A Novel Stretch Energy Minimization Algorithm for Equiareal Parameterizations

Abstract

Surface parameterizations have been widely applied to computer graphics and digital geometry processing. In this paper, we propose a novel stretch energy minimization (SEM) algorithm for the computation of equiareal parameterizations of simply connected open surfaces with a very small area distortion and a highly improved computational efficiency. In addition, the existence of nontrivial limit points of the SEM algorithm is guaranteed under some mild assumptions of the mesh quality. Numerical experiments indicate that the efficiency, accuracy, and robustness of the proposed SEM algorithm outperform other state-of-the-art algorithms. Applications of the SEM on surface remeshing and surface registration for simply connected open surfaces are demonstrated thereafter. Thanks to the SEM algorithm, the computations for these applications can be carried out efficiently and robustly.

A Novel Algorithm for Volume-Preserving Parameterizations of 3-Manifolds

“…Numerical methods for computing surface equiareal parameterizations have been gradually getting much more attention in recent years because equiareal mappings preserve the density of the vertices well. Some efficient numerical methods, including the stretch-minimizing method [60,74], the Lie advection method [81], discrete OMT [80,68], density-equalizing mapping [17], and the stretch energy minimization (SEM) [76] have been developed to achieve this purpose.…”

“…The boundary spherical equiareal parameterization is carried out by applying the SEM algorithm [76]. The original SEM algorithm [76] aims to compute disk-shaped equiareal parameterizations of simply connected open surfaces by minimizing the stretch energy functional…”

“…Similarly as [58,73,76] and based on Proposition 4.1, the cotangent formula (4.14) can be fairly changed according to the image of a mapping f by…”

“…An angle-preserving parameterization is called a conformal parameterization, which has been applied in the texture mapping of surfaces [40,50] and the image analysis of proximal femur surfaces [55]. On the other hand, an area-preserving parameterization is called an authalic or equiareal parameterization, which has been applied in the surface registration and the remeshing [17,76]. More details for methods and applications of surface parameterizations can be found in the survey papers [30,64,42,10,33,39].…”

“…First, the initial boundary mapping is chosen to be a spherical conformal parameterization f : S → S 2 , which is computed by minimizing the discrete Dirichlet energy [22]. Then, the boundary mapping is deformed into an equiareal parameterization, which is iteratively computed by minimizing the stretch energy [76]. Finally, the volume-preserving parameterization with the prescribed boundary mapping is computed by minimizing the volumetric stretch energy.…”