VisCo Grids: Surface Reconstruction with Viscosity and Coarea Grids
Authors: Albert Pumarola
Artsiom Sanakoyeu
Lior Yariv
Ali Thabet
Yaron Lipman
Venue: NeurIPS 2022 Accept (2022)
Keywords: surface reconstruction
implicit neural representations
signed distance functions

Abstract: Surface reconstruction has been seeing a lot of progress lately by utilizing Implicit Neural Representations (INRs). Despite their success, INRs often introduce hard to control inductive bias (i.e., the solution surface can exhibit unexplainable behaviours), have costly inference, and are slow to train.  The goal of this work is to show that replacing neural networks with simple grid functions, along with two novel geometric priors achieve comparable results to INRs, with instant inference, and improved training times. To that end we introduce VisCo Grids: a grid-based surface reconstruction method incorporating Viscosity and Coarea priors. Intuitively, the Viscosity prior replaces the smoothness inductive bias of INRs, while the Coarea favors a minimal area solution. Experimenting with VisCo Grids on a standard reconstruction baseline provided comparable results to the best performing INRs on this dataset.

VisCo Grids: Surface Reconstruction with Viscosity and Coarea Grids
Albert Pumarola"), Artsiom Sanakoyeu"! Lior Yariv?, Ali Thabet', Yaron Lipman" Meta AI, WWizmmann Institute of Science
Abstract
Surface reconstruction has been seeing lot of progress lately by utilizing Implicit Neural Representations (INRs). Despite their success, INRs often introduce hard control inductive bias (i.e., the solution surface haviours), have costly inference, and are slow to train. The goal of this work is to show that replacing neural networks with simple grid functions, along with novel geometric priors achieve comparable results to INRs, with instant inference, and improved training times. To that end introduce VisCo Grids: grid-based surface reconstruction method incorporating Viscosity and Coarea priors. Intu- itively, the Viscosity prior replaces the smoothness inductive bias of INRs, while the Coarea favors minimal area solution. Experimenting with VisCo Grids on a standard reconstruction baseline provided comparable results the best performing INRs on this dataset.
exhibit unexplainable be-
Introduction
Reconstructing 3D surfaces from sparse point clouds is a long standing prob- lem in both computer vision and graph- ics [7]. Methods tackling this prob- lem aim estimate 3D surfaces given as input unordered point sets (point clouds), with or without corresponding normals. Surface representations can be divided two groups: parametric and implicit. Parametric methods rep- resents the surface using some para- metric domain, while implicit methods represent the surface as some level-set, (p)) = c], of volumet-
S = (p ric function f R + R. While para- metric methods can casily sample the sur- face, implicit methods can readily adapt to topological changes of the reconstructed surface. Parametric methods include, e.g., meshes and spline surfaces, while implicit methods use, e.g., volumetric data structures such voxel grids, Radial Basis Functions (RBFs), or (recently) neural networks.
VisCo
Viscosity
w/o Coarea
Figure 1: The VisCo prior (left) incorporating viscosity and Coarea; Middle and right shows ablations on each.
Implicit Neural Representation (INR) 29 35 13 3 16 18 4 (37] is categorized as an implicit method using a neural networks to define the implicit function f. INRs build upon the inherit inductive bias in neural networks and their optimization process to provide smooth yet flexible and expressive
These authors contributed equally this work.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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surface reconstructions. INRs have several disadvantages: First, the neural inductive bias is hard to control, often introducing undesired or unexplained surface behaviours. In fact, a considerable research effort is dedicated to fix/change/control this bias [45 4 27 26]. Second, INRs have an increased deployment cost, requiring many network evaluations for surface contouring, e-g. with marching cube based methods (29 35], or direct rendering [49] 31. Lastly, although using high optimized solvers, INRs are still slow to train.
