Inferring Neural Signed Distance Functions by Overfitting on Single Noisy Point Clouds through Finetuning Data-Driven based Priors

Authors: chao  chen
yu-shen  liu
zhizhong  han
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-06 10:07:03

Abstract: It is important to estimate an accurate signed distance function (SDF) from a point cloud in many computer vision applications. The latest methods learn neural SDFs using either a data-driven based or an overfitting-based strategy. However, these two kinds of methods are with either poor generalization or slow convergence, which limits their capability under challenging scenarios like highly noisy point clouds. To resolve this issue, we propose a method to prompt pros of both data-driven based and overfitting-based methods for better generalization, faster inference, and higher accuracy in learning neural SDFs. We introduce a novel statistical reasoning algorithm in local regions which is able to finetune data-driven based priors without signed distance supervision, clean point cloud, or point normals. This helps our method start with a good initialization, and converge to a minimum in a much faster way. Our numerical and visual comparisons with the stat-of-the-art methods show our superiority over these methods in surface reconstruction and point cloud denoising on widely used shape and scene benchmarks. The code is available at https://github.com/chenchao15/LocalN2NM.

Main paper: 
 introduction: 
 It is an important task to estimate an implicit function from a point cloud in computer graphics, computer vision, and robotics. An implicit function, such as a signed distance function (SDF), describes a continuous 3D distance field to indicate distances to the nearest surfaces at arbitrary locations. Since point clouds are easy to obtain, they are widely used as an information source to estimate SDFs, particularly without using normals that are not available for most scenarios. The challenge for SDF estimation mainly comes from the difficulty of bridging the gap between the discreteness of point clouds and the continuity of implicit functions.Recent methods [62, 64, 29, 14, 95, 80, 58, 74] overcome this challenge using either a data-driven based or an overfitting-based strategy. To map a point cloud to a signed distance field, the data-driven based methods [60, 27, 36, 45, 81, 79, 22, 42, 92, 83] rely on a prior learned with signed distance supervision from a large-scale dataset, while the overfitting-based methods [28, 1, 102, 2, 99, 4, 21, 50, 18, 88] do not need signed distance supervision and just use the point cloud to infer a signed distance field. However, both of the two kinds of methods have pros and cons. The data-driven based methods can do inference fast but suffers from the need of large-scale training samples and poor generalization to instances that are unseen during training. Although the overfitting-based methods have a better generalization ability and do not need the large-scale signed distance supervision, they usually require a much longer time to converge during inference. The cons of these two kinds of methods dramatically limit the performance of learning neural SDFs under challenging scenarios like highly noisy point clouds. Therefore, beyond pursuing higher accuracy of SDFs, how to balance the generalization ability and the convergence efficiency is also a significant issue.To resolve this issue, we propose to learn an SDF from a single point cloud by finetuning data-driven based priors. Our key idea is to promote the advantages of both the data-driven based and the overfitting-based strategy to pursue better generalization, faster inference, and higher accuracy. Our method overfits a neural network on a single point cloud to estimate an SDF with a novel loss without using signed distance supervision, clean point, or point normals, where the neural network was pretrained as a data-driven based prior from large-scale signed distance supervision. With finetuning priors, our method can generalize better on unseen instances than the data-driven based methods, and also converge much more accurate SDFs in a much faster way than the overfitting-based methods. Moreover, our novel loss for finetuning the data-driven based prior can conduct a statistical reasoning in a local region which can recover more accurate and sharper underlying surface from noisy points. We report numerical and visual comparisons with the state-of-the-art methods and show our superiority over these methods in surface reconstruction and point cloud denoising on widely used shape and scene benchmarks. Our contributions are summarized below,• We introduce a method which is capable of funetuning a data-driven based prior by minimizing an overfitting-based loss without signed distance supervision, leading to neural SDFs with better generalization, faster inference, and higher accuracy.• The proposed overfitting-based loss can conduct a novel statistical reasoning in local regions, which improves the accuracy of neural SDFs inferred from noisy point clouds.• Our method produces the state-of-the-art results in surface reconstruction and point cloud denoising on the widely used benchmarks. 
related works: 
 Learning implicit functions has achieved promising performance in various tasks [62, 64, 29, 14, 95, 80, 58, 74, 30, 31, 33] . We can learn neural implicit representations from different supervision including 3D supervision [61, 69, 59, 17] , multi-view images [78, 44, 38, 101, 46, 94, 63, 41, 98, 97, 25, 86, 100, 89, 84, 85] , and point clouds [92, 43, 60, 27] . We briefly review the existing methods related to point clouds below.In 3D supervision, many techniques utilize a data-driven approach to learning priors, and then apply these learned priors to infer implicit models for unseen point clouds. Some strategies focus on acquiring global priors [60, 27, 36, 45, 81, 79, 22, 42] at the shape level, whereas others aim to boost the generalization of these priors by learning local priors [92, 83, 11, 37, 6, 51] at the component or patch level. These learned priors facilitate the marching cubes algorithm [47] to reconstruct surfaces from implicit fields. The effectiveness of these methods often rely on extensive datasets, but they may not generalize well when facing with unseen point clouds that significantly deviate in geometry from training samples.In an effort to enhance generalization, some methods concentrate on precisely fitting neural networks to single point clouds. These methods incorporate innovative constraints [28, 1, 102, 2, 99, 4, 21] , utilize gradients [50, 18, 88] , employ differentiable Poisson solvers [70] , or apply specially tailored priors [51, 54] to learn either signed [50, 28, 1, 102, 2, 15, 56, 13] or unsigned distance functions [18, 104, 103] . Despite achieving significant advances, these approaches typically require clean point clouds to accurately determine distance or occupancy fields around the point clouds. The key to accurately reconstructing surfaces on noisy point clouds is to minimize the effect of noise in inferring implicit functions. PointCleanNet [73] was developed to filter out noise from point clouds through a data-driven approach. GPDNet [72] incorporated graph convolution based on dynamically generated neighborhood graphs to enhance noise reduction. Some other methods leveraged point cloud convolution [6] , alternating latent topology [90, 57] , semi-supervised strategy [106, 19] , dual and integrated latent [76] , or neural kernel field [91, 35] to reduce noise from point clouds. On the unsupervised front, TotalDenoising [10] adopts principles similar to Noise2Noise [40] , utilizing a spatial prior suitable for unordered point clouds. DiGS [3] employs a soft constraint for unoriented point clouds. Noise2NoiseMapping [52] leverage statistical reasoning among multiple noisy point clouds with specially designed losses. Some methods using downsample-upsample frameworks [48] , gradient fields [49, 9, 16, 68, 65] , convolution-free intrinsic occupancy network [67] , intra-shape regularization [66] , eikonal equation [96, 23] , neural Galerkin [34] and neural splines [93] have been implemented to further diminish noise in point clouds. Our method falls in this category, but we aim to promote the advantages of both the data-driven based and the overfitting-based strategy to pursue better generalization, faster inference, and higher accuracy. 
