Implicit-ARAP: Efficient Handle-Guided Neural Field Deformation via Local Patch Meshing
Authors: Daniele Baieri
Filippo Maggioli
Emanuele Rodolà
Simone Melzi
Zorah Lähner
Venue: NeurIPS 2025 poster ()
Keywords: neural fields
sdf
deformations
arap
geometry
implicit
Primary area: applications
TL;DR: An efficient method for neural fields deformation based on local surface meshing.

Abstract: Neural fields have emerged as a powerful representation for 3D geometry, enabling compact and continuous modeling of complex shapes. Despite their expressive power, manipulating neural fields in a controlled and accurate manner -- particularly under spatial constraints -- remains an open challenge, as existing approaches struggle to balance surface quality, robustness, and efficiency. We address this by introducing a novel method for handle-guided neural field deformation, which leverages discrete local surface representations to optimize the As-Rigid-As-Possible deformation energy. To this end, we propose the local patch mesh representation, which discretizes level sets of a neural signed distance field by projecting and deforming flat mesh patches guided solely by the SDF and its gradient. We conduct a comprehensive evaluation showing that our method consistently outperforms baselines in deformation quality, robustness, and computational efficiency. We also present experiments that motivate our choice of discretization over marching cubes. By bridging classical geometry processing and neural representations through local patch meshing, our work enables scalable, high-quality deformation of neural fields and paves the way for extending other geometric tasks to neural domains.

Implicit-ARAP: Efficient Handle-Guided Neural Field Deformation via Local Patch Meshing
Daniele Baieri University of Milano-Bicocca daniele. baieri @unimib.
Filippo Maggioli
Pegaso University
filippo. maggioli wwwplliiiiiiiaaaaiee
Emanuele Rodolà
Simone Melzi University of -iino-- s simone melzi Sunimib it
Sapienza University of Rome rodolašdi. uniromal.it
Zorah Lähner
University of Bonn, Lamarr Institute laehner@uni boon. de
Abstract
Neural fields have emerged as powerful representation for 3D geometry, enabling compact and continuous modeling of complex shapes. Despite their expressive power, manipulating neural fields in a controlled and accurate manner particu- larly under spatial constraints remains an open challenge, existing approaches struggle to balance surface quality, robustness, and efficiency. We address this by introducing a novel method for handle-guided neural field deformation, which lever- ages discrete local surface representations to optimize the As Ss-Rigdd-As-Possibee deformation energy. To this end, we propose the local patch mesh representation, which discretizes level sets of neural signed distance field by projecting and deforming flat mesh patches guided solely by the SDF and gradient. We conduct comprehensive evaluation showing that our method consistently outperforms baselines deformation quality, robustness, and computational efficiency. We also present experiments that motivate choice of discretization over marching cubes. By bridging classical geometry processing and neural representations through local patch meshing, our work enables scalable, high-quality deformation of neural fields and paves the way for extending other geometric tasks to neural domains.
Introduction
Implicit inresentationnswhhee surface is defined not explicitly but, for example, the zero level set of a signed distance mannction-hvvee long been used in computer graphics. However, they have recently gained renewed attention, particularly due to advances in neural rendering techniques such as NeRF [36]. Neural fields provide compact representation for implicit representations in the weights of a neural network, and offer several advantages: they support flexible topology, avoid predefined discretization, and integrate naturally with gradient-based optimization methods. These properties make neural fields well-suited for reconstruction tasks. As this representation becomes more widespread, the demand for tools that enable direct analysis and manipulation of implicit surfaces continues to grow.
The traditional representation used in geometry processing applications is the polygonal mesh, in which the surface is explicitly modelled by collection of connected polygons. This representation
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Moving handles
Our pipeline
Main competitor
Static handles
Relative EL
0%
5%
10%
Impr.
Speedup
x33
Volume
x0.1170
Error
Arca
x0.1194
Input
Implicit-RRAp
NFGP S7
Edges
x0.5675
Faces
x0.6981
Figure 1: Given a neural surface and deformation handles, we employ our local patch mesh represen- tation to optimize a neural deformation field via the ARAP loss. Our method introduces significant improvements over previous state of the art while employing 0.75% of the time. We show the nor- malized change edge lengths (before/after deformation) as colormap on the deformed surfaces.
provides high level of interpretability and allows for fine-grained manipulation by artists. Addi- tionally, local surface properties can be easily computed because the neighborhood information is explicitly encoded. Due to its wide adoption and easy manipulation, number of editing methods have been developed for explicit representations: one of them is optimizing the (ARAP) energy to deform an object while satisfying user-defined handle constraints and at the same time preserving the surface geometry (in ferms of its edges) to a reasonable extent. This yields natural deformations and has been widely adapted in various applications 7 37 23
Evaluating energies such as ARAP directly on implicit surfaces is challenging: properties like edge lengths cannot be computed without applying a discretization scheme. and equivalent properties do not necessarily exist for isosurfaces. Alternating between both representations is certainly possible 35.. but it is expensive and inherits the sensitivity to discretization choices [57]. To counter this, we propose a new patch- based meshing for implicit representations which is 1) efficient to compute even at high resolution, 2) not sensitive to the specific choice of discretization, and 3) can be used on all isosurfaces to cover the complete geometric information of the neural field. Along with local patch meshing, our work proposes multiple other significant contributions:
Extending the ARAP estimation of the thin-shell energy to neural representations;
Providing an efficient method for handle-guides deformation of neural fields (see Figure D: Introducing a robust and efficient alternative for deforming high reghhresolution meshes.
2
Related work
Handle-Based Deformations Mesh-based shape editing methods have long history in geometry processing due to the wide acceptance and explicit representation of meshes (58) Among these, methods which use handles to indicate the preferred deformation are intuitive for human users to understand and generate. The p-risidd----posiblle (ARAP) energy [5]] has become popular due to its straight-forward interpretation and easy optimization while satisfying handle constraints. However, while not complex, its global nature still prevents processing of high-resolution shapes and sensitive to the mesh discretization. The second aspect was overcome in SR-ARAT (26) with a smoothed and rotation-enhanced ARAP version. While the rigid version is widely used in variety of applications 77 37 23), there exist similar formulations other energies related to shape deformation, for example conformal 442 (55) and elastic deformation energies [9]. Instead of using pre-defined physical energy, it is also possible to learn a set of handles from a collection of shapes 29] 4T]. However, these require suitable set of training data and are restricted to the space of deformations learned initially.
The idea of parametrizing shape deformations with neural networks has been previously explored in recent research: two contributions which are particularly close to our work are Neural Jacobian Fields 2 and Neural Shape Deformation Priors (53]. NFJ is mesh morphing (i.e., transfer of pose between two given shapes) model which can be trained to inject several different priors, including ARAP deformations. NSDP, on the other hand, is data-driven handle- based mesh deformation

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model. While it allows for very fast deformation of low/medium mameedium-resoliiee resolution meshes, it is limited to the deformation prior and the handle set observed in the training data, limitations overcome by Implicit-RRAP
While all these methods designed for polygonal meshes, there various types of shape represen- tations for which other types of deformation approaches are better suited. One such example which is especially relevant to our work is spatial deformation tasks, which apply to solid models [46] 477)
radial basis functions 5
or cages for character animation 24
Neural Fields Editing Energies like ARAP act directly on surface deformations, making them naturally suited to explicit representations such as triangle meshes. In contrast, editing implicit surfaces is more challenging due their global coupling local changes the surface can affect distant regions of the field. Even locating the surface space can be non-trivial without certain assumptions. The recent popularity of NeRF-like methods [36] has renewed interest in integrating deformations into implicit frameworks. Dynamic NeRFs have been implemented by introducing time parameter 45 or optimizing deformation fields 8 Other works modify the MLP weights
of neural fields to generate 17 or edit 3 shapes directly, though this lacks efficient formulations of explicit energies like ARAP and leads to slow optimization. Mehta al. [35] alternate between explicit and implicit representations to enable deformations, but require costly conversions at each step. Novello et al. 3 supports diverse deformations but lacks handle-based constraints. Neural fields with spatial features allow shape editing through feature-space interpolation (00 I. Similarly to our approach, Esturo et al. [8] optimize all isosurfaces simultaneously using divergence- free fields an approach also applicable beyond implicit functions [I5]. Recent work [14) shows cage-based deformations can be modeled with neural fields by learning mappings to barycentric coordinates. Text-driven editing of implicit or hybrid representations (e.g.. Gaussian splatting) has also been
explored 30 56 19 TO 40).
The method closest to ours, due to Yang et al. [57]. optimizes deformation losses sampled from implicit representations but is highly inefficient, requiring several hours. In contrast, we leverage full neural field information efficiently by using small, randomly placed discrete patches. Our use of mesh-based ARAP objectives supported by 11]. which highlights issues with higher-order neural field derivatives, and (28) which advocates finite differences over analytical gradients.
Other efforts to simplify local SDF representations include enhancing marching cubes with Bézier patches (54) and composing surfaces from implicit primitives [31]. However, unlike our approach, these methods do not yield local representations that are both explicit and structurally simple.
3 Method
Our method employs a local patch model (Sec. 3.I to sample the surface of a neural signed distance field, which then deformed according an ARAP-like energy (Sec. 3.2 with an efficient optimization scheme (Sec. 3.3
3.1 Local patch meshing
Given 3D surface S represented implicitly by neural signed distance field fo: R3 + R. we require a discrete local representation in order to compute the ARAP energy induced by a given deformation field d: 3 R We achieve this by generating local patches for multiple isosurfaces of fe. The procedure is separated into sampling and projection:
Sampling. We start by sampling k points V = [1,)4 from disk with radius p the 2D plane (including the origin) and computing the planar Delaunay triangulation. There are several possible distributions to sample the 2D disk: we describe and visualize some possible options in the supplementary material. However, our experiments showed that the particular choice has little influence our method, as we show in Figure 9
Projection. Once we obtain the 2D patch P = (V, F). we can discretize each isosurface by projecting the patch and fitting it to the local geometry. To that end, we approximate a uniform sampling of n points O {or)2 from the zero level set So using the rejeetion/projeccion algorithm proposed by Yang et al. [57] Additionally, sample points from other level sets by sampling
3