The goal of this work it show that network-free grid-based implicit representations can achieve INR-level reconstructions when incorporating suitable priors. To that end, we present VisCo Grids: a grid-based surface reconstruction algorithm that incorporates well-defined geometric priors: Viscosity and Coarea. In short, VisCo, see Figure |I right. The viscosity loss, is replacing the Eikonal loss used in INRs for optimizing Signed Distance Functions (SDF). The Eikonal loss posses
[8 4 many bad minimal solution that are avoided in the INR setting due to the network's 's inductive bias, but are present in the grid parametrization, see e.g., Figure I middle. The viscosity loss, uses the notion of vanishing viscosity to regularize the Eikonal loss and provide well defined smooth solution that converges to the "correct" viscosity SDF solution. The viscosity loss provides smooth SDF (right). Therefore,
solution but do not punish excessive or "ghost" surface parts, see e.g., Figure [ second useful prior is the coarea loss, directly controlling the surface's ' area, and encourages it to be smaller. The coarea loss is defined using a "squashing" function applied to the viscosity SDF making it upproximately an indicator function, and then integrates its gradient norm over the domain. Integrating the gradient norm of a function is called the Total Variation loss 12 27 and is measuring the perimeter of indicator functions, which in case approximates area (S) ). VisCo grids (as other grid methods) have instant inference, and even with our current rather naive implementation are faster train than INRs. Considerable training time improvement expected with more efficient implementation.
We tested VisCo Grids standard 3D reconstruction dataset, and achieved comparable accuracy to the state-of-the-art INR methods. Through ablations, we demonstrate the properties and benefit the VisCo prior.
2 Related Work
3D Surface Reconstruction Classical approaches for surface reconstruction from point clouds 7 are either parametric [2] or implicit with mostly linear function bases, e.g., grids radial basis functions [10] [4]]. Recent works have developed methods for surface reconstruction using neural networks, which consist of a non-linear function space, making these methods non-convex. Those
methods differ by how they choose to represent the 3D reconstruction. [20] 46 21 employ a parametric point of view. Such discretizations do not yield watertight reconstruction, and/or lack topological detail. A more flexible solution is the Implicit Neural Representation (INR). INRs based methods 335 29
27 6 show great progress in leveraging the inductive bias of MLPs to represent smooth surfaces, using additional losses and regularizers. For example, [27] introduce a perturbed Dirichlet loss (i.e., norm of gradient) to push for unique and regular occupancy solution; [6] incorporates Divergence loss (i.e., absolute value of the divergence of the gradient of trained distance field) for encouraging the learned field resemble gradient field of true distance functions. Neural Spline [47) does not use neural networks directly, rather derive a kernel based formulation arising from infinitely-wee wide shallow networks. Shape As points (SAP) 37 represent the surface using a differentiable Poisson solver and contouring process.
13
18
43
Grid-based representations Recent works suggested reduce, completely or partially, the use of neural networks in implicit representations and replacing it with grid-based data structure. This due to the heavy computational resources required in optimizing and evaluating neural networks. Plenoxels [1] propose a dieeddependent sparse voxel model and show comparable results to NeRF 31 and speedup of two orders of magnitude. Neural Geometric Level of Detail 44 uses an octree-based feature volume and small MLP to represent SDF. (32 shows fast training of INR's using a small neural network augmented by a multiresolution hash table with trainable features.
Similar to DeepSDF (35), both work used 3D supervision for learning the SDF.

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3 Method
We consider the 3D euclidean space 12 with points p (z, y, E R.. We discretize the unit cube C = [0, 1 with a 3D voxel grid G = {pr). with nodes ps indexed by / = (i. k). i, k E [n] { [L....,R]) .R.. i.e. pr (Zijk, Mijk, Z.jk). We denote by h nー and by N n the total number of nodes. We represent our reconstructed surface as a zero level of a scalar function f defined over the cube C. f is defined by prescribing its values the grid's nodes fi and trilinear interpolating in each voxel. We will denote by f fp) the interpolated value at point
Given an input point cloud consisting of points qk E R$ with without (unit norm) normals ve E RR, & E [m], our goal is to compute f so that its zero level set approximates the unknown surface, i.e.,
S = c f(p) = 0}. Our approach to compute f is to minimize loss function of the form
(1)
L = Ldata + Eprior
(2)
where
An Vf(g)) - nx
Ldata
If(())1
+
(3)
m7
where ||-II is the standard euclidean norm in R', / E R is the gradient of f sampled at point Note that V f is defined in interior of voxels, which is generically where the input points ya resides. fama is the standard data loss encouraging the zero level to pass through the input points le. and its normals (defined by gradients of f) to coincide with input normals nt.
