Amit Bracha

David Bensaïd, Amit Bracha, Ron Kimmel

Evaluating the similarity of non-rigid shapes with significant partiality is a fundamental task in numerous computer vision applications. Here, we propose a novel axiomatic method to match similar regions across shapes. Matching similar regions is formulated as the alignment of the spectra of operators closely related to the Laplace-Beltrami operator (LBO). The main novelty of the proposed approach is the consideration of differential operators defined on a manifold with multiple metrics. The choice of a metric relates to fundamental shape properties while considering the same manifold under different metrics can thus be viewed as analyzing the underlying manifold from different perspectives. Specifically, we examine the scale-invariant metric and the corresponding scale-invariant Laplace-Beltrami operator (SI-LBO) along with the regular metric and the regular LBO. We demonstrate that the scale-invariant metric emphasizes the locations of important semantic features in articulated shapes. A truncated spectrum of the SI-LBO consequently better captures locally curved regions and complements the global information encapsulated in the truncated spectrum of the regular LBO. We show that matching these dual spectra outperforms competing axiomatic frameworks when tested on standard benchmarks. We introduced a new dataset and compare the proposed method with the state-of-the-art learning based approach in a cross-database configuration. Specifically, we show that, when trained on one data set and tested on another, the proposed axiomatic approach which does not involve training, outperforms the deep learning alternative.

Saar Huberman, Amit Bracha, Ron Kimmel

A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface and estimating distances on the polygon. We show that exact geodesic distances restricted to the polygon are at most second-order accurate with respect to the distances on the corresponding continuous surface. By order of accuracy we refer to the convergence rate as a function of the average distance between sampled points. Next, a higher-order accurate deep learning method for computing geodesic distances on surfaces is introduced. Traditionally, one considers two main components when computing distances on surfaces: a numerical solver that locally approximates the distance function, and an efficient causal ordering scheme by which surface points are updated. Classical minimal path methods often exploit a dynamic programming principle with quasi-linear computational complexity in the number of sampled points. The quality of the distance approximation is determined by the local solver that is revisited in this paper. To improve state of the art accuracy, we consider a neural network-based local solver which implicitly approximates the structure of the continuous surface. We supply numerical evidence that the proposed learned update scheme provides better accuracy compared to the best possible polyhedral approximations and previous learning-based methods. The result is a third-order accurate solver with a bootstrapping-recipe for further improvement.

Amit Bracha, Thomas Dagès, Ron Kimmel

While dealing with matching shapes to their parts, we often apply a tool known as functional maps. The idea is to translate the shape matching problem into "convenient" spaces by which matching is performed algebraically by solving a least squares problem. Here, we argue that such formulations, though popular in this field, introduce errors in the estimated match when partiality is invoked. Such errors are unavoidable even for advanced feature extraction networks, and they can be shown to escalate with increasing degrees of shape partiality, adversely affecting the learning capability of such systems. To circumvent these limitations, we propose a novel approach for partial shape matching. Our study of functional maps led us to a novel method that establishes direct correspondence between partial and full shapes through feature matching bypassing the need for functional map intermediate spaces. The Gromov Distance between metric spaces leads to the construction of the first part of our loss functions. For regularization we use two options: a term based on the area preserving property of the mapping, and a relaxed version that avoids the need to resort to functional maps. The proposed approach shows superior performance on the SHREC'16 dataset, outperforming existing unsupervised methods for partial shape matching.Notably, it achieves state-of-the-art results on the SHREC'16 HOLES benchmark, superior also compared to supervised methods. We demonstrate the benefits of the proposed unsupervised method when applied to a new dataset PFAUST for part-to-full shape correspondence.