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the 3D interval |-; uniformly at random. We project patch for each sampled point as V2 for R3 (t (1y))) j=1' where t maps 2D points 3D as t: (a, v) + (x, y, 0), 0 the sampled origin points, and R2 the rotations aligning patch and surface normals at each Oi which we obtain
as the normalized gradient of Se (first column of Figure 2 Then, map each patch vertex onto level set fo: via the SDF closest surface point formula [57 12 which
maps any 3D point its nearest neighbor on level set / as
(p) p'=p- - || f()]]
(1)
In practice, when f is a neural SDF, this formula has to be applied recursively to yield accurate results (i.e. points whose signed distance is approximately 1). Note that the radius p of the tangent disk specifies measure locality which is highly dependent on the underlying surface S, Figure 2 shows examples of successful projection and radius overestimation.
By sampling origin points from multiple isosurfaces of the signed distance field fa. we enable representing any local region of the neural field with patch mesh. We describe how this structure can be applied to shape deformation in Section 3.3] Moreover, we compare meshing algorithm Marching Cubes [32 33 in terms of its benefits for handle-guie guided deformation Section 4.2
the classic SDF
Figure 2: Examples of patch projection in favourable case (top two rows) and with overestimated radius.
3.2 Deformation model
we represent our deformation We model via MLP gø RR + SO(3) R mapping 3D coordinates to roto- itto-rrasslaiion, with parameters d. We use Rş and to to refer to the rotation and the translation fields separately. The output layer predicts a 6D vector, where the first three components interpreted Euler angles and converted rotation matrix. We define the complete deformation as do (a0))((((e)))))) (R x) + (z)).
Following previous work in neural field deformation 8 continuous function of the embedding space d: R R
39
45
57
=
3.3 Optimization
100
We optimize our model similarly to Yang et al. [57] we summarize the process in the supple- mentary material. The goal is to optimize for tar- get handle positions while regularizing the com- puted deformation to have some desired proper- ties. Given a set of handles as with souree-target position pairs H {s,, | (where s; t, in the case of static handles), we fit via sim- ple MSE loss Lhandle = 1. Et=1l/d - 2. The key part of our loss function is the As- Rigid-As-Possinge
L
50
0
Lamp
0
16
L static
bandle
0
(ARAP) energy introduced by Sorkine and Alexa (51). While this formu- lation was previously adopted as regularizer for generative neural models [6 23), our work is the first to employ it in the setting of implicit geometry processing. We aim to ensure that our map de deforms the level sets of fo as-rigidly- as-possible. We evaluate this in a Monte-Carlo fashion, by sampling a set of points {2x 12 where m n points are sampled uniformly from the zero level set of fa and
100
moving bandle
50
0
10
100
1000
Step
Figure 3: Loss curves for single Implicit-ARAD training (Figure first column). We show separate handle losses for static and moving handles for visualization purposes.
points

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uniformly from the bounded volume |-1; A surface patch is computed at each of these points via the algorithm presented in Section 3.1 yielding a patch- representation {(Ve, F))== Then, the ARAP regularization is computed as
Larap
Σ 1= (ELR))EEE
Whi (Idd (1e) do (vz)) (Ro (v.) (ve wy))1l
Where Es are the mesh edges for patch mesh (Va, F) and Wi,j are the cotangent Laplacian edge weights. By optimizing the deformed edges (left hand side of the difference) to be as close as possible the rotational part (right hand side), we effectively mitigate the action of the translation field. The original ARAP formulation only used vertex-wise rotations, as handles were fit as pre-pro pre-proessinng step via Laplacian smoothing of the handle function over the surface. This operation is non-trivial for implicit surfaces, therefore we include a translation field, which allows any given handles set to be fit. The network 9o is optimized with ADAM 25 steps until convergence of the loss function L= T + X2 Jaaap- The entire procedure for computing the Lamp loss, which we described in this section, is repeated at each iteration, including all handle points as part of the surface sample. Nonetheless, we have observed convergence to be extremely quick, typically in the order of few hundreds of iterations, as showed in Figure our experiments, we usually trained our model for total of 1000 steps.
Inpot Moin [/V e 1- 1.000
Ours
NEGP
Only zero level Mean Vo o - 1.124
Only handles loss
Mean IV o 4 1.21
Mean [V/ o1 1.099
Meua fe dz 1 1.678
STD o 000074
STD IV "1,1=.0000001 0
STD [V "d 0.201
STD 1% =0.3
STD fo | 1.380
Figure 4: Ablation study for our method. We compare mean and standard deviation of the deformed SDF gradient norm for our method against: NFGP, our method with zero level set patches only, and our method without ARAP loss. The best results, terms of both visual quality and preservation of the SDF field, are obtained by our full pipeline, proving the benefits of its individual components.
4 Experiments
The training procedure described in Section 3.3 can be applied seamlessly for both neural field and mesh deformation, by simply changing the MLP architecture (further details may be found
in Section 6.2.1. We provide results for neural fields here, while we refer to Section 7.2 for an evaluation of high-resolution mesh deformation.
Data and baselines We employ two datasets in our evaluation: the first one (TFD) is obtained by designing set of hand-crafted deformation experiments using mesh data from Thingi Ok (59 and
the Stanford 3D scanning repository The second one is the DeFAUST dataset introduced in 34 We this data evaluate the performance of method against several baselines. In our comparison, we consider the original As-Rigid-Ass As-Possible deformation method proposed by Sorkine and Alexa [5I] (ARAP). We also include the spokes and rims variant introduced by Chao al. ( (Elastic), who propose the inclusion of an clasticity model in the ARAP optimization, as well as the smooth rotations alternative (SR-ARAP) (26). where the ARAP energy is modified to improve smoothness and volume preservation. The last baseline is the neural field deformation approach introduced by Yang et al. [57], who propose use the continuous formulation of the thin shell energy at random samplings of the implicit surface to optimize the deformation field, similarly to our method. For this baseline, we will indicate in the following whether the deformation applied on the SDF field via its inverse (NFGPspF) on the original input mesh (NFGPMesh). Although the Neural Shape Deformation Prior (NSDP) approach proposed by Tang et (53) achieves natural looking mesh deformations
with interactive performance, it is data-driven approach that requires the specification of consistent
5

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Input
sDGP
OUrSSDF
ARAP
Elastic
SR-ARAP
MLP OurssDF
Figure 5: Qualitative results of our method in comparison multiple baselines for neural field deformation. Discrete methods [51 9 26 are applied by extracting the marching cubes mesh at resolution I 512 and applying remeshing step, suggested in (77] We show true implicit deformations in green, while deformations of the zero-level set meshes are rendered in blue.
handles at training time; consequently, it is not suitable for more general framework, and cannot be applied to our set of experiments. We conduct a separate comparison on mesh deformation between our method and NSDP Section 7.3 Implicit-ARAP may be applied in multiple settings:
MLP Ours
regular MLP applied to input mesh, trained on its SDF representation
Iny OurssDr)
invertible MIP applied to input SDF field
OLrs
regular MLP applied the zero level mesh of input SDF
Hardware and hyperparameters All our experiments were run on a desktop computer with 12GB NVIDIA RTX4070Ti GPU. Achieving efficiency in this setting allows us to show that our method is suitable for consumer grade hardware (and therefore end-user applications). The hyperparameters used for the deformation tasks and the SDF fitting are listed in Section 7. We sample patch points via the sphere random uniform distribution (see Figure I0P.
Metrics To evaluate meshing accuracy with respect to some continuous implicit surface, we leverage the SDF fo and compare the respective level set value to fo(p) for surface points p the set of patches {P: 1P))
Eparch
máx max fo(p) - ls]. #=1 pEp,
In practice, the innermost max is estimated by evaluating the point-wise enror for several points sampled from the triangles of the patch mesh P; = (V,, F). We chose to aggregate via maximum rather than mean, because single outlier can create severe artifacts in the patch.
We use four metrics to quantitatively evaluate the computed deformations, considering both global and local aspects of the geometry. First, we consider the percent error in volume and area of the deformed geometry with respect to the original one. Given a surface and its deformed version S' these computed
Vs
Vss Vs
As
Asi As
Eyol
Furea
where Vx and A indicate volume and surface area of shape X. In order evaluate the distortion induced on the input geometry, we use two distinct local criteria: edge lengths and face angle errors. To obtain consistent values across all experiments, provide the former percentage of the