The prior, Cprioe. is the main contribution of this work, where we combine two novel losses,
Syrirr X Aviscusity + de Aecurrea
Intuitively, the viscosity loss optimizes for a smooth Signed Distance Function (SDF) solutions, avoiding auxiliary bad minima of the Eikonal equation, while the coarea loss strives to minimize the area of the zero level surface. Our loss has 4 hyper-parameters Ap: An, Av, dc. We provide more details on these priors next.
3.1 Viscosity Loss
The goal of the viscosity loss is to make f approximate an SDF over C. Given boundary conditions asking f to vanish on some closed compact surface S, the SDF solves the Eikonal equation PDE, i.e., [f) = 1, in certain well defined sense (viscosity). This motivated some previous work to
directly optimize the Eikonal loss 43
Leikonal = 11) 2 dp Unfortunately, the Eikonal loss has many undesirable minima which are not SDFs. Figure Saauree@shone shows a ID example: both depicted solutions (denoted f) vanish at the input points 91, 92 (black points) and globally minimize the Eikonal loss over the grid (grid points are shown in blue).
(5)
The INR works mentioned above use neural networks for representing I which injects an inductive bias avoiding these bad minima, however on grids, minimizing equation 5 cannot avoid these solutions. See, e.g., middle column in Figure
pr
Q2
More classical Eikonal solvers do work with grids however use mostly fast marching or sweeping methods [33] 41 50, IT] Namely, use a special discretization of the Eikonal equation favoring the viscosity solution of the Eikonal [40]. and update node values according to moving front [41). Since this discretization is up- up-wind (will only propagate values in one direction) and requires choosing the maximal among its solution, its success in adaptation to loss is not clear.
Figure 2:
Two global min- imizers of the Eikonal loss over a grid in ID. Top solu- tion is not SDE
We use a different approach to build a loss favoring SDF solutions over grids motivated by the vanishing viscosity method [15]. Namely, adding to the Eikonal PDE a small perturbation of the
3

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Laplacian of f (denoted by if). -V = O, makes the PDE semi- linear elliptic [ and hence with suitable boundary conditions it is uniquely solvable inside S with a smooth solution, approaching the viscosity positive distance function to the boundary as -> O. Similarly, for V [p)) - e (p) = O the solution approaches the negative distance function inside the domain. Motivated by the vanishing viscosity principle we suggest the following viscosity loss:
-fo (IIFF())
1)sign - caf(r)) 2
Cviscosity
dp
We discretize this loss over the grid G by replacing the first order derivatives and second order derivatives with symmetric finite differences, i.c.,
DaSI = DofigA JILII - fi-
DarT= Dal fiti,j - -2frj. + fi-1.jt
2h
1
and similarly for Dy and D2- We use these discrete operators to approximate the gradient V f(r) = (Def1. Dyfi, Dzfi) and Laplacian Af(pr) = D D D2fr. The discretized viscosity loss now takes the form
Σ ((I))(p))) N
1 - -e =āf(Pi))
Viscooiiyy
3.2 Coarea loss
The coarea loss is approximating the area the zero level set, and therefore incorporating it in the optimization pushes the reconstrueted surface to be economic in area.
