Title: Aligning Gradient and Hessian for Neural Signed Distance Function
Venue: NeurIPS 2023
Accepted: yes
Keywords: implicit neural representation

signed distance function

shape operator
Abstract: The Signed Distance Function (SDF), as an implicit surface representation, provides a crucial method for reconstructing a watertight surface from unorganized point clouds. The SDF has a fundamental relationship with the principles of surface vector calculus. Given a smooth surface, there exists a thin-shell space in which the SDF is differentiable everywhere such that the gradient of the SDF is an eigenvector of its Hessian matrix, with a corresponding eigenvalue of zero. In this paper, we introduce a method to directly learn the SDF from point clouds in the absence of normals. Our motivation is grounded in a fundamental observation: aligning the gradient and the Hessian of the SDF provides a more efficient mechanism to govern gradient directions. This, in turn, ensures that gradient changes more accurately reflect the true underlying variations in shape. Extensive experimental results demonstrate its ability to accurately recover the underlying shape while effectively suppressing the presence of ghost geometry.

Aligning Gradient and Hessian for Neural Signed Distance Function

Abstract

The Signed Distance Function (SDF), as an implicit surface representation, provides a crucial method for reconstructing a watertight surface from unorganized point clouds. The SDF has a fundamental relationship with the principles of surface vector calculus. Given a smooth surface, there exists a thin-shell space in which the SDF is differentiable everywhere such that the gradient of the SDF is an eigenvector of its Hessian matrix, with a corresponding eigenvalue of zero. In this paper, we introduce a method to directly learn the SDF from point clouds in the absence of normals. Our motivation is grounded in a fundamental observation: aligning the gradient and the Hessian of the SDF provides a more efficient mechanism to govern gradient directions. This, in turn, ensures that gradient changes more accurately reflect the true underlying variations in shape. Extensive experimental results demonstrate its ability to accurately recover the underlying shape while effectively suppressing the presence of ghost geometry.

1 Introduction

In recent years, the neural signed distance function (SDF) has demonstrated its capability to represent high-fidelity geometry [37, 16, 8, 53, 1, 19]. Existing approaches primarily employ neural networks to map coordinates to their corresponding signed distance values. Depending on whether or not supervision is used, these approaches can be classified into two categories: learning-based methods and optimization-based methods, where the former fits data samples to their corresponding groundtruth implicit representations [37, 16, 8, 22, 21] and the latter directly infer the underlying SDF from point clouds [1, 19, 5, 3, 48] or multi-image [45, 29, 35]. Despite significant advancements in SDF-based surface reconstruction, both types of methods have their drawbacks. The supervised reconstruction methods may not generalize well [42] on shapes or point distributions that are not present in the training data. The optimizing-based methods, on the other hand, struggle to resolve the ambiguity of the input point cloud, particularly when normal information is absent. In this paper, our

Yunxiao Zhang

focus is primarily on optimization-based reconstruction techniques for point clouds without normal information.

The majority of optimization-based approaches utilize first-order constraints to regulate the variation of the SDF. For example, the Eikonal term [19, 40, 53] is commonly employed to ensure that the gradients are unit vectors. However, as pointed out in [6], the Eikonal term is weak in its ability to regulate the direction of the gradients. As a result, it is challenging to prevent the emergence of ghost geometry (the level sets of the SDF are highly disordered) and unnecessary shape variations even when the Eikonal term is enforced. To address these issues, various second-order smoothness energy formulations [54, 6] are proposed to steer the direction of the gradients to change toward a desirable configuration. However, it entails considerable difficulty to regulate the extent to which the smoothness term is enforced. A commonly seen artifact is that the resulting surface may be excessively smooth.