method: 
 Overview. We aim to infer a neural SDF f from a single point cloud with noises M . Our method includes two stages shown in Fig. 1 , one is to learn a prior f ′ in a data-driven manner, the other is to infer a neural SDF f on unseen noisy point cloud M . At the first stage, we learn a prior by training a neural SDF using ground truth signed distances of clean meshes indicated by embeddings c ′ j . At the second stage, we finetune the learned prior f ′ to infer a neural SDF f of M using our proposed local noise to noise mapping, where the embedding c indicating M is also learned. We can use the marching cubes algorithm [47] to extract the zero-level set of f as the mesh surface of M .Neural Signed Distance Function. We leverage an SDF f to represent the geometry of a shape. An SDF f is an implicit function that can predict a signed distance s for an arbitrary location q, i.e., s = f (q). The latest methods usually train a neural network to approximate an SDF from signed distance supervision or infer an SDF from 3D point clouds or multi-view images. A level set is an iso-surface formed by the points with the same signed distance value. For instance, zero-level set is a special level set, which is formed by points with a signed distance of 0. On the zero-level set, the gradient ∇f (q) of the SDF f at an arbitrary location q is also the surface normal at q. Data-driven Based Prior. As shown in Fig. 1 , we employ an auto-decoder similar to DeepSDF [69] for learning a prior f ′ in a data-driven manner and inferring a neural SDF f for single point clouds with noises, respectively. We employ a data-driven strategy to learn a prior f ′ from clean meshes first. Specifically, we learn f ′ with an embedding c ′ j as a condition of queries. For each shape, we sample queries q around a shape represented by c ′ j , and establish the signed distance supervision by recording the signed distance s to the ground truth mesh. Thus, we learn the prior f ′ by minimizing the prediction errors to the ground truth signed distances,EQUATIONwhere c ′ i is a learnable condition for the i-th training shape, q j is the j-th query that is randomly sampled around the i-th shape, and s j i is the ground truth signed distance. We also add a regularization term on the learned embeddings c ′ i , and α is the balance weight. Signed Distance Inference. With the learned prior f ′ , we infer a neural SDF f for a single point cloud with noises M . We do not require ground truth signed distances, clean point clouds, or even point normal during the inference of f . Specifically, we infer f by finetuning parameters of f ′ with a learnable embedding c indicating the single point cloud with noises. The finetuning relies on a novel statistical reasoning algorithm on local regions.The advantage of our method lies in the capability of conducting the statistical reasoning in local regions. Comparing to the global reasoning method [52] , our method is able to not only infer more accurate geometry but also significantly improve the efficiency. Our method starts from randomly sampling a local region m n on the shape M . We randomly select one point on M as the center of m n , and set up its K nearest noisy points as a local region m n . Then, we randomly sample U queries {q u } U u=1 around m n , and also randomly select U noisy points {p v } U v=1 out of m n for statistically reasoning the surface in each iteration.Our key idea of inferring a neural SDF f is to estimate a mean zero-level set that is consistent to all points in the local region m n . To this end, we use the U sampled queries {q u } to represent the zero-level set in this area using f , and minimize the distances of the U noisy points {p v } to the zero-level set in each iteration. Statistically, the expectation of the zero-level set should have the minimum distance to all the noisy point splitting in region m n . Specifically, we first project the U sampled queries {q u } onto the zero-level set of f using a differentiable pulling operation [50] . For each query qu , its projection on the zero-level set is,EQUATIONwhere q′ u is the projection of qu on the zero-level set, s = f (q u , c), ∇f (q u , c) is the gradient of f at the location qu , and c is the learnable embedding that represents the noisy point cloud M .With the pulling operation, we can use projections {q ′ u } of queries {q u } to approximate the zero-level set in region m n . With a coarse zero-level set estimation, we expect this zero-level set can be consistent to various subsets of noises {p v } sampled from m n . Thus, we minimize the errors between the {q ′ u } U u=1 and a subset of points {p v } U v=1 on area m n in each optimization iteration,EQUATIONwhere we learn f through finetuning the prior f ′ and learning the embedding c representing the noisy point cloud M . The expectation is over the local regions m n that randomly sampled from the noisy point cloud M , and the subset patch p v randomly sampled from each m n . We follow the method [52] to use the EMD to evaluate the distance between the two sets of points, which leads the neural SDF f to converge on the specific noisy point cloud M .Initialization. The network architecture of f is the same to the one of prior f ′ . We learn f with the parameters of f ′ as the initialization, representing the prior that we learned. For the embedding c that represents M , we initialize c as the center of the embedding space learned by the prior f ′ in Eq. 1, i.e., c = 1/I I i=1 c ′ i . This initialization is important for the accuracy and efficiency of learning f for single noisy point cloud M . This finetuning of parameters of f ′ also shows advantages over the auto-decoding [69] in terms of generalization and efficiency. We will justify these advantages in our experiments.Implementation Details. We randomly select one point from noisy point cloud M as a center, and select its K = 1000 nearest points to form a local region m n . We also randomly sample U = 1000 queries around the K noisy points for statistically reasoning. Specifically, we adopt a method Metrics PSR [39] PSG [24] R2N2 [20] COcc [71] SAP [70] OCNN [87] IMLS [45] introduced by NeuralPull [50] to sample queries around each one of the K noisy points. We use a Gaussian distribution centered at each point and set the standard deviation as the distance to the 51th nearest neighbor in the point cloud. We run the marching cubes for surface reconstruction at a resolution of 256 for shapes, and 512 for large-scale scenes.The length of the embedding c or c ′ is set to 256. We use Adam optimizer for learning a neural implicit network, which is an auto-decoder similar to DeepSDF [69] . For training, we use an initial embedding learning rate of 0.0005 for updating embeddings and an auto-decoder learning rate of 0.001 for optimizing the prior network. Both learning rates are decreased by 0.5 for every 500 epochs. We train the prior network f ′ for 2000 epochs. For inference, we finetune the network f ′ for each noisy point cloud in 4000 iterations with a learning rate of 0.0001. 