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longest edge in the source mesh. Specifically, the error is computed as
l| e1 [d(u) d(v)
(5)
E
m
(u,o)E
For the face angles, we compare the corresponding inner angles of source and deformed triangular faces:
FA
1
cos-1
u.v ||u|||a|||
cos -1
d(u) d(v) -(())
(6)
F JEF (W,))EE())
[Id(w)]](d()))
Both metrics require consistent connectivity between vertex sets. To implement them for the neural field pipeline, we extract the zero level set mesh from and forward-defone its vertices.
Table Average runtime, angle error (FA) and percent errors volume, area, and edge lengths (EL) for our neural field deformation experiments. Explicit ARAP baselines run using the zero level set marching cubes mesh as input.
TFD
DeFAUST
Ours
NFGP
ARAP
Elastic
SR-ARAP
Ours
NFGP
ARAP
Elastic
SR-ARAP
Volume 0.00%
6.66%
9.40%
9.01%
5.60%
0.00%
15.15%
17.36%
17.33%
12.09%
Area
0.85%
5.30%
0.29%
0.30%
3.97%
0.63%
20.02%
16.44%
16.41%
8.67%
EL 3.26%
3.34%
0.33%
0.38%
3,44%
2.41%
8.71%
5.72%
5.73%
8.45%
FA
4.541"
3.516"
0.358
0.379
4.615
3.399
7.577
1.677"
L.683
6.833"
Time
2m:48s
14h:26m
9m::1s
9m:444
10m:27s
5m:36s
14h:26m
8m: Is
8m:38s
7m:06s
4.1
Neural field deformation
The problem of deforming neural fields with handle guidance was first introduced by Yang al. [57], but the literature is missing follow-up proposals of significant improvements over their work. Berzins et al. B hint at neural shape editing as one of the applications of their method, but complete implementation is not available. Other baselines are obtained by applying explicit methods on the zero level set of neural field, although these methods do not preserve the neural field information.
Neural Field Pipeline
Architecture Following past literature 57
38 we define the deformed
field as h(x) = So (1 (xr)). Intuitively, we obtain the SDF value by mapping the query point in the deformed space x to its image x' (s.t. (x')) in the source space, via the inverse of the deformation. Therefore, we employ an invertible MLP architecture based on coordinate splitting, originally proposed by Cai et al. [8]. Since this architecture is derived from the NICE model [13. it retains its volume-preservation property, notoriously useful prior for deformations (see Table [T). The network is composed of 6 coordinate splitting layers, where the individual coordinate processing blocks are implemented as 3-layer MLPs with Softplus activation. a hidden dimensionality of 256, and 6-frequencies Fourier features encoding [52]. We provide additional details about the invertible MLP architecture in Section 6.2.
H
fe
Deformation training d
Deform
Results We show averaged quantitative results for method and baselines in Table For true implicit methods (Ours and NFGP), we compute volume and area directly from the deformed field h. Since the EL and FA metrics require consistent connectivity between source and deformed shape, we compute them by deforming the zero level set mesh of fo: the same holds for the discrete methods ARAP, Elastic and SR-ARAP. We used marching cubes resolution R = 512. From the data presented in the table, we observe that our method achieves optimal volume error due the volume-prec -rreeerving network architecture. Combining the results in Table [ with the visualizations Figure we can appreciate how our method reliably yields plausible results a fraction of the time required by the baselines, especially NFGP. Moreover, the results over DeFAUST show, we can easily double the training iterations to achieve even better results in more challenging cases. The visible handle error in some of our outputs stems from our choice of loss balancing weights,
fo o d
7

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which we set to achieve the best results on average: if a more accurate handle fit is needed, the loss weights can be balanced, as we show in Section 7.6 Lastly, Figure showss the benefits both computing local patches for all level sets and employing the ARAP regularization for our deformation network. We use the deformed SDF gradient norm as a measure of preservation the signed distance properties. We further discuss this aspect in Section 7.5
4.2 Local patch meshing
The final section of our experiments devoted evaluating our local patch meshing algorithm. We begin by highlighting that our method should not be considered as a drop-in replacement for marching cubes: even by sampling a very large number of patches, it is unlikely cover the entire surface, and a set of largely overlapping patches is not in general a useful representation for the surface. Instead, our method generates discretizations of local surface regions, and we are interested to a) verify how accurately these patches represent the underlying geometry and b) provide indications on how to select useful radius and density values for deformation tasks.
10-1
Patches
MC
MC
10-2
Epuch
10-3
10
Patches
100
200
300
400
500
R (voxel grid resolution)
R
32
64
128
256
512
Figure 6: Accuracy evaluation of local patching against marching cubes. We construct the patch meshes in this experiment to match the total vertex count and average edge length of the corresponding marching cubes triangulation. The per-patch vertex count fixed 30 for all resolutions.
Reconstruction Ability In Figure O we compare marching cubes meshes to local patch meshes constructed with similar vertex count and average edge length. The visualizations show the change in "coarseness" of our representation wrt the resolution, while the graph provides some insights to the accuracy of local patch meshing. The line plot shows the average approximation error Epatch = - EppP,, [o(p) - 4. with the shaded areas covering the entire region between Eparch and the maximum error Eatch (Equation (3)). For MC, these metrics can be computed by considering the entire mesh as a patch (i.e., n = 1 and Pi is the marching cubes mesh). Despite the marching cubes line hinting at lower deviation from the mean, even for coarse patches. This is because the patch vertices are mapped exactly onto the surface, while marching cubes places triangles based on how the surface crosses the sampled voxels; therefore, error only depends on the overall size of the patch relative to the "flat-ness" the approximated local surface region.
method achieves much lower average
100
Vollmne
Area
Error (%)
Edge Lengths
Error (%)
10
0
5
Volume
p
Edge Lengths
0.001
0.01
0.1
50
100
150
200
Patch radius
Patch density
Figure 7: Variation of deformation error metrics wrt patch radius (left, fixed density 30) and density (right, fixed radius 0.03), averaged over TFD dataset.

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Approximation Error and Deformation Quality The inset figure (top right) shows how patch radius and density impact the approximation error (average over TFD shapes). We observe negligible change when varying the patch density, where values above 50 do not result in significant accuracy improvements. This is also the case for deformation eπror metrics (Figure 7 right). Given that increasing density results in additional cost in deformation time and memory (inset figure, bottom right), a conservative choice ap- pears to be the best one. In our experiments, radius values around 0.01 would usually provide geomet- rically meaningful patches without significant arti- facts (we use absolute units for the radius since input shapes normalized in the unit cube). On the other hand, while smaller radius values reduce the approx- imation error, the patches tend collapse single points as their radius approaches zero. This results in less expressive representation of the local geome- try, which negatively affects the deformation results as showed in the left plot of Figure / Lastly, overly large patches also cause degradation in performance due the increase approximation error. In Sec- tion 778] we provide visualizations of Implicit-ARAP outputs as the patch radius and density change. The results are visually consistent with the quantitative evaluation presented here.
10-1
Exanch
10-2
10-3
50
Putch density
100
10
150
10-2
200
10
Patch radius
Triinig Liee (e)
Training VRAM
0
100
200
Patch density
Patches
MC 128
MC
MC
Density:
Radius: 0.03
Random
2
Time: Im:49s
Time: 44m:77
Time: 5m:18s 1.56CB
Time: Omn:45s
VRAM: 1.41BB
Figure 8: LPM and MC as underlying triangulation for deformation tasks. Our sampling is both memory efficient and expressive.
EL: 0.83%
EL: 0.88%
EL: 0.87%
0.95%
Area: 6 16%
Area: 6.5 6.57%
Area: 6.30%
Area: 63%
Volume: 6.98%
Volume:
7.22%
Volume:
Figure 9: Mesh deformation results for multiple patch distributions. The final results are qualitatively identical and the variations in metrics negligible in most cases. See Figure for the input shape.
Discretization In Figure g we present qualitative results of mesh deformation using both patches and marching cubes (MC) underlying discretizations for ARAP energy. We show three different variations for MC: zero level set only with resolution R = 256, zero level set only with resolution R 128, and meshing all level sets with R 128 to make the computed energy similar to that of our method. The last option leads to meshes with very high triangle count especially for higher SDF values, which increases the memory requirement and limits the resolution the optimization converges to rigid transformation, probably due to the coarse-ness of the level sets'
128. Consequently,

