Coulomb shapes: using electrostatic forces for deformation-invariant shape representation

Canonical shape analysis is a popular method in deformable shape matching, trying to bring the shape into a canonical form that undoes its non-rigid deformations, thus reducing the problem of non-rigid matching into a rigid one. The canonization can be performed by measuring geodesic distances between all pairs of points on the shape and embedding them into a Euclidean space by means of multidimensional scaling (MDS), which reduces the intrinsic isometries of the shape into the extrinsic (Euclidean) isometries of the embedding space. A notable drawback of MDS-based canonical forms is their sensitivity to topological noise: different shape connectivity can affect dramatically the geodesic distances, resulting in a global distortion of the canonical form. In this paper, we propose a different shape canonization approach based on a physical model of electrostatic repulsion. We minimize the Coulomb energy subject to the local distance constraints between adjacent shape vertices. Our model naturally handles topological noise, allowing to 'tear' the shape at points of strong repulsion. Furthermore, the problem is computationally efficient, as it lends itself to fast multipole methods. We show experimental results in which our method compares favorably to MDS-based canonical forms.