# A Physicist's View on Partial 3D Shape Matching
## 1 Introduction
They first define a set of well-chosen landmarks or keypoints on the surfaces of the shapes, then assign \"signatures\" to those keypoints (i.e., their coordinates in the metric parameterizing domain), and finally determine a correspondence between these points, using the similarities of their signatures. Such a strategy has become standard for comparing 2D images. Methods such as SIFT [19], SURF [20] or ORB [21] are commonly used to detect keypoints within 2D images and assign them signatures. Those keypoints are then matched using techniques such as RANSAC or the iterative closest point (ICP) algorithm. The problem of keypoint detection and signature assignment is harder for 3D objects. Many methods for assigning a signature to a keypoint have been proposed [22], such as those that extrapolate the 2D signatures for images by building multiscale representations of the neighborhood of the keypoint [23; 24; 25; 26], those that rely on the properties of the Laplace-Beltrami operator defined on the surface of interest [27; 28; 29], and those that rely on conformal mapping to a standard domain [14; 30; 31]. Matching the points based on their signatures is performed using the concept of \"bag-of-features\" to search for similar shapes, following the idea of Google search [32], the concept of shape distributions [33], by directly comparing top correspondences between shapes [29], or by using optimal transport [14; 34; 35; 36]. The second approach consists of relaxing the requirements that the correspondence be point-wise by considering soft correspondence [16; 37; 38; 39]. Finally, we briefly mention the data-driven approaches that take advantage of modern machine learning frameworks. We refer readers interested in those techniques to recent papers and surveys [22; 40; 41; 42; 43; 44; 45]. Our aim in this paper is to provide an alternate framework for shape comparison that falls under the shape correspondence category, but that uses methods derived from statistical physics to measure the similarities between shapes, with a special focus on partial matching (see Figure 1 for an overview). We consider shapes that are defined by their surface, usually represented by a triangulated mesh characterized with vertices, edges, and triangles. We consider all vertices in a mesh as keypoints. We test two types of signatures for the vertices, one based on a representation of their neighborhood, and one based on the properties of the Laplace Beltrami operator for the mesh. The former is based on the idea of scale-invariant spin images adapted to triangular meshes, the LD-SIFT signatures [46]. The latter is based on the idea of solving the Shrodinger equation on the surface to characterize how waves travel on this surface and therefore capture its geometry. The corresponding signatures are referred to as wave kernel (WK) signatures [29]. The mapping between the vertices is generated from the transport plan that solves either the optimal balanced transport (OT) problem or the optimal unbalanced transport (UOT) problem. We consider the unbalanced versions of the OT problem as it is expected to solve partial matching problems. We use a statistical physics approach to solve these OT problems [47; 48]. The cost associated with the optimal plan defines a distance between the two shapes. This paper does not stand on its own. The concepts of spin images and WK signatures for meshes have been proposed before. The idea of using optimal transport to compute correspondences between points describing shapes has been described in detail in the pioneering work of [14], and applied in different settings [31; 34; 35]. The novelty of this paper is to integrate those different components into a global physics-based approach for solving the full and partial shape registration problems. Our report should not be expected to be exhaustive: we limit ourselves to two shape signatures and two optimal transport techniques, but provide in-depth analyses of their strengths and weaknesses.