In this research, we re-examine the problem of surface reconstruction based on surface vector calculus, and introduce a novel loss to facilitate the inference of a faithful SDF directly from raw point clouds without oriented normals. An interesting observation is that within the narrow thin-shell space of the underlying surface, where the real SDF is differentiable everywhere in the narrow thin-shell space, the gradient of the SDF is an eigenvector of its Hessian matrix and the corresponding eigenvalue is zero. Specially, when a point is situated on a surface, the normal vector transforms into an eigenvector of the Hessian of the SDF, with the corresponding eigenvalue being zero. In essence, the gradient and Hessian of the SDF must be aligned within a thin-shell region surrounding the underlying surface. This alignment allows for more effective control over the direction of the gradients. Building on the key observation, we develop a new loss function that promotes the alignment of the gradient and Hessian of the SDF, rather than relying on smoothness energy. We have conducted a comprehensive evaluation of our proposed approach on a variety of benchmarks and compared it to recently proposed reconstruction methods. Extensive experimental results show that our approach outperforms the state-of-the-art techniques, whether overfitting a single shape or learning a shape space. It can not only accurately recover the underlying shape but also suppress the occurrence of ghost geometry, which verifies the effectiveness of the gradient-Hessian alignment.

2 Related Work

2.1 Traditional Methods

Traditional reconstruction methods can be classified into two categories: explicit and implicit. Explicit methods focus on establishing direct connections between input points, while implicit methods aim to fit an implicit field that conforms to the given points and normals. Roughly speaking, popular explicit reconstruction techniques utilize computational geometry methods, such as Delaunay triangulation or Voronoi diagrams, to infer connections between points. These methods can produce well-tessellated triangle mesh surfaces [26, 14, 47]. However, they may struggle to ensure manifoldness, particularly when the input point cloud contains defects such as irregular point distribution, noise, or missing parts. In contrast, implicit reconstruction techniques [25, 23, 39] are capable of generating manifold and watertight surfaces. However, the majority of these methods necessitate that the input point cloud is equipped with normal information.

In the event that the provided point cloud is devoid of normals, it is necessary to either estimate the normals and orientations prior [52] to surface reconstruction or devise a novel reconstruction framework that can operate in the absence of normal information [20, 27]. In essence, all of them have to estimate normals either prior to or during the process of surface reconstruction. As a result, these approaches remain heavily reliant on the quality of the estimated normals. In cases where the provided point cloud is of poor quality, it becomes challenging for these methods to overcome the inherent ambiguity introduced by the absence of normal information.

2.2 Supervision-based Learning Methods

Learning-based methodologies have demonstrated a superior capacity for reconstruction, particularly in the context of neural implicit function [51]. They optimize the network to implicitly encode a signed distance field or occupancy field with the supervision of ground truth. Preceding methodologies primarily employ object-level priors [37, 34, 18] to encode shapes, which are referred to as global

priors. However, the utilization of global priors restricts the generation ability as the network is unable to deal with shapes that are absent from the training set. Subsequently, numerous methodologies concentrate on local data priors [22, 16, 8, 21] to enhance generalization capacity by utilizing small receptive fields. The approaches for acquiring local data priors encompass regular grid [22], KNN [17, 16, 8], and octree [46, 44] and so on. Although local data priors enhance the generation ability of learning-based methodologies, the priors derived from limited data may result in excessively smoothed surfaces. At the same time, the reconstruction performance may further deteriorate if the distribution of the test point cloud significantly deviates from the training data [42].

2.3 Optimization-based Learning Methods

To enhance the generalization ability, the direct fitting of 3D representations from raw point clouds without supervision has been extensively investigated in recent years, enabling end-to-end prediction of the target surface. The majority of optimization-based reconstructions necessitate the fitting of individual shapes with specific network parameters, where the parameters are acquired by imposing additional constraints. SAL/SALD [1, 2] employs unsigned distances to facilitate sign-agnostic learning. IGR [19] and Neural-Pull [5] primarily utilize Eikonal terms to constrain the field to be an SDF. Additionally, several methodologies [3, 4, 55] have been modified from Neural-Pull in pursuit of enhanced quality. DiGS [6] incorporates Laplacian energy into neural implicit representation learning for unoriented point clouds. In summary, the preponderance of existing methodologies constrain the norm of gradients or minimize smoothness energy to attain equilibrium between geometric details and overall simplicity, but this inevitably results in ghost geometry or over-smoothed reconstruction outcomes. In this paper, our emphasis is on the alignment of the gradients and the Hessian of the SDF to more effectively regulate the direction of the gradients.