First, simiarly
48
we use the centered Laplace CDF
exp 1 2 exp
8 0
(.3(s)
(8)
220
to transform the SDF to a smooth approximation of the indicator function:
Xa(p) v)((-((p)
(9)
As 3 + 0, Xa converges to an indicator function leading to 1 inside S and 0 outside. The coarea loss is defined as
Leuarea -f [VVxx(P))I dp
(10)
To understand why this loss approximates the area we can use the coarea formula [39]
j [ixx(p) dp L area (s))ds,
(11)
where X3 (s) = {p xp())= $} the preimage of the value s. Since X- E [0, 1] the r.h.s. inte- gral can be restricted to the interval [o, 1]. and therefore the coarea loss averages the area the level sets of Xa. Next,
XØ = (p1=-11 s (p15() -3 (0)
which shows that the level set E (0. of Xa is the level set - -V) of the SDF f. As 3 + 0, - v (s) + 0 for all E (0, 1) (and uniformly in (-) for fixed
0). Therefore the average of the level set area (i.e., the r.h.s. of equation II converges the area of (0) = Figure [ (right) shows how removing the coarea loss introduces an extraneous zero level set, and hence results a recon-
Figure 3:
Reconstruction of a semisphere point cloud (white dots) without (left) and with (right) coarea loss.
an undesired surface part. Figure 3 shows a comparison struction of semisphere with and without coarea. In the experiments section we provide more ablation tests with the coarea and viscosity losses.

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To discretize the coarea loss we let wy denote the centers of grid's voxels, and note that Vxs(wy) = > (= where
1
|s|
(s(s)
exp
2/
the PDF of the Laplace distribution, and V f(w)) is computed as linear combination of the voxel's corner values fi...., fta, see more details in the Appendix. We end up with the discretized loss:
Econren
Σ (x(-f(wr)) [V((w))l
(12)
This loss is usually incorporated with a small hyper-parameter x with the purpose of eliminating redundant surface parts.
4 Experiments
In this section we extensively evaluate VisCo grids. First, we evaluate on two standard surface
reconstruction benchmarks [46] 22 (Sec. 4.1) against large variety of state-of-the-art methods:
Poisson Surface Reconstruction 22 DGP 46), IGR 18] SIREN 43 FFN
45
NSP
[47]
PHASE [27] GD 14, BPA H. SPSR (25 RIMLS (34
SALD
5 IGR
19
OccNet
30)
DeepSDF (36, LIG [23], Points2Surf (7). DSE (8), IMLSNet [28] and ParseNet 422
We then
perform an ablation study (Sec. 4.2), and conduct detail examination of the main components the
model, namely the viscosity and coarea losses. Finally, we discuss the model's ability to reconstruct
sparse point clouds (Sec. 4.3) using scans from Stanford 3D Scanning Repository.
4.1 Surface reconstruction benchmarks
We next evaluated our model on two benchmarks: Surface Reconstruction Benchmark (46] and Surface Reconstruction from Real-Scans [22]. Each containing challenging object with complex shape and topology. Importantly, we use same hyper-paramenters for all meshes of all benchmarks with extensive hyper-parameter search.
Surface Reconstruction Benchmark
This benchmark 466 consists noisy range scans, each
containing point cloud and normal data. We evaluate method against current state of the art methods on this benchmark: Deep Geometric Prior (DGP) [46]. Implicit Geometric Regularization
(IGR) 18 SIREN [43], Fourier Feature Networks (FFN) 455. NSP (4) and PHASE (27). We additionally compare to the classical method of Poisson Surface Reconstruction (24). Quantitative results are summarized in Table [ We report the Chamfer (dc) and Hausdorff (dz) distances
between the reconstructed meshes and the ground-truth point clouds. Furthermore, we report their
corresponding one sided distances (d and d'²) between the reconstructed meshes and the input noisy point cloud. Representative qualitative results are shown in Figure []] Note that we achieve comparable results to the current state-of-the-an INR methods.
Surface Reconstruction from Real-Scans This benchmark 22 consists of 21 noisy range scans of real objects. We evaluate our method against: GD
14, BPA 8) SPSR 255 RIMLS 34, SALD (5).
IGR 19). OccNet [30], DeepSDF [36]. LIG (23] Points2Surf 17.