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representation. Using only the zero level set, the model seems unable to properly apply the ARAP prior to the deformation, for resolutions R 256 and R 128. This is likely due using a single discretization during the optimization process: without seeing multiple possibilities (e.g. ranaddom patches), the continuous deformation model may find local optima that "cheat" the ARAP loss without actually resulting in local rigidity. Overall, our patching approach appears more stable, reliable, and efficient. Moreover, the results presented in Figure Juggess that the variation in error metrics due to the choice in patch point sampling is negligible.
Conclusions
Discussion We presented novel way to apply as-rigid-as-a -aii--ss-ppssss as-possible deformations to neural fields which is highly efficient and more flexible and robust than previous work. To this end, we proposed to mesh patches from several isosurfaces those to regularize a deformation field encoded in neural network. This has important advantages because it detaches the computational complexity from the resolution and allows for regularization that includes properties of the embedding space, e.g., the The core idea can be applied seamlessly in the context of deforming high resolution meshes and neural fields; in the latter case, we employ an invertible deformation which allows to define the output neural field, at the cost of generality. The combination of these properties - directly inferring the new SDF and general deformation space is hard obtain due to the unpredictable possible changes in the SDF from an unconstrained deformation, but it would make for a challenging future work. In the context of mesh deformation, we believe that employing more efficient neural SDF representations provides an interesting direction for future investigation. Nevertheless, we believe work is valuable step in the direction of efficient and flexible editing of neural fields, and that our local discretization could be applied to solve more geometric problems in the implicit domain.
signed distance field and then compute the energy on
of our invertible model.
Limitations The general framework of our deformation method ("transporting" the input neural field along the inverse deformation) is adopted from previous contributions in the literature. As such, our method inherits the common limitation of lacking exact guarantees on the preservation of any properties of the input field (for instance, we discuss gradient norm preservation in Section 77.5). Additionally, the optimization of the ARAP loss is very dependent on the discretization: while our experiments showed that our patches are valid choice, would be beneficial to improve this representation with the goals of a) removing hyperparameters and b) increase surface coverage. Lastly, concerning mesh deformation (see Section 7.2 our method's constant time scaling is only beneficial for high resolution meshes: further improving the optimization time would allow our method to also target low/medium resolution meshes and improve its generality.
Acknowledgments
This work was partially supported by the Italian Ministry of Education, Universities and Research under the grant Dipartimenti di Eccellenza 2023-2027 of the Department of Informatics, Systems and Communication of the University of Milano-Bicoced and by the PRIN 2022 project GEOPRIDE Geometric primitive firting on 3D data for geometric analysis and 3D shapes. We acknowledge the support of NVIDIA Corporation with the RTX A5000 GPUs granted through the Academic Hardware Grant Program to the University of Milano Milano-Bicocca for the project Learned representations for implicit binary operations on real-world 2D-3D data. This work is further supported by the MUR FIS2 grant n. PIS-2023-00944 "NEXUS" (cup B53C25001030001) and by Sapienza University of Rome via the Seed of ERC grant "MINT.AI" (cup B83C25001040001)
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Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Answer: [Yes]
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21

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Contents
Introduction
2
Related work
3
Method
3.1
Local patch meshing
3.2
Deformation model
3.3
Optimization
Experiments
4.
Neural field deformation
4.2
Local patch meshing
5
Conclusions
10
6
Additional details
23
6.1
Local patch meshing
23
6.2
Model
23
6.2.1
Network architectures
6.3
Training procedure
24
6.4
Implementation
25
7
Additional experiments
25
7.1
Hyperparameters/
25
7.2
High resolution mesh deformation
................................
26
7.3
Comparison to NSDP
28
................................
7.4
Ground truth evaluation
................................
28
7.5
SDF preservation
29
7.6
Loss balancing
29
7.7
Matching ARAP resolution
30
7.8
Correlation deformation results and patch parameters
8
Data license
31
22

---

Additional details
6.1 Local patch meshing
We visualize four possible types of discretiza- tion in Figure [T]] such uniform sampling in polar coordinates, normalized normal sampling, linear (deterministic) sampling, and combining uniform coordinates with normal radius. Even though the patches are visibly different both in terms of point distribution and triangle appear- ance, the specific choice has little influence on our method, as we show in Figure g
Uniform 2D Normal
Lincar
Mixed
Figure 10: Different distributions for sampling the 2D disk (100 pts per patch).
Algorithm T presents the rejection/projection algorithm sample the zero level set of SDF introduced by Yang et al. [57]. In our implementation, several parts of the algorithm are parallelized for efficiency: for instance, the inner loop samples large number of points æ at once, retaining and projecting those with absolute distance < т. By performing enough sampling attempts in a single iteration, the algorithm can frequently terminate in single step (i.e. by retaining at least N points among those that were sampled).
Algorithm 1 SDF zero level set sampling.
1: procedure REIBCTPROJECTSS.MPPIIGGG. N, (max)
2: S 5t Ø
3: while |S| < N do
4:
Rejection step > Sample from bounded 3D domain Ensure close to surface
5: ~ ((--111))
6: while (x) > do
7:
x ~ ~(1-1111})
8:
end while
9:
Projection step Iterate closest surface point
for
i 1 mmx do
V (o(
11:
t x (x)
[z)))]
12:
end for
13:
x S (x)
14:
end while
15:
return 5
16:
end procedure
6.2
Model
6.2.1
Network architectures
Shape model To represent the input shape internally to our deformation algorithm, we adopt neural SDF model proposed in previous literature [50 52]. This model is suitable to our application due to its efficiency on consumer-grade hardware and robustness with respect to the input geometry. The SDF is represented via MLP network with 8 layers, hidden dimensionality of 256, residual connection the fourth layer, 6-frequencies Fourier features encoding, and Softplus activation. This network is optimized via eikonal training, originally proposed by Gropp et al. [21), which employs the following four losses:
Lygo
=
Ezes, (o(z))
(7)
Treikonal = EZER*
[|V](())]| 1|1
(8)
(1-
V fo(r) n(x)
Loormals = ExESo
(9)
V [[(
Lpenally ExERa exp (-alfo(x))
(10)
23

---

Defined on a discretization of the represented surface Sg. Intuitively, these respectively constrain the network to: 1) vanish on surface points (sampled from the input mesh triangles) 2) have unitary norm of gradient 3) have gradient aligned with surface normals (indicated by n(x) for surface point at), and 4) have minimal zero level set, to avoid artifacts due to under-determination. We list the values for loss weights and a which we employed in our implementation in T The Adam optimizer runs for a total of 10000 steps and uses a starting learning rate of 10 and scheduler which halves at steps 1000, 2000, and 5000.
Deformation model. For the invertible net- work employed in the neural field deforma- tion pipeline, we use 6 coordinate splitting lay- (2,
Rot (taxte)
ers, where the individual coordinate process- ing blocks are implemented as 3-layer MLPs with Softplus activation, hidden dimensional- ity of 256, and 6-frequencies Fourier features encoding [52]. Each of these layers splits and combines coordinates according to the layer index i, by selecting the "focus" coordinate w = Pi mod 3. where p = (I, v)z is the in- put vector. Each layer predicts a translation of the "focus" coordinate and 2D roto-translation of the two others, which we progressively ag- gregate to obtain (z) and (z). The layer architecture for this model is visualized in Fig- ure 11 Contrarily to the Lipschizz-cotinnouus MI.P used in NFGP (57) this architecture al- lows for an analytic expression of its inverse and thus more efficient, does not require fixed point iterations for inversion.
1
L
L
Ooaaton
(u, -)
(a,
8
Rus
Cuedeee
D
(f
Figure 11: Diagram for the forward (top) and in- verse (bottom) passes of the invertible MLP archi- tecture we employ in our neural field deformation pipeline.
For the mesh deformation pipeline, we use standard MLP composed of 8 linear layers with hidden dimensionality of 256 and Softplus activation. We apply Fourier features encod- ing with 6 frequencies at the input layer and residual connection at the 4th layer. For both networks, we adopt the neural deformation ini- tialization scheme of 43 [8. which allows the initial state of the model to predict the identity transformation without causing symmetry break- ing failures or numerical instability interacting with the Adam optimizer (which are common when the weight matrices are entirely initalized to zero).
Ours (MIP))
Ours (Inv)
R/L ARAP
NEGP
Figure 12: Results for method and baselines for a non-bijective deformation, which invertible net- works (Inv NFGP) cannot represent.
6.3 Training procedure
Algorithm 2 shows all the high-level operations performed in a single step of Implicit-ARAP training. First, a single flat patch is constructed via the sampling procedure described in Section 3.1 Then, we construct set of origin points for surface patches: we include 1) the handle sources, 2) a quasi-uniform sampling of the zero level set (see Algorithm [T and 3) uniform sampling of the bounded embedding space. The for loop lines 7-10 aligns the flat patch at each origin and deforms it according to the local geometry. This operation is performed in parallel for all patches to avoid bottleneck. Finally, the patch mesh is constructed as the union of all surface patches, and its vertices deformed via dg. The computation of the ARAP and handle losses is described in Section 3.3
24