3 Method

3.1 Neural signed distance function

Given an unoriented point cloud P restricted in the range Ω : [ -1 , 1] × [ -1 , 1] × [ -1 , 1] , our task is to find a neural signed distance function (SDF) f θ : R 3 ↦→ R , such that f θ predicts a signed distance value for an arbitrary query point q ∈ Ω , where the neural function f θ is parameterized by θ and can be assumed to be C 2 -continuous everywhere in the domain. We use S l to denote the level-set surface at the value of l , i.e.,

The underlying surface can be naturally obtained by extracting the zero level-set surface S 0 , through the use of contouring algorithms, such as Marching cubes [30]. The task of this paper is to fit the high-fidelity implicit representation without supervision.

3.2 Surface vector calculus related to SDF

In order to fit the SDF, there are typically three types of boundary conditions for constraining f θ : (1) Dirichlet condition f θ ( p ) = 0 that requires each point p ∈ P to be situated on S 0 as far as possible, (2) Eikonal condition ∥∇ f θ ∥ 2 = 1 that enforces f θ be a distance field with unit gradients, and (3) Neumann condition ∇ f θ = N that aims to align the gradients with the normal field N . Existing neural implicit representations incorporate the aforementioned boundary conditions as constraints, either explicitly [19, 40] or implicitly [5, 1]. However, it is important to note that the Neumann condition cannot be enforced due to the absence of oriented normals.

According to the surface vector calculus [32], the SDF has a fundamental relationship with the

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differential properties of a smooth surface. Let J ∇ f θ denote the Jacobian matrix of the gradient ∇ f θ ,

or equivalently the Hessian matrix H f θ of f θ . Since this matrix is symmetric and diagonalizable, we can find the eigenvectors and eigenvalues of H f θ for any point where f θ is differentiable (a narrow thin-shell space surrounding the base surface). By differentiating both sides of the identity ∥∇ f θ ∥ 2 = 1 , we get:

which implies that ∇ f θ ( q ) is exactly an eigenvector of the Hessian H f θ ( q ) , with a corresponding eigenvalue of 0. To be more detailed, when q lies on the surface, the gradient represents the normal vector at q . Consequently, we have

Additionally, the other two eigenvectors of H f θ ( q ) define the principal directions at q [32, 36], with the eigenvalues respectively being the opposite of the corresponding principal curvatures. If q is off the surface but in the differentiable region, the three eigenvectors reflect the differential properties of the closest surface point of q , i.e., the projection of q onto the surface. In summary, the eigenvectors of the Hessian can disclose the fundamental properties of the underlying surface.

A similar property holds in the 2D setting. In Fig. 1, we provide a 2D circle as a toy example to illustrate this property. It's evident that in the vicinity of the base surface, the Hessian matrix of the SDF consistently exhibits two eigenvectors: one aligned with the gradient, and the other orthogonal to the gradient. Moreover, we visualize the distribution of the eigenvalues using a color-coded style, with the eigenvalue corresponding to the gradient being 0.

3.3 Gradient-Hessian Alignment

It should be noted that even if the underlying surface is infinitely smooth, the SDF may not necessarily be smooth everywhere. From a geometric perspective, the SDF is non-differentiable at medial-axis points with at least two nearest projections onto the surface. The neural implicit function f θ , being at least C 2 -smooth, is unlikely to be identical to the real SDF. We can make a reasonable assumption that f θ resembles the real SDF in the differentiable region that in the non-differentiable region. In light of this, it is reasonable to enforce the alignment between the gradients and the Hessian for points within a thin-shell space of appropriate width.

Based on the aforementioned analysis, we propose a loss function to promote the alignment of the Gradient-Hessian:

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Difference from the Eikonal term. By enforcing the Eikonal term, we can prevent the SDF from degenerating into a trivial field, such as f θ = 0 almost everywhere. But for a distance field rooted at any surface, ∥∇ f θ ∥ = 1 always holds except at the non-differentiable points. In practice, ∥∇ f θ ∥ = 1 is evaluated at discrete points, and the number of network parameters may significantly exceed the number of sample points. Consequently, specifying only ∥∇ f θ ∥ = 1 can result in ghost artifacts in the reconstructed surface, such as bulging effects, redundant parts, or excessive surface variations. For instance, consider the last row of Fig. 2. The reconstructed surfaces may differ significantly from the desired surface, even when ∥∇ f θ ∥ = 1 holds almost everywhere. This observation motivates us to seek more effective control over the gradient direction.