DSE 38] IMLSNet 28 and
ParseNet 42 Quantitative results are summarized in Table We report Chamfer Distance (dc),
F-score, Normal Consistency Score (NCS) 30), and Neural Feature Similarity (NFS) 22 distances between the reconstructed meshes and the ground-truth point clouds. Furthermore, we report the one sided distances (dj) and dz)) between the reconstructed meshes and the input noisy point cloud. Note that we achieve 1-st to 3-rd place
Representative qualitative results are shown in Figure across all categories, with top F-score.
4.2 Ablation study
We provide an ablation study of the main components of our model Table 3 and Figure Specifically we compared with the following alternatives: i) Eikonal loss without the viscosity term that prevents undesirable non-SDF solutions, i.e., = 0 in equation [ ii) removing the coarea loss enforcing minimal surface area. i.c., X O; and iii) removing the normal loss, i.c., An 0. Note
5

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BOON
BOON
OOEK
DEEN
Poisson IGR
PHASE
VisCo
compared to existing methods. Note how Figure Qualitative results for surface reconstruction 46 VisCo Grids achieve comparable level of details when compared to other baselines.
Poisson
DGP
IGR
SIREN
FFN
NSP
PHASE
Ours (30 mins)
Ours (8 mins)
de
0.60
0.22
0.32
0.31
0.22
0.21
0.21
0.28
dæ
14.89
8.82
4,71
8.19
4,49
4.65
4.29
3.00
5.69
Anchor
2g
dg
14.89
2.79
1.32
2.43
0.10
1.11
1.23
1.07
1,15
0.44
0.20
0.25
0.21
0.34
0.21
0.18
0.26
0.25
dy
7.24
4.30
4.15
Daratech
dz
0.44
0.04
0.08
0.09
0.10
0.088
0.08
0.14
0.13
dz
7.24
1.89
1.59
1.77
0.10
1.14
1,80
1.76
1.78
de 0.27
0.18
0.17
0.15
0.20
0.14
0.15
0.15
0.15
dæ 3,10
4.31
2.22
2.18
2.87
1.35
2.52
2.22
2.23
DC
2p
d 3.10
2.53
2.61
2,76
0.12
2.75
2,78
2.76
2.78
0.26
0.21
0.16
0.17
0.22
0.16
0.16
0.17
0.17
du 6.8
5.98
3.52
4.64
5.04
3.20
3.14
4.40
4.45
Gargoyle
0.26
0.06
0.06
0.08
0.09
0.08
0.07
0.11
0.11
4
6.80
3.41
0.81
0.91
2.75
1.09
0.96
0.98
de
0.20
0.14
0.12
0.17
0.35
0.12
0.11
0.12
0.13
dı
4.61
3.67
1.17
0.82
3.90
0.69
0.96
1.06
1.14
Lerd Quas
dg
dz
4.61
2.03
0.98
0.76
0.06
0.62
0.96
0.64
0.68
Table 1: Surface reconstruction results the benchmark of [46]. We show reconstruction results for each model for our method at 256 grid resolution with 30 minute and 8 minute time budget. We also show results from comparative methods. Bold numbers signify top performance. We report Chamfer and Hausdorff distances using ground truth scans (dc. d and input scans (dz d ). Note that VisCo Grids achieve comparable results to SOTA INRs, and matches it in terms of Chamfer distance in 3 out of 5 meshes.
that without coarea and viscosity the reconstruction tends have holes and discontinuities near the surface boundaries. Only combination of all the components results good surface reconstruction.
Learning with viscosity. We further provide a more in depth discussion of the proposed viscosity loss. Figure compares reconstructions with different levels of the viscosity parameter, i.e.. e in equation 6 As can be inspected from this figure, viscosity affects the smoothness of the reconstructed surface. For a low viscosity parameter the zero level sets become noisy. This can be explained by the viscosity eikonal loss (i.c.. equation 6 becoming numerically too close to the eikonal loss in

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Input
SPSR
SALD
IGR
VisCo
Figure 5: Qualitative results for surface reconstruction of real objects 22 compared existing methods. Note how VisCo Grids does not over-extend the surface in the bottom row example. The competing methods meshes were provided by the benchmark organizers.
