Isometry-Aware Preconditioning for Mesh Parameterization

Abstract

This paper presents a new preconditioning technique for large-scale geometric optimization problems, inspired by applications in mesh parameterization. Our positive (semi-)definite preconditioner acts on the gradients of optimization problems whose variables are positions of the vertices of a triangle mesh in R2 or of a tetrahedral mesh in R3, converting localized distortion gradients into the velocity of a globally near-rigid motion via a linear solve. We pose our preconditioning tool in terms of the Killing energy of a deformation field and provide new efficient formulas for constructing Killing operators on triangle and tetrahedral meshes. We demonstrate that our method is competitive with state-of-the-art algorithms for locally injective parameterization using a variety of optimization objectives and show applications to two- and three-dimensional mesh deformation.