Selective sampling with Gromov–Hausdorff metric: Efficient dense-shape correspondence via Confidence-based sample consensus

The demand for 3D environment sensing and understanding technologies has rapidly grown in recent years, leading to various essential applications, such as autonomous driving, AR/VR, and animations that require millisecond processing capabilities. One of the most significant and complex problems in this field is the dense point correspondence between partially rigid and nonrigid 3D shapes. The goal is to obtain a point-to-point mapping between two different point clouds or meshes of two figures at distinct positions. Robust optimization methods such as RANSAC have been crucial in developing stable and sustainable computer vision algorithms for 3D rigid alignment. Although many improvements have been introduced to the RANSAC algorithm, including learning-based proxy measures, few methods have extended the search space analysis proposed by RANSAC, making it difficult to adopt these methods in production systems. Recent works in the field of dense-shape correspondence for spectral decomposition of shapes use the Laplace–Beltrami–Operator (LBO) eigenbasis with remarkable performance. By projecting per-point features onto an LBO basis, these models can compute functional mapping based on LBO, which has led to a significant step forward in dense-shape correspondence tasks, thereby enriching the research in this field.

Training the spatial feature module is a time-consuming process, particularly in non-supervised datasets, which makes these methods impractical in many real-world applications. Furthermore, solving for the functional maps can be an unstable process because it requires matrix inversion or a least-squares approach. If the spatial features are incorrect or the spectral maps are too far from one another, the chances of the functional map being coherent become extremely low.

Despite these challenges, functional-map-based methods remain powerful tools for dense-shape correspondence tasks. However, to make them more practical for real-world applications, more efficient and stable solutions are required for rapid and accurate training.

One of the advantages of the proposed selective sampling with the Gromov–Hausdorff metric (SSGHM) is that it can filter out unsatisfactory features during both the training and testing phases. In functional mapping, poor features can be compared to "rotten apples", that harm the overall alignment process. With our method, the best subsets of the point cloud are selected at each step, ignoring bad points that can negatively affect the alignment.

Spatial techniques often struggle with symmetries, mirrored points, and local deformations because local descriptors have difficulty distinguishing between them.

The key contributions of this study can be summarized as follows:

• Selective sampling: A novel method based on a neural network is proposed for the selective sampling of point-cloud data, which allows the selection of the best subsets of points for alignment. By ignoring bad points that only harm the alignment and selecting the most informative ones, the method improves alignment accuracy and reduces computation time.

• Gromov–Hausdorff metric: The Gromov–Hausdorff metric is introduced into point-cloud alignment as a consensus-sampling technique, to provide a global distance measure that considers topological differences between point clouds or meshes.

• Computation time: Convergence time and stability introduce a faster training scheme that disregards incorrect or non-informative points, utilizing informative structures as the driving force for training.