Why not smoothness energy. In previous literature, various forms of smoothness energy, such as Dirichlet Energy [28], Hessian energy [9, 54], and Laplacian energy [6], have been employed. Generally, the inclusion of smoothness terms can encourage smoothness in the SDF, reducing variations, and thereby mitigating the emergence of ghost geometries.

However, for these smoothing-based approaches, while they can reduce the occurrence of ghost geometry, they often sacrifice the ability to accurately represent geometric details. In contrast, our regularization term enforces the alignment between the gradient and the Hessian, which significantly differs from enforcing simple smoothness. As shown in Fig. 2, we created a 'L' shape with 100 boundary points, following the same experimental setup as DiGS [6], to illustrate the contrast between various approaches. Smoothing-based approaches tend not to reduce the smoothness energy to zero, even at termination. In contrast, our regularization term can approach zero more closely. Moreover, aligning the gradient and the Hessian follows an inherent property of the SDF, independent of the specific case. This sets our approach apart from smoothing-based methods that require case-by-case tuning of smoothing weights. As a result, our approach exhibits superior feature-preserving ability.

We conducted two ablation studies in Section 4.3, which clearly demonstrate that our alignment loss term not only more effectively suppresses the occurrence of ghost geometry but also yields more faithful reconstruction results than traditional smoothing-based approaches.

3.4 Loss function

Based on the aforementioned discussion, our neural function f θ has to be at least C 2 continuous. In this paper, we consider two types of activation functions. The first activation function is Sine, which facilitates learning high-frequency information as pointed out in SIREN [40]. Obviously, Sine is C inf continuous and satisfies our requirements. Additionally, Neural-Pull [5] leverages the SoftPlus activation function, which is a smooth version of ReLU and can also serve our purpose.

To this end, we establish the entire loss function by integrating the alignment loss into the existing loss configuration such that the network optimization can be formulated as

where E align ( f θ ) is given by

and E old ( f θ ) is the original loss function of existing works (the details will be discussed in the experiments). We leverage SIREN [40] without the normal version and Neural-Pull [5] in our experiment. Since the alignment effect between gradients and Hessian should be more emphasized near the underlying surface, we leverage an adaptive weighting scheme inspired by [15]:

we set δ = 10 by default. At the same time, we set the default value of α to 6. More details can be checked in our supplementary material.

4 Experiments

Metrics. The indicators for comparison include normal consistency, chamfer distances, and F-Score, where normal consistency (%, abbreviated as 'Normal C.') reflects the degree to which the normals of the reconstructed surface agree with the normals of the ground-truth surface, chamfer distance (scaled by 10 3 , using L 1 -norm) measures the fitting tightness between the two surfaces, and F-Score (%) indicates the harmonic mean of precision and recall (completeness). We set the default threshold of F-Score to 0.005. All meshes are uniformly scaled to [ -0 . 5 , 0 . 5] , and 100K points are sampled from each mesh for evaluation.

4.1 Optimization-based Surface Reconstruction

Surface Reconstruction Benchmark (SRB). The Surface Reconstruction Benchmark (SRB) [49] contains five shapes, each of which has challenging features, e.g., missing parts and rich details. The approaches for comparison include screened Poisson surface reconstruction (SPSR) [23], SIREN [40], Shape as points (SAP) [38], iPSR [20], Predictive Context Priors (PCP) [3], CAP-UDF [55] and DiGS [6]. Note that SPSR leverages normals given by the input scans. As shown in Tab. 1, our method outperforms the existing methods, as evidenced by superior results in both Chamfer distance and F-score metrics. In particular, the visual comparison presented in Fig. 3 demonstrates that our method is capable of accurately recovering the hole feature of the Anchor model, despite the absence of points on the inner wall. Furthermore, it successfully preserves the adjacent gaps of the Lord Quas model. In addition, we compared our approach with DiGS using SRB, following the evaluation settings of DiGS. Detailed performance statistics can be found in the supplementary material.