Prior
Method
de (x10-") 4
Fiscure (%)
NCS (x10-
NES (x107")
GD 1
31.72
87.51
88.86
82.20
Triangulation baved
BPA
40.37
80.95
87.56
68.69
SPSR
31,05
87.74
94.94
89.38
Smoothness
RIMLS
32, 80
87.05
91.97
85.19
Ours
32.11 (3rs)
88.52 )
94.20 (3rd)
89. (2nd)
SALD
31.13
87.72
94.68
86.86
Modeling
IGR
19
32. 20
87.18
95.99
89.10
232.71
Learning Semanties
DeepSDF 36
263,92
19.83
77.95
40.95
LIG
AA]
48. 75
83.76
92.57
81.48
Local Learning
Points2Surl
DSE
A8
32.
86.88
87.20
76.81
Hybird
IMLSNet 28
38.46
82.44
93.31
85.30
ParseNel
Table 2: Surface reconstruction results on the 20 real-scanned benchmark 22 meshes. We report Chamfer Distance (dc), F-score, Normal Consistency Score (NCS) (30), and Neural Feature Similar- ity (NFS) [22). Methods are grouped according to surface geometry priors, originally defined in the benchmark. Our method achieves top F-score and 1-st to 3-rd place in all scores.
equation Fndd and the solution deviates from the viscosity SDF solution. This leads to artifacts across the surface, similar the limit case (c 0) where only the eikonal loss is used, the second column from the left, For high viscosity parameter, and as expected with the addition of a non-vanishing Laplacian term, the surface becomes over-smoothed.
Learning with coarea. Similarly, we also provide an analysis of the proposed coarea loss. In Figure [ we show the effect of changing the parameter weight of the coarea loss, Ag. As can be observed the figure, a low coarea weight leads the presence excessive surface area in reconstruction. In contrast, a high weight will strive to minimize the surface area. In a sense, the coarea serves surface tension parameter; stronger tension will ignore points, weaker tension will overfit.
7

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VisCo
Normals
w/o Coarea
w/o Viscosity
Figure 6: Ablation for the main components of method. Removing elements of our loss leads to subpar reconstructions. We can observe these artifacts in the level sets shown in this figure. Removing viscosity results in discontinuities in the final surface, while coarea produces excess surface area.
Baseline
wro normals
w/o viscosity
wio coarea
dc
0.21
0.55
0.72
dy
3.00
7.82
10.83
10.24
Anchor
dz
0.15
0.36
47
1.07
7.84
1.44
9.68
de
0.26
0.24
0.24
0.23
dy
Daratech
0.14
0.13
0.12
0.13
1.76
2.69
1.77
1,77
de
0.15
0.15
0.15
0.34
du
2.22
2.24
2.24
6,58
DC
dz
0.09
0.08
0.08
0.16
47
2.76
2.76
2.79
2,82
de
0.17
0.58
0.47
0.59
d
4.40
6.35
Gargoyle
de
0.11
0.07
0.26
0.38
d
0.96
2.39
1.34
1.25
de
0.12
0.12
0.12
0.58
d
1.06
1.38
1.04
6,05
Lord Quas
dz
0.64
0.69
0.64
3.73
Table 3: Ablations study. We show the contribution of each component of VisCo Grids. Baseline is the full method. The remaining columns correspond optimizing without normal loss, viscosity loss and coarea loss, respectively. We show results for each mesh the benchmark 46. The results justify the use of the different components in VisCo Grids.
VisCo (1e:2)
High (1e-1)
Low (1e3)
(0.0)
Figure 7: Viscosity loss ablation. Setting a high viscosity loss parameter, e, leads to oversmoothing In contrast, setting it too low leads to noise and discontinuities in the surfaces, similarly to removing by setting e = 0.

