Title: NeuralGF: Unsupervised Point Normal Estimation by Learning Neural Gradient Function
Venue: NeurIPS 2023
Accepted: yes
Keywords: Point Clouds

Normal Estimation

Neural Gradient
Abstract: Normal estimation for 3D point clouds is a fundamental task in 3D geometry processing. The state-of-the-art methods rely on priors of fitting local surfaces learned from normal supervision. However, normal supervision in benchmarks comes from synthetic shapes and is usually not available from real scans, thereby limiting the learned priors of these methods. In addition, normal orientation consistency across shapes remains difficult to achieve without a separate post-processing procedure. To resolve these issues, we propose a novel method for estimating oriented normals directly from point clouds without using ground truth normals as supervision. We achieve this by introducing a new paradigm for learning neural gradient functions, which encourages the neural network to fit the input point clouds and yield unit-norm gradients at the points. Specifically, we introduce loss functions to facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning a global surface representation of the data. Meanwhile, we incorporate gradients into the surface approximation to measure the minimum signed deviation of queries, resulting in a consistent gradient field associated with the surface. These techniques lead to our deep unsupervised oriented normal estimator that is robust to noise, outliers and density variations. Our excellent results on widely used benchmarks demonstrate that our method can learn more accurate normals for both unoriented and oriented normal estimation tasks than the latest methods. The source code and pre-trained model are publicly available.

NeuralGF: Unsupervised Point Normal Estimation by Learning Neural Gradient Function

Abstract

Normal estimation for 3D point clouds is a fundamental task in 3D geometry processing. The state-of-the-art methods rely on priors of fitting local surfaces learned from normal supervision. However, normal supervision in benchmarks comes from synthetic shapes and is usually not available from real scans, thereby limiting the learned priors of these methods. In addition, normal orientation consistency across shapes remains difficult to achieve without a separate postprocessing procedure. To resolve these issues, we propose a novel method for estimating oriented normals directly from point clouds without using ground truth normals as supervision. We achieve this by introducing a new paradigm for learning neural gradient functions, which encourages the neural network to fit the input point clouds and yield unit-norm gradients at the points. Specifically, we introduce loss functions to facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning a global surface representation of the data. Meanwhile, we incorporate gradients into the surface approximation to measure the minimum signed deviation of queries, resulting in a consistent gradient field associated with the surface. These techniques lead to our deep unsupervised oriented normal estimator that is robust to noise, outliers and density variations. Our excellent results on widely used benchmarks demonstrate that our method can learn more accurate normals for both unoriented and oriented normal estimation tasks than the latest methods. The source code and pre-trained model are available at https://github.com/LeoQLi/NeuralGF.

1 Introduction

Normal vectors are one of the local descriptors of point clouds and can offer additional geometric information for many downstream applications, such as denoising [70, 45, 44, 43], segmentation [6365] and registration [62, 68]. Moreover, some computer vision tasks require the normals to have consistent orientations, i . e ., oriented normals, such as graphics rendering [10, 21, 61] and surface reconstruction [26, 31-33]. Therefore, point cloud normal estimation has been an important research topic for a long time. However, progress in this area has plateaued for some time, until the recent introduction of deep learning. Currently, a growing body of work shows that performance improvements can be achieved using data-driven approaches [24, 8, 40, 39], and these learning-based methods often

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give better results than traditional data-independent methods. However, they suffer from shortcomings in terms of robustness to various data and computational efficiency [36]. Besides, they usually rely on a large amount of labeled training data and network parameters to learn high-dimensional features. More importantly, the presence of varying noise levels, uneven sampling densities, and various complex geometries poses challenges for accurately estimating oriented normals from point clouds.

Most existing point cloud normal estimation methods aim to locally fit geometric surfaces and compute normals from the fitted surfaces. Recent surface fitting-based methods, such as DeepFit [8], AdaFit [86] and GraphFit [38], generalize the n -jet surface model [15] to learning-based frameworks that predict pointwise weights using deep neural networks. They obtain surface normals by solving the weighted least squares polynomial fitting problem. However, as shown in Fig. 1(a), their methods can not guarantee consistent orientations very well, since n -jet only fits a local surface, which is ambiguous for determining orientation. Furthermore, their methods need to construct local patches based on each query point for surface fitting. This results in the algorithm needing to traverse all patches in the point cloud, which is time-consuming. Most importantly, they rely on the priors learned from normal supervision to fit local surfaces, while such supervision is not always available.