---

Algorithm 2 Implicit-RRAP training loop.
procedure TRAIN(T, Å1, A2 n, m, p. k. fo. de, ((s,, (=)
2:
for f - to I do
V, F. <- DISKSAMPLE(p, k)
P Section 3.1, "Sampling"
4:
O {s:]e
>> Handle sources as origins
5:
O U REJECTIONSAMPLE/I
h)
6:
O O U UNIFORMSAMPLE((
m))
7:
for j
n do
> Section 3.1, "Projection"
8:
V, + ALIGN(V, 0j, S0) + PROJECT(V
9:
10:
end for
11:
P + FV
12:
P' + P)),
P. Section 3.2
13:
Larpp + ARAPLOSS(P, P')
>> Section 3.3
14:
Laandle + T. Σx ((da(#;) 4:l²
D. Section 3.3
L Lnartee + X2 Lraap
16:
OPTIMIZE(L; ()
> Compute gradients, ADAM step
17:
end for end procedure
18:
6.4 Implementation
We implemented our algorithm in Python, relying on PyTorch [44] for neural network primitives, linear algebra and automatic differentiation. In addition, we used Polyscope [49] for visualization, extending its GUI with functionalities for easy point picking, which we used design deformation experiments. While our viewer renders 3D triangle mesh extracted with marching cubes for the sake of efficiency, the points selected on the shape by the user are mapped exactly onto the implicit surface via iterations of the SDF closest point equation (Equation (.) allowing to select an arbitrary set of accurate handles. For given set of points, the user can then specify an affine transformation and save both the resulting handle transforms and the original positions. Our codebase is available at this ur!
7 Additional experiments
7.1 Hyperparameters
Where unspecified, all of deformation experiments using the hyperparameters showed in T To train the neural SDFs, we used the architecture described Section 6.2.1 with the hyperparameters listed in Table
Table 2: Hyperparameter values for the deformation procedures. k and p refer to patch density and radius, respectively, while A, are the loss balancing weights, n and m define the number sampled patches per training step (see Algorithm 2
Mesh
Field
Data
TFD
DeFAUST
TFD
DeFAUST
Steps
1000
2000
1000
2000
LR
10-3
10-3
10-3
10-3
A
103
103
103
103
A2
101
104
10a
104
k
30
30
30
30
P
0.01
0.03
0.01
2000
2000
2000
2000
1000
1000
1000
1000
25

---

Table Hyperparameter values for the SDF fitting procedure.
Aero
Aeikonal
Anommal
penalty
100
3000
100
50
3000
Input
ARAP
Elastic
SR-ARAP
NFGP Mesh
MLP Ours Mesh
Figure 13: Qualitative results of our method in comparison to multiple baselines for mesh deformation, on TFD examples.
7.2 High resolution mesh deformation
Our pipeline be applied for classic mesh deformation, where the trained network is used to deform the vertices of an input mesh (internally converted to a neural SDF for optimization).
Architecture For the mesh deformation pipeline, we do not require invertibility as we are simply interested in the forward deformation of set of points in the zero level set of fe. Therefore, we use standard MLP composed of linear layers with a hidden dimensionality of 256 and Softplus activation. We apply Fourier features encoding with 6 fre- quencies at the input layer and residual connection at the 4th layer. As mentioned in the main manuscript, the network is again initialized pre- dict the identity transformation [43] [8. This architecture 1.53x more efficient to query than the invertible one, resulting in faster runtime for the deformation procedure (excluding the internal SDF fitting). Figure 12 is also related to this section, it shows how the standard MLP we in this task may represent deformations such that bijections between source and deformed shape are not possible (e.g. changes of topology).
Mesh Pipeline
M = (V,F)
SDF rossustractinn
H
Sp
Deoemmmmmiiigg
d
Results Well-established explicit methods [51
are very efficient for most use-cases, but they do not scale to meshes with millions of vertices due to super-linear time and memory costs. By representing the input geometry implicitly in the weights of neural network, and only computing Laplacian edge weights for small triangle patches, our method achieves runtime and VRAM usage independent of the input size.
Detoom
M' (do(
26

---

As highlighted in Table Implicit-RRAP's runtime is comparable to that of the explicit baselines. However, the time scaling of our method constant. therefore it is expected remain efficient even resolutions higher than those employed in our evaluation. Moreover, most (~72%) of our runtime spent fitting neural SDF to the input mesh: employing a faster procedure for this step could greatly improve our performance. The remainder of Table 4 summarizes the performance of our method: Implicit-ARAP yields a clear improvement over SR-ARAP and NFGP. However, ARAP and Elastic achieve much better metrics than our method for area, EL and FA. Since these two methods provide the best solution (in least-squares sense) for local rigidity of the given input mesh under the handle constraints, this is to be expected. However, including useful priors such as smoothness (SR-ARAP) immediately aligns the scale error values to ours and NFGP.
Combining these results with the qualitative evaluation we present in Figures 13 and 14 provides clear picture of the accuracy, robustness and efficiency of our method, which consistently yields accurate results with minimal artifacts. We point out that the results of ARAP baselines for the cubes experiment (Figure 13) are correct, as the 2nd and 4th cubes are unconstrained and may be mapped arbitrarily. Implicit-ARAP and NFGP, on the other hand, exploit the spectral bias of neural networks to propagate the handle maps smoothly over the whole domain. However, NFGP deforms the individual cubes into trapezoids more evidently than our method. The last advantage in using an implicit method like ours lies in its independence from the quality the input shape's connectivity: for example, the CGAL implementation of explicit baselines failed to run using the buddha mesh (see Figure
Input
ARAP
Elastic
SR-ARAP
NFGP Mesh
MLP OurS
Figure 14: Qualitative results of our method in comparison to multiple baselines for mesh deformation, on DeFAUST examples.
27

---

Table 4: Average runtime, angle error (FA) and percent errors volume, area, and edge lengths (EL) for our high-resolution mesh deformation experiments. The time for our method and NFGP includes 4m:44s required for the initial neural SDF fit, meaning our deformation phase runs in Im:49s on average for 1000 steps.
TFD
DeFAUST
Ours
NFGP
ARAP
Elastic
SR-ARAP
Ours
NFGP
ARAP
Elastic
SR-ARAP
Volume
4.27%
8.41%
11.82%
11.42%
8.24%
0.99%
15.02%
12.41%
12.25%
11.69%
Area
2.83%
7.04%
0.23%
0.25%
3,46%
0.82%
20.25%
0.45%
0.45%
5,36%
EL
0.76%
0.87%
0.12%
0.12%
0.89%
1.06%
8.71%
0.37%
0.38%
3.39%
FA
3.448
4.178"
0.486*
0,495
4.743
1.517
8.859
0.517
0.523
5.628
Time
6m:33s
14h:31m
8m:06s
8m:00s
9m:07s
8m:22s
14h:31m
7m:15s
8m:27s
7m: Is
7.3
Comparison to NSDP
In this section, we compare Implicit-ARAP to NSDP 53 recent contribution in neural methods for 3D mesh deformation. We highlight the main differences between the two methods in Table Differently from Implicit-RRAP, NSDP is data-driven, therefore to ensure fairness we carry out this evaluation on the original test set employed by the authors, the Deforming DeforminggTingss41 Things4D dataset [27]. We observe that Implicit-ARAP, leveraging the local rigidity
The results are available in Table 5 prior, achieves a clear advantage for deformation quality metrics. However, on the low/medium resolution data available in DT4D, NSDP can compute deformations faster than Implicit-ARAP, by pre- vre-training the model on large amounts data. This comes at the cost of generality, as NSDP requires that the handles employed inference time be "semantically" the same as those used during training (e.g., limb extremities of humanoid character).
Table 5: Comparison of Implicit-ARAP and NSDP
53
performance on DT4D 27
Volume
Area
EL
FA
Time
VRAM
Ours
0.58%
2.04%
2.03%
4.728
6m:33s
1.3GB
NSDP
7.80%
5.76%
3.59%
6.603
0m:18s
2.IGB
Table 6: Highlights of differences between our method and Neural Shape Deformation Priors 53
Implicit-ARAP
NSDP
Training Handle set
Trained for each input
Pretrained on large dataset
Any
Defined at training time
Evaluation data resolution
Medium/high (50k-500k) Local rigidity (ARAP)
Medium/low (10k-50k)
Deformation prior
Linear blend skinning (data prior) No
Inference-time optimization
Yes
7.4 Ground truth evaluation
Throughout our evaluation, we ran deformation experiments which were obtained from 3D shape pairs (i.e., shape pairs from FAUST [4, animations from DT4D (27). In these cases, one may leverage the "target" shape information to evaluate how closely the algorithm approximates the deformation prior showed in the data, by computing the difference between the deformed shape and the target shape [53]. This provides an additional criterion of evaluation for deformation algorithms, which while not an "objective" metric of correctness, can provide additional insight into the behavior of particular method. The results are available in Table
Table 7: Evaluation of Implicit-ARAPP ability to approximate target shapes, when available for our deformation experiments. For reference, NSDP averaged 7.79 on DT4D.
DeFAUST
DT4D
3.54 10 3.44 10
28

---

7.5 SDF preservation
We are interested in evaluating to what degree Implicit-ARAP deformations preserve the signed distance properties of the imput field. Like past contributions, our method lacks theoretical guarantee of exact preservation, and obtaining such a guarantee an open problem. In Figure we provide an ablation experiment showing that our formulation is beneficial unitary gradient norm preservation. We complete that result with Table which presents the average of the same data over all our experiments.
To give practical measure of how well the SDF properties are preserved, we provide sphere- traced [22] renders of SDFs deformed with Implicit-ARAP in Figure [5] We used the sphere tracing implementation from Polyscope 44), We note that, while ad-hoc rendering methods for deformed fields such as [8] could be used, the scope of this work allows us to be satisfied with these basic results. The results show that the deformed fields are rendered without significant artifacts.
Table 8:
SDF gradient norm preservation, averaged over all our experiments. Only true implicit methods are evaluated. The mean and STD of the deformed field gradient norm evaluated over dense uniform sampling (IM points) of 1-1; 1j.. Recall that for exact SDFs
| =
Mcan fa o d
STD V de
Ours
1.016
0.146
NFGP
1.029
0.290
Input
0.999
0.077
Figure 15: Sphere- renders of SDFs deformed with Implicit-RRAP. The deformation inputs may be found in Figures] and 13
Input
Ours
NEGP
Only zero level
Only handles loss
Figure 16: Alternative visualization of the ablation study presented in Figure
7.6 Loss balancing
As we mentioned the main manuscript, our neural representation setting we cannot ensure 100% accurate handle fit, we have to optimize the input handles via gradient descent, balancing their contribution with that of the ARAP loss. result, especially difficult cases with substantial deformations, the optimization may favor better preservation of local rigidity and sacrifice some
29

