Texture Mapping Using Surface Flattening via Multidimensional Scaling

We present a novel technique for texture mapping on arbitrary surfaces with minimal distortions, by preserving the local and global structure of the texture. The recent introduction of the fast marching method on triangulated surfaces [9], made it possible to compute geodesic distances in O(~ n lg ~ n) where ~ n is the number of triangles that represent the surface. We use this method to design a surface flattening approach based on multi-dimensional scaling (MDS). MDS is a family of methods that map a set of points into a finite dimensional flat (Euclidean) domain, where the only given data is the corresponding distances between every pair of points. The MDS mapping yields minimal changes of the distances between the corresponding points. We then solve an ‘inverse’ problem and map a flat texture patch onto the curved surface while preserving the structure of the texture.