Soft Maps Between Surfaces

The problem of mapping between two non-isometric surfaces admits ambiguities on both local and global scales. For instance, symmetries can make it possible for multiple maps to be equally acceptable, and stretching, slippage, and compression introduce difficulties deciding exactly where each point should go. Since most algorithms for point-to-point or even sparse mapping struggle to resolve these ambiguities, in this paper we introduce soft maps, a probabilistic relaxation of point-to-point correspondence that explicitly incorporates ambiguities in the mapping process. In addition to explaining a continuous theory of soft maps, we show how they can be represented using probability matrices and computed for given pairs of surfaces through a convex optimization explicitly trading off between continuity, conformity to geometric descriptors, and spread. Given that our correspondences are encoded in matrix form, we also illustrate how low-rank approximation and other linear algebraic tools can be used to analyze, simplify, and represent both individual and collections of soft maps.