
(a) Visualization of the output meshes with and without divergence-free loss. From the same source point cloud, left side mesh is slim com- pared to the target point cloud while the right side mesh can fit perfeetly.
Error with different sparsity levels. The struction accuracy increases along with the number of GT correspondenees. However, even with 1% ground truth correspondence, our method can get small CD and HD errors.
Figure Divergence-free constrains ablation (left) and Sparsity correspondence ablation (right).
Sparsity of the correspondences Our method utilizes certain amount of ground truth correspon- dence. In this section, explore the influence of correspondence numbers the deformation qualities. We sample correspondences in different proportions to the point cloud numbers: 1%, 5%, 10%, 20% to test the recovered intermediate mesh quality.
Noisy correspondences ablation We test the robustness of our method in different ways. First, we test against local noise ground-truti truth correspondences. The test data contains 5% correspondences with respect to the total number of input points. We sample 5%. 10%, and 20% of the correspondences and swap them with their 5th nearest neighbor correspondences (see Fig. TO.. We also test against.

---

Published as a conference paper at ICLR 2025
the global noise, where randomly swap them with other correspondences, regardless of whether the swapped correspondences are neighboring. This represents an extreme case for noise simulation. Due to page limitation, only show the local cases in the main paper. For more ablations please refer to the appendix.
10%
5%
Figure 10: We add noise to the correspondences by randomly choosing different portions of the GT correspondences and swap them with its 5th nearest neighbor correspondences. Qualitative results show that our method stable up to 10% correspondences and still gives relatively reasonable results up to 20% misaligned correspondences.
Combining other methods to obtaín correspondences Our proposed method integrates seamlessly with existing point-registration techniques when ground-truth correspondences are unavailable. In this section, we present results using prior method[Cao et al. (2024a) to first obtain correspondence vertices, followed by the application our method. Many point-registration approaches provide only sparse correspondence pairs with small deviations. However, due to the robustness of our method and the fact that we require only around 10% of correspondences to achieve strong results, we can effectively utilize most of their output. Fig. [T iillustrats the results of our method built on top of matching technique. For more results using obtained correspondences, please refer to Appendix A
Pr
Figure 11: Results using correspondences obtained by other methods Cao et al. 2024a). The method offers 5000 correspondences to each pair the shapes.
DISCUSSION
Limitations and future works Even though work is self-supervised, still need sparse correspondences of the point clouds. Moreover, due to the lack neighboring information, our method struggles with large deformations and may introduce artifacts around the reconstructed surfaces (c.f. Appendix A Fig. 23). In the future, we will explore dealing with large deformations and extend work to dynamic implicit surface generation.
Conclusion this paper, we introduce a method to recover the implicit surface of two given point clouds based on sparse correspondences, while also generating a natural and physically plausible intermediate deformation. Our approach does not require any intermediate shape supervision beyond the provided source and target point clouds. Our method directly deforms the implicit field using an explicitly estimated velocity field, enabling to estimate deformations directly from the point cloud input. This approach allows for the representation of more flexible topologies and can handle more challenging scenarios. Our method also broadens the application of implicit representations from static objects to dynamic objects.
10

---

Published as a conference paper at ICLR 2025
BIBLIOGRAPHY
Mare Alexa, Daniel Cohen-Or, and David Levin. As-sigid-as-posiiile shape interpolation. In Seminal Graphics Papers: Pushing the Boundaries, Volume 2, pp. 165-172. 2023.
Vladimir Armll. Ordinary differential equations. Springer Science & Business Media, 1992.
Seung-Year Yeob Baek, Jeonghun Lim, and Kunwoo Lee. Isometric shape interpolation. Computers & Gruphics, 46:257-263, 2015.
Mirza Faisal Beg, Michael Miller, Alain Trouvé, and Laurent Younes. Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61:139-157, 02 2005.
Federica Bogo, Javier Romero, Matthew Loper, and Michael J. Black. FAUST: Dataset and evaluation for 3D mesh registration. In CVPR, 2014.
Dieter Bothe, Mathis Fricke, and Kohei Soga. Mathematical analysis of modified level-set equations. Mathematische Annalen, pp. 1-41, 2024.
Mario Botsch, Mark Pauly, Markus H Gross, and Leif Kobbelt. Primo: coupled prisms for intuitive surface modeling. In Symposium Geometry Processing, 2006.
Aljaž Božič, Pablo Palafox, Michael Zollhöfer, Justus Thies, Angela Dai, and Matthias Nießner.
Neural deformation graphs for globally-consistent non-rigid reconstruction. CVPR, 2021.
James Bradbury. Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Necula, Adam Paszke, Jake VanderPlas, Skye Wanderman-Milne and Qiao Zhang. JAX: composable transformations of Python+NumPP programs, 2018. URL
htttp://gitten wttp///oo////oo/oooooooooooooott
Dongliang Cao, Marvin Eisenberger, Nafie Amrani, Daniel Cremers, and Florian Bernard. Spectral meets spatial: Harmonising 3d shape matching and interpolation. In CVPR, 2024a.
Wei Cao, Chang Luo, Biao Zhang, Matthias Nießner, and Jiapeng Tang. Metion2vecsets: 4d latent vector set diffusion for non-rigid shape reconstruction and tracking. In CVPR, 2024b.
Guillaume Charpiat, Pierre Maurel, J-P Pons, Renaud Keriven, and Olivier Faugeras. Generalized gradients: Priors on minimization flows. International journal of computer vision, 73:325-344, 2007.
Xu Chen, Yufeng Zheng, Michael Black, Otmar Hilliges, and Andreas Geiger. Snarf: Differentiable forward skinning for animating non-rigid neural implicit shapes. In ICCV, 2021.
Luca Cosmo, Emanuele Rodola, Michael M Bronstein, Andrea Torsello, Daniel Cremers, Y Sahillioglu, al. Shrec' 16: Partial matching of deformable shapes. In Eurographics Workshop on 3D Object Retrieval, EG 3DOR. 2016.
Luca Cosmo, Antonio Norelli, Oshri Halimi, Ron Kimmel, and Emanuele Rodolà. LIMP: Learning Latent Shape Representations with Metric Preservation Priors, pp. 19-35. Springer International Publishing. 2020.
Boyang Deng, JP Lewis, Timothy Jeruzalski, Gerard Pons-Moll, Geoffrey Hinton, Mohammad Norouzi, and Andrea Tagliasacchi. Neural articulated shape approximation. In ECCV. Springer, August 2020.
Yu Deng, Jiaolong Yang, and Xin Tong. Deformed implicit field: Modeling 3d shapes with learned dense correspondence. In CVPR, 2021.
Paul Dupuis, Ulf Grenander, and Michael Miller. Variational problems on flows of diffeomorphisms for image matching. Quarterly of applied mathematics, pp. 587-600, 1998a.
Paul Dupuis, Ulf Grenander, and Michael Miller. Variational problems on flows of diffcomorphisms for image matching. Quarterly of applied mathematies, pp. 587-600, 1998b.