Yaniv Wolf, Amit Bracha, Ron Kimmel

Recently, 3D Gaussian Splatting (3DGS) has emerged as an efficient approach for accurately representing scenes. However, despite its superior novel view synthesis capabilities, extracting the geometry of the scene directly from the Gaussian properties remains a challenge, as those are optimized based on a photometric loss. While some concurrent models have tried adding geometric constraints during the Gaussian optimization process, they still produce noisy, unrealistic surfaces. We propose a novel approach for bridging the gap between the noisy 3DGS representation and the smooth 3D mesh representation, by injecting real-world knowledge into the depth extraction process. Instead of extracting the geometry of the scene directly from the Gaussian properties, we instead extract the geometry through a pre-trained stereo-matching model. We render stereo-aligned pairs of images corresponding to the original training poses, feed the pairs into a stereo model to get a depth profile, and finally fuse all of the profiles together to get a single mesh. The resulting reconstruction is smoother, more accurate and shows more intricate details compared to other methods for surface reconstruction from Gaussian Splatting, while only requiring a small overhead on top of the fairly short 3DGS optimization process. We performed extensive testing of the proposed method on in-the-wild scenes, obtained using a smartphone, showcasing its superior reconstruction abilities. Additionally, we tested the method on the Tanks and Temples and DTU benchmarks, achieving state-of-the-art results.

Amit Bracha, Thomas Dagès, Ron Kimmel

When matching parts of a surface to its whole, a fundamental question arises: Which points should be included in the matching process? The issue is intensified when using isometry to measure similarity, as it requires the validation of whether distances measured between pairs of surface points should influence the matching process. The approach we propose treats surfaces as manifolds equipped with geodesic distances, and addresses the partial shape matching challenge by introducing a novel criterion to meticulously search for consistent distances between pairs of points. The new criterion explores the relation between intrinsic geodesic distances between the points, geodesic distances between the points and surface boundaries, and extrinsic distances between boundary points measured in the embedding space. It is shown to be less restrictive compared to previous measures and achieves state-of-the-art results when used as a loss function in training networks for partial shape matching.

Amit Bracha, Noam Rotstein, David Bensaïd et al.

Depth estimation is a cornerstone of a vast number of applications requiring 3D assessment of the environment, such as robotics, augmented reality, and autonomous driving to name a few. One prominent technique for depth estimation is stereo matching which has several advantages: it is considered more accessible than other depth-sensing technologies, can produce dense depth estimates in real-time, and has benefited greatly from the advances of deep learning in recent years. However, current techniques for depth estimation from stereoscopic images still suffer from a built-in drawback. To reconstruct depth, a stereo matching algorithm first estimates the disparity map between the left and right images before applying a geometric triangulation. A simple analysis reveals that the depth error is quadratically proportional to the object's distance. Therefore, constant disparity errors are translated to large depth errors for objects far from the camera. To mitigate this quadratic relation, we propose a simple but effective method that uses a refinement network for depth estimation. We show analytical and empirical results suggesting that the proposed learning procedure reduces this quadratic relation. We evaluate the proposed refinement procedure on well-known benchmarks and datasets, like Sceneflow and KITTI datasets, and demonstrate significant improvements in the depth accuracy metric.

Noam Rotstein, Amit Bracha, Ron Kimmel

Reconstruction of geometric structures from images using supervised learning suffers from limited available amount of accurate data. One type of such data is accurate real-world RGB-D images. A major challenge in acquiring such ground truth data is the accurate alignment between RGB images and the point cloud measured by a depth scanner. To overcome this difficulty, we consider a differential optimization method that aligns a colored point cloud with a given color image through iterative geometric and color matching. In the proposed framework, the optimization minimizes the photometric difference between the colors of the point cloud and the corresponding colors of the image pixels. Unlike other methods that try to reduce this photometric error, we analyze the computation of the gradient on the image plane and propose a different direct scheme. We assume that the colors produced by the geometric scanner camera and the color camera sensor are different and therefore characterized by different chromatic acquisition properties. Under these multimodal conditions, we find the transformation between the camera image and the point cloud colors. We alternately optimize for aligning the position of the point cloud and matching the different color spaces. The alignments produced by the proposed method are demonstrated on both synthetic data with quantitative evaluation and real scenes with qualitative results.