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

Abstract

Manifold parameterizations have been applied to various fields of commercial industries. Several efficient algorithms for the computation of triangular surface mesh parameterizations have been proposed in recent years. However, the computation of tetrahedral volumetric mesh parameterizations is more challenging due to the fact that the number of mesh points would become enormously large when the higher-resolution mesh is considered and the bijectivity of parameterizations is more difficult to guarantee. In this paper, we develop a novel volumetric stretch energy minimization algorithm for volume-preserving parameterizations of simply connected 3-manifolds with a single boundary under the restriction that the boundary is a spherical area-preserving mapping. In addition, our algorithm can also be applied to compute spherical angle-and area-preserving parameterizations of genus-zero closed surfaces, respectively. Several numerical experiments indicate that the developed algorithms are more efficient and reliable compared to other existing algorithms. Numerical results on applications of the manifold partition and the mesh processing for three-dimensional printing are demonstrated thereafter to show the robustness of the proposed algorithm.

3D brain tumor segmentation using a two-stage optimal mass transport algorithm

“…Algorithm 1 computes the discrete OMT on . The steps 1–10 of Algorithm 1 provide an initial spherical area-measure-preserving map , similar to the stretch energy minimization algorithm 31 . To minimize the energy functional ( 4 ) and achieve the spherical image constraint, we apply the stereographic projection to map the spherical image of onto the extended complex plane .…”

“…Let be the uniform partition of the interval [0, 1] into p subintervals. For , we compute the interior map by solving the linear system where and is the mass-weighted Laplacian matrix, similar to the volumetric stretch energy minimization algorithm 31 , such that in which, as in the literature 31 , 35 – 37 , is the modified cotangent weight, where is the dihedral angle between and in tetrahedron . Then, the map is the desired OMT map .…”

“…However, there are infinitely many volume-preserving maps between and . The resulting map by 31 is, in general, not an OMT map from to .…”

“…However, the first step in 29 for the computation of a harmonic map from to is not satisfied with an OMT map. Very recently, in 2019, Yueh et al 31 proposed a novel algorithm for the volume-preserving parameterization from to by minimizing the volumetric stretch energy by modifying the Laplacian matrix with the current volume-stretching factor at each iteration step. However, there are infinitely many volume-preserving maps between and .…”

A Novel 2-Phase U-net Algorithm Combined with Optimal Mass Transportation for 3D Brain Tumor Detection and Segmentation

“…Based on Brenier's approach and the variational principle 22 , Su et al 23 developed a volume-preserving parameterization from a 3-manifold M with a spherical boundary to a unit ball B 3 . Recently, Yueh et al 24 proposed a novel algorithm to compute a volume-preserving parameterization from M to B 3 by modifying the denominators of the coefficients of the corresponding Laplacian matrix by imposing the local volume stretch factor at each iteration step and adopted the projected gradient method (PGM) combined with the homotopy technique in 25 to find the OMT map between M and B 3 . In addition, a two-stage OMT (2SOMT) procedure from M to B 3 and from B 3 to a cube was efficiently developed by Lin et al 19 and applied prior to U-net training and inference in 3D brain tumor segmentation.…”