Adaptive Block Coordinate Descent for Distortion Optimization

Abstract

We present a new algorithm for optimizing geometric energies and computing positively oriented simplicial mappings. Our major improvements over the state‐of‐the‐art are: (i) introduction of new energies for repairing inverted and collapsed simplices; (ii) adaptive partitioning of vertices into coordinate blocks with the blended local‐global strategy for more efficient optimization and (iii) introduction of the displacement norm for improving convergence criteria and for controlling block partitioning. Together these improvements form the basis for the Adaptive Block Coordinate Descent (ABCD) algorithm aimed at robust geometric optimization. ABCD achieves state‐of‐the‐art results in distortion minimization, even under hard positional constraints and highly distorted invalid initializations that contain thousands of collapsed and inverted elements. Starting with an invalid non‐injective initial map, ABCD behaves as a modified block coordinate descent up to the point where the current mapping is cleared of invalid simplices. Then, the algorithm converges rapidly into the chosen iterative solver. Our method is very general, fast‐converging and easily parallelizable. We show over a wide range of 2D and 3D problems that our algorithm is more robust than existing techniques for locally injective mapping.