ShapeNet. The ShapeNet dataset [13] comprises a diverse collection of CAD models. We follow the splitting of [50] for the 13 categories of shapes with total of 260 shapes. Our comparison is performed using 3K points. The baseline methods include Screened Poisson Surface Reconstruction (SPSR) [23], NSP [50], SAL [1], IGR [19], SIREN [40], DiGS [6], OSP [4], iPSR [20] and PGR [27]. Note that SPSR and NSP require normal inputs. To ensure the validity of our experiments, we provide

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ground-truth normals to those that require normal information. Additionally, we include the learnable baselines Shape as Points (SAP) [38] and POCO [8] for comparison and retrain them from scratch on ShapeNet with 3K points. Based on the quantitative results presented in Tab. 2 and the visual comparison in Fig. 4. Our method effectively suppresses unnecessary surface variations and adapts the implicit representation to the inherent complexity encoded by the point cloud. Furthermore, our reconstruction accuracy is comparable to supervised methods. Specifically, our method outperforms POCO in Normal Correctness, Chamfer, and F-Score, but lags slightly behind POCO in terms of IOU.

ABC and Thingi10K. The ABC dataset [24] contains a variety of CAD meshes, while Thingi10K [57] comprises shapes with rich geometric details. We follow [16] to perform splitting for each dataset and randomly sample 10K points from each mesh. Each dataset contains 100 shapes. The baseline methods include Screened Poisson Surface Reconstruction (SPSR) [23], SAL [1], IGR [19], SIREN [40], Neural-Pull [5], Shape as Points (SAP) [38], DiGS [6], iPSR [20], and PGR [27]. Note that we found that the supervised version of SAP does not generalize well to shapes not present in the training set (ShapeNet) when using a global PointNet-based encoder. Therefore, we compare our approach against the unsupervised version of SAP. We also include the supervised method POCO [8] for comparison. To assess the generalization ability of supervised methods, we retrain them using 10K points on ShapeNet. The quantitative comparison results are presented in Tab. 3. Furthermore, the visual comparison on ABC [24] and Thingi10K [57] (see Fig. 4) demonstrates that our method can effectively recover CAD features such as small holes and thin plates and has ability to accurately recover high-fidelity geometric details.

3D Scene. Finally, we conducted tests on a 3D scene dataset [56] by sampling 20K points for each model. The quantitative comparison results are presented in Tab. 4 and visual comparison in Fig. 5. Our method, when combined with either SIREN [40] or Neural-Pull [5], yields significant improvements even with only 20K points for the scene-level reconstruction.

Notably, our method effectively captures both the details and overall shape of these challenging scenes.

4.2 Shape Space Learning

The D-Faust [7] dataset contains high-resolution raw scans (triangle soups) of 10 humans with various poses. We follow DualOctreeGNN [46] to perform splitting that 6K scans are used for training and

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2K scans for testing. In the training phase, we adopt the encoder following Convolutional Occupancy Networks [41]. Besides, we adopt the FiLM conditioning [11] that applies an affine transformation to the network's intermediate features as SIREN is weak in handling high-dimensional inputs [11, 33]. The baseline approaches include IGR [19], SAL [1], SALD [2], DualOctreeGNN [46] and DiGS [6]. Also, we further demonstrate the results of IGR, as well as the results of DualOctreeGNN trained without normals. As shown in Fig. 6, although IGR can generate well details, it yields spurious planes away from the input. SAL, supervised with unsigned distance, can only produce smooth results. Additionally, SALD, with the support of normal supervision, can generate more details, but its reconstruction accuracy is even worse than SAL since it suffers from a large systematic misalignment that does not respect input poses.

By contrast, DualOctreeGNN gets the most impressive results since the well-designed octree network can capture local prior for details. However, the performance of either IGR or DualOctreeGNN is compromised without normals.

DiGS cannot yield reliable results though it is also based on SIREN. To summarize, our method can learn shape space without requiring input normals or additional supervision but still produces faithful shapes. It's important to note that the resulting surfaces of DualOctreeGNN for point clouds without input normals are not watertight, despite the small Chamfer distances.

4.3 Ablation Study

Our ablation studies were conducted on the ShapeNet dataset [13], which comprises 13 categories of shapes, with 20 shapes per category for a total of 260 shapes, each represented by 3K points. We use SIREN as the activation function for our baseline.