Inspired by neural implicit representations [6, 7, 46, 49, 16], we introduce neural gradient functions (NeuralGF) to approximate the global surface representation and constrain the gradient during the approximation. Our method includes two basic principles: (1) We formulate the neural gradient learning as iteratively learning an implicit surface representation directly from the entire point cloud for global surface fitting. (2) We constrain the derived surface gradient vector to accurately describe the local distribution and make it consistent across iterations. Accordingly, we implement these two principles using the designed loss functions, resulting in a model that can provide accurate normals with consistent orientations from the entire point cloud (see Fig. 1(b)) through a single forward pass, and does not require ground truth labels or post-processing. Our method improves the state-of-the-art results while using much fewer parameters. To summarize, our main contributions include:

Introducing a new paradigm of learning neural gradient functions, building on implicit geometric representations, for computing consistent gradients directly from raw data without ground truth.

Implementing the paradigm with the losses designed to train neural networks to fit global surfaces and estimate oriented normals with consistent orientations in an end-to-end manner.

Reporting the state-of-the-art performance for both unoriented and oriented normal estimation on point clouds with noise, density variations, and complex geometries.

2 Related Work

Unoriented Normal Estimation . Point cloud normal estimation has been extensively studied over the past decades and widely used in many downstream applications. In general, most of these studies focus on estimating unoriented normals, i . e ., finding more accurate perpendiculars of local surfaces. The Principle Component Analysis (PCA) [26] is the most commonly used algorithm that has been widely applied to various geometric processing tasks. Later, PCA variants [1, 59, 54, 35, 27], Voronoi

diagrams [3, 18, 2, 51] and Hough transform [11] have been proposed to improve the performance by preserving the shape details or sharp features. Moreover, some methods [37, 15, 23, 56, 4] are designed to fit complex surfaces by containing more neighboring points, such as moving least squares [37], truncated Taylor expansion ( n -jet) [15] and spherical surface fitting [23]. These methods usually suffer from the parameters tuning, noise and outliers. To achieve more robust capability, they usually choose a larger neighborhood size but over-smooth the details. In recent years, deep learning-based normal estimation methods have been proposed and achieve great success. The regression-based methods predict normals from structured data or raw point clouds. Specifically, some methods [12, 66, 43] transform the 3D points into structured data, such as 2D grid representations. Inspired by PointNet [63], other methods [24, 83, 25, 9, 81, 82, 40, 41, 19, 39] focus on the direct regression for unstructured point clouds. In contrast to regressing the normal vectors directly, the surface fitting-based methods aim to train a neural network to predict pointwise weights, then the normals are solved through weighted plane fitting [36, 14] or weighted polynomial surface fitting [8, 86, 84, 80, 38] on local neighborhoods. Although these deep learning-based methods perform better than traditional data-independent methods, they require ground truth normals as supervision. In this work, we propose to use networks to estimate normals in an unsupervised manner.

Consistent Normal Orientation . Estimating normals with consistent orientations is also important and has been widely studied. The most popular and simplest way for normal orientation is based on local propagation, where the orientation of a seed point is diffused to the adjacent points by a Minimum Spanning Tree (MST). Most existing normal orientation approaches are based on this strategy, such as the pioneering work [26] and its improved methods [34, 69, 72, 67, 76, 29]. More recent work [53] integrates a neural network to learn oriented normals within a single local patch and introduces a dipole propagation strategy across the partitioned patches to achieve global consistency. These propagation-based approaches usually assume smoothness and suffer from local error propagation and the choice of neighborhood size. Since local consistency is usually not enough to achieve robust orientations across different inputs, some other methods determine the normal orientation by applying various volumetric representation techniques, such as signed distance functions [50, 55], variational formulations [71, 2, 28], visibility [30, 17], active contours [75], isovalue constraints [74] and winding-number field [77]. In addition to the above approaches, the most recent researches [24, 25, 73, 39] aim to learn oriented normals directly from point clouds in a data-driven manner. Similar to the unoriented normal estimation task, these deep learningbased methods perform better than traditional methods, but they require expensive ground truth as supervision. In contrast, our method achieves better performance in an unsupervised manner.

Neural Implicit Representation . Recent works [52, 57] propose to learn neural implicit representation from point clouds. Later, one class of methods [6, 7, 46, 60, 85, 47, 48] focuses on training neural networks to overfit a single point cloud by introducing newly designed priors. SAL [6] uses unsigned regression to find signed local minima, thereby producing useful implicit representations directly from raw 3D data. IGR [22] proposes an implicit geometric regularization to learn smooth zero level set surfaces. Neural-Pull [46] uses the signed distance to move a point along the gradient for finding its nearest path to the surface. SAP [60] proposes a differentiable Poisson solver to represent the surface as an oriented point cloud at the zero level set of an indicator function. These methods focus on accurately locating the position of the zero iso-surface of an implicit function to extract the shape surface. However, they ignore the constraints on the gradient of the function during optimizing the neural network. We know that the gradient determines the direction of function convergence, and the gradient of the iso-surface can be used as the normal of the extracted surface. If the gradient can be guided reasonably, the convergence process can be more robust and efficient, avoiding local extremum caused by noise or outliers. Motivated by this, we try to incorporate neural gradients into implicit function learning to achieve accurate oriented normals.