experiments and analysis: 
 We compare our method with the latest methods in terms of numerical and visual results on synthetic point clouds and real scans in surface reconstruction.Datasets and Metric. We use eight datasets including shapes and scenes in the evaluations.For shapes, we conduct experiments under five datasets including ShapeNet [12] , ABC [22] , FA-MOUS [22] , Surface Reconstruction Benchmark (SRB) [92] and D-FAUST [5] . For scenes, we conduct experiments under three real scan datasets including 3D Scene [105] , KITTI [26] , Paris-rue-Madame [75] , and nuScenes [8] . We leverage L1 Chamfer Distance (CD L1 ), L2 Chamfer Distance (CD L2 ) to evaluate the error between the reconstructed surface and ground truth. We also use Normal Consistency (NC) [59] and F-Score [82] with a threshold of 1% to evaluate the normal accuracy of the reconstructed surface. In the ablation study, we also report time consumption to highlight the superiority of our data-driven based prior. For KITTI and Paris-rue-Madame datasets, due to their lack of ground truth meshes, we only report visual comparisons.Noisy Input ConvOcc POCO ALTO N2NM Ours GT We first report our results on shapes from ShapeNet. We report evaluations by comparing our method with the latest prior-based and overfitting-based methods in Tab 1. For prior-based methods, we compare our method with PSG [24] , R2N2 [20] , COcc [71] , OCNN [87] , IMLS [45] , POCO [7] , and ALTO [90] .All of these methods are pretrained to learn priors using shapes with noises in training set of ShapeNet. We also follow these methods to use the same set of training shapes to learn our prior. For overfitting-based methods, we compare our method with PSR [39] , SAP [70] , and N2NM [52] . These methods did not need to learn a prior, and have the ability of inferring neural implicit functions on each shape in the testing set. We also follow these methods and report our results by finetuning our prior through overfitting on each testing shape. All the shapes for testing are corrupted with noises with a variance of 0.005. The comparisons in Tab. 1 indicate that our method can infer much more accurate neural implicit functions than the prior-based methods. The improvement comes from the ability of conducting test time optimization with the learned prior and inferring signed distances using the local noise to noise mapping. Moreover, our local statistical reasoning not only achieves better ability of recovering geometry from noisy points than overfitting-based methods but also significantly reduces the time complexity during the test time overfitting procedure with our prior. Different from prior-based methods, our ability of conducting testtime optimization with our local statistical reasoning loss can significantly improve the generalization ability on unseen shapes. Tab. 2 shows that our method can infer neural implicit functions on single shapes much faster than the overfitting-based methods. We also demonstrate our advantages in visual comparisons in Fig. 2 . Evaluation on ABC. We also report our evaluations on ABC dataset in Tab. 3. We learn priors from shapes in training set, and finetune this prior for each single shape in the testing set. The numerical comparisons are conducted on the testing set of ABC dataset released by P2S [22] . It includes two versions with different noise levels. Similarly, we also report comparisons with prior-based methods and overfitting-based methods. With our local noise to noise mapping, we achieve the best performance over all baselines. Compared to prior-based methods, such as P2S [22] , COcc [71] , and POCO [7] , our loss can infer more accurate geometry during the test time overfitting procedure. Also, the ability of finetuning the prior can also provide a coarse estimation and a good start for inferring neural implicit from single noisy points. Besides the accuracy, we also observe improvements on efficiency. Fig. 3 demonstrates the improvements over the baselines in terms of surface completeness and edge sharpness. Evaluation on FAMOUS. We report evaluations on more complex shapes on FAMOUS dataset. Similar to SRB, we also use the prior learned from ShapeNet. We evaluate the performance on two kinds of noises in Tab. 5. We can see that our method can recover more geometry details and achieve higher accuracy and smoother surfaces. We also report visual comparisons in Fig. 5 , which also highlights our improvements in Dataset PSR [39] P2S [22] COcc [71] NP [50] IMLS [45] PCP [55] POCO [7] OnSurf [53] Evaluation on D-FAUST. Finally, we report our results on non-rigid shapes, i.e., humans. Different from rigid shapes in the previous experiments, humans are with more complex poses. We learn a prior from the training set, and finetuning the prior on unseen humans with different poses. We mainly compare our method with overfittingbased methods in Tab. 6 . We can see that our method achieves the best performance in CD, F-Score, and comparable performance to N2NM [52] but with faster inference speed. We further show the visual comparison in Fig. 6 . We can see that our method can recover more accurate geometry and poses.Input IMLS P2S NeuralPull Ours GTSince we have a limited number of scenes for training, we use the prior learned from ShapeNet as the pretrained prior in our experiments on scenes. Specifically, we conduct experiments on four different scene datasets: 3D Scene [105] , KITTI [26] , Paris-rue-Madame [75] and nuScenes [8] , where the results on nuScenes are reported in the appendix.Evaluation on 3D Scene. We further evaluate our method in surface reconstruction for scenes in 3D Scene [105] . We follow previous methods LIG [37] to randomly sample 1000 points per m 2 . We compare our method with the latest methods including COcc [71] and LIG [37] , DeepLS [11] , NeuralPull (NP) [50] and Noise2NoiseMapping (N2NM) [52] . For prior-based methods COcc [71] and LIG [37] , we leverage their released pretrained models to produce the results, and we also provide them with the ground truth point normals. For overfitting-based methods DeepLS [11] , NP [50] and N2NM [52] , we overfit them to produce results with the same noisy point clouds. We follow LIG [37] to report CD L1 , CD L2 and NC for evaluation. We report the comparisons in Tab. 7. The results demonstrate that our method outperforms both kinds of methods with learned priors such as LIG [37] and overfitting-based N2NM [52] . The visual comparisons in Fig. 7 show that our method can reveal more geometry details on real scans, which justifies our capability of handling noise in point clouds. Dataset PSR [39] NP [50] IMLS [45] LPI [15] PCP [55] POCO [7] OnSurf [53] GP [16] Table 6 : Accuracy of reconstruction on D-FAUST dataset in terms of CD L1 , NC and F-Score. Evaluation on KITTI. Following GridPull [16] , we further evaluate our method on KITTI [26] odometry dataset (Sequence 00, frame 3000 to 4000), which contains about 13.8 million points, which are split into 15 chunks. We reconstruct each of them and concatenate them together for visualization. We compare our method with the latest methods SAP [70] and GridPull [16] . As shown in Fig. 8 , our method is robust to noise in real scans, successfully generalizes to large-scale scenes, and achieves visual-appealing reconstructions with more details.Evaluation on Paris-rue-Madame. Following N2NM [52] , we further evaluate our method on Paris-rue-Madame [75] , which contains much noises. We split the 10 million points into 50 chunks each of which is used to learn a neural implicit function. We compare our method with LIG [37] and N2NM [52] . For LIG [37] , we produce the results for each chunk with released pretrained models. For N2NM [52] , we overfit on all chunks until convergence. As shown in Fig. 9 , we achieve better performance over LIG [37] and N2NM [52] in large-scale surface reconstruction, which highlight our advantages in reconstructing complete and detailed surfaces from noisy scene point clouds.We conduct ablation studies on the ABC dataset [22] to justify each module of our method. Prior. We conduct experiments to explore the importance of data-driven based prior. We first replace our learned embedding c and parameter with randomly initialized embedding and parameter, or only replace c with randomly initialized embedding. As shown in Tab. 9, The degenerated result of "Without Prior" and "Without Embed" indicates that directly inferring implicit functions without our prior or learned embedding makes it difficult to accurately learn the surfaces of the noisy point clouds, and also slows the convergence. Then we fix the learned parameters and only optimize the embedding c, similar to auto-decoding.The results also get worse, as shown in "Fixed Param". Local Region Splitting. We further validate the effectiveness of local region splitting strategies. We employ three different splitting strategies in Tab. 10. We first split the whole space where the noisy point cloud is located uniformly into multiple voxel blocks, as shown by the result of "Voxel". The severely degenerated results indicate that this splitting strategy is even worse than the global method N2NM [52] , as it results in many empty voxel blocks. Then we randomly select a point from the noisy point cloud as a center to sample all points within a radius of 0.1 as a local region. The result of "Sphere (Fixed Size)" slightly degenerates due to some of the spheres containing too few points. In contrast, our splitting strategy, as shown by the result of "Sphere (KNN)", ensures that each local region has enough points to help achieve superior performance.Metrics COcc [71] LIG [37] DeepLS [11] Local Region Size. We further validate the effectiveness of local region sizes (points number in a local region) in Tab. 12. We use different local region sizes including {500, 1000, 3000, 5000}. The results show that 1000 is the best.Metric Random Square Sphere (SAL) Ours Time 8.3min 7.1min 5.5minTable 13 : The effect of SDF initialization.SDF initialization. We further validate the effectiveness of different SDF initializations in Tab. 13 and Fig. 10 , including random initialization, geometry initialization [1] , initialization to a simple square shape, and ours.Random Square Sphere Ours GT Noise Type. We report our performance with various noise types, i.e., impulse noise, quantization noise, Laplacian noise, and Gaussian noise. Visual comparison in Fig. 12 justifies that we can also handle other types of noise quite well. Moreover, we also tried more challenging cases with nonuniform noises which do not have a zero expectation across a shape, like a shape with only a half of points having noises or a shape with several patches having noises. The result in Fig. 11 shows that our method can also handle nonuniform noises well. Noise Level. We report our performance on point clouds with different levels of noise. As shown in Tab. 14, the noise levels of middle and max come from the ABC dataset [22] .The middle indicates noises with a variance of 0.01L, where L is the longest edge of the bounding box. The max indicates noises with a variance of 0.05L. Our extreme noise comes with a variance of 0.07L. The CD L2 comparison shows that our results slightly degenerate with max and extreme noise, but still outperform N2NM [52] . The visual results in Fig. 13 indicates that our method is more robust to noises even when the noise variance is as large as 7%. Sparsity. We report the effect of the sparsity of noisy point clouds. We downsample the noisy point clouds to 25% and 50% of their original size to validate the impact of sparsity.The CD L2 results in Tab. 15 and visual comparisons in Fig. 14 indicate that our method can handle sparsity in noisy point clouds better than N2NM [52] . Since our data-driven based prior can help to learn a more complete surface and reduce the impacts brought by the sparsity.10% 30% 50% 70% 100% (3k) GT Time Consumption. Since our method can handle sparsity and require less time as the point number decreases, we conduct an experiment with downsampled noisy points in Tab. 16 . Fig. 14 indicates that we can work well on much fewer points, and also provide an alternative of improving efficiency. Optimization. We visualize the optimization process in Fig. 15 . We reconstruct meshes using the neural SDF f learned in different iterations. We see that the shape is updated progressively to the ground truth shapes. 
conclusion: 
 We propose a method to resolve the key problem in inferring SDFs from a single noisy point cloud. Our method can effectively use a data-driven based prior as an initialization, and infer a neural SDF by overfitting on a single noisy point cloud. The novel statistical reasoning successfully infers an accurate and smooth signed distance field around the single noisy point cloud with the data-driven based prior. By finetuning data-driven based priors with statistical reasoning, our method significantly improves the robustness, the scalability, the efficiency, and the accuracy in inferring SDFs from single point clouds. Our experimental results and ablations studies show our superiority and justify the effectiveness of the proposed modules. 
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [Yes] Justification: We discuss the application and potential positive impact of our method in the introduction.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.• Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations. • The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)? Answer: [NA] Justification: There is no such risk to the paper. Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort. 12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected? Answer: [Yes] Justification: We use open-source datasets and code under their licence. Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided. • If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. 
 
 Appendix: Our method is still limited to too large noises. For noises that corrupted shapes too much, our method still produces bad results. One direction for our future work is to improve our prior, so that we could have a better sense of a shape even under large noises.We detail our evaluations on each scene in 3D scene dataset in Tab. 17. The comparisons highlight our advantages in each scene.Metrics COcc [71] LIG [37] DeepLS [11] We visualize more surface reconstruction results under ShapeNet [12] , ABC [22] , Surface Reconstruction Benchmark (SRB) [92] , FAMOUS [22] , D-FAUST [5] and nuScenes [8] in Fig. 16 , Fig. 17 Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [Yes] Justification: We provide our demonstration code as a part of our supplementary materials. We will release our source code, data and sufficient instructions upon acceptance. Guidelines:• The answer NA means that paper does not include experiments requiring code. Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 


Decision: Initially, the reviewers had mixed responses, but after the rebuttal and the discussions, the majority of the reviewers voted in favour of acceptance. Except for Reviewer 76DM, who never returned after the initial review despite both the authors and the AC trying to reach the reviewer, all reviewers recommended accepting the paper. Reviewer AdmT's rating currently shows up as borderline reject, but in the discussions recommended acceptance, hence this seems to be just that the reviewer did not update the rating after the discussions.

In short, there were concerns about the limited novelty, but reviewers agreed that the claims made in the paper have been adequately verified, and the paper also shows a significant performance gain. The AC agrees with this recommendation and recommends accepting the paper. The authors are, however, strongly encouraged to incorporate the discussions during the rebuttal to the main paper, as it played a pivotal role in the paper's acceptance.

Reviews: review: paper_summary: paper_summary: The paper proposes 3D shape reconstruction given a noisy point cloud using a test-time optimization approach. The paper proposes a new loss function that can work directly on noisy point clouds without the need of ground truth normals or SDF. The approach involves learning a network (DeepSDF) to predict SDFs and later fine-tune the network on a single shape during test. The learned priors reduce the test-time fitting and leads to better reconstruction results.
strength_weakness: strength_weakness: 1. In the introduction, the test-time optimization is hailed as a novel thing, however previous methods (including DeepSDF) has already proposed it.
2. The main contribution is the loss function defined using "pulling" operation, which forces the surface defined by the projected points close to the input points be similar to the input point clouds. However, why this loss function should work in a noisy point cloud setting is not well motivated and well explained.
3. The writing of the method section can be significantly improved. For example, the symbol $f$ seems to represent both the neural network that predicts the signed distance field and the field it self. More about it in the Questions section.
4. The use of the phrase "statistical reasoning"is quite vague. The phrase is used throughout the manuscript but does not convey anything meaningful.
5. It is not clear whether the baselines in the experiment section are also trained on noisy data to make them robust.
6. Comparison with Points2Surf [21] method is missing. I think it work quite well on shapes in Figure 4.
7. What does "extreme noise" mean in the Table 13? What does it translate to in percentage of the scale of the scene.
8. How do you achieve 100% sparsity in Table 14? Does it imply no points?
9. Noise level of 0.5% is quite low to claim that the method is robust.
questions: questions: 1. Please improve the notations in the method section. The vectors should be bold. Use different symbols for parameterized function and output.
2. The motivation of the loss function is not clear. Perhaps a diagram would enhance the understanding for a reader.
limitations: limitations: 1. Limitations regarding pose variations and incomplete data (holes) are not discussed.
paper_summary: paper_summary: The authors propose an implicit surface reconstruction method from point clouds that uses a learned prior to initialize the optimization of a neural SDF. First, a neural SDF generator based on DeepSDF [66] is trained on a dataset of shapes to learn a prior over shapes. Given a point cloud, both the shape code used as input for the generator, as well as the parameters of the neural SDF are then optimized to fit the point cloud, using an approach based on Noise to Noise Mapping (N2NM) [50], but focusing on local patches instead of the full global point cloud. The authors show that this approach performs better than N2NM as well as several state-of-the-art methods in an extensive evaluation on several datasets.
strength_weakness: strength_weakness: - While the idea of using a learned prior as initialization of an optimization based method has not been done for surface reconstruction as far as I can tell, it is not very surprising that this can be done and is also relatively straight-forward to do. Both stages are essentially the same as existing work [66] and [50], with some deviation from [50] in using local patches. So all-in-all, the technical contribution does not seem huge, although possibly still large enough for acceptance.
- While local patches do seem to improve performance, it is not fully clear why this is the case, and the exact details of using local patches are missing some details in the exposition, making it unclear exactly how they are implemented.
- A few recent methods are missing both from the related work and the comparisons (see below).
- The exposition could use some additional details and clarifications in some parts (see below).

Most of the weaknesses, apart from the first one, can be addressed with text changes to some extent. Given the good performance of the method and the extensive evaluation, I lean towards acceptance.

Details:
- The field of surface reconstruction from point clouds is quite vast, so the authors missed a few works:
  - Neural-Singular-Hessian: Implicit Neural Representation of Unoriented Point Clouds by Enforcing Singular Hessian, Wang et al., TOG 2023
  - Iterative Poisson Surface Reconstruction (iPSR) for Unoriented Points, Hou et al., TOG 2022
  - 3DShape2Vecset: A 3D Shape Representation for Neural Fields and Generative Diffusion Models, Zhang et al., TOG 2023
  - Geoudf: Surface reconstruction from 3d point clouds via geometry-guided distance representation, Ren et al., TOG 2023

- Given good normals, surface reconstruction becomes much easier - the normal computation could be followed by Poisson reconstruction based on the normals for example. Therefore papers to compute oriented normals are relevant, such as:
  - Orienting Point Clouds with Dipole Propagation, Metzer et al., TOG 2021
  - Globally Consistent Normal Orientation for Point Clouds by Regularizing the Winding-Number Field, Xu et al., TOG 2023
  - SHS-Net: Learning Signed Hyper Surfaces for Oriented Normal Estimation of Point Clouds, Li et al., CVPR 2023

- The exposition could use some additional details and clarifications:
  - The approach for using local patches is missing some details: How many patches are used during optimization? In which order are patches evaluated? One after the other? Or is a random patch chosen in each iteration? This should be clarified.
  - The difference of the second stage to [50] could use some more discussion. The authors describe the application of [50] to local patches as main difference (as shown in Eq. 3). But since the expectancy in Eq. 3 is taken over random samples near random patches, how is this different than taking the expectancy over random samples near the whole shape (as in [50])? This could use some  discussion. (Or is the SDF actually optimized separately per patch? From reading I did not get this impression.)
  - [50] also uses a geometric consistency constraint as regularization, although such a regularization is not described by the authors. If it is not used, a discussion why the authors removed it might be useful.
  - c seems to be initialized to a fixed constant vector: the average of all embeddings in the training set, while the original DeepSDF paper uses a c randomly sampled from a Gaussian. Why did the authors use a different initialization for c here? Did the average empirically perform better? This could be discussed.
  - In the paragraph starting at line 129, It should be stated clearly that the method described here is the method proposed in [50] (although evaluated at random samples near random local patches, rather than random samples near the whole shape).

- The evaluation is quite extensive, but would benefit from a few improvements:
  - In the evaluation, it would be good to hear more about how the time used to fine-tune each optimization-based method was chosen. Was each result optimized until convergence? Or otherwise, how was the number of iterations/optimization time chosen?
  - It would be good to add DeepSDF to the comparison, as it is the closest to the initialization used for the optimization stage, and would show how much the fine-tuning  improves accuracy.
  - In the ablation, using a large local region seems worse than using a global approach without local regions. This should be discussed.

- References [52] and [53] are duplicates
questions: questions: - A clarification of the unclear details regarding local patches would help.
- A clarification of how timings for methods were chosen would help as well.
limitations: limitations: The authors have not discussed clear limitations unfortunately.
paper_summary: paper_summary: The presented work tackles the problem of reconstructing a shape from a noisy point cloud (PC) into an implicit representation, using the signed distance function (SDF). Recent methods are categorised into i) data-driven approaches that learn a shape prior with a dataset of training shapes, with poor generalization but generally faster, or ii) approaches that overfit on a single PC at a time, with better generalization but slow convergence. The paper proposes a middle ground: finetuning a pre-trained shape prior model to a single PC, using a novel approach that performs "local statistical reasoning": supervising local parts of the shape with multiple sampled noisy PC. The method is experimentally validated against multiple baselines, either data-driven or overfitting-based, and on multiple datasets showing better reconstruction accuracy at lower convergence times. A final ablation study shows that the prior and locality of the loss play a role in the method's performance.
strength_weakness: strength_weakness: 1. The main pipeline is fairly simple (fine-tuning a pre-trained model). I believe the locality of the "statistical reasoning" is perhaps the main novelty, and the ablation shows that it improves accuracy and convergence time (§ at L.342), but it is not discussed or explained why using a local approach instead of a global one improves convergence and accuracy.
2. The writing could be improved. More specifically, the experimental results on the different datasets all make more or less the same points and so could be condensed into something more compact. This can free up some space to expand other parts and increase their clarity, e.g., the related works and the "Neural Signed Distance Function." paragraph (L.101).
3. Throughout the work, the noise is assumed (additive) Gaussian. While this is not a weakness in itself, this still is an assumption that should be explained and made clear:
   1. Why is a Gaussian noise model good? Are Lidar data, for instance, usually with gaussian noise and what real use-cases does it apply to? Generally speaking, the motivation of the work could be expanded beyond "reconstructing a PC to an SDF".
   2. "Statistically, the expectation of the zero-level set should have the minimum distance to all the noisy point splitting." (L.131-132) This is true assuming Gaussian noise with expectation 0. With other noise types the loss may not perform as well (e.g., "Text removal" in Noise2Noise [1]).

[1] Lehtinen et al., Noise2Noise: Learning Image Restoration without Clean Data, ICML 2018.
questions: questions: 1. What is the reasoning behind using the locality in the "statistical reasoning" and its better performance?
2. For the ShapeNet dataset, is it per-category and then averaged or trained on multiple categories combined? (and which ones?) Same questions regarding the SRB/FAMOUS experiments with ShapeNet prior.
3. What are the PC sizes? It's hard to appreciate the other parameters without this info. What are the datasets sizes in number of training and test shapes?
4. How does the prior help if it apparently does not need to be from the same dataset as the test shapes (e.g., SRB/FAMOUS experiments)? Is it actually pre-training on any valid SDFs that helps? If so, how do you think it compares to the simple SAL [2] geometric initialization?
5. Why are the metrics (CD-L2/L1, NC, F1-score) not used consistently through the datasets and results?

