A Scale Space for Contour Registration Using Minimal Surfaces

Abstract

Previously, we presented a method for contour registration using minimal surfaces. This method involves embedding each of two unregistered two-dimensional contours into two parallel planes separated in three-dimensional space. The minimal surface is then computed between the two contours via mean curvature flow. We then evolve the rigid registration of one of the two contours which in turn changes the minimal surface. Mean curvature flow of the surface and evolution of the curve registration both support a consistent energy functional, i.e., area of the connecting surface. We review the implementation details and show an example registration.

In this paper we concentrate on developing this method as a registration scale space. The separation of the two contour planes serves as a scale space parameter, larger separations producing coarser registrations. At the finest scale, which occurs as the separation distance approaches zero, this registration method is identical to minimizing the set-symmetric difference between the interiors of the contours. Thus, this method can be viewed as a geometric generalization of set-symmetric difference registration. We explain the scale space properties of this registration method theoretically and experimentally. Through examples we show how at increasingly coarser scales, our method overcomes increasingly coarser local minima apparent in set-symmetric difference registration. In addition, we present sufficient conditions for existence of the minimal surface connecting two contours. This condition yields an upper bound for the separation distance between two contours and gives an estimate for the coarsest registration scale.

This work was supported by NSF grant CCR-0133736 and NIH grant R01-HLS0004-01A1.