3 Method

Preliminary . Neural implicit representations are widely used in representing 3D geometry and have achieved promising performance in surface reconstruction. Benefiting from the adaptability and approximation capabilities of deep neural networks [5], recent approaches [57, 22, 46, 48] use them to learn signed distance fields as a mapping from 3D coordinates x to signed distances. These approaches represent surfaces as zero level sets of implicit functions f , i . e ., S = { x ∈ R 3 | f ( x ; θ ) = 0 } , where f : R 3 → R is a network with parameter θ , which is usually implemented as a multi-layer perceptron

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(MLP). If the function f is continuous and differentiable, the normal of a point p on the surface can be represented as n p = ∇ f ( p ) / ∥∇ f ( p ) ∥ , where ∥·∥ denotes the euclidean L 2 -norm and ∇ f ( p ) indicates the gradient at p , which can be obtained in the differentiation process of f . Specifically, there are some important properties of the gradient: (1) The maximum rate of change of the function f is defined by the magnitude of the gradient ∥∇ f ∥ and given by the direction of ∇ f . (2) The gradient vector ∇ f ( p ) is perpendicular to the level surface f ( p )=0 .

Overview . We aim to train a neural network f with parameter θ from input points x to obtain the gradient ∇ f during the optimization of f . The optimization encourages f to fit x and the gradient ∇ f to approach the normal of the implicit surface. We sample a query point set Q = { q j } N Q j =1 and a surface point set G = { g j } N G j =1 from raw point cloud P . Specifically, Q is sampled through some probability distribution D from P [13, 46]. To explicitly indicate the underlying geometry, we sample the point set G ⊂ P that may be distorted by noise, and we verify that it can facilitate the learning of f (see Sec. 4.2). As shown in Fig. 2, our optimization pipeline is formulated as an iterative moving operation of the union of points Q and G , more specifically, with I steps of position movement. During normal estimation, the entire point cloud P is fed into the network to derive the gradient embedded in the learned function. To learn the neural gradient function based on the implicit surface representation of f , we consider a loss of the form

where T : R × R → R is a differentiable similarity function, and we adopt the standard L 2 (Euclidean) distance. V ( x ; θ ) and F ( x ; θ ) represent the approximated gradient vector and global surface representation of the underlying geometry P , respectively. V ( x ) and F ( x ) denote the corresponding measures, respectively. We will introduce them in the following two sections. λ > 0 is a weighting factor. The expectation E is made for certain probability distribution x ∼ D in 3D space.

3.1 Learning Global Surface

The n -jet surface model represents a polynomial function that maps points ( x, y ) ∈ R 2 to their height z that is not in the tangent space over a local surface [8, 86, 38]. In contrast, we aim to learn an implicit global surface representation of the point cloud. The gradient indicates the direction in 3D space in which the signed distance from the surface increases the fastest, so moving a point along the gradient will find its nearest path to the surface. According to this property, we adopt a moving operation introduced in [46] to project a query point q to q ′ , and q ′ = q -f ( q ; θ ) · n q , n q = ∇ f ( q ; θ ) / ∥∇ f ( q ; θ ) ∥ . We also use ∇ f to denote the normalized gradient for simplification. For the point q ∈ Q , we expect the function f could provide correct signed distance f ( q ; θ ) and gradient ∇ f ( q ; θ ) that can be used to move q to the nearest point ˆ q over the surface, and we minimize the error ∥ q ′ -ˆ q ∥ . In this way, the function f can be used as an implicit representation of the underlying surface and the zero level set of f will be a valid manifold describing the point cloud. In

this work, the moving operation for input point set { Q,G } is formulated as

where { Q ′ , G ′ } is the position of point set { Q,G } after moving in multiple steps I (see Fig. 2), and { ˆ Q, ˆ G } is the target position. f i represents the signed distance of points during the i -th position movement step. In practice, we find the nearest point of Q and G from the raw point cloud P as the target position. Different from previous works [46-48], we introduce a multi-step movement strategy to optimize the point set { Q ′ , G ′ } to cover the surface instead of single-step movement. The insight of our design is that it is always difficult to directly reach the target in one step if some points are distributed far from the surface or distorted by noise. At this time, the optimization of the moving operation may go through many twists and turns to converge.