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O
High (le-5)
Low (fe77)
(lo4)
Figure 8: Coarea loss ablation. Coarea loss favors solution with low surface area. Here, note how larger coarea weight tends to fill the cavities and close the gaps in shape. contrast, very low weight fails drive the optimization procedure towards solution with smaller surface area.
1%
10%
25%
Figure VisCo reconstructions from sparse point cloud inputs.
4.3 Reconstructing from sparse point clouds
We evaluate model ability reconstruct surfaces in the challenging of sparse input point clouds. For this experiment we use point clouds from the Stanford 31) Scanning Repository and downsample them at different levels: 1%, 10%, and 25%. In Figure 1 we visualize the VisCo reconstructions. Note that VisCo can reconstruct the shapes even with sparse input. This provides further validation for the proposed geometrical priors.
Conclusions
We introduced VisCo Grids, novel grid-based surface reconstruction approach that leverages two novel geometrically motivated priors: viscosity and coarea. We advocate VisCo's prior for grid functions as an alternative to the implicit inductive bias of implicit neural representations for the task of surface reconstruction. One important limitation of our method, shared by all grid methods, is that its degrees of freedom, namely nodes' location, are set a-priori. In contrast, using non-linear function spaces, such as neural networks, allows a more flexible usage the degrees of freedoms in the model and can adjust areas with more detail. Nevertheless, and direct priors, incorporated with modern computing power, are valuable add-ons to the surface reconstruction toolbox, providing few clear benefits over neural networks, such well-understood control over surface properties, instant inference time, and faster training.
still believe that grid functions
We thank the anonymous reviewers for their constructive comments. LY was supported by grant from Israel CHE Program for Data Science Research Centers.
Acknowledgements.
Social impact. We don't see any immediate negative societal impact from our work. However, we acknowledge that high quality 3D reconstruction in general can be used in malicious settings.

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12


Reviews:
[{"review_id": "w_w2PypbqIb", "reviewer": "Reviewer_5amW", "summary": "The paper proposes a new method for representing surfaces on grids using prior-based losses: viscosity loss and Coarea loss. They have shown better results than neural network SDFs in reconstructing surfaces on the Surface Reconstruction Benchmark.", "questions": "For the comparison, each sample is \"optimized\"/ \"learned\" on a separate grid? Does it downgrade to only solving a PDE, not a learning problem? It seems not so fair to compare with the neural network-based method, which can extend to \"arbitrarily\" resolution, while the grid is fixed.", "limitations": "The limitation mainly comes from the grid-based representation. It seems can only represent one model after optimization, unlike other network-based can use a latent code as an additional condition. ", "rating": 5, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Pro:\n\n1. Instead of using a neural network as a bridge to build implicit surface representation, the authors propose to go back to using the grid-based method. It is easier to train than a neural network and is easier to find the zero surface.\n\n2. The authors bring two new priors to the domain, viscosity and Coarea loss, to replace Eikonal loss and control the surface's area. They use the finite difference to compute the gradient or the Laplacian.\n\nCon:\n\n1. It does not directly compare how Viscosity loss and Coarea loss would perform with the neural network. The comparison with other works seems not fair.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "t_RnaspGj__", "reviewer": "Reviewer_WtxG", "summary": "The paper proposes a method for SDF based surface reconstruction from point clouds using grid based functions, rather than the recently popular INR approaches. They use standard INR losses along with a new prior loss that has two components, a viscosity prior (for smoothness in the SDF) and a coarea prior (for minimal surface area). They show quantitative results on the SRB dataset, which are comparative to SoTA, and qualitative results on the Stanford 3D scanning repository.", "questions": "PHASE [18] is quite similar (a method with a direct prior based on smoothness and minimal area) and gets better (or similar) performance. Why is this the case? Is it just because of it using a neural network vs this work being network free, or could it possibily be better direct priors?