---

Published as a conference paper at ICLR 2025
Iyy Eckstein, J-P Pons, Yiying Tong, C-CJ Kuo, and Mathieu Desbrun. Generalized surface flows for mesh processing. In Proceedings of the fifth Eurographics symposium on Geometry processing, 2007.
M. Eisenberger and D. Cremers. Hamiltonian dynamics for real-world shape interpolation. In ECCV, 2020.
Marvin Eisenberger, Zorah Lähner, and Daniel Cremers. Divergence-free shape interpolation and correspondence. arXiv preprint arXiv:/806. 10417, 2018.
Marvin Eisenberger, David Novotny, Gael Kerchenbaum, Patrick Labatut, Natalia Neverova, Daniel Cremers, and Andrea Vedaldi. Neuromorph: Unsupervised shape interpolation and correspondence in one go. In CVPR, 2021.
Mathis Fricke, Tomislav Marić, Aleksandar Vučković, Ilia Roisman, and Dieter Bothe. A lo- cally signed-distance preserving level set method (sdpls) for moving interfaces. arXiv preprini 1:Xiv:2208.01266 2022.
Kyle Genova, Forrester Cole, Daniel Vlasic, Aaron Sama, William T Freeman, and Thomas Funkhouser. Learning shape templates with structured implicit functions. In ICCV, 2019.
Amos Gropp, Lior Yariv, Niv Haim, Matan Atzmon, and Yaron Lipman. Implicit geometric regular- ization for learning shapes. In ICML, 2020.
Zekun Hao, Hadar Averbuch-Elor, Noah Snavely, and Serge Belongie. Dualsdf: Semantic shape manipulation using a two-level representation. In CVPR, 2020.
José A. Iglesias, Martin Rumpf, and Otmar Scherzer. Shape-aware matching of implicit surfaces based thin shell energies. Foundations of Computational Mathematics, 18(4):891-927, June 2017.
Mark Jones, Andreas Bærentzen, and Milos Sramek. 3d distance fields: A survey of techniques and applications. IEEE transactions on visualization and computer graphics, 12:581-99, 08 2006. doi: 10.1109/TVCG..206.55
Erwin Kreyszig, Herbert Kreyszig, and E. J. Norminton. Advanced Engineering Mathematics. Wiley, Hoboken, NJ, tenth edition, 2011.
Yang Li, Hikari Takehara, Takafumi Taketomi, Bo Zheng, and Matthias Nießner. 4dcomplete: Non-rigid motion estimation beyond the observable surface. In ICCV, 2021.
Hsueh-Ti Derek Liu, Francis Williams, Alec Jacobson, Sanja Fidler, and Or Litany. Learning smooth neural functions via lipschitz regularization. In ACM SIGGRAPH 2022 Conference Proceedings, pp. 1-13, 2022.
Matthew Loper, Naureen Mahmood, Javier Romero, Gerard Pons-Moll, and Michael J. Black. SMPL: A skinned multi-person linear model. ACM Trans. Graphics (Proc. SIGGRAPH Asia), 34(6): 248: 1-248:16, October 2015.
Baorui Ma, Zhizhong Han, Yu-Shen Liu, and Matthias Zwicker. Neural-pull: Learning signed distance functions from point clouds by learning to pull space onto surfaces. CoRR, abs/201 13195,,, 2020.
Baorui Ma, Yu-Shen Liu, and Zhizhong Han. Reconstructing surfaces for sparse point clouds with on-surface priors. In CVPR, 2022.
Ishit Mehta, Manmohan Chandraker, and Ravi Ramamoorthi. A level set theory for neural implicit evolution under explicit flows. In ECCV, 2022.
Lars M. Mescheder. Michael Oechsle, Michael Niemeyer, Sebastian Nowozin, and Andreas Geiger. Occupancy networks: Learning 3d reconstruction function space. CoRR, abs/1812.03828, 2018.
Ken Museth, David Breen, Ross Whitaker, and Alan Barr. Level set surface editing operators. Level Set Surface Editing Operators, SIGGRAPH, 21, 05 2002.
12

