Parameterization of Triangular Meshes with Virtual Boundaries

Parameterization of a 3D triangular mesh is a fundamental problem in mesh processing, such as texture mapping, multiresolution modeling, and smooth surface fitting. The convex combination approach is widely used for parameterization because it has good properties such as fast computation and little distortion of embedded triangles. However, the approach has one drawback: most boundary triangles have high distortion in the embedding compared with interior ones. In this paper, we present an extension of the convex combination approach that resolves the drawback by using a virtual boundary. We construct a virtual boundary near the real boundary of the 3D mesh and fix the virtual boundary onto a given convex polygon. Due to the virtual boundary, the real boundary triangles can better reflect the shape of the corresponding 3D triangles. We also make the shape of the 2D boundary closer to that of the original 3D meshes by projecting the 3D boundary onto an appropriate 2D plane and obtaining its convex hull. With the proposed approach, we can obtain a parameterization of a 3D mesh with less distortion than with the original convex combination approach.