\n\nThe following work might be of interest to you: https://arxiv.org/abs/2206.03087v1, it is a concurrent work (accepted to CVPR2022) with similar priors (they have them as regularizations): a Hessian regularization for smoothness and a minimal surface regularization.", "limitations": "Yes, limitations in conclusion, and social impact in subsection in conclusion.", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n- Agree that general direct priors are still important for surface reconstruction given how important the task is and how various the shapes can be, and the paper gives two interesting priors\n- Mathematical explanation, intuition and visualisations of the two components of the prior all show their value\n- Results with normals are comparative to SoTA\n- Good ablations of the components\n- Paper is well written\n\nWeaknesses\n- Limited quantitative results (5 shapes). While SRB is a well established baseline, having this as the only quantitative results is not great.\n- Given that the method is not using a neural network and is grid based, a major strength of the method should be lower optimisation time, with the weakness being storage of the learned INR. A discussion of both is missing, with no comparison of optimisation time or learned storage to other methods.\n- The use of minimising the Laplacian as a prior has been explored in INR work, such as DiGS [* 1] and NSP [32], and has connections to PHASE [18]. This is not discussed at all (and should be in the related work), and DiGS is not even cited.\n- Poor results without normals, especially given the contribution is meant to be a good prior for SDFs and thus should give results comparable to PHASE [18] or DiGS [* 1].\n\n[* 1] https://arxiv.org/abs/2106.10811", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "LzgA3XJUuwl", "reviewer": "Reviewer_ubX9", "summary": "The proposed method tries to improve the surface reconstruction quality by introducing two losses.\nThe viscosity loss tries to imporve the smoothness of the surface part without points.\nThe coarea loss tries to approximate the area of the zero level set.\nThe experiments show the proposed method can get better surface reconstruction results, especially with sparse point inputs.", "questions": "1. Could the authors explain why the coarea loss is introduced more clearly?\n2. Could the authors do some experiments on point clouds by MVS, which have more noises than the data used in the paper?", "limitations": "The authors discuss the limitation in the conclusion briefly. The work does not have potential negative societal impact.", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n1. The paper is generally well written. All the figures and equations are clear.\n2. The authors clearly explain why the viscosity loss should be introduced.\n3. The experiments results on sparse point inputs are improved.\n\nWeaknesses\n1. The main contribution of the paper is the introduction of two losses. There is no neural network sturctue contribution.\n2. The motivation of coarea loss is not explained clearly. We can only know its effectness by the ablation study.\n3. The quantitive results on benchmarks and of the ablation study are not improved significantly. On several data of the benchmark, the top results are not generated by the proposed method. In the ablation study, the baseline method cannot beat other settings.\n4. The experiments are performed on point clouds without many noises. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "B5MtKadefYr", "reviewer": "Reviewer_tYGS", "summary": "This submission proposes 2 new geometric prior terms for fitting regular grids of SDFs to point clouds. The 2 priors are simulating the viscosity of the field to smooth it, and minimizing the area of the 0 level set.", "questions": "- How are gradients computed from the grid in eq. (3)?\n- Why is Fig(2) showing a problem? the 0-isosurface is the same...\n- In Tab. 2, why is Daratesh better without any additional loss term?\n- 4.2 ablates different weights for the viscosity term. How hard is it to tune in practice?", "limitations": "A limitation not mentioned by the authors is that it is not possible to learn a shape prior with this formulation: only single shape fitting is possible.", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- technical soundness\n- nice presentation with clear figures\n- great ablation of the 2 added priors and their respective effects.\n\nWeaknesses:\nMAJOR - experiments are only conducted on 5 shapes only, no precise information is given about the optimization strategy (grid resolution, gradient descent steps...), and no normal consistency is provided.\nMINOR - the introduction, the too short related work section, and the general scoping of the paper wrongly confuses 3D shape reconstruction from a point cloud with 3D shape representation with neural networks. Both are orthogonal research topics.\nMINOR - the paper claims that using a grid of SDF has \"clear benefits\", but what are they? And the proposed 2 loss terms could be used on implciit networks as well..", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept