Estimating dense point-to-point correspondences between two isometric shapes represented as 3D point clouds is a fundamental problem in geometry processing, with applications in texture and motion transfer. However, this task becomes particularly challenging when the shapes undergo non-rigid transformations, as is often the case with approximately isometric point clouds. Most existing algorithms address this challenge by establishing correspondences between functions defined on the shapes, rather than directly between points, because function mappings admit a linear representation in the spectral domain. State-of-the-art methods compute this linear representation using the eigenfunctions of the Laplace-Beltrami Operator (LBO) along with a small set of initial corresponding functions between the shapes. However, for approximately isometric point clouds, two key issues arise: (1) the eigenfunctions of the LBO may become misaligned, and (2) the initial corresponding functions may include outliers, both of which degrade the quality of the resulting correspondences. In this work, we propose an efficient approach to align the spectra of the LBOs of the two shapes, enabling the eigenfunctions to remain compatible even for approximately isometric 3D point clouds. Additionally, we introduce a technique to make function correspondence estimation robust to outliers. We validate our approach by comparing it with state-of-the-art 3D shape-matching algorithms on benchmark datasets, demonstrating its effectiveness.

Implicit 3D surface reconstruction of an object from its partial and noisy 3D point cloud scan is the classical geometry processing and 3D computer vision problem. In the literature, various 3D shape representations have been developed, differing in memory efficiency and shape retrieval effectiveness, such as volumetric, parametric, and implicit surfaces. Radial basis functions provide memory-efficient parameterization of the implicit surface. However, we show that training a neural network using the mean squared error between the ground-truth implicit surface and the linear basis-based implicit surfaces does not converge to the global solution. In this work, we propose locally supported compact radial basis functions for a linear representation of the implicit surface. This representation enables us to generate 3D shapes with arbitrary topologies at any resolution due to their continuous nature. We then propose a neural network architecture for learning the linear implicit shape representation of the 3D surface of an object. We learn linear implicit shapes within a supervised learning framework using ground truth Signed-Distance Field (SDF) data for guidance. The classical strategies face difficulties in finding linear implicit shapes from a given 3D point cloud due to numerical issues (requires solving inverse of a large matrix) in basis and query point selection. The proposed approach achieves better Chamfer distance and comparable F-score than the state-of-the-art approach on the benchmark dataset. We also show the effectiveness of the proposed approach by using it for the 3D shape completion task.

Autonomous assembly of objects is an essential task in robotics and 3D computer vision. It has been studied extensively in robotics as a problem of motion planning, actuator control and obstacle avoidance. However, the task of developing a generalized framework for assembly robust to structural variants remains relatively unexplored. In this work, we tackle this problem using a recurrent graph learning framework considering inter-part relations and the progressive update of the part pose. Our network can learn more plausible predictions of shape structure by accounting for priorly assembled parts. Compared to the current state-of-the-art, our network yields up to 10% improvement in part accuracy and up to 15% improvement in connectivity accuracy on the PartNet [23] dataset. Moreover, our resulting latent space facilitates exciting applications such as shape recovery from the point-cloud components. We conduct extensive experiments to justify our design choices and demonstrate the effectiveness of the proposed framework.

Reflection symmetry is ubiquitous in nature and plays an important role in object detection and recognition tasks. Most of the existing methods for symmetry detection extract and describe each keypoint using a descriptor and a mirrored descriptor. Two keypoints are said to be mirror symmetric key-points if the original descriptor of one keypoint and the mirrored descriptor of the other keypoint are similar. However, these methods suffer from the following issue. The background pixels around the mirror symmetric pixels lying on the boundary of an object can be different. Therefore, their descriptors can be different. However, the boundary of a symmetric object is a major component of global reflection symmetry. We exploit the estimated boundary of the object and describe a boundary pixel using only the estimated normal of the boundary segment around the pixel. We embed the symmetry axes in a graph as cliques to robustly detect the symmetry axes. We show that this approach achieves state-of-the-art results in a standard dataset.

We propose an energy minimization approach to detect multiple reflection symmetry axes present in a given image representing fronto-parallel view of a scene. We perform local feature matching to detect the pairs of mirror symmetric points, and in order to formulate an energy function, we use the geometric characteristics of the symmetry axis. That is, it passes through the midpoint of line segment joining the two mirror symmetric points and is perpendicular to the vector joining two mirror symmetric points. We propose a novel k-symmetry clustering algorithm to minimize this energy function in order to efficiently find all the symmetry axes present in the given image. We evaluate the proposed method on the standard datasets and show that we get comparable and better results than that of the state-of-the-art reflection symmetry detection methods.

Reflection symmetry is present in most of the man-made or naturally formed objects. In computer vision, real-world scenes are represented by dense 3-D models or by 2-D projections, such as images captured by cameras. Most of the existing methods either detect reflection symmetry from dense 3-D models or 2-D projections. However, generating a dense 3-D model is a computationally expensive process and reflection symmetry may not be evident in any of the 2-D views obtained through projections. In this letter, we propose an energy minimizationbased approach to detect the reflection symmetry present in the object from its multiple 2-D projections captured from different viewpoints and the sparse 3-D model obtained using these projections. The proposed approach only estimates the sparse 3-D model and utilizes content of the images in terms of local scale invariant features. The energy minimization problem reduces to the problem of finding the eigenvector corresponding to the smallest eigenvalue of a small matrix, thereby leading to reduction in computations.