Patrice Koehl , University of California, Davis, Davis
Joel Hass , University of California, Davis, Davis
A new algorithm is presented that provides a constructive way to conformally warp a triangular mesh of genus zero to a destination surface with minimal metric deformation, as well as a means to compute automatically a measure of the geometric difference between two surfaces of genus zero. The algorithm takes as input a pair of surfaces that are topological 2-spheres, each surface given by a distinct triangulation. The algorithm then constructs a map f between the two surfaces. First each of the two triangular meshes is mapped to the unit sphere using a discrete conformal mapping algorithm. The two mappings are then composed with a Möbius transformation to generate the function f. The Möbius transformation is chosen by minimizing an energy that measures the distance of f from an isometry. We illustrate our approach using several "real life" datasets. We show first that the algorithm allows for accurate, automatic, and landmark-free non-rigid registration of brain surfaces. We then validate our approach by comparing shapes of proteins. We provide numerical experiments to demonstrate that the distances computed with our algorithm between low-resolution, surface-based representations of proteins are highly correlated with the corresponding distances computed between high-resolution, atomistic models for the same proteins.
Systems of equations, Curve, surface, solid, and object representations, Size and shape, Surface fitting
P. Koehl and J. Hass, "Automatic Alignment of Genus-Zero Surfaces," in IEEE Transactions on Pattern Analysis & Machine Intelligence.