We optimize L d ( θ ) in an end-to-end manner based on the moved point set { Q ′ , G ′ } . For the points G ⊂ P in the first step, we expect them to be on the underlying surface. For the points { Q ′ , G ′ } moved after the first step, we expect them to be as close to the underlying surface as possible. Hence, we leverage a regression loss term on the predicted distances f Q i and f G i of point set { Q,G } , i . e .,

3.2 Learning Consistent Gradient

A standard way for estimating unoriented point normals is to fit a plane to the local neighborhood N ( p, P ) of each point p [37]. Given a neighborhood size K , we let V ( p ) ∈ R K × 3 denote the set of centered coordinates of points in that neighborhood. Then the plane fitting is described as finding the least squares solution of

However, the inner product in the formula allows the distribution of normals on both sides of the surface (positive or negative orientations) to satisfy the optimization. Therefore, this approach cannot determine the normal orientation with respect to the surface, and a consistent orientation needs to be solved by post-processing, such as MST-based orientation propagation. In this work, we directly estimate oriented normals by incorporating neural gradients into implicit function learning [6, 7],

where n is the unit gradient (normal) of each point on the surface. The above formula aims to measure the minimum deviation of each point from the underlying surface. Specifically, the gradient is perpendicular to the surface and the distance from the surface increases the fastest along the direction of the gradient. Therefore, we measure the deviation from P to Q by finding the optimal gradients. Generally, raw point clouds may contain noise and outliers that can severely degrade the accuracy of the surface approximation. To reduce the impact of inaccuracy in P , we introduce multi-scale neighborhood size { K s } N K s =1 for V ( x ) based on a statistical manner, that is

where N K s ( q, P ) denotes the K s nearest points of q ∈ Q in P . V ( q ) is built as a vector from the averaged position ¯ x = ∑ K s k =1 N K s ( q, P ) /K s on surface to query q . Then, the objective of Eq. (5) is turned into the aggregation of the error at each neighborhood size scale, i . e .,

where i =1 , · · · , I is the number of steps. In practice, we only use f 1 of the first step and its ∇ f 1 to measure the deviation, since we want the optimization to converge as fast as possible, which also implies that the position Q ′ after the first move step in Eq. (2) is as close to the surface as possible.

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In addition, we pursue better gradient consistency across iterations, i . e ., make the gradient directions in each move step parallel. To achieve this, we align the gradient ∇ f i after the first step with the gradient ∇ f 1 of the first step. Meanwhile, we also consider the confidence of each point with respect to the underlying surface. Thus, we introduce confidence-weighted cosine distance to evaluate the gradient consistency at points { Q,G } , with the form of

where ⟨·⟩ denotes cosine distance of vectors. w =exp( -ρ · | f 1 | ) is an adaptive weight that indicates the significance of each input point based on the predicted distance, making the model pay more attention to points near the surface. In summary, our method combines an implicit representation and a gradient approximation with respect to the underlying geometry of P . We not only consider the gradient consistency at q during iterations, but also unify the adjacent regions of q to compensate for the inaccuracy of P . This is important because the input point cloud may be noisy and some points are not on the underlying surface.

Loss Function . We optimize global surface approximation and consistent gradient learning in an end-to-end manner by adjusting the parameter θ of f using the following loss function

where λ 1 = 0 . 01 , λ 2 = 0 . 1 and λ 3 = 10 are balance weights and are fixed across all experiments.

Neural Gradient Computation . The above losses require incorporating the gradient ∇ f ( x ; θ ) in a differentiable manner. As in [22, 7], we construct ∇ f ( x ; θ ) as a neural network in conjunction with

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f ( x ; θ ) . Specifically, each layer of the network has the form y ℓ +1 = ϕ ( Wy ℓ + b ) , where θ =( W , b ) is the learnable parameter of ℓ layer with output y ℓ +1 , and ϕ : R → R is a non-linear derivable activation function. According to chain-rule, the gradient is given by [22]

where ϕ ′ is the derivative of ϕ , and diag ( z ) arrange its input vector z ∈ R m on the diagonal of a square matrix R m × m . In practice, we use the automatic differentiation module of PyTorch [58] to implement the computation of the gradient ∇ f ( x ; θ ) in the forward pass.

4 Experiments

Architecture . To learn the neural gradient function, we use a simple neural network similar to [6, 46, 22]. It is composed of eight linear layers and a skip connection from the input to the intermediate layer. Except for the last layer, each linear layer contains 512 hidden units and is equipped with a ReLU activation function. The last layer outputs the signed distance for each point. The parameters of the linear layer are initialized using the geometric initialization [6]. After optimization, we use the network to derive pointwise gradients from the data P (see Fig. 2).

































































































Distribution D . We make the distribution D concentrate in the neighborhood of P in 3D space. Specifically, it is set by uniform sampling points p from P and placing an isotropic Gaussian N ( p, σ 2 ) centered at each p , with standard deviation parameter σ is adaptively set to the distance from the l -th nearest point to p [6, 7]. We set l to 25 for the PCPNet dataset and 50 for the FamousShape dataset.

