A Splitting Scheme for Flip-Free Distortion Energies

SIAM Journal on Imaging Sciences

Oded Stein, MIT

Jiajin Li, The Chinese University of Hong Kong

Justin Solomon, MIT

Figure 1. Minimizing distortion energies in a variety of applications using our splitting method: UV mapping (left, computing a distortion-minimizing map from the surface to R2), shape deformation (center, fixing control points to deformed position and find the distortion-minimizing map), volume correspondence (right, finding the distortion-minimizing map between the interior of two different surfaces). Our method produces a flip-free result, unlike methods based on energies such as ARAP, which can exhibit flips when performing the same operation (flipped elements in red).


We introduce a robust optimization method for flip-free distortion energies used, for example, in parametrization, deformation, and volume correspondence. This method can minimize a variety of distortion energies, such as the symmetric Dirichlet energy and our new symmetric gradient energy. We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel Alternating Direction Method of Multipliers (ADMM) algorithm to deal with the non-convex, non-smooth nature of distortion energies. The scheme results in an efficient method where the global step involves a single matrix multiplication and the local steps are closed-form per-triangle/per-tetrahedron expressions that are highly parallelizable. The resulting general-purpose optimization algorithm exhibits robustness to flipped triangles and tetrahedra in initial data as well as during the optimization. We establish the convergence of our proposed algorithm under certain conditions and demonstrate applications to parametrization, deformation, and volume correspondence.

Cite as

author = {Stein, Oded and Li, Jiajin and Solomon, Justin},
title = {A Splitting Scheme for Flip-Free Distortion Energies},
journal = {SIAM Journal on Imaging Sciences},
volume = {15},
number = {2},
pages = {925--959},
year = {2022}


This work is supported by the Swiss National Science Foundation’s Early Postdoc.Mobility fellowship. The MIT Geometric Data Processing group acknowledges the generous support of Army Research Office grant W911NF2010168, of Air Force Office of Scientific Research award FA9550-19-1-031, of National Science Foundation grant IIS-1838071, from the CSAIL Systems that Learn program, from the MIT–IBM Watson AI Laboratory, from the Toyota–CSAIL Joint Research Center, from a gift from Adobe Systems, from an MIT.nano ImmersionLab/NCSOFT Gaming Program seed grant, and from the Skoltech–MIT Next GenerationProgram.