---

measure of handle accuracy. In Figure 17 we show that it is possible to re-balance the loss term and achieve accurate handle the cost less natural-looking deformation.
100
500
1000
X2
Figure 17: Decreasing the weight of the ARAP loss (A2) allows to balance handle satisfaction and deformation quality, Å1 1000 as for all other experiments.
7.7 Matching ARAP resolution
Furthermore, explore the connection between method and the original ARAP through the results in Figure T8 ARAP results are generally sensitive to the input mesh resolution: different discretizations of the same shape will result in outputs which, while technically correct, exhibit different properties. For low resolution meshes, local rigidity will be optimized between large triangles, often resulting in behaviours typical of skeleton-based linear blend skinning deformations. On the other hand, high resolution meshes will exhibit local rigidity in more localized regions. This apparent from the top row of Figure 18
To determine if Implicit-ARAP ARAP would exhibit the same phenomenon, we ran our method varying radius and density so that the average edge length of training patches would approximate that of the ARAP input meshes. Our results show that this is not the case, as the variations in our outputs are insubstantial.
Res. 2.8%
16%
30%
44%
58%
72%
86%
100%
ARAP
[V] 10
IV 6 10*
[V] 1.1 10a
[V] 1.6 10 7mx42s
[V 2.1 102 10m:28s
V
2.6 10"
[V] 3.1 10
[V| 3.6 10
4m:574
16m:20s
Ours
1: (= 0m:31s
À: = = 3m:37s
114 == 5m:144
A: == 6m:15s
190
= 219 = 9tm:59
(= 2mn:02s
== 7mm:22
8m::40s
Figure 18: Comparison of our method ARAP with matching resolutions (showed as a percentage of full resolution). The training patches are constructed with radius 0.03 (avoiding degenerate patches) and manually selecting the density & which best approximates the average edge length of the corresponding ARAP input mesh (which we obtained by decimating the sculpture mesh at multiple resolutions). We also show the total runtime (without SDF fit in our case).
7.8 Correlation of deformation results and patch parameters
Lastly, Figure 19 completes the discussion about correlation of patch parameters and deformation performance Section 4.2) we anticipated from the quantitative evaluation, we observe that small patches (radius below 0.01) result in artifacts due to the insufficient influence of the ARAP energy in
30

---

the optimized deformation. Viceversa, when the patches are too large, artifacts arise from interaction of geodesically distant regions (e.g., the front left paw with the body, or the neck with the spine). Finally, coherently with the results in Figure 7 variations in patch density do not have significant impact on visual quality.
Patch radius
10
30
-00
150
210
Patch density
Figure 19: Qualitative results for Implicit-ARAP, varying patch density and radius.
8
Data license
For our experiments, we gathered a small collections of 3D shapes from Thingi 10k [59] and the Stanford 3D scanning repository. Then, we designed various deformation experiments for each of these shapes. Finally, we included a set of deformation experiments automatically generated from the we list the license name of each of these assets.
FAUST 4 dataset. Table
Asset
Reference
License
DeFAUST shapes
Figure 14
FAUST License
Armadillo (Stanford)
Figures
12
1
and
15
CCBY-NC 4.U
Dragon (Stanford)
Figures
1 and
CC BY-N 4.0
Buddha (Stanford)
Figures
5
and 15
CC BY-NC 4.0
Piranha Plant (Thingi10k)
Figures
and
CC BY-NC 4.0
Hand (Thingi 10k)
Figure
CCBY-NC-SA 4.0
Sculpture (Thingil0k)
Figures 6
and
18
CC 4.0
Cat (Thingi 10k)
Figure
CC 4.0
Octopus (Thingil0k)
Figures and
17
CCBY 4.0
Table 9: References in the paper and license for all 3D meshes we employed in our experiments.
31