Implementation . We choose the neighborhood scale set { K s } N K s =1 as { 1 , K/ 2 , K } , K = 8 , and set the parameter ρ of adaptive weight to 60 . We select N Q = 5000

points from the distribution D and N G =2500 points from P to form the input { Q,G } . The number of steps is set to I =2 for all evaluations. As for metric, we use normal angle Root Mean Squared Error (RMSE) to evaluate the estimated normals and Percentage of Good Points (PGP) to show the normal error distribution [40, 39].

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4.1 Performance Evaluation

Evaluation of Oriented Normal . In this evaluation, we not only compare with one-stage baseline methods, including PCPNet [24], DPGO [73] and SHS-Net [39], but also compare with twostage baseline methods that combine representative algorithms based on different design philosophies. Specifically, we use the various combinations of unoriented normal estimation methods (PCA [26], AdaFit [86] and HSurf-Net [40]) and normal orientation methods (MST [26], SNO [67] and ODP [53]). Note that PCPNet, DPGO, SHS-Net, AdaFit and HSurf-Net rely on supervised training with ground truth normals. We report quantitative evaluation results on datasets PCPNet and FamousShape [39] in Table 1. Our method has better performance under the vast majority of data categories (noise levels and density variations), and achieves the best average results. It is worth

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mentioning that our method achieves a large performance improvement on the FamousShape dataset, which contains different shapes with more complex geometries than the PCPNet dataset. We provide a visual comparison of the estimated normals in Fig. 3, which shows normal vectors of a column section with a gear-like structure. In Fig. 4, we show the normal error map on point clouds with uneven sampling and multi-branch structure. The normal error distribution is shown in Fig. 5. It is clear that our unsupervised method has advantages over baselines in all categories when the angle threshold is larger than 50 ◦ . The evaluation results demonstrate that our method can consistently provide good oriented normals in the presence of various complex structures and density variations.

Evaluation of Unoriented Normal . In general, estimating unoriented normals is easier than estimating oriented normals as it does not need to explore more information to determine the orientation, but only focuses on finding the perpendicular of a local plane or surface according to the input patch. Moreover, existing deep learning-based methods for unoriented normal estimation all rely on supervised training with ground truth normals. In contrast, our method is designed for estimating oriented normal in an unsupervised manner. To evaluate the unoriented normals, we compare with the baselines using our oriented normals and ignore the orientation of normals. In Table 2, we report the quantitative evaluation results of supervised and unsupervised methods on datasets PCPNet and FamousShape. We can see that, compared to unsupervised methods, our method achieves significant performance gains on almost all data categories of these two datasets and has the best average results. The supervised methods learn from ground truth normals, and their supervision provides perfect surface perpendiculars, leading to superior results. Our better results than PCPNet highlight the strong learning ability of our method without labels.

Surface Reconstruction . For surface reconstruction from point clouds, we compare our method with the latest methods in two different ways. (1) We determine the zero level set of the learned implicit function f via signed distances, and use the marching cubes algorithm [42] to extract the global surface representation of the point cloud. A visual comparison of the extracted surfaces on point clouds with different noise levels is shown in Fig. 6, where our surfaces are much cleaner and more complete under the interference of noise. (2) Based on the estimated oriented normals, we employ

the Poisson reconstruction algorithm [33] to generate surfaces from wireframe point clouds, and the results in Fig. 7 show that our method can handle extremely sparse and uneven data. The reconstructed surfaces of unsupervised methods on LiDAR point clouds of the KITTI dataset [20] are shown in Fig. 8, where our undistorted surface exhibits a more realistic road scene of real-world data.

Moreover, we repurpose some implicit representation methods, which are designed for surface reconstruction, to estimate oriented normals from point clouds. The comparison of normal RMSE on some selected point clouds in the datasets PCPNet and FamousShape is reported in Table 3. A visual comparison of the reconstructed surfaces on noisy point clouds is shown in the supplementary material. We can see that our method has clear advantages, especially on noisy point clouds. We provide more evaluation results on different data in the supplementary material.

4.2 Ablation Studies

We aim to achieve the best performance on average in both oriented and unoriented normal estimation. We provide the ablation results on the FamousShape dataset in Table 4 (a)-(d), which are discussed as follows.

(a) Loss . We do not use one of the four loss terms in Eq. (9) and the weight w in Eq. (8), respectively. We also try incorporating the weight w into L v ( θ ) of Eq. (7). These ablations do not provide better performance in both oriented and unoriented normal estimation. We observe that the performance drops a lot without using L v ( θ ) , and that L con ( θ ) has a larger impact on performance. Note that L d ( θ ) plays a more important role on the PCPNet dataset than the FamousShape dataset in our experiments.

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(b) Input . We do not sample a subset G from P , and take Q as the input instead of { Q,G } . The results verify the positive effect of G on explicitly indicating the surface and improving performance.

(c) Iteration . We set the number of steps I to 1 and 3 , respectively. More iterations bring better performance, but we still choose I =2 after weighing the running time and memory usage.

