On generalized distance transformation of digitized pictures

In this paper, we present a generalization of the distance transformation of a digitized picture in two different aspects. First, we define the generalized distance transformation of a binary picture (GDTB). A subclass of GDTB, called a local minimum filter family of GDTB (LMF-GDTB), characterized by a series of local minimum filters with varying neighborhoods, is discussed in detail. A skeleton is defined for LMF-GDTB, and it is proved that any binary picture can be reconstructed exactly from its skeleton with the distance value on it. Second, the gray weighted distance transformation (GWDT) is extended to a generalized GWDT (GGWDT) by introducing an arbitrary initial picture. After the fundamental equation of GGWDT and its solution are derived, it is proved that an arbitrary gray picture is generated by iterative application of GGWDT from a uniquely determined elementary picture and a sequence of initial value pictures.