Harmonic fields based surface and volume mapping

• Main Author: Xia, Jiazhi
• Other Authors: He Ying
• Format: Theses and Dissertations
• Language: English
• Published: 2011
• Subjects: DRNTU::Engineering::Computer science and engineering::Computer applications::Computer-aided engineering
• Online Access: https://hdl.handle.net/10356/45771
• Institution: Nanyang Technological University

Harmonic fields are widely used in computational science and engineering for their computational efficiency and promising properties. Rad´o theorem strongly supports the application of harmonic fields in parameterization by proving that harmonic maps from 2D Riemannian manifold to convex planar domain are diffeomorphism. But the limitations of harmonic maps are also obvious: first, there are a lot of real applications requiring the mapping constraints to be set inside the domain. However, in such cases, harmonic maps could not be guaranteed to be one-to-one. Second, Rad´o theorem holds true only in the 2D case. Volumetric harmonic maps could not be guaranteed to be a diffeomorphism. In this thesis, we systematically study the harmonic fields and their applications in surface and volume parameterization by overcoming the two aforementioned drawbacks technically. In the first part of the thesis, we present an editable polycube map framework in Chapter 4, that, given an arbitrary high-resolution polygonal mesh and a simple polycube representation plus optional sketched features indicating relevant correspondences between the two, provides a uniform, regular and artist-controllable quads-only mesh with a parameterized subdivision displacement scheme. The proposed method is based on a divide and conquer strategy. The mesh surface is divided into patches according to the feature constraints. A diffeomorphism is built between each pair of topological disk patches. After that, a global smoothing step is adopted to improve the continuity along the segmentation boundaries. In Chapter 5, we also develop another method to overcome the drawback of harmonic maps in parameterizing 3D facial expressions. With the salient features (such as the eyes, mouth and nose) in the captured expression, viii we first compute a geodesic mask with the user-specified radius to segment the facial expression. Then we cut the 3D faces along a geodesic curve connecting the eyes, nose and mouth. As a result, each 3D face is topologically equivalent to an annulus. Next, we solve a harmonic function using the Dirichlet boundary condition. Finally, we compute the consistent mapping among all the captured frames by tracing the integral curves that follow the gradient field of the harmonic function. Observing that both the geodesic and harmonic functions are intrinsic and independent of the embedding, the proposed method is invariant to the expression, and also guarantees the exact correspondences of the salient features.