Smoothly deformable spheres

Abstract

Existing shape models with spherical topology are typically designed either in the discrete domain using interpolating polygon meshes or in the continuous domain using smooth but non-interpolating schemes such as NURBS. Polygon models and subdivision methods require a large number of parameters to model smooth surfaces. NURBS need fewer parameters but have a complicated rational expression and non-uniform shifts in their formulation. We present a new method to construct deformable closed surfaces, which includes the exact sphere, by combining the best of two worlds: a smooth and interpolating model with a continuously varying tangent plane and well-defined curvature at every point on the surface. Our formulation is simpler than NURBS while it requires fewer parameters than polygon meshes. We demonstrate the generality of our method with applications ranging from intuitive user-interactive shape modeling, continuous surface deformation, reconstruction of shapes from parameterized point clouds, to fast iterative shape optimization for image segmentation.