Reviews:
[{"review_id": "ACByxd0MEt", "reviewer": "Reviewer_GZaa", "summary": "The paper proposes a local patch based meshing technique for the purpose of neural field deformation. It is positioned in contrast to Geometry Processing for Neural Fields (GPNF) paper in the sense that it is solving a more specific problem of handle guided neural deformation -- which is does in a fraction of time (and with higher quality) compared to GPNF. The local patch based meshing avoids the need for repeated marching cubes steps that make optimization very costly. Moreover the local mesh structure naturally provides the topology information to apply ARAP which is not possible in a pure neural field setting. Experiments are conducted large scale datasets such as Thingi10K and DeFAUST which corroborate the paper claims. Ablations are provided for various design choices made in the method which help in understanding contributions of individual components.", "questions": "1. Given that the paper is solving a specific part of the larger geometry processing problems handled by GPNF, I am curious to know what prevents the local patch based meshing to handle tasks other than ARAP? Perhaps a detailed discussion on this would be helpful to the reader. \n\n2. Can you throw some clarity on why GPNF takes 14hrs?", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: \n\n1. For the task of ARAP, the approach taken by the paper makes total sense as local patching provides a natural means to apply ARAP loss. I am also hopeful that this approach can be explored in future works for other geometry processing/ graphics tasks like UV mapping, etc. \n\n2. The performance gains achieved viz time saved and improved accuracy (especially in volume preservation) are noteworthy. I find the ablations quite useful. \n\nWeakness:\n\n1. The difference in method is not very clear with respect to GPNF. I find several similarities -- GPNF too seems to not require MC for its optimization. It looks like the use of an invertible neural network model for approximating the deformation field is also present in GPNF. So what are key differentiations other than the fact that a local mesh structure allows for ARAP like losses to be applied? \n\n2. I don't get why GFNP's volume loss is also not zero given that it too uses an invertible network which in theory should also be volume preserving as is the case in this paper?\n\n3. I am also not very clear about the reason for ARAP (mesh based) to have such artefacts as shown in Figure 4. Why is the paper theoretically superior in approximating deformations? Is this supposed to do with the discretisation of ARAP? \n\n4. I believe the paper would benefit from giving a more gentle introduction to ARAP, and the baselines considered while highlighting the key differences.", "was_revised": true, "final_justification": "The authors have provided sufficient clarity to my questions. Moreover the comments made by other reviewers have also been answered satisfactorily. Overall I find this paper a useful contribution and hence I vote an accept (score 5).", "rebuttal": "We thank the reviewer for their positive and constructive feedback about our work. We answer each point individually:\n\n**Application to other geometry processing tasks**: in this paper, we decided to target shape deformation specifically, to address the lack of a method solving this relevant task in a feasible time scale. There is no obstacle in applying our local patch meshing method to the other simpler tasks addressed in GPNF (smoothing and sharpening); however, the solutions proposed by the authors for these other applications are already quite efficient (~10 minutes on a TitanX GPU). Additionally, our general “framework” (constructing local patch meshes > neural deformation > optimizing the deformation by some property of the local geometry) could adapt naturally to several other tasks (e.g. UV mapping).\n\n**Differences to GPNF + GPNF inefficiency**: we agree that in the paper, we did not provide extensive details about GPNF as a deformation method. We provide them here, and, given the opportunity, we will improve this part in the camera-ready revision. The overall framework of GPNF is similar to ours, i.e., a neural deformation is applied to a neural field and optimized while leaving the shape’s parameters untouched. However, the authors of GPNF chose to formulate the shape deformation problem in a way that is coherent with the continuous structure of neural fields, directly optimizing elastic energy (stretching and bending terms) at different sampling of the zero level set throughout the optimization. These energy terms depend on 2nd degree derivatives of the deformed neural field, which are expensive to compute and backpropagate, and result in a rather complex optimization problem (resulting in 14hrs on average, close to the ~10hrs reported by the authors in the GPNF paper). While this formulation is elegant and well-principled, we employ the ARAP formulation (which is a discretization of the elastic energy) by constructing local discretizations of the underlying surface, and achieve continuous propagation over the surface by sampling different patches throughout the optimization. This is not only more efficient to compute and backpropagate, but the optimization converges in significantly fewer iterations, coherently with ARAP for meshes.\n\n**GPFN non-zero volume error**: the neural architecture employed in GPNF is invertible, but it is not volume preserving. The only guarantee coming with invertibility of the deformation is the topological consistency between the source shape and the deformed shape. To achieve volume preservation by construction, more restrictive conditions need to be met: in the supplementary material (A.2.1, Deformation model), we provide the details of the neural architecture we employ, which meets these additional criteria.\n\n**ARAP artefacts**: one of the main advantages of our method (and neural fields-based methods in general) is not being bound to a particular discretization of the input geometry. On the other hand, ARAP and its variants are rather sensitive to triangulation and exhibit these types of artefacts, especially with marching cubes triangulations (which was also previously noted by the authors of GPNF). Furthermore, these mesh-based methods will provide visibly different results for different resolutions of the same geometry, as we show in Figure 17 in the supplementary materials.\n\n**Baseline descriptions**: we agree that the paper would benefit from a paragraph describing the employed baselines, including GPNF. We will include this in the camera-ready revision."}, {"review_id": "iKVNNDc15v", "reviewer": "Reviewer_YEpM", "summary": "This paper address the existing challenge of deforming shapes represented by neural fields. It leverages local mesh representation to be deformed with guidance by the SDF and its gradient. Experimental comparison with baselines demonstrates the superiority in terms of deformation quality, robustness, and computational efficiency.", "questions": "- is it possible to approximate the deformed SDF with a new SDF, rather than composing the R and f with two separated models?", "limitations": "There is paragraph discussing limitations in the conclusion.", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n\n- The technique is well developed. It leverages the ARAP in an elegant way to solve the deformation of SDF. The deformed shape can be represented as a invertible MLP of the deformation along with the original sign function.\n- Experimental results study the robustness of the technique to the samples local patch. \n- The comparison against baselines shows superiority.\n\nWeaknesses\n- The Fig.5 is challenging to look at, especially for the layers, although it delivers the information that the proposed method is better.", "was_revised": true, "final_justification": "My original concerns are minor. The authors provided responses on those points. I will maintain my positive rating.", "rebuttal": "We thank the reviewer for their positive comments on our work and are glad to see it was well received. We address issues raised by the reviewer individually:\n\n**Figure 5 presentation**: we agree with the reviewer that the presentation of the figure is not optimal. Unfortunately, it is somewhat complex to show the level sets of a neural field in a compact way; one possible alternative is to show a single slice of the non-zero level sets as a background render, overlapped with the 3D shape of the zero level set. Since this alternative limits visibility of the zero level set, we preferred the visualization currently used in the paper. If the reviewer deems it necessary, we can update the visualization for the camera-ready revision. \n\n**New SDF after deformation**: this is an interesting question that deserves further exploration. In general, our method does not provide a new SDF “out of the box”: a trivial way to obtain one would be to deform the zero level set and optimize a new SDF from scratch. However, this could easily be done using any other deformation method, meaning it is not a particular advantage of our pipeline. To our knowledge, there are no deformation methods that edit the neural SDF parameters to satisfy some deformation constraints while maintaining SDF properties. We thank the reviewer for highlighting this point, which could be an interesting direction for future research."}, {"review_id": "aYFDYnH1rE", "reviewer": "Reviewer_A7SK", "summary": "This paper proposes a local patch meshing strategy as a bridge between neural implicit fields and mesh-based shape manipulation. Instead of relying on explicit iso-surface extraction (e.g., via Marching Cubes), the method projects and deforms predefined patch meshes using signed distance function (SDF) values and gradients. By randomly sampling patch meshes from various level sets of a neural SDF, the approach enables the use of mesh-based deformation priors such as ARAP loss for shape manipulation.\nThey show reliable and efficient mesh deformations, offering better computational efficiency compared to full-surface extraction and remeshing pipelines. In addition, the paper includes extensive ablation studies on hyper-parameters such as patch size, density, sampling strategies.", "questions": "The current mesh deformation evaluation primarily focuses on geometric preservation metrics such as edge length, volume, and face angle. However, it might be beneficial to include vertex-wise error metrics between the deformed and target mesh, especially if temporal mesh sequences are available. In such cases, a subset of vertex displacements could serve as handle movements to be used as input constraints for the deformation method.\nThe paper could be further strengthened by including comparisons against two more recent methods,  Neural Jacobian Fields (NJF) and Neural Shape Deformation Priors (NSDP), which addresses similar challenges in non-rigid deformation.", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\nThe work is well-motivated and presents an interesting idea by introducing ARAP-based deformation constraints into neural field representations, without requiring explicit mesh extraction. The proposed local patch meshing strategy provides a flexible and generalizable way to represent local surface geometry across arbitrary level sets of the SDF. It also demonstrates robustness to variations in patch structure, patch density, and input mesh resolution.\nFigures 13 and 14 show that the method produces more plausible and stable mesh deformations on the TDF and DFAUST datasets compared to existing baselines, highlighting the practical advantages of the approach in real-world scenarios.\n\nWeaknesses:\n1)\tFrom the quantitative results in Tables 1 and 4, the method is inferior to some baselines regarding Area, EL, and FA.\n2)\tMissing comparison against two recent works:\nneural network optimization-based method -- NJF\nlearning-based shape manipulation method -- NSDP\nAigerman, et al. Neural Jacobian Fields: Learning Intrinsic Mappings of Arbitrary Meshes. SIGGRPAH 2022\nTang, et al. Neural Shape Deformation Priors. NeurIPS 2022", "was_revised": true, "final_justification": "Thanks for rebuttal and response. Most of my concerns are addressed.  I keep my score of Accept 5.", "rebuttal": "We thank the reviewer for their positive and constructive comments about our work. We individually address each of the points raised in the review:\n\n**Missing baselines**: we thank the reviewer for pointing out NSDP and NJF as additional possible baselines for our evaluation. We will include both in our related work. Regarding the comparison to these two, we remark that:\n1. NSDP’s architecture is bound to the handles used for training. In particular, the pretrained model released by the authors was trained on human/animal data with a fixed set of handles at the tip of each limb and on top of the head. This is a limitation of NSDP, which prevents us from evaluating it even on TFD and DeFAUST data, since the deformations in our datasets have variable amounts (and semantics) of handles. Therefore, the best option for a fair comparison is to evaluate Implicit-ARAP on NSDP’s test data separately from the other baselines.\n2. NFJ does not support handle-guided deformation as a task, and we think it would not be possible to do an even comparison: as a morphing model, its “interface” requires two shapes A and B, to output the morphing of A into B. While it is true that the NFJ model can be trained to represent an ARAP deformation space, at inference time, it requires the pre-deformed shape to yield its output, and there is no way to condition the model’s output on the handles alone. For our evaluation, we require methods which take a shape and a set of handles and output the deformed shape, and including NFJ would result in an inconsistent comparison.