---

Published as a conference paper at ICLR 2025
Tiago Novello, Vinícius da Silva, Guilherme Schardong, Luiz Schirmer, Hélio Lopes, and Luiz Velho. Neural implicit surface evolution. ICCV, 2023.
Stanley Osher and Nikos Paragios. Geometric Level Set Methods in Imaging, Vision, and Graphics, volume xxix. 01 2003.
Sida Peng, Junting Dong, Qianqian Wang, Shangzhan Zhang, Qing Shuai, Xiaowei Zhou, and Hujun
Bao. Animatable neural radiance fields for modeling dynamic human bodies. In ICCV, 2021.
L Sang, A Saroha, M Gao, and D Cremers. Weight-aware implicit geometry reconstruction with curvature-guided sampling. arXiv preprint (eXii:2306002099 2023.
James A Sethian. A fast marching level set method for monotonically advancing fronts. proceedings of the National Academy of Sciences, 93(4):1591 1996.
Vincent Sitzmann, Julien N.P. Martel, Alexander W. Bergman, David B. Lindell, and Gordon Wetzstein. Implicit neural representations with periodic activation functions. In NeurlPS, 2020.
Olga Sorkine and Mare Alexa. As-rigid-as-possible surface modeling. In Symposium Geometry processing, volume 4, pp. 109-116. Citeseer, 2007.
Mark Sussman and Emad Fatemi. An efficient, interface-preserving level set redistancing algorithm and its application interfacial incompressible fluid flow. SIAM Journal scientific computing. 20(4): 169, 1999.
Mark Sussman, Peter Smereka, and Stanley Osher. A level set approach for computing solutions to incompressible two-phase flow. Journal of Computational physics,
1994.
Shantanu Vyas, Ting-Ju Chen, Ronak R Mohanty, Peng Jiang, and Vinayak Krishnamurthy. Latent embedded graphs for image and shape interpolation. Computer-Aided Design, 140:103091, 2021.
Wenbo Wang, Hsuan-I Ho, Chen Guo, Boxiang Rong, Artur Grigorev, Jie Song, Juan Jose Zarate, and Otmar Hilliges. 4d-dress: A 4d dataset of real-world human clothing with semantic annotations. In Proceedings of the IEEE Conference on Computer Vision and Patern Recognition (CVPR). 2024.
Yiming Wang, Qin Han, Mare Habermann, Kostas Daniilidis, Christian Theobalt, and Lingjie Liu. Neus2: Fast learning of neural implicit surfaces for multi-view reconstruction. In ICCV, 2023.
Guandao Yang. Serge Belongie, Bharath Hariharan, and Vladlen Koltun. Geometry processing with neural fields. In NeurlPS, 2021.
Lior Yariy, Jiatao Gu, Yoni Kasten, and Yaron Lipman. Volume rendering of neural implicit surfaces. In NeurlPS, 2021.
Huizhen Zhao, Fuxian Liu, Longyue Li, and Chang Luo. novel softplus linear unit for deep convolutional neural networks. Applied Intelligence, 48: 48:1707-1720, 2018.
Wenbo Zhao, Xianming Liu, Zhiwei Zhong, Junjun Jiang, Wei Gao, Ge Li, and Xiangyang Ji. Self- supervised arbitrary-scale point clouds upsampling via implicit neural representation. CVPR, 2022.
Silvia Zuffi, Angjoo Kanazawa, David Jacobs, and Michael J. Black. 3D menagerie: Modeling the 3D shape and pose of animals. In CVPR, 2017.
13