(d) Scale Set . The scale set { K s } N K s =1 in Eq. (6) is chosen to be { 1 , K/ 2 , K } , K =8 in our implementation. Here we change the scale set to different sizes, e . g ., { 1 } , { K } and { 1 , K } , K =8 , or different values of K , e . g ., 4, 16 and 32. Some of these settings have advantages in a single evaluation, but do not give better results for both oriented and unoriented normal estimation, such as { K } and { 1 , K } .

5 Conclusion

In this work, we propose to learn neural gradient functions from point clouds to estimate oriented normals without requiring ground truth normals. Specifically, we introduce loss functions to facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning a global surface representation of the data. Meanwhile, we incorporate gradients into the surface approximation to measure the minimum signed deviation of queries, resulting in a consistent gradient field associated with the surface. Extensive evaluation and ablation experiments are provided, and our excellent results in both unoriented and oriented normal estimation demonstrate the effectiveness of innovations in our design. Future work includes adding more local constraints to further improve the accuracy of normals.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a method to learn normals for point clouds using a neural representation. The idea is to perform a multi-step point moving and compute a series of losses to constrain the gradient of the field to describe shape locality with consistency. The paper proposes a large series of comparisons with the existing methods, showing promising results, although the training time is particularly slow.\n\n== POST-REBUTTAL ==\n\nAfter the discussion, the authors addressed my concerns. From other reviews, I also see a general consensus for acceptance, and the general recommendations are about clarifying some minor aspects. The only negative reviewer seems unwilling to defend the rejection position and does not point to significant weaknesses. Hence, I decided to increase my score and lean toward acceptance. Authors already agreed to incorporate the suggested changes, which are indeed essential; especially, I would stress that the two figures that show partial sections of complete shapes should be clearly described so as to not be deceptive.\n\nI wish authors best of luck with their work.\n\nSTRENGTHS:\n1) The paper obtained good reconstruction results; the shown shapes have varying numbers of different geometries, and the method seems to outperform the competitors.\n2) While the method part is not straightforward to understand, the attached code and the implementation details should provide enough details for replicability.\n\nWEAKNESSES:\n1) The paper does not convey a clear and precise message about its contribution. I am not an expert in this specific field, and I have quite a hard time understanding what makes this work different from previous ones. Accordingly to the conclusion, the main contribution seems in introducing a new loss/optimization schema. However, from the ablation study (which is also not easy to inspect, given the number of experiments and the lack of a best in bold), the \"full\" method is not always the clear winner, and it makes unclear the contribution of the losses. From my understanding, the main advancement is in providing multi-scale neighbour consistency, which helps orient all the normals in the same direction. I suggest clearly stating the main insights and adding more structure to 3.2 (e.g., with paragraph titles). \n\n2) Some works seem missing: [A] proposes a smoothness regularization for implicit representations; [B] involves a differentiable Poisson Shape reconstruction. I think a proper discussion would be useful, and especially fostering this kind of discussion in the related works. For example, at the moment, the two paragraph end highlighting that the method is unsupervised, and achieves better performance than the previous works. But how is this obtained? What is the key aspect that enables such advancement and is not considered in the previous works (if any)?\n\n\n\n[A]: Implicit Geometric Regularization for Learning Shapes, Implicit Geometric Regularization for Learning Shapes, Gropp et al., ICML 2020\n[B]: Shape As Points: A Differentiable Poisson Solver, Peng et al., NeurIPS 2021\n\nQUESTIONS:\nA) From 3.2 I cannot completely understand how the method obtains the correct normal orientation between the two possible directions. Is it obtained thanks to the multi-scale neighbourhood size?\nB) From supplementary material, the training time seems dramatically slower than other methods. What is the main cause of that?\n\nLIMITATIONS:\nLimitations are sufficiently discussed in the supplementary material", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-06-19"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a method for estimating oriented normals from a given point cloud by utilizing a neural networkto model a sign distance field (SDF). The proposed approach involves training the SDF representation, which allows for easy querying of gradients at the positions of the point cloud. The method employs a set of loss functions that simulate an iterative process of moving query points to match target points based on the neural gradients. Experiments are conducted on unsupervised oriented normals estimation from input point clouds, which may contain noise, outliers, and density variations.\n\nSTRENGTHS:\n- The authors clearly highlight the limitations of previous works, providing a good motivation for their proposed method. The promising results showcased in Figure 1 further support this motivation.\n- The introduction and related work sections are comprehensive, providing the necessary background for understanding the normal estimation task. This makes the paper self-contained.