Novel theory and algorithm for fuzzy distance transform and its applications

This paper describes the theory and algorithms of fuzzy distance transform (FDT). Fuzzy distance is defined as the length of the shortest path between two points. The length of a path in a fuzzy subset is defined as the integration of fuzzy membership values of the points along the path. The shortest path between two points is the one with the minimum length among all (infinitely many) paths between the two points. It is demonstrated that, unlike in the binary case, the shortest path in a fuzzy subset is not necessarily a straight-line segment. The support of a fuzzy subset is the set of points with nonzero membership values. It is shown that, for any fuzzy subset, fuzzy distance is a metric for the interior of its support. FDT is defined as the process on a fuzzy subset that assigns at each point the smallest fuzzy distance from the boundary of the support. The theoretical framework of FDT in continuous space is extended to digital spaces and a dynamic programming-based algorithm is presented for its computation. Several potential medical imaging applications are presented including the quantification of blood vessels and trabecular bone thickness in the regime of limited special resolution.