Comparison with smoothness energies In Figure 7, we increased the weighting coefficient of the regularization term to 1 × 10 3 , 1 × 10 5 , and 1 × 10 7 , respectively, for both our approach and those based on smoothing. It's evident that even with a large weight, our regularization term consistently produces high-fidelity surfaces, in contrast to the smoothing-based approaches, whereas their reconstruction results may become overly smooth or even fail (Note that double layers for Dirichlet energy).

A similar trend is observed in the 3D context. As illustrated in Figure 9, our method excels in reconstructing faithful and high-fidelity shapes from input data with 100K points, while the other approaches tend to produce over-smoothed results.

We also conducted an ablation study to compare the quantitative performance of our energy term with that of Dirichlet Energy [28] and Hessian Energy [6]. The results presented in Tab. 8 demonstrate that our energy term E align ( f θ ) significantly improves the performance. It's worth noting that the comparison between DiGS [6] (with a Laplacian energy) and ours has been made in previous subsections, which shows that ours has a superior performance. Therefore, in this subsection, we do not include Laplacian energy for comparison.

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Effect of weights of L align . We conducted an ablation study to evaluate the effect of the weight α of L align. By selecting different values, i.e., α = { 0 . 06 , 0 . 6 , 6 , 60 } , we keep the statistics in Tab. 6, which demonstrates that the best performance was achieved when the weight was set to 6.

Effect of adaptive per points weights.

We demonstrate the effectiveness of the adaptive weight β q for each query in Tab. 7. We also compare the performance of different decay parameters δ used to compute β q in Eq. 8, and find that δ = 10 yields the best results.

Comparison to LSA. Recently, LSA [31] proposed enforcing alignment between the gradient of each level set and the gradient of the 0-level set to account for the direction of the gradient. However, it is important to note that the gradient of the 0-level set may not align with the true surface normals for its results. As such, using

the gradient of the 0-level set as a reference may not be particularly effective. We present results for SIREN [40] using both our loss and LSA under six shapes from the Stanford Scanning dataset with 2 million points same as the settings of LSA in its main paper. As shown in Fig. 8, our method effectively eliminates ghost geometry in empty areas due to incorrect gradient direction, while LSA does not. Tab. 9 presents the comparison statistics for three different approaches: 1) our methods combined with SIREN [40], 2) original SIREN and 3) LSA.

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5 Conclusion

In this paper, we propose a novel approach for surface reconstruction from unoriented point clouds by aligning the gradient and the Hessian of the Signed Distance Function (SDF). Unlike existing smoothness energy formulations that minimize SDF volatility, our approach offers more effective control over gradient direction, allowing the implicit function to adapt to the inherent complexity of the input point cloud. Comprehensive experimental results demonstrate that our approach effectively suppresses ghost geometry and recovers high-fidelity geometric details, surpassing state-of-the-art methods in terms of reconstruction quality.

