Reversible Harmonic Maps between Discrete Surfaces

Abstract

Information transfer between triangle meshes is of great importance in computer graphics and geometry processing. To facilitate this process, a smooth and accurate map is typically required between the two meshes. While such maps can sometimes be computed between nearly-isometric meshes, the more general case of meshes with diverse geometries remains challenging. We propose a novel approach for direct map computation between triangle meshes without mapping to an intermediate domain, which optimizes for the harmonicity and reversibility of the forward and backward maps. Our method is general both in the information it can receive as input, e.g. point landmarks, a dense map or a functional map, and in the diversity of the geometries to which it can be applied. We demonstrate that our maps exhibit lower conformal distortion than the state-of-the-art, while succeeding in correctly mapping key features of the input shapes.ACM Reference format:

Elastic Correspondence between Triangle Meshes

“…However, these methods still require extracting a point-wise map from the result, for applications such as joint triangular remeshing. Finally, perhaps closest to our approach, is the method by Ezuz et al [ESBC19]. It minimizes the harmonic energy of the forward and backward maps coupled with a reversibility constraint, which promotes bijective maps.…”

“…The first term ∑ t∈Tâ t tr Gt is sensitive to changes in edge lengths and coincides with the Dirichlet energy used in [ESBC19]. The second term penalizes variations in triangle areas.…”

Dense Point‐to‐Point Correspondences Between Genus‐Zero Shapes

“…Map refinement Another approach is to start with partial correspondence information, either in the form of sparse constraints or soft maps, and to transform these into dense point‐to‐point correspondences. Examples of such approaches include generalizing Euclidean affine combinations to surfaces [PBDSH13], optimizing the transport plan between surfaces [MCSK*17], and minimizing an energy that tries to preserve the given correspondences while penalizing the (approximate) Dirichlet energies of both the forward and inverse maps [ESBC19].…”

Consistent Shape Matching via Coupled Optimization

“…In comparison, our method directly optimizes for a dense map and thus we avoid the error prone task of map recovery from a given functional map. Recently, [ESBC19] proposed an optimization framework which is similar in nature to ours, solving jointly for the correspondence and its inverse. Their approach is intrinsic to the shape as our method, but they utilize a different energy functional and obtain a global optimization problem, whereas ours is local.…”