Lifted bijections for low distortion surface mappings

Abstract

Figure 1: The algorithm presented in this paper generates low-distortion bijective mappings between surfaces from a sparse set of landmarks (visualized as colored spheres here). The maps are visualized by transferring the texture of the visible part in the left mesh of each pair to the right mesh, using the computed mappings. For example, the right pair shows a mapping of a horse to a giraffe; note how the map stretches gracefully at the neck area. AbstractThis paper introduces an algorithm for computing low-distortion, bijective mappings between surface meshes. The algorithm recieves as input a coarse set of corresponding pairs of points on the two surfaces, and follows three steps: (i) cutting the two meshes to disks in a consistent manner; (ii) jointly flattening the two disks via a novel formulation for minimizing isometric distortion while guaranteeing local injectivity (the flattenings can overlap, however); and (iii) computing a unique continuous bijection that is consistent with the flattenings. The construction of the algorithm stems from two novel observations: first, bijections between disk-type surfaces can be uniquely and efficiently represented via consistent locally injective flattenings that are allowed to be globally overlapping. This observation reduces the problem of computing bijective surface mappings to the task of computing locally injective flattenings, which is shown to be easier. Second, locally injective flattenings that minimize isometric distortion can be efficiently characterized and optimized in a convex framework. Experiments that map a wide baseline of pairs of surface meshes using the algorithm are provided. They demonstrate the ability of the algorithm to produce high-quality continuous bijective mappings between pairs of surfaces of varying isometric distortion levels.

Continuous semantic description of 3D meshes

“…It intrinsically encodes a smooth semantic mapping between two shapes that allows for a smart automatic geometry transfer. Note that such correspondence may also be computed using surface mapping algorithms [42,43]. However, these algorithms are clearly not suited for the real-time interactions required by design applications.…”

Seamless surface mappings

“…In [Kraevoy and Sheffer 2004;Bradley et al 2008;Schreiner et al 2004] coarse meshes are used as the base domain. In [Aigerman et al 2014] the mappings to the common planar domain are not required to be injective but rather only locally-injective, however they also specify the correspondences along geodesics connecting the input landmarks, thereby prescribing the image of the seams. [Steiner and Fischer 2005] generate seamless parameterizations of a mesh.…”

“…More generally, the map f can be defined also in cases where Φ, Ψ are locally, but not globally injective. In these cases, after applying the cut-and-paste operations and reaching the situation where the images of Φ, Ψ coincide, the map f can be defined via the implicit relation Φ = Ψ • f , and can be computed using a lifting algorithm, similar to the one detailed in Aigerman et al, [2014]. That algorithm sequentially computes the map piece by piece by using the fact that locally the flattenings are injective and can be inverted.…”

“…(8) while Φ is fixed. For the optimization of E (Φ; c) as a function of Φ we employ the convexification method of [Aigerman et al 2014]. For completeness, in Appendix B we provide all the necessary details of the optimization of E (Φ; c).…”

“…The surface mapping is then induced according to the overlay of the flattened disks of one mesh over the flattened disks of the other mesh. This method often requires prescribing correspondences also along the cuts [Aigerman et al 2014] or alternatively facing the challenging problem of optimizing also the diskboundary correspondence and/or the cuts directly over the surfaces [Schreiner et al 2004;Kraevoy and Sheffer 2004]. In most cases the choice of cuts affects the resulting mapping, and quite often the cut area will exhibit visible artifacts such as non-smoothness, distortion bias or wrong correspondences.…”