Volumetric Harmonic Map

Abstract

We develop two different techniques to study volume mapping problem in Computer Graphics and Medical Imaging fields. The first one is to find a harmonic map from a 3 manifold to a 3D solid sphere and the second is a sphere carving algorithm which calculates the simplicial decomposition of volume adapted to surfaces. We derive the 3D harmonic energy equation and it can be easily extended to higher dimensions. We use a textrehedral mesh to represent the volume data. We demonstrate our method on various solid 3D models. We suggest that 3D harmonic mapping of volume can provide a canonical coordinate system for feature identification and registration for computer animation and medical imaging. 1. Introduction. With the rapid development of imaging technology, there has been an explosive growth of three-dimensional (3D) image data collected from all kinds of physical sensors. The technique to process and analyze these image data becomes important. In this paper, we develop two different techniques to study volume mapping problem in Computer Graphics and Medical Imaging fields. One of our techniques is to compute a harmonic map from a 3 manifold to a 3D solid sphere. To the best of knowledge, it is the first work to practi- cally compute 3D harmonic map between 3D volumes. The other technique is to calculate the simplicial decomposition of volume adapted to surfaces. Compared with existed Finite Ele- ment Method (FEM) mesh generation, our new technique emphasizes geometry and topology sanity maintenance. The functional space on a manifold is determined by the manifold geometric character- istics. The harmonic spectrum can reflect many global geometric information of the mani- fold. Some applications on computer graphics and medical imaging fields, such as volume registration, shape analysis, etc. can be carried out by examining the behavior of special dif- ferential operators on it. In the literature, some researchers used harmonic map for surface matching (1) and the construction of conformal map for genus zero surfaces (2, 3). For 3D brain volume transformation research, Gee (4) studied brain volume matching with a gener- alized elastic matching method within a probabilistic framework. The approach can resolve issues that are less naturally addressed in a continuum mechanical setting. Ferrant et al. (5) presented an algorithm for non-rigid registration of 3D MR intraoperative image sequences showing brain shift. The 3D anatomic deformation field, in which surfaces are embedded, is then inferred from the displacements at the boundary surfaces using a biomechanical finite element model for the constituent objects. In 2D case, a harmonic map between two convex planar regions is diffeomorphic if and only if the restriction on the boundary is diffeomorphic. 3D harmonic map is much more