---

Published as a conference paper at ICLR 2025
A APPENDIX
A.1 MATHEMATICAL PROOFS
Existence of the velocity field We assume that our velocity field is the solution of Eq. 4 The existence of the solution for the differential equation is given by Picard-Lindelöf Theorem AreoremAAmold (1992) which states that for Φ R x I" + R", for continuous and Lipschitz continuous V. then there exist [1 e, + e]. As we choose V smooth enough, the requirement satisfied.
such that the initial value problem Eq. (4) have unique solution V on interval
Velocity fields that generate diffeomorphism Dupuis et al. (1988)) proved the existence of the smooth trajectory generated by Eq. (4) depends the smoothness constraint in the vector field vector
V. They also proved that choosing V such that V is a smooth and compactly-supports field with an inner product defined by a differential operator L ensures the solution the space of diffeomorphism.
Smooth operator The differentiable operator & introduced in Eq. (6 is chosen to have the type - + yI, where a enforces the smoothness and y >> 0 ensures the operator is non- non-singus -ingula In the experiments, we set a 0.01 and operator L with norm
1. The velocity field is Hilbert space defined by the
||'ly (V, LV)
(17)
Beg et al. (2005) proved that this type of choice for operator L stratifies the requirement that é is diffeomorphism Dupuis et al. (1998b).
Level-set equation
Following Eq. 9 and smoothness assumption of f and f we have
df(o(x, t) di dx 0,
(18)
we have df(e(x.t) 0 almost everywhere. Compute the derivatives, we have
df(o(x.t) = 0ft + 0fx061 = 0.
(19)
dt
Together with Eq. and 0xf = ff, we have
df(x(x,1) =
+ - V=
(20)
dt
which is the original level-set equation.
Modified Level-set equation To ensure the Eikonal constraint on continuous time-space for any t, it is equivalent to solving an additional initial problem of the PDE, i.e.
{
d [V t)ll = 0, EI, de [|V 0)1 1.
(21)
The function above ensures that [x, t)ll for any E I. Eq. 11 holds on the zero-crossing
surface 0? because the function value f is zero the zero-cross surface and the two term 0f. + V. V.V. V
and R. both equal zero in the surface domain Q. is more compact to solve Eq. (33 than solve Eq. (10)
plus Eq. (21) separately since the later solves two PDEs [Bgo et al. (2014).
Note that Eq. 13 needs two initial value [(x 0) fd and ||V [x 0) = 1. We explain in
the Appendix A.2 how we avoid pre-training network to satisfy the initial condition.
A.2 TRAINING STRATEGY
We implement method using Jax Bradbury al. 2018) and set the learning rate 0.005 with a decay rate 0.5 within interval 2000. We initialize Implicit-Nett weights and bias such that represents a sphere at step 0, following the method proposed in Gropp al. (2020).
14

---

Published as a conference paper at ICLR 2025
-10-3
LipMLP'CD)
LIPMLPHHD)
NISBICDI
NISEAI
Ouri(CD)
OursHiD
G1
(a) Visualization of the middle step meshes for fox datasets. We visualize the deformed meshes for middle step 5, 15, 35. and 45 NISE Novello et al.] (2023). LipMLP et al. (2022) and our method together with the ground truth (GT) meshes (bottom row). The middle steps usually have the highest error in quantitative evaluation. The figure shows that our method can still keep all details on the middle step meshes.
10
20
30
10
50
(b) Error plot of the fox datasct.
Figure 12: Quantitative and qualitative evaluation of the deformed shapes. Chamfer Distance (CD)
scaled by 10 and Hausdorff Distance (HD) scaled by 102 for 55 intermediate shapes.
Invertible Lipschitz Positional Encoding We adopt the invertible Lipschitz positional encoding same as Yang et al. (2021) cooperate with MLP in both velocity net and implicit net to produce stable output.
cos cos(22 x.;;))
sin(2" x.,)
cos(2"
Sin(2"Tmaa)
(x)
v 2m + (2),
(22)
20x
2
2 π
2m
Oriented point cloud Our method does require an oriented point cloud (point cloud with normal). However, if the normal information {n}', for i = {0, 1} is available for the given point clouds, the second-order constraints on matching loss can be added according to Eq. (2). The normal loss term is
/ P.
V
Ln
1
n'),,,, for e {0, 1}
(23)
|
L, term can accelerate the convergence speed of Implicit-Net at time & = 0 and 1.
Network initialization We initialize Implicit-Net such that it represents unit sphere at time 0 Gropp et al.] (2020). Thus it is valid signed distance field that satisfies the initial condition Eq. (21). satisfy the initial condition Eq. (3) without pre-train the net on f f'(x, 0), we set Am
much larger than Xy such that the network first converges at time ( to fit f(x, 0) = 0 on the given input point cloud Po. Thus, for the experiment showed on the paper, we set A = 100, Am = 200, X, 20 and λ 10.
Warm up training As described in Sec. we first freeze the implicit network, i.e. we set X X, = 20, Am 100 for first 2000 epochs, then we gradually increase the it using A
20200 100 2-2000
for A < n, and \, = 100 for 2 n, where is the kth-epoch, and = 5000. As we observe that velocity field convergence is faster than the implicit field, decrease the velocity loss to train only implicit net after certain epoch, i.e. Xo = 0 for k > 8000.
A.3 QUANTITATIVE RESULTS
We show the error plot for fox datasets in which the average error number is reported in Fig. 5a Even with small deformation between each key mesh, the comparison methods still report high errors the middle step meshes. We show the ground truth meshes with the middle steps in Fig. 12a and the error plot for each mesh in Fig. 12b
15