[2] Atzmon and Lipman, SAL: Sign Agnostic Learning of Shapes from Raw Data, CVPR 2020.
limitations: limitations: Limitations are briefly mentioned in the appendix regarding noise level. What kind of noise level is considered too big? It might also be worth commenting on the noise types (see Weakness 3.), is the method working only for Gaussian noises?
paper_summary: paper_summary: The paper presents a method for learning neural signed distance functions (SDFs) from noisy point clouds. This approach integrates the advantages of data-driven and overfitting-based methods to enhance generalization, accuracy, and inference speed. They employ statistical reasoning within local regions to refine the estimation of SDFs without requiring clean point clouds or signed distance supervision.
strength_weakness: strength_weakness: 1. There are a lot of typos in the manuscript, e.g., In the Abstract and Introduction, "stat-of-the-art" should be "state-of-the-art"; Line 57: "Learning implicit functions have achieved" should be "Learning implicit functions has achieved"; Line 39: "overfits an neural network" should be " a neural network", "Querys", and so on. In the Abstract, the "prompt" might be "promote". It would be best to check out all the typos.
2. Dependency on initial conditions: The performance might heavily rely on the quality of the initial data-driven priors, which might limit the method's effectiveness when such priors are not well-tuned or applicable.
3. The paper does not extensively discuss the computational demands or scalability when applied to larger datasets (e.g., nuScenes and Waymo) or more complex scenarios.
questions: questions: 1. How does the method perform under limited computational resources? When scaling to larger datasets, is there a significant trade-off between accuracy and computational efficiency?
2. Can the method maintain its performance across different types of noise or corruption that were not part of the training set?
3. How does the method perform when such priors are unavailable, or how sensitive is it to the initial conditions set by these priors?
limitations: limitations: See the "Questions" and "Weaknesses" sections above.

Comments: title: We are glad to know our rebuttal addressed your concerns
We value the opportunity to discuss with you
comment: Dear reviewer 76DM,

As the reviewer-author discussion period is about to end, can you please let us know if our rebuttal addressed your concerns? If it is not the case, we are looking forward to taking the last minute to make further explanation and clarification.

Thanks,

The authors
Dear reviewer AdmT,

As the reviewer-author discussion period is about to end, can you please let us know if our rebuttal addressed your concerns? If it is not the case, we are looking forward to taking the last minute to make further explanation and clarification.

Thanks,

The authors
Hi, it's near the end of the rebuttal period. Could you please respond to the rebuttal?
Hi, could you please chime in? It's near the end of the discussion period!
Thanks for your comments and advice. We will add this discussion in our revision.


Best,

The authors
Thank you for the added clarifications.
Indeed, EMD may match some queries and points that are far away on the shape because of the one-to-one mapping and inherent noise in the problem. I see now how local patches can help alleviate this issue by only considering a smaller part of the shape at a time.

I would suggest adding these explanations to the paper to strengthen the use of local patches, if the authors find it relevant.
Dear reviewers,

As the reviewer-author discussion period is about to end, we are looking forward to your feedback on our rebuttal. Please let us know if our responses address your concerns. We would be glad to make any further explanation and clarification.

Thanks,

The authors
**1. Ablation Studies on Shape Codes**

We did two ablation studies related to shape codes. One is the code length in Tab.8, the other is the code initialization in Tab.9. In Tab.9, we tried random initialization for shape code in “Without Prior” (using network parameters that are learnable and randomly initialized), “Without Embed” (using network parameters that are learnable and initialized with our prior), and “Fixed Param” (using network parameters that are not learnable and initialized with our prior). 

Moreover, as we mentioned in c) in “G3. Traditional test-time optimization vs ours” in the rebuttal above, we also tried various normalizations based on the random initialization, which produces too bad results to get reported in the paper. In addition, we also tried some other variations like our averaged code in our preliminary results, such as randomly selecting one shape code from the training set as the initialization. Similarly, randomly selecting also produces too bad results to get reported in the paper.

We will follow your advice to reorganize our ablation study on shape codes in our revision. Please also let us know if you would like to see any other alternatives on code initialization.

**2. How about CD?**

You raised a great point. Actually, we followed [50] to use EMD to report our results. As a main contribution, the authors of [50] did a pretty good job to find that EMD works while CD does not work, please see Fig.4 and ablation studies “CD” in Tab.11 in their paper for more. But we are different, and we make everything happen in a patch rather than over the whole shape, which changes the results with CD a little bit. In our preliminary results, we found that the CD can produce much better results in our method than in [50], but the results are still much worse than the ones with EMD. Another consideration of using EMD is for fair comparison with [50]. 

Although CD can handle density difference between noises and queries and produces more reasonable results in local patches, it does not have one thing that is the key for the success as EMD, which is the one to one correspondence. CD involves many to one correspondence which is not efficient and targeted in statistical noise to noise mapping.

We will follow your advice and add a discussion on this in our revision.

**3. Why does averaged shape code work?**

The “Object-like” SDF is basically a shape, since the code is averaged over all training shapes. With the pretrained network parameters, that is our prior, at the very beginning, this averaged shape code can produce a shape that may be very similar to some shapes in the training set, which however is not necessarily similar to the noisy point cloud. This means that it is not nearer to the optimal point or closer to the target shape, but provides a relatively stable area for the optimization.

The reason why we use averaged shape code is three fold. 

i) One is that it can not only initialize an SDF with inside and outside, which is basically the same purpose as the geometry initialization (initializes network parameters so that a sphere like SDF is produced) introduced in SAL, but also inherit some geometry details from the training dataset. 

ii) Another one is that it provides a great scope for the optimization to search for a result. This is because the area where the averaged shape code is gathers more shape codes that represent meaningful shapes in the training set than other areas in the shape code space. This area provides a large enough candidate pool for our inference loss, which avoids the optimization to fly away due to the large change of shape code updated by the inferred information, such as signed distances in our case, which is fluctuating iteration by iteration, even at the same locations.

iii) Based on ii), the network parameters which saw various shapes during training can work with the shape code to fast update the SDF according to the inferred information from the loss.

These three folds make our averaged shape code is much better than the random initialization which provides a start where there may be no other learned shape codes at all, which makes it hard to find optimization directions in the following. 

We will follow your advice and add this discussion in our revision.
Thanks for your comments. Please find our explanation and clarification below. 

