Variational Sphere set Approximation for Solid Objects

We approximate a solid object represented as a triangle mesh by a bounding set of spheres having minimal summed volume outside the object. We show how outside volume for a single sphere can be computed using a simple integration over the object’s triangles. We then minimize the total outside volume over all spheres in the set using a variant of iterative Lloyd clustering that splits the mesh points into sets and bounds each with an outside volume minimizing sphere. The resulting sphere sets are tighter than those of previous methods. In experiments comparing against a state-of-the-art alternative (adaptive medial axis), our method often requires half as many spheres, or fewer, to obtain the same error, under a variety of error metrics including total outside volume, shadowing fidelity, and proximity measurement.