Learning the identity with basic morphological operators (potato peeler algorithm)

摘要

Abstract: Let f denote an image and F a morphological operator depending on a set of parameters $LFBC@m$-j$/$RTBC@. The purpose of this paper is to find a set of parameters which satisfies the equation F(f,m) $EQ f where F is one of the two basic morphological operator: erosion or dilation. It is obvious that there exists a trivial solution m of f $EQ F(f,m), which is given by m(x) $EQ 0 if x $EQ 0 and m(x) $EQ $MIN$INF otherwise. Our approach is to consider this problem as an optimization problem: `find the best set of parameters which minimizes the error between the desired image f and the output filtered image F(f,m)'. Among various possible approaches, we have chosen a specific one introduced by Ph. Salembier for adaptative structuring elements. In our problem, we prove that Salembier algorithms always converge toward a solution distinct from m.!14