Biharmonic Volumetric Mapping Using Fundamental Solutions

Abstract

Abstract-We propose a biharmonic model for cross-object volumetric mapping. This new computational model aims to facilitate the mapping of solid models with complicated geometry or heterogeneous inner structures. In order to solve cross-shape mapping between such models through divide and conquer, solid models can be decomposed into subparts upon which mappings is computed individually. The biharmonic volumetric mapping can be performed in each subregion separately. Unlike the widely used harmonic mapping which only allows C 0 continuity along the segmentation boundary interfaces, this biharmonic model can provide C 1 smoothness. We demonstrate the efficacy of our mapping framework on various geometric models with complex geometry (which are decomposed into subparts with simpler and solvable geometry) or heterogeneous interior structures (whose different material layers can be segmented and processed separately).

Impact of Early and Late Visual Deprivation on the Structure of the Corpus Callosum: A Study Combining Thickness Profile with Surface Tensor-Based Morphometry

“…1) is called the discrete harmonic energy (Wang et al 2004a). In our prior work (Xu 2013), we proved that the discrete harmonic energy is consistent with the traditional harmonic energy. Here the string coefficient $\frac{1}{12}$ is chosen to ensure that the quadratic form in Eqn.…”

“…We model the CC with 3D tetrahedral meshes and combine CC area and thickness measures in a vector at each vertex to be used as a metric for the statistical analysis of CC shape morphometry. To calculate thickness, we apply the volumetric Laplace-Beltrami operator proposed in our prior work (Wang et al 2004a), which is now the de facto standard in volumetric harmonic map research (Wang et al 2004b; Li et al 2007; Tan et al 2010; Pai et al 2011; Paillé and Poulin 2012; Wang et al 2012a; Xu et al 2013a; Li et al 2013; Wang et al 2013b). By solving the Laplace’s equation, we construct a harmonic field on each of the CC tetrahedral meshes.…”

Towards a Holistic Cortical Thickness Descriptor: Heat Kernel-Based Grey Matter Morphology Signatures

“…It was widely used in volumetric harmonic map research (e.g. Wang et al, 2004b; Chern et al, 2015; Li et al, 2007; Tan et al, 2010; Pai et al, 2011; Li et al, 2010; Paillé and Poulin, 2012; Wang et al, 2012; Xu et al, 2013; Li et al, 2013; Aigerman and Lipman, 2013; Shi et al, 2015). As shown in Fig.…”

Measure controllable volumetric mesh parameterization

“…Martin et al constructed trivariate spline for cylindrical volumes by computing harmonic volumetric parameterization in [7]. Xu et al [8] designed a biharmonic map applying a multiple fundamental solutions system for fast computation. Using Green's functions, Xia et al [9] parameterized star-shaped volumes.…”