Determining the Geometry of Boundaries of Objects from Medial Data

We consider a region in R2 or R3 with generic smooth boundary B and Blum medial axis M, on which is defined a multivalued “radial vector field” U from points x on M to the points of tangency of the sphere at x with B. We introduce a “radial shape operator”Srad and an “edge shape operator”SE which measure how U bends along M. These are not traditional differential geometric shape operators, nonetheless we derive all local differential geometric invariants of B from these operators. This allows us to define from (M,U) a “geometric medial map”on M which corresponds, via a “radial map” from M to B, to the differential geometric properties of B. The geometric medial map also includes a description of the relative geometry of B. This is defined using the “relative critical set”of the radius function r on M. This set consists of a network of curves on M which describe where B is thickest and thinnest. It is computed using the covariant derivative of the tangential component of the unit radial vector field. We further determine how these invariants are related to the differential geometric invariants of M and how these invariants change under deforming diffeomorphisms of M.