Discrete conformal maps and ideal hyperbolic polyhedra

We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring M\"obius invariance, the definition of discrete conformal maps as circumcircle preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to addresses the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.Comment: 62 pages, 22 figures. v2: typos corrected, references added and updated, minor changes in exposition. v3, final version: typos corrected, improved exposition, some material moved to appendice

Second-order Shape Optimization for Geometric Inverse Problems in Vision

We develop a method for optimization in shape spaces, i.e., sets of surfaces modulo re-parametrization. Unlike previously proposed gradient flows, we achieve superlinear convergence rates through a subtle approximation of the shape Hessian, which is generally hard to compute and suffers from a series of degeneracies. Our analysis highlights the role of mean curvature motion in comparison with first-order schemes: instead of surface area, our approach penalizes deformation, either by its Dirichlet energy or total variation. Latter regularizer sparks the development of an alternating direction method of multipliers on triangular meshes. Therein, a conjugate-gradients solver enables us to bypass formation of the Gaussian normal equations appearing in the course of the overall optimization. We combine all of the aforementioned ideas in a versatile geometric variation-regularized Levenberg-Marquardt-type method applicable to a variety of shape functionals, depending on intrinsic properties of the surface such as normal field and curvature as well as its embedding into space. Promising experimental results are reported