---

Published as a conference paper at ICLR 2025
Adi
Po
Adi O
04
Figure 13: Full visualization of Fig.
9a in main paper.
Adtie
Adi
Po
Aniv O
Agv
Pa
P
Visualization of divergence- free constraint on Faust dataset. With Adi > meshes), the deformed shapes still slim and only adopt the movement of the target point cloud Pi, while Adiv = (blue meshes) the deformed meshes have the same gesture and body shape of the target point cloud
Figure 14:
(yellow
A.4 DIVERGENCE- FREE CONSTRAINT ABLATION
In this section, visualize the deformed meshes under two different settings: with divergence-fiet term (Adiv > 0) and without divergence-free term (Adiv = 0). In Fig. 13) the recovered deformation meshes stay slim and thin when Adiv > 0 (top row) and only adopted features such the shape the mouth of the target point cloud. When Adiv 0 (bottom row), the deformed meshes can perfectly fit the target point cloud, which means the volume expanded compared to the source point cloud. Fig. shows another example of the volume persevering effect.
A.5 LAPLACIAN CONSTRAINT ABLATION
In this section, show the visual ablation of Laplacian constraint equation 6 Our smoothness ablation on the velocity field ensures spatial smoothness over the integration domain, which is particularly helpful for very sparse correspondences.
16

---

Published as a conference paper at ICLR 2025
Po
Figure 15: Our smoothness ablation on the velocity field ensures spatial smoothness over the integration domain, which is particularly helpful for very sparse correspondences.
A.6 MODIFIED LEVEL SET EQUATION ABLATION
In this section, show additional visualization results of our proposed modified level set equation (MLSE) with original level set equation (OLSE). Comparing MLSE to OLSE, enforcing the Eikonal loss intermediate time steps challenging with OLSE. This process involves moving the points using velocity and then enforcing the Eikonal constraint on the moved points, which can cause coupling effect that leads to the degeneration of the implicit field or velocity field. As shown in Fig. 16 the first two rows illustrate that, while the final mesh fits the target, artifacts are created in the intermediate steps. The bottom two rows demonstrate a topology change the point cloud (e.g., the crossed legs of the cat separate later). OLSE degenerates in the middle steps, and due to the continuity of the function, it retains the degenerated legs even when fitted to the target point cloud.
GT
OLSB
MLSB
GT
MLSE
Po
Figure 16: Additional ablation results for MLSE and OLSS.
A.7 DETAIL PRESERVING
We show the results for meshes containing more complicated details. We deform the original Armadillo mesh using Blender to create the target shape and run our method to interpolate the intermediate shapes. As shown, our method can preserve most of the complicated geometry details.
A.8 CORRESPONDENCE SPARSITY ANALYSIS
In this section, present the qualitative results of our method across varying numbers of ground truth correspondences. We generated input point clouds by sampling 20, 000 points and assessed deformation quality approximately 1%, 5%, and 10% correspondence levels. As illustrated
17

---

Published as a conference paper at ICLR 2025
Pa
[ 0
Pi
Figure 17: We show extra results on Armadillo for showing the detail- persevering of our method. Our method can preserve most of the complicated geometry details while producing physically plausible intermediate shapes.
1%
5%
P
Figure 18: We explore the impact of the number of correspondences the quality of the final deformation. When only small percentage of correspondence is used, artifacts tend to appear in the intermediate shapes. Remarkably, our method achieves reasonable estimations with few as approximately 1% ground-truth correspondences. Furthermore, when more than approximately 10% correspondences are available, our proposed method consistently delivers high-quality results.
in Fig. 18 we display the intermediate shapes produced using different quantities of correspondences during training. Here. S, and S are the ground truth meshes. Po and Pi are the sampled point cloud inputs. Our method effectively handles different sparsity levels of correspondences and delivers high-quality results when approximately 5% of the correspondences are available.
A.9 NOISY CORRESPONDENCES ANALYSIS
In this section, we show additional visualization of the local noise correspondence ablation together with global noisy analysis.
For global noise on ground-truth correspondences. The test data contains 5% correspondences relative to the total number of input points. We sample 1 1%, 5%, and 10% of the correspondences and randomly swap them with other correspondences, regardless of whether the swapped correspondences are neighboring. This represents an extreme case for noise simulation. In this scenario, as shown in Fig. 19 method produces satisfactory results with 5% wrong correspondences and still gives reasonable deformation with 10% misaligned correspondences.
Additionally, show correspondences with respect to the total number of input points. We sample 5%, 10%, and 20% of the correspondences and swap them with their 5th nearest neighbor correspondences (see Fig. 20
results local noise ablations. The test data contains 1%, 5%
18