\n- The method is presented in a well-structured manner, starting with a high-level overview before delving into the details. This organization helps to keep the reader engaged and informed throughout the paper.\n\nWEAKNESSES:\n- The mathematical definitions and notations in the method section need to be revisited. Some crucial definitions, such as $f_i$ in Equation 2, are mentioned without being properly defined, making Sections 3.1 and 3.2 harder to understand. Additionally, the notation $ \\{ Q, G\\} $ is confusing as the operation involving this set containing two sets of points is not clearly defined or highlighted. The notation $f_i^G$ also requires clarification. Furthermore, the unit gradient $\\boldsymbol{n}$ mentioned in Line 164 lacks clear information about its 3D positions.\n- The authors refer to the post-processing step of computing the gradients of the learned SDF as \"inference\". However, \"inference\" typically refers to the process of applying a trained model on unseen inputs for generalization. This usage of terminology can be misleading and confusing.\n- Figure 2 does not fully capture the presented method as it introduces notations that are only defined in the text. It is unclear from the figure alone, even with the caption, what the input to the model is.\n- A comparison to other neural SDF methods that learn an SDF from an input point cloud, such as SAL (https://arxiv.org/abs/1911.10414) or IGR (https://arxiv.org/abs/2002.10099), is missing. These methods could be relevant for extracting normals easily, as in the suggested method.\n- The paper lacks a reference to SAP (https://pengsongyou.github.io/sap), which provides a scenario of reconstructing normals and a surface from an input point cloud.\n- It is recommended to include illustrations for the different losses mentioned, similar to the visualization in Figure 2 (i) at the bottom.\n\nQUESTIONS:\nPlease see concerns and questions in weaknesses. Few further questions:\n\n- Are the loss coefficients tuned for different noise levels, or are they fixed throughout the experiments?\n- Can the presented method also yield the complete negative solution, i.e., $-\\boldsymbol{n}$, which is equivalent up to a global sign? Does this depend on the initialization or other factors?\n\nLIMITATIONS:\nThe authors discussed limitation, however the limitation discussion and figure is shown soley in the appendix.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work the authors present an unsupervised framework for predicting globally consistent, accurate normals given a point cloud as input. The crux of the method is to predict an implicit surface representation (a signed distance function to be exact) from the input point cloud by leveraging the fact that the gradient of the Neural SDF of surface points gives the surface normal at that point. The authors distinguish their method from existing similar approaches by overcoming drawbacks like lack of global consistency by designing a loss function that penalizes incorrect signed predictions and also considers multi-scale neighborhoods. Furthermore, they consider a multi-step iterative approach for refining their surface normal estimations. They show impressive results in terms of RMSE of unoriented/oriented normals across challenging datasets across different levels of sampling densities and noise.\n\nSTRENGTHS:\n1. The proposed method achieves SoTA performance compared to other unsupervised methods on the PCPNet and FamousShape dataset. \n2. In the presence of noisy input points, their method is quite competitive even against supervised baselines which is a big plus.\n3. Accurately oriented point clouds are hugely sought-after in downstream applications like surface reconstruction and their results on single object surface reconstruction demonstrates the value of their method in this important application.\n\nWEAKNESSES:\n1. Comparison with other implicit representations. At its core, the proposed method learns a Neural SDF for the shape represented by the input point cloud and the estimated normals are simply the gradients computed from this Neural SDF using automatic differentiation. As such I believe there should be more comparison drawn to similar methods like Neural-Pull [1], SIREN [2], SAL [3], and Neuralangelo [4]. In particular, [4] uses a finite-difference-based approach to estimate the surface normal during training and I would be curious to see if a similar technique can get similar results in lesser resources for normal estimation.\n\n\n2. Hyperparameter tuning. Since the method is optimization based, one potential risk is mishandling the hyper-parameter tuning, which turns out to be very important. For example, how does the method choose the hyper-parameter for each shape? Does a different input point set require different hyper-parameters or one hyper-parameter can be used for a wide range of shapes? I think tuning hyper-parameter of a neural field fitting procedure can drastically change the performance, and arguably with proper hyper-parameter tuning, one can probably find a way to curate smooth solutions to different input instances. As a result, it's very essential for this kind of test-time optimization-based method to have a rigorous hyper-parameter tuning procedure reported to eliminate the risk of accidentally adding human judgment into producing the results. \n\nReferences:\n\n[1] Neural-Pull: Learning Signed Distance Functions from Point Clouds by Learning to Pull Space onto Surfaces. https://arxiv.org/abs/2011.13495\n[2] SIREN. https://arxiv.org/abs/2006.09661\n[3] SAL. https://arxiv.org/abs/1911.10414 \n[4] Neuralangelo: High-Fidelity Neural Surface Reconstruction. https://research.nvidia.com/labs/dir/neuralangelo/\n[5] NKSR. https://research.nvidia.com/labs/toronto-ai/NKSR/\n\nQUESTIONS:\n1. The density-based experiments are unclear to me. Could the authors please explain what the “gradient” and “stripe” settings exactly imply? The supplementary shows a few qualitative examples but can you explain how they were generated? It also seems that supervised methods perform comparably w.r.t. the proposed method. Could the authors give an intuition for why their method suffers from density variation but not on noisy point clouds (see Table 2 and Fig. 5, especially at low threshold angles)? Which hyperparameters can be tuned to address noise vs. density variation and what is the trade-off for the same? This would help with a better understanding of the authors’ contributions.