**1. Too far in “we avoid noisy points and queries that are far away from each other” is not related to sparsity.**

When we say too far, that is really far, and not caused by sparsity. Taking humans in D-FAUST dataset for example, if we sample some queries near the human’s head, the queries may get paired with noisy points on the human’s feet in the global reasoning method, due to the total randomness over the shape. Obviously, this kind of paires should be avoided since the statistical reasoning on these pairs does not contribute to the learning of SDFs, in contrast, it wastes some time and adds some additional burden on optimization. To relieve this burden, the optimization also requires some kinds of regularization, like the one in [50], to constrain the movement in the pulling, so that the field is inferred as an signed distance field. Please read more about the regularization in our reply to reviewer fwt4’s 5th question. 

We establish queries and noisy points pairs in local patches, which filters out unwanted pairs and does not need additional regularization or other things.

**2. Ablation on Global and Local in L342**

The experimental setting is the same except how we establish noisy points and queries. For the “Global”, we keep the sampling and pairing strategies the same as [50], which randomly samples a set of queries and a set of noisy points without the patch limits. The unwanted pairs in “Global” degenerates the accuracy and slows down the convergence.

**3. Difference to DeepSDF test-time optimization in Fig.8 and Fig.9**

i) We can infer signed distances but DeepSDF can not infer but using inferred or GT signed distances

DeepSDF requires GT or inferred signed distances to conduct the test-time optimization. It does not infer signed distances as ours, but just uses the ones that are ready to use. As stated in the second paragraph in the left column on Page 7 in the DeepSDF paper, they generate SDF point samples using depth maps and normals to sample points with known distances, which establish the GT signed distances. Then, DeepSDF uses maximum a posterior estimation to find the shape code (using a prior represented by fixed network parameters) that best explains a partial shape observation in Fig.8 and Fig.9, using Eq.10 with a loss function in Eq.4.

Ours does not require depth images and normals to estimate signed distances, but directly infer signed distances from noisy points. This means if you only have a piece of noisy point cloud, DeepSDF can not conduct the test-time optimization but we can handle it quite well.

ii) During test-time optimization, our can handle huge uncertainty while DeepSDF can not

Because of i), DeepSDF uses constant signed distance values as supervision, while we use signed distances inferred by the loss which produces fluctuating signed distances iteration by iteration, even at the same location.

iii) For prior generalization, we fine-tune the network while DeepSDF uses fixed network parameters

DeepSDF uses the same kind of supervision during both training and testing, such as signed distances, which makes prior generalization relatively easy. Thus, DeepSDF just uses fixed network parameters for prior generalization during test-time optimization.

However, we usually do not have signed distances for most scenarios during testing, but just raw observations, like noisy points. Thus, we need novel ways to infer signed distances, while turning using fixed parameters for prior generalization problems. Our method fine-tunes the network with a prior initialization for prior generalization.
Thanks to the authors for the detailed replies.

Regarding the contributions and novelty, what I meant with the paper using the learned prior as initialization is not that the optimization is like in DeepSDF, but that (apart from the local patches), the optimization is very similar to Noise-to-Noise, but using DeepSDF as initialization for the neural SDF that is being fitted during the optimization. While the choice of initial shape code c is interesting, this choice is not ablated as far as I can tell (adding an ablation would be good actually, especially if it is claimed as contribution), and also does not add a very large contribution by itself. Overall though, given the good performance, I still think the contribution is probably large enough for acceptance.

Regarding the discussion why local patches improve performance, I think the discussion provided in the rebuttal seems reasonable to some extent. If point cloud and query densities do not match across the shape, then queries may be matched to more distant points, and the local patches essentially stratify the problem by confining it to local patches. At this point though - why not use the Chamfer Distance instead? The CD would not have this problem with different point and query densities. Is it because we still want the behavior of EMD with its longer distance matches rather than CD in local patches? So overall, while this explanation is already reasonable if clarified a bit, the discussion of this point could also go a bit more in depth, especially since this is one of the main claimed contributions. The discussion of this point (whether it is a clarified version of the current discussion, or a more in-depth discussion) should be added to the paper.

Regarding the discussion why using an average shape code for c is better than a random shape code, the only relevant information I can find in the response (apart from references to the ablation) is that the average shape code gives a more 'object-like' appearance. This could some more elaboration. What does 'object-like' mean? Is the average shape code c typically closer to the optimum? Why is this the case? Maybe because the distribution of shape codes used by typical shapes is different from a Gaussian distribution and the average is better centered in this actual distribution of shape codes, so the initial shape output by the neural SDF is already closer to the target shape compared to a random shape code? A somewhat more elaborate discussion of this point would improve the motivation of this design choice.
Thanks for letting us know our rebuttal addressed your concerns.

We will response to your comments in the rebuttal above.

Best,

The authors
**Local patches work better**
> This is because we avoid noisy points and queries that are too far away from each other,

From my understanding, this is affected by the point cloud density: sparser PC might have fewer noisy points where the queries are projected. If this is correct, then using a local patch against the global shape should not differ much in that regard, *if* the number of sampled points and queries $U$ is scaled appropriately (ignoring memory restriction for the sake of the discussion). Is the local patch approach leveraging something else?

In the “Global and Local.” ablation (L.342), how are sampled the queries and noisy points for the global approach?

**DeepSDF test-time optimization.**

Note that DeepSDF has also been shown to work to reconstruct partial and noisy point clouds in its work, see Section 6.3 and Fig. 8 and 9 in their work. It does however require some pseudo-SDF to be constructed from the PC.
I thank the authors for their answers and clarifications to my questions, and for the added results in the rebuttal PDF.

I believe my weakness #3 regarding Gaussian noise has been mostly answered in the common rebuttal above, with the additional experiment in the rebuttal PDF. Additionally, I had misunderstood that all experiments were using additive Gaussian noise, while some are:
> real scans with unknown noise types, such as our results on SRB in Tab.4 and Fig.4, D-FAUST in Tab.6 and Fig.6, 3D scene in Tab.7 and Fig.7, KITTI in Fig.8, and Paris-rue-Madame in Fig.9.

Please, see my other comment in the common rebuttal for additional points.
Dear reviewers,

Thanks for your valuable comments. We will be happy to take any questions. Please let us know if our rebuttal address your concerns.

Thanks,
The authors