---

Published as a conference paper at ICLR 2025
10%
3%
74
19%
1056
596
7% 196
Figure 19: The input data contain around 5% ground-truth correspondences. We add noise to the correspondences by randomly swapping 1%, 5%. 10% of the correspondences globally. Qualitative results show that method is stable up to 5% error and still gives relatively reasonable results up to 10% error. Note that this is an extreme situation as mismatching happens globally.
A. COMPARE WITH MESH- BASED METHODS
Mesh-based surface deformation is a well-explored area, many papers have done mesh deformation with [Eisenberger Cremers (202)) Alexa et al. 2023); Vyas et al. (202T) or without corre- spondence Eisenberger et al. (2021) Cao et al. (2024b a). In contrast to implicit-based methods, mesh-based methods enable more stable, artifacts free results as no surface fitting or estimation is needed. We compare our method against the state-of-the-ard mesh method, SmS Cao et al.) (2024a). SmS does not require ground truth correspondences and is capable of producing physically plausible intermediate shapes. Our method achieves results comparable to SmS. As illustrated in Fig, 21 However, while our approach successfully preserves all the fine details, it tends to create artifacts around the surfaces and may result in less physically accurate deformations when the deformation is too large, we show some failure cases in Fig. [3] Moreover, our method reconstructs smoother meshes compared SmS and GT meshes, because our implicit representation allows ren- der higher-resolution meshes. On the contrary, mesh-based methods, like SmS keep the original resolution (same vertices, triangles, and faces) of input meshes.
However, in some situations, such as inconsistent topology or incomplete shape without ground truth complete shape, our method can handle these challenging scenarios. Mesh-based methods struggle in these situations. Fig. 22 shows some challenging cases. In the cat (top row) example, the source and the target point cloud are sampled from meshes that have holes in the meshes. The source and target meshes are incomplete in different areas. The second example centaur shows that case of complete source mesh but incomplete target mesh The third example even though the source and target meshes are complete, because of the overlapped feet, and overlapped arm in the target shape, the topology of the meshes is different. These three challenging examples are not feasible for the mesh-based methods. The mesh-based methods cannot handle them because the vertices and faces are not one-to-one matches anymore, even with ground truth correspondences. Since the topology of the deformed mesh is fixed, it is not trivial to deform to shape that has different topology. Our
19

---

Published as a conference paper at ICLR 2025
20%
10%
5%
1%
20%
10%
3%
Pr
1%
Figure 20: The input data contain around 5% ground-truth correspondences. We add noise to the correspondences by randomly choosing 1%, 5%. 10%, 20% of the GT correspondences and swap them with its 5/h nearest neighbor correspondences. Qualitative results show that our method is stable up to 10% correspondences and still gives relatively reasonable results up 20% misaligned correspondences.
method, on the other hand, uses implicit representation and does not define mesh topology explicitly. Thus, our method can handle these cases.
A.11 FAILURE CASES
As mentioned in Sec. our work has some limitations. Here, present some failure cases and discuss potential future improvements. Compared to mesh- based methods, our approach struggles with large deformations. Fig. illlustrates scenario where mesh-based methods succeed, but
20

---

Published as a conference paper at ICLR 2025
SmS
So
8
Ours
Po
SmS
So
S1
Ours
Figure 21: Comparison results against state-of-the-an mesh-based method SmS Cao et (2024a). We produce comparable results with the meshed-based methods. However, mesh-based methods preserve more details such as human fingers. Implicit methods, on the other hand, enable rendering arbitrary resolution meshes.
Figure 22: The proposed method can deal with inconsistent topology input, such different incomplete shapes or self-intersect shapes.
method produces unsatisfactory results (top two rows). the second scenario (bottom row), although mesh-based methods fail, our method also produces artifacts around the feet due to insufficient local constraints in those areas. Another limitation occurs when there are large missing areas in the source target point clouds. Unlike smaller holes (as shown in Fig. and Fig. 22), substantial missing parts result in failure cases because our Velocity-Ner cannot correctly move the points to the appropriate locations.
21

---

Published as a conference paper at ICLR 2025
Figure 23:
Failure cases the proposed method. When the deformation is too large, our method tends to create artifacts the surfaces.
CD (x10*) JHD (x10?) (SAo(x11)) iP RP-RMSE (x10)
NFGP Yang et al.
2021
0.272
0.025
0.075
X
LipMLP Liu et al. (222
14.99
2.125
1.252
X
NISE Novello 2023 Ours
6,588
2.167
0.321
X
0.270
0.047
0.023
0.024
Table I: We evaluate our method with comparison methods on the 4d-Dress Wang et al. (2024) dataset where the intermediate shapes are given. We report the average Chamfer Distance (CD) and Hausdorff Distance (HD) over 5 interpolated meshes. We also report the standard deviation of the surface area SAo to indicate the changes mesh-area over the deformation. Moreover, evaluate our velocity field, we compute the per-por pernpoint Euclidean distance as Root Mean Square Error (P-MSE) over the ground truth meshes. For the comparison methods that cannot compute the per-point distance error, we mark it as X.
A.12 PHYSICAL PLAUSIBLE QUANTITATIVE RESULTS
To quantitatively evaluate our method, we use the 4D-Dress dataset Wang et al.) (2024), which comprises high-frequency meshes capturing human movements. For evaluation, select mesh pair with four intermediate meshes as source and target inputs. The four intermediate meshes serve as ground truth. We compute the Chamfer Distance (CD) and Hausdorff Distance (HD) quantitatively assess the accuracy of our method. Additionally, to demonstrate that our approach preserves volume, report the standard deviation of the surface area across all interpolated meshes, verifying that the meshes do not overstretch or shrink during deformation. The results are summarized in Tab. [ with visualizations provided Fig. 24
For the comparison method NFGP Yang et al. 2021). we manually identified handle points for five time steps, defined the rotation and translation between these points, and trained the model for each pair sequentially, requiring five separate training iterations.
A. CHANGE GENUS EXAMPLE AND DIRECTLY DEFORMING TRIANGLE MESHES
In this section, we demonstrate that our method effectively handles genus-changing deformations. Additionally, show that our approach directly deform triangle meshes by treating mesh
22