\n\n2. In the analysis of un-oriented point clouds, we can see that LRR is the second-best performing unsupervised normal estimation method. Yet, we do not see an application of LRR with techniques for introducing orientation like MST in the study on the estimation of oriented normals. Adding LRR + (some orientation method) type of methods to Table 2 and Fig. 5 would make the results a bit more stronger. Basically this can help address the question of why don't we just pick a SoTA unoriented unsupervised point cloud method and do some simple orientation method?\n\nLIMITATIONS:\nLimitation section in the supplementary. Potential other limitations can include shapes with open surface and shapes without volume.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis work proposes to learn neural gradient functions from point clouds to estimate oriented normal in an unsupervised manner. Specifically, this method introduces several loss functions to constrain query points to iteratively fit the underlying surface, which is defined by the sampled points. Meanwhile, the local gradients are incorporated into the surface approximation to measure the minimum signed deviation of queries, resulting in a consistent normal field associated with the surface. Lastly, some evaluations demonstrate the superior performance of the proposed method over existing approaches.\n\nSTRENGTHS:\n1. This paper is well organized and nicely written. The presentation of motivation is clear and smooth.\n2. The work is well motivated. The idea of learning neural gradients for normal estimation is inspiring. \n3. Some visual and quantitative evaluation are promising.\n\nWEAKNESSES:\n1. I am not sure why we need a distribution D. \n2. I do not see any running efficiency statistics, which would be important for a fair assessment.\n3. I think there is a missing comparison with [73]. Also, some challenging cases from [73] should be included.\n\nQUESTIONS:\n1. I believe that the designed loss functions can facilitate query points to iteratively reach the moving targets and aggregate onto the approximated surface, thereby learning a global surface representation of the data. However, I am doubting that if the incorporate gradients can achieve a consistent normal field, especially handling some challenging cases. Is it possible to give more details?\n2. Both point sets of Q and G are sampled from input point cloud, so in the test stage, how to compute the normal for all points?\n\nLIMITATIONS:\n1. I do not see any failure cases in the paper. It is essential to show some failure cases for the reader to investigate the failure model to benefit future research.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2023-07-07"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a method that predicts oriented surface normals given a 3D point cloud. The method is based on SDF surface reconstruction. Given a potentially noisy point cloud, a SDF is fitted using an approach similar to Neural-Pull [45]. The surface normal at each point can then be computed based on the gradient of the SDF. Since a global SDF of the shape is recovered, the orientation of the predicted surface normal is also globally consistent. Additionally, the paper proposed various improvements on top of Neural-Pull to improve normal prediction quality: a multi-step movement strategy during optimization, and a multi-scale neighborhood size strategy.\n\nSTRENGTHS:\n* The proposed method is unsupervised -- It is optimized on a per-shape basis and unlike [23,8,39,38], it does not need to be trained on a large collection of shapes, which also avoids any training-evaluation domain gap.\n* Unlike local fitting methods [8,82,37,15], the proposed method produces globally consistent surface orientation thanks to the use of a global SDF.\n* The proposed method achieved state-of-the-art performance on both oriented and unoriented surface normal estimation. It worked well especially for noisy point clouds.\n* The paper has included very comprehensive ablations to show the effects of various design decisions as well as the new components such as multi-step supervision and multi-scale neighbor selection.\n\nWEAKNESSES:\n* The proposed method resembles many existing works on point cloud 3D reconstruction, such as SAL [6], Neural-Pull [45], Shape As Points (Peng et al.), NDF (Chibane et al.). In fact, recovering the normal can be considered as the side effect of surface reconstruction -- once the oriented surface is obtained, the oriented normal can be obtained naturally. As there is no comparison on surface normal quality with such methods in the paper, it is not clear if the proposed method has any significant benefit over these methods from  surface reconstruction community.\n* Compared to feed-forward methods such as PCPNet [23], the proposed method requires per-shape optimization, which can be computationally expensive.\n* The paper will be easier to follow if it can devote some paragraphs to the connections between state-of-the-art surface reconstruction and normal estimation literature.\n\nQUESTIONS:\n* How does the proposed method compare with previous works in terms of speed?\n* It is possible to directly repurpose surface reconstruction methods for normal estimation? How would they perform?\n\nLIMITATIONS:\nThe limitation and societal impact of the paper is adequately addressed in the supplemental material.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2023-07-07"}