\n\nDue to these reasons, we did not consider a comparison in the setting proposed in our evaluation. We will include this discussion in the paper. The table below shows an evaluation of Implicit-ARAP and NSDP: the two methods are evaluated on DeformingThings4D, the dataset used by NSDP for training, using the test split provided in the NSDP codebase. We combined unseen identities and unseen motions to construct the test set. The 3D shapes contained in this dataset range from \\~10k to \\~50k vertices. We will include these results, as well as some visualizations, in the camera-ready revision.\n\n|    | Volume  | Area | EL  | FA   | Time | VRAM |\n|----|---------|------|-----|------|------|------|\n|**Ours**|**0.58%**    |**2.04%** |**2.03%**|**4.728°**|6m:33s|**1.3GB** |\n|NSDP|7.80%    |5.76% |3.59%|6.603°|**.18s**  |2.1GB |\n\nFor completeness, we also provide the following table, highlighting differences between our method and NSDP.\n\n|    | Ours  | NSDP |\n|----|---------|-----|\n|Handles | Any       | Defined at training time     |\n|Evaluation data resolution| Medium/high (50k-500k) | Medium/low (10k-50k) |\n|Deformation prior | Local rigidity | Data-driven (LBS animations) | \n|Training | Trained for each input | Pretrained on large dataset |\n|Inference-time optimization| Yes | No |\n\n**Performance gap in quantitative evaluation**: it is true that in some cases our method performs worse than mesh-based alternatives for the three metrics listed by the reviewer. This is motivated by the ARAP baselines using the same set of edges for optimization and metrics evaluation. This is not the case for our method and NFGP, which have no notion of the input discretization. In the neural field setting, our method still achieves results orders of magnitude better than the state of the art in the neural field domain (NFGP).\n\n**Vertex-wise error metrics**: we agree with the reviewer that some form of ground truth would be extremely helpful in evaluating deformation methods. However, even if temporal mesh sequences were available, using the vertex positions as “objective” would be incorrect, as their variation depends on the particular priors of the employed animation (such as linear blend skinning vs elastic simulations). While it is interesting to evaluate how closely a deformation method approximates (e.g.) LBS, it is not a metric of its performance. Furthermore, the data we employed does not have temporal mesh sequences, as the deformation examples in DeFAUST are obtained from FAUST, which only consists of a few poses of the subjects without any temporal or sequential relation."}, {"review_id": "JAam2uMp7a", "reviewer": "Reviewer_BfYx", "summary": "This paper presents a technique for handle-driven deformation of shapes represented as implicit surfaces. To enable plausible deformations, the authors propose a method for evaluating the widely used As-Rigid-As-Possible (ARAP) energy directly on implicit surfaces. Since ARAP traditionally relies on information such as edge lengths and surface connectivity—data that are not readily available in implicit representations—the paper introduces a strategy to approximate these quantities in the implicit domain. Specifically, the proposed method projects a set of planar meshes onto various level sets of the implicit function approximating the target surface. These projected meshes are then used to compute the ARAP regularization loss by leveraging the connectivity between their vertices. Combined with soft handle constraints, this regularization loss guides the optimization of a deformation field parameterized by a neural network.", "questions": "1. Have the authors considered using volumetric meshes when applying discretization-based deformation methods? Since the proposed method benefits from the volume preservation property of the NICE model, while many baselines operate solely on surface meshes, this comparison may seem somewhat unbalanced.\n2. Could the authors provide statistics (e.g., mean and standard deviation) for the vertex and face counts of the meshes used in the comparisons? This includes both the raw outputs from Marching Cubes and their re-meshed versions.\n3. While Table 1 reports that the proposed method achieves zero volume error, Figure 9 shows that all patch construction methods exhibit some degree of volume error. Could the authors clarify this discrepancy? Additionally, please provide dataset details—specifically, which shapes and how many were sourced from the Thingi10K and DFAUST datasets?\n4. In the “Data and baselines” paragraph, the authors describe three variants of the proposed method. Which variant corresponds to “Ours” in Table 1?", "limitations": "Yes. The authors discussed the limitations of their work.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 1, "originality": 1, "strengths_and_weaknesses": "This paper presents a straightforward technique for estimating ARAP energy in the context of deforming implicit surfaces. The approach involves projecting a set of 2D mesh patches onto the target surface and leveraging their connectivity to approximate the ARAP energy.\n\nWhile the core idea is interesting, I have some concerns regarding its practical applicability and use cases. A rich set of tools for processing implicit surfaces—such as mesh extraction, topology optimization, and mesh-based deformation—has been extensively studied and is already well-established in the computer graphics literature. As a result, while the paper demonstrates that deformation in the implicit domain is feasible, it remains unclear whether this approach offers tangible practical benefits over existing methods. In particular, the requirement to train and store a separate neural network for each deformation step could lead to inefficiencies, especially in interactive design scenarios where multiple deformations are applied in sequence. Addressing this limitation or proposing ways to mitigate it would further strengthen the work.\n\nAlthough the paper targets a well-defined problem, the overall scope of the contribution feels relatively narrow—especially when compared to related efforts such as [1], which address a broader range of surface properties (e.g., normals, curvature) and additional applications like smoothing and sharpening.\n\nLastly, the experiment section feels somewhat unorganized. For example, acronyms such as “EL” and “FA” (Line 228) are used without prior introduction, and some results referenced in the text—such as those on the DFAUST dataset mentioned in Line 234—are not presented in the main paper. Clarifying these points would improve the clarity and completeness of the experimental evaluation.\n\n[1] Geometry processing with neural fields, Yang et al., NeurIPS 2021", "was_revised": true, "final_justification": "The authors have addressed the concerns raised in my initial review by providing appropriate justifications and additional analyses. While the level of technical novelty appears modest, the proposed method—despite its simplicity—demonstrates clear performance improvements over the baselines. I also had concerns regarding the clarity and organization of the experimental section, but I believe these issues can be addressed in the final revision.", "rebuttal": "We thank the reviewer for their constructive feedback about our work. We agree with the reviewer about the lack of clarity in some passages of the paper, which we attempt to clarify in this rebuttal.\n\n**Applicability and use cases**: the usage of neural fields in vision and graphics pipelines has grown more and more prominent in recent years, due to the massive efforts in this research area. One main limitation of these new representations is the lack of *native* tools for their manipulation and analysis: while most of them allow conversions to the mesh representation, this pattern nullifies the advantages of neural representations, such as unbounded resolution and independence from a particular discretization. We believe this motivates our research and several other efforts [1-6], solving problems directly in the neural field domain. Regarding interactive editing, the reviewer raises an interesting point: however, we did not propose interactive editing as a possible application because, while our method is ~130 times faster than the previous state of the art, it still is not fast enough for interactive usage. We believe further research in this direction would be extremely valuable, and that our work could be a starting point for such exploration. \n\n[1] Haque, Ayaan et al. \"Instruct-NeRF2NeRF: Editing 3D Scenes with Instructions\" Proceedings of the IEEE/CVF International Conference on Computer Vision. 2023.\n\n[2] Chen, Xu, et al. \"Snarf: Differentiable forward skinning for animating non-rigid neural implicit shapes.\" Proceedings of the IEEE/CVF International Conference on Computer Vision. 2021.\n\n[3] Tao, Yuanyuan, et al. \"Neural Implicit Reduced Fluid Simulation\". SIGGRAPH Asia 2024 Conference Proceedings.\n\n[4] Yang, Lingchen, et al. \"Implicit neural representation for physics-driven actuated soft bodies.\" ACM Transactions on Graphics (ToG) 41.4 (2022): 1-10.\n\n[5] Aigerman, Noam, et al. \"Neural jacobian fields: Learning intrinsic mappings of arbitrary meshes\". SIGGRAPH 2022 Conference Proceedings.\n\n[6] Tang, Jiapeng, et al. \"Neural shape deformation priors.\" Advances in Neural Information Processing Systems 35 (2022): 17117-17132.\n\n**Additional applications**: ARAP is a fundamental tool in geometry processing (it also received the test of time award in SGP 2023). GPNF would provide important tools for geometry processing on neural fields, but ARAP-like deformations can only be obtained with GPNF in an unfeasible time scale (\\~10hrs on a TitanX GPU). For this reason, we consider ARAP-like deformations a worthy task to which we specifically dedicated our effort. By contrast, the methods for smoothing and sharpening proposed in GPNF are already quite efficient (\\~10mins on a TitanX GPU). Extending our method to include smoothing is extremely simple: once a patch is constructed and deformed, we may optimize the smoothness of the patch mesh directly. If the reviewer deems it necessary, we could include some qualitative results for smoothing experiments, which we expect to have more or less the same runtime as our deformation method.\n\n**Improving organization of experiments section**: we agree that the organization of this section could be improved. About the two particular points raised in the review: \n1. The evaluation metrics were not described in the main paper due to space constraints. We refer to the supplementary material in L205 for a detailed description. However, for the camera-ready revision, we could move the full introduction of the employed metrics to the main text.\n2. The DeFAUST results we reference in L234 are those in Table 1, linking to the previous sentence in the text. We will update the text to highlight this connection.\n\n**Figure 9 volume error**: we understand the confusion about this point. The deformation network employed in this experiment is a standard MLP (not volume preserving), like the one used for the experiments in Fig. 13 and 14. The motivation for this choice is that we aimed to evaluate the change in error metrics with respect to the choice of patch sampling, and the volume-preserving network would have prevented us from seeing the change in the volume error metric (as it would always have been 0%). We will clarify this in the camera-ready revision. The requested information about the experimental data is reported below.\n\n**Other questions**\n\n1: To our knowledge, the baselines we employed in the evaluation of our method do not support volumetric meshes. Our method exploits the NICE properties to achieve volume-preserving deformation of surfaces.\n\n2: The statistics for the resolution of the employed data are reported below.\n\n4: We thank the reviewer for catching our mistake. The variant used in Table 1 is `Ours_SDF^Inv`, while the one used in Table 4 is `Ours_Mesh^MLP`.\n\n\n**Table R.1**: Statistics of evaluation data.\n\n| Dataset | Mean vertex count | STD vertex count | Mean face count | STD face count |\n|---------|-------------------|------------------|-----------------|----------------|\n|TFD original |197761|183709|395210|367031|\n|TFD marching cubes |181573|71397|363189|142839|\n|DeFAUST original |172850|13198|345697|26396|\n|DeFAUST marching cubes|277480|18649|554956|37299|\n\n**Data**: the DeFAUST dataset was introduced in [7], and it is obtained by employing sparse vertices of the FAUST shapes as deformation handles. The deformations are defined over a subset of per-subject shape pairs (same person in different poses) in the FAUST dataset, totaling 80 deformation experiments. The input shapes are also subdivided to achieve medium-high resolution.\nSince it is hand-crafted, the TFD dataset is of a smaller scale and counts 10 shapes, with 3 deformation experiments each. We list the shapes from Thingi10k and the Stanford 3D scanning repository below.\n* Buddha, https://graphics.stanford.edu/data/3Dscanrep/\n* Dragon, https://graphics.stanford.edu/data/3Dscanrep/\n* Armadillo, https://graphics.stanford.edu/data/3Dscanrep/\n* Piranha plant, https://www.thingiverse.com/thing:3625381\n* Octopus, https://www.thingiverse.com/thing:159217\n* Sculpture, https://www.thingiverse.com/thing:21126\n* Samurai, https://www.thingiverse.com/thing:6570837\n* Cat, https://www.thingiverse.com/thing:34965\n* Hand, https://www.thingiverse.com/thing:46891\n* Dino, from ARAP and NFGP experiments\n\n[7] Maggioli, Filippo et al. \"SShaDe: scalable shape deformation via local representations\". arXiv, 2024."}]

Author rebuttal:

Meta-review: The paper received 4 reviews. The authors-reviewers discussion clarified all points raised in the initial reviews. A consensus that the paper makes a useful contribution and can be finalised for publication was reached amongst the reviewers. The paper can be accepted as poster.
Decision: Accept (poster)