Our current algorithm still has a few drawbacks. Firstly, it struggles with processing super sparse inputs, such as sketch point clouds or LiDAR data (as shown in Figure 10). The inherent challenge arises from completing extensive missing parts and closing the gaps between the input stripes. Secondly, it faces difficulties when dealing with large-scale scenes, like those from Matterport3D [12], mainly due to the neural networks' catastrophic forgetting issue. Representing large-scale scenes within a single network becomes exceedingly challenging. In our future work, we plan to address these challenges. To handle sparse inputs more effectively, we are considering implementing a supervision mechanism. Additionally, we aim to adopt a sliding window strategy, similar to approaches like DeepLS [10] and BlockNeRF [43], to mitigate the catastrophic forgetting issue.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a new smoothing loss to better regularize the estimated Signed Distance Field (SDF) from unoriented point clouds. By aligning the SDF's gradient and its Hessian, the quality of reconstructed surfaces improves significantly. Compared with commonly used Eikonal loss, the proposed loss is more effective. Experiments validate the effectiveness of the proposed method as well as not so sensitive to input noise and hyper-parameters.\n\nSTRENGTHS:\n1. The proposed loss is novel and theoretically reasonable.\n2. The loss could be easily worked with different methods as a plugin.\n3. The experiments are almost complete and big improvements validate the effectiveness.\n\nWEAKNESSES:\nThe paper is poorly written. The authors write the paper more like from a mathematic point of view rather than computer vision point of view. Many details are missing, and some said \"obviously\" and \"evidence\" are not clear at all. For a computer vision paper, Figures are much more important for illustration and detailed explanation of Figures and Equations, especially where to look at in the figure and why an equation intuitively works.\n\nHowever, if the authors could provide details in the rebuttal process and the final version, I believe that the paper could be accepted. If not, it should be rejected, as this is not a math paper.\n\nQUESTIONS:\n1.\tThe first paragraph in Sec 3.3 requires figure illustration. It’s difficult to understand why Equ. 4 intuitively works. What is Equ. 4 enforcing? What do you mean by \"alignment\"? A few figures are necessary. It is not clear at all where Equ. 4 comes from.\n2.\tIn Figure 2, The 100 points inputs are different for three methods? Please show the same inputs’ results as comparisons; otherwise, it is  not convincing.\n3.\tIn Figure 3, it is NOT evidence that the alignment loss can suppress the ghost geometry as mentioned in Line 173. The authors need clearly show which region in Figure 3 is problematic, and why the last column is good. It seems the alignment loss works similar to Dirichlet loss, and I do not see any benefit. Are the black lines zero-isosurface? Then the last column’s result seems quite bad.\n4.\tThe big Q in Equ. 7 is not defined.\n5.\tTable 5’s caption needs explanations of each row, especially the difference between same name’s rows.\n\nLIMITATIONS:\nThe current experiments only show surface reconstruction from point clouds. But I think the loss could benefit other scenarios, like incorporated into Neus, but the authors do not show its effectiveness. Also, whether the loss works for unsigned distance field (UDF) also? This could be an interesting future topic. The current point cloud scenario is somewhat limited.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper under review proposes a method for signed distance function (SDF) reconstruction from point cloud data without the use of normal data at the points.  The paper proposes a new term as a regularization to complement the EIkonal loss, which is a necessary condition for an SDF.  The purpose of this regularization is to reduce \"ghosting\" effects.  The term is obtained by differentiating the Eikonal constraint, resulting in a second-order constraint, namely that the Hessian times the gradient is zero.  The paper shows experimental comparison to SOA on multiple benchmarks with favorable results.\n\nSTRENGTHS:\n- New regularization term for neural SDFs; original to best of my knowledge; neural SDFs are a very relevant research topic of interest in vision/graphics\n- Paper is mostly clear\n- Experiments on many benchmark datasets, with favorable results to the authors' approach\n\nWEAKNESSES:\n- Experimental results on SRB and ShapeNet use different metrics than recent papers in area, e.g., DiGS[6] - not clear why they have been changed.  Please report results with previous protocols.\n- A new sampling strategy is reported in supplementary for the alignment loss.  Not clear what the effect of this is, and why it is needed.\n- Theoretical justification is weak, e.g., line 156-174; there is no clear understanding of this new term - no clear explanation of why the ghosting is reduced with this term.\n--The authors attribute ghosting to in practice only applying the Eikonal constraint to a sampling of points; unclear if this is true due to random sampling or this is only part of the problem.  The Eikonal equation does not have a unique solution which is in part why there is ghosting.\n--Unclear why the smoothness term not being min zero means it's difficult to regulate. \n--It is stated that the norm and direction of the gradient is de-coupled.  However, a minimization of the loss is being considered - the optimizing flow for both the Eikonal and alignment will both change the direction & norm of the gradient during minimization.\n- Line186-187: the constraint is true a.e. so why is the adaptive weighting needed?\n\nQUESTIONS:\nAddress the weaknesses.\n\nLIMITATIONS:\nNot discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-02"}

comments: SUMMARY:
The authors provide a novel regularizing term for training of implicit neural SDF representations, motivating the function gradient to be a zero eigenvector of the Hessian near the zero level set (the surface). They demonstrate that it can be used to reduce the appearance of ghost geometry and increase quality of the representations.

STRENGTHS:
- A principled novel regularization term that seems to significantly improve the learned representations.
- Is tackling an important application domain, as neural geometry representations are broadening the application domain of deep learning techniques.

