Stable Region Correspondences Between Non‐Isometric Shapes

Abstract

We consider the problem of finding meaningful correspondences between 3D models that are related but not necessarily very similar. When the shapes are quite different, a point‐to‐point map is not always appropriate, so our focus in this paper is a method to build a set of correspondences between shape regions or parts. The proposed approach exploits a variety of feature functions on the shapes and makes use of the key observation that points in matching parts have similar ranks in the sorting of the corresponding feature values. Our algorithm proceeds in two steps. We first build an affinity matrix between points on the two shapes, based on feature rank similarity over many feature functions. We then define a notion of stability of a pair of regions, with respect to this affinity matrix, obtained as a fixed point of a nonlinear operator. Our method yields a family of corresponding maximally stable regions between the two shapes that can be used to define shape parts. We observe that this is an instance of the biclustering problem and that it is related to solving a constrained maximal eigenvalue problem. We provide an algorithm to solve this problem that mimics the power method. We show the robustness of its output to noisy input features as well its convergence properties. The obtained part correspondences are shown to be almost perfect matches in the isometric case, and also semantically appropriate even in non‐isometric cases. We provide numerous examples and applications of this technique, for example to sharpening correspondences in traditional shape matching algorithms.

Region-Based Correspondence Between 3D Shapes via Spatially Smooth Biclustering

“…Regarding RBC-specific techniques, the most relevant work is the recent stable region correspondences (SRC) approach [10], which uses a power iteration scheme. We can directly compare the corresponding regions obtained by S 4 B and SRC with respect to the ground-truth mapping between the two shapes.…”

“…In this area, our work is most closely related to [10], where it was recently proposed to cast RBC as a biclustering problem. That work also introduced an efficient biclustering algorithm based on a power iteration scheme.…”

“…These techniques are better suited to the general shape correspondence problem, as they do not seek precise (e.g., bijective) maps between points, easily accommodating significant sampling, geometric, or even topological changes [35]. Furthermore, those soft maps can also be used as input to more refined point-based correspondence methods, to help improve the robustness and accuracy thereof [10].…”

“…This problem is different from another well-studied task, called shape co-segmentation, since in RBC the goal is not to find meaningful semantic segments in various shapes (e.g., limbs in animal shapes), but rather to determine regions in the two shapes that are in correspondence. As recently mentioned in [10], RBC is closely related to biclustering [21], in particular if the points on the shapes can be endowed with a similarity measure (e.g., based on on some descriptor). Given such a measure, a similarity matrix may be built and the goal of biclustering is to simultaneously cluster both the rows and columns of this matrix [21].…”

“…However, maybe surprisingly, biclustering has not been used for shape correspondence, with the notable exception of [10]. This is arguably due to the two following challenges: Typical 3D shapes in computer vision and graphics contain considerable geometric information, which is typically not taken into account in biclustering methods.…”