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.

Analytic Eigensystems for Isotropic Distortion Energies

“…Other distortion energies were addressed using block coordinate descent methods (Fu et al 2015;Horman and Greiner 1999) and Gauss-Newton solvers (Eigensatz and Pauly 2009). More recently, Kovalsky et al (2016) showed that the Laplacian matrix is a reliable quadratic proxy for many distortion energies, while Claici et al (2017) advocated the discrete Killing operator as an isometry-aware proxy. Alternatively, Rabinovich et al (2017) proposed an iterative reweighting scheme of the Laplacian based on gradient residuals.…”

“…To overcome this issue, many strategies have been developed for modifying the Hessian matrix using a variety of convex proxies. For instance, some techniques replace the Hessian with a Laplacianlike preconditioning matrix at the cost of degrading convergence to first-order (Claici et al 2017;Kovalsky et al 2016). Other methods alter the Hessian by removing negative eigenvalues numerically (Stomakhin et al 2012;Teran et al 2005) or via 2D composite majorization (Shtengel et al 2017).…”

Accelerating ADMM for efficient simulation and optimization

“…[2017] proposed a scalable approach to compute locally injective mappings, via local-global minimization of a reweighted proxy function. Claici et al [2017] proposed a preconditioner for fast minimization of distortion energies. Shtengel et al [2017] applied the idea of majorizationminimization [Lange 2004] to iteratively update and minimize a convex majorizer of the target energy in geometric optimization.…”

Progressive embedding

“…These non-linear energies are difficult to minimize, stemming a series of methods specifically targeting this problem. They include coordinate descent Labsik et al 2000], parallel gradient descent [Fu et al 2015], Anderson Acceleration [Peng et al 2018], as well as other quasi-newton approaches [Claici et al 2017;• 32:3 Kovalsky et al 2016;Rabinovich et al 2017;Shtengel et al 2017;Smith and Schaefer 2015;Zhu et al 2018].…”