Parametrization, alignment and shape of spherical surfaces

We develop parametrization and alignment techniques for shapes of spherical surfaces in 3D space with the goals of quantifying shape similarities and dissimilarities and modeling shape variations observed within a class of objects. The parametrization techniques are refinements of methods due to Praun and Hoppe and yield parametric mesh representations of spherical surfaces. The main new element is an automated technique to align parametric meshes for shape interpolation and comparison. We sample aligned surfaces at the vertices of a dense common mesh structure to obtain a representation of the shapes as organized point-clouds. We apply Kendall’s shape theory to these dense point clouds to define geodesic shape distance, to obtain geodesic interpolations, and to study statistical properties of shapes that are relevant to problems in computer vision. Applications to the construction of compatible texture maps for a family of surfaces are also discussed.