---

Published as a conference paper at ICLR 2025
UEMP
NPGP
Dura
Oura
Velee Paed
Figure 24: Visualization results of 4D-Dress Wang et al. (2024). The dataset consists of high- frequency meshes with deformations. We sample points from the first and fifth meshes as the source and target inputs, respectively. and recover the intermediate shapes. Correspondences are visualized using colored points (Po and P1), with the bottom row showing the ground truth SMPL Loper et al. (2015) model. Additionally, visualize our velocity field sequence of point cloud deformations in the second row from the bottom.
23

---

Published as a conference paper at ICLR 2025
Ours Deformed Meshes
Implicit Field
Fo
Velocity Field
Figure 25: Change genus case. In this example, we present a case of genus change, showcasing how the implicit representation enables arbitrary topological transformations. Furthermore, demonstrate that our method can directly deform triangle meshes using velocity network by treating mesh vertices as point cloud points. Notably, there clear resolution difference between the deformed meshes and the reconstructed meshes produced by our implicit network.
vertices as points in a point cloud. Notably, the original triangle meshes contain 6, 890 vertices. Leveraging implicit network, which supports arbitrary resolution for output meshes, rendered the deformed meshes with approximately 106, 000 vertices.
A.14 No CORRESPONDENCES AND PARTIAL CORRESPONDENCES
In this section, we highlight scenarios where no correspondences or only partial correspondences available. The results in Fig. 26 demonstrate that our velocity field remains consistent, and the implicit network does not degenerate, unlike NISE Novello et al. (2023) or LipMLP et al. (2022). Remarkably, even without correspondences,
method can still recover reasonable deformations between the two input point clouds, as shown in Fig. 27] We attribute this robustness to the joint training of our implicit network with strong physical constraints. However, observe that, in the ab- sence of correspondences, the deformations are less smooth compared to cases with correspondences, and some artifacts tend appear around the recovered surfaces.
24

---

Published as a conference paper at ICLR 2025
Ours Impliiit Field
Oura Velecity Fleid
A
P
Ours Velecity Field
P
Figure 26: No correspondences case. In this example, we present a case with no correspondences. Remarkably, method can still handle certain deformations, thanks to the volume-preserring constraint, smoothness constraint, and joint training with the implicit network.
Implicit Field
Po
Velocity Field
P
Corres
Ours Implicit Field
Ours
Velocity Field
E6
Po
Corres Error
Figure 27: Partial shape interpolation. We demonstrate partial correspondences using a partial target point cloud. The correspondence error is measured the misaligned geodesic distance relative to the ground truth, with the total mesh area normalized to 1. Despite the incomplete target shape, our method successfully recovers the intermediate meshes.
25