WEAKNESSES:
- It seems unclear to me how the method is specifically designed to handle unoriented point clouds. If there is a claim that the regularizer somehow motivates consistent alignment of resulting normals, it'd be nice to have some commentary on this.
- The evaluation metrics are a little strange to me:
   - It's unclear how they are comparing normals, as there would need to be a correspondence between the reconstructed surface and the ground truth surface.
   - F-score seems a bit of an odd choice. I initially assumed that they are using the interior of the isosurface to specify points classified as inside the volume, but the default threshold 0.005 seems quite low for this. So perhaps they are asking for the network output to classify points on the shape, but I am not sure of how they do this either, as nearly all points will evaluate to nonzero values. 
  - In my opinion, Hausdorff would be good to include, as it would give a sense of the maximum error.

QUESTIONS:
See above weaknesses.

LIMITATIONS:
The authors adequately address the limitations of the methods, which are mostly that there is still room for further improvement in their output.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2023-07-05

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the task of surface reconstruction from point clouds without normals and uses neural signed distance functions. They consider previous smoothness losses, and come up with their own by taking the derivative of the Eikonal equation, thus guaranteeing that their loss constrains a fundamental property of SDFs. They motivate this loss in various other ways (which to me are fairly unclear). They also show good results on many datasets, and compare to many different types of approaches.\n\nSTRENGTHS:\n- Using second order information derived from the Eikonal equation makes sense\n- Lots of experiments and comparisons to many types of methods\n\nWEAKNESSES:\n- One of the benefits of the split of the ShapeNet dataset is that you use is that you can compute IoU with it, which is a very different but important measure to IoU.\n- Your explanation for why L_{align} is different from the Eikonal term is essentially that it works better for finitely many points, which is not an explanation. You should explain that with finitely many points constraining second order information reduces the number of possible solutions, and maybe allude to the toy problem in DiGS (or even construct the same toy problem with your loss term).\n- Your explanation for why not smoothness energy is not very compelling either. Your main argument is that other energies are not 0 at the optimum, hence it is hard to choose how regularisation to apply. A much better argument is your earlier point that the loss is derived from a fundamental property of SDFs, while the others are general smoothing losses that are not specifically guiding towards a proper SDF and will be smoothing in unwanted ways.\n\nOverall great idea and results, not great presentation/soundness. Your explanation for your loss needs to be more clear, keeping it to it is a fundamental property of an SDF works just fine. A toy problem to back this up could be to see what happens when you regularise the 2D shapes with high weight with the different smoothness energies, mostly likely other methods would stop obeying the points and go for a smoother surface while yours would still obey the input points.\n\nQUESTIONS:\n- Is there a reason you are using most of DiGS' datasets but not using their version of the metrics?\n- Your explanation of the implication is Rodigues' formula is not clear. I couldn't find the formula name in your citation [34], and looking else online it seems that the formula states that one of the eigenvalue-eigenvector pairs of the Hessian should be a principle curvature, principle direction pair, not all eigenvalue-eigenvector pairs? Why does it imply equation (3)?\n- What is Figure 3 trying to show us?\n\nLIMITATIONS:\nNot discussed.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel approach to learning the SDF directly from point clouds without the use of surface normals. The key insight behind this is that aligning the gradient of the SDF with its Hessian allows for better training of the distance field. Through extensive experiments, the paper demonstrates the effectiveness of the proposed approach in accurately recovering the underlying shape and effectively reducing geometry artifacts.\n\nSTRENGTHS:\n- The paper exhibits a good writing style, effectively conveying the ideas and concepts to the readers. The content is clear, concise, and well-structured, allowing for easy understanding of the proposed method.\n\n- The paper introduces a novel method for learning the signed distance function directly from unoriented point clouds. \n\n- The paper presents extensive experimental results that showcase the effectiveness of the proposed approach.\n\nWEAKNESSES:\n- The paper does not explicitly discuss potential limitations or challenges of the proposed method. Addressing and acknowledging these limitations would provide a more comprehensive understanding of the method's applicability and potential areas for improvement.\n\n- The paper lacks a time analysis of the proposed method. While the experimental results demonstrate the effectiveness of the approach, the paper does not provide insights into the computational efficiency or time complexity of the method. Understanding the time requirements would be valuable for assessing its practicality and potential for real-time applications.\n\nQUESTIONS:\nCan the proposed method guarantee the reconstruction of watertight surfaces? For large-scale scenes, are the reconstructed surfaces watertight?\n\nLIMITATIONS:\nThe paper lacks a discussion of potential limitations and the negative social impact of the proposed method.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}