Reviews:
[{"review_id": "TR3JifHHYW", "reviewer": "Reviewer_hBUT", "summary": "The paper proposes a method to simultaneously recover the implicit neural representation of two given point cloud inputs, together with time-varying intermediate shapes between them. The point movement is modeled using an explicit velocity field regularized with smoothness and divergence-free constraints, and the shape is implicitly deformed using the modified level-set equation. The method is evaluated on four datasets, demonstrating its ability to model both pose and non-rigid deformation, as well as its robustness to incomplete and sparse input.", "questions": "- What is the main motivation for this setting ?\n- How  robust is the method to the failures of point-registration techniques, when ground-truth correspondences are not available ?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- **The modified level-set equation**: The paper modifies the Level-set equation by introducing an additional term (Equ.12). This modification ensures both the velocity field and implicit function are supervised to enforce the Eikonal constraint at each step. This approach is novel and effective in this context.\n- **Performance**: The method demonstrates a better capability to recover physically plausible intermediate shapes compared to the given baselines.\n- **Robustness to Incomplete and Sparse Input**: The proposed method is robust to the size of the target input point cloud. Unlike other methods, it can accurately recover the target shape and generate plausible intermediate shapes, even when the input is ten times sparser.", "weaknesses": "- **Motivation**: Please clarify the motivation for this setting. Why is it important to recover physically plausible intermediate shapes? Why is it reasonable to assume that correspondances are available?\n- **Correspondances**: While an ablation of the sparsity of the correspondences is available in the supplementary materiel, it’s not clear what happens when correspondances are not available i.e without the last term of Equ.15. On the other hand, can other methods rely on the correspondance information? For example, could this constraint be embedded in NISE and how does it perform in this case ?\nOn the other hand, one numerical result that can strenghten the paper is the average error of the method using correspondences from Cao et al. (2024a) . This should be added to table (a). In addition, it would be helpfull to also show the result with GT correspondences  in Fig.11 to better see the difference. \n- **Ablation of the modified level-set equation ablation**: Figure 8 is a bit surprising. Why is the OLSE baseline not able to recover the final shape while this constraint is explicitly enforced in the loss function ? Is the volume preserving term used in this experiment ?", "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": "1ruqrkE3mo", "reviewer": "Reviewer_qpsz", "summary": "This paper proposed a self supervised method for frame interpolation between two given point clouds. Technically, the proposed method models the point movement using an explicit velocity field and transform the time-varying implicit field through modified level-set equation. Experiments on various datasets show its effectiveness.", "questions": "* Can the proposed method be used to process non-watertight surfaces, such as clothes, by replacing the SDF with UDF?\n* Should Eq. 4 consist of $\\phi(x,t)=x^1$, where $x^1$ is the endpoint of the trajectory.\n* Why not include Eq. 3 in the loss function?\n* Given two shapes, there are many methods to calculate the flow of each vertices/points. Once the flow calculated, the frames between them are easy to interpolate. What is the advantages over these methods?\n* Can the proposed method be used for triangle mesh directly?\n* If replace the marching cubes with dual countouring, would the reconstruction increase?", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* The writing is easy to read and follow.\n* The problem modeling is clear and the formula derivations are also correct.\n* The proposed method achieves better performance than baseline methods at different datasets.", "weaknesses": "* As shown in Fig. 4, the trajectory of each point is assumed as a line segment, making the shape different than the tow given shapes.\n* The datasets used in the experiment do not consists of complex diformation. Can the proposed method deal with these conditions?\n* There is no efficiency analysis about the proposed methods and baseline methods, which is important in practice.\n* There is no experiment about reconstruction accuracy under different network sizes.", "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": "ZMxUB4O7Nc", "reviewer": "Reviewer_RFX6", "summary": "This paper focuses on the task of learning continuous intermediate surface deformations between two given 3D point clouds. An end-to-end framework built upon fluid mechanics and physical constraints is designed to learn the time-varying implicit field via an explicit velocity field. Various comparison experiments and ablation studies are conducted to show the effectiveness and characteristics of the proposed learning approach.", "questions": "As for the divergence-free velocity field, is it reasonable to constrain that the total mass inside surface always stay the same, especially when we deform two objects of distinct types and/or categories?\n\nFigure 6 presents an interesting example of handling incomplete inputs. What would happen if the input model becomes \"broken,\" for example, if one paw is separated from the dog's leg? Is your method robust enough to handle such scenarios?\n\nI suggest adding visulizations of the learned velocity fields.\n\nSince the method is SDF-driven, it cannot handle surfaces with large boundaries, such as the clothes models in the DeepFashion3D dataset. Nevertheless, it would be beneficial to discuss this issue.\n\nCan your method handle inputs of different topologies? For example, a standing man with genus 0 and another person with hands on the waist, which is of genus 2.\n\nThe authors claim that the intermediate shapes are physically plausible; however, all evaluation metrics provided are geometric. Additionally, the velocity loss, which includes a smoothness term and a divergence-free term, is not directly related to physical principles. Therefore, this claim requires better justification using quantitative physical measures.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The theoretical formulations are well-motivated and technically sound, and the learned deformation results look smooth and reasonable.\n- The method is able to handle rigid and non-rigid deformations without intermediate shape supervision.", "weaknesses": "- One critical issue is the setting that a certain ratio of ground-truth point-to-point correspondences must be provided (although might be partially noisy). In fact, this is a very strong prior, especially when the points that are given GT correspondences are scattered around the whole shape surface. For example, during training 20,000 points are sampled from the surface, 5%-20% of them (1000-4000 points) are corresponded. This prior information makes the task itself much easier. From this perspective, I do not agree with the claim that the proposed learning framework is fully “unsupervised”.\n\n- The experimental results suggest that the input shapes have similar geometries or poses and the method cannot handle very large deformations.", "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": "OIrzQrYUoT", "reviewer": "Reviewer_NAfz", "summary": "Leveraging the Lagrangian approach in fluid mechanics, the authors introduce a novel method for implicit surface deformation. It models point trajectories during surface deformation using explicit velocity fields, enabling the recovery of physically plausible intermediate shapes through a modified level-set function. Unlike previous implicit methods, this approach requires only a sparse set of correspondences between the source and target point clouds in the unsupervised loss computation, eliminating the need for inefficient supervision from intermediate shapes. Experiments cover both intra- and inter-category shape deformations, as well as point clouds of different quality (sparse/incomplete).", "questions": "- How are the sparse correspondences selected for each deformation? Do they have to cover the entire shape, or is the method still effective if the correspondences are nested in certain regions of the surface?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- The paper is well-written, clear, and easy to follow.\n\n- The authors provide detailed derivations for formulating the Modified Level-Set Equation from the ordinary differential equation of the velocity field.  \n\n- The proposed method deforms source and target surfaces using a loss function that effectively avoids dependence on intermediate shapes, and requires only a sparse set of correspondences between the source and target point clouds.\n\n- The method demonstrates efficiency in handling pose deformation within the same category and non-rigid deformation across different categories, and is flexible to incomplete and sparse point cloud inputs. \n\n- Ablation studies are provided to validate robustness of the method to different correspondence qualities.\n\n- Multiple visualizations are provided for qualitative evaluations of the method.", "weaknesses": "- Correspondences between the source and target shape have to be provided, which are not always available. Establishing correspondences between cross-category shapes might be challenging. \n\n- It is more informative if the correspondences utilized in deforming each surface pair can be visualized as well in some figures. \n\n- The readability of Figure 2 and Figure 7 should be improved.", "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)