Two simply connected sets that have the same area are IP-equivalent

摘要

A pair of neighboring, opposite-valued pixels in a two-valued digital image is called interchangeable if reversing their values preserves the topology of the image. It was conjectured in Rosenfeld, Saha, Nakamula, Pattern Recognition 34 (2001) 1853–1865 that if two digital images have the same number of 1's, and their sets of 1's S,T are simply connected, then S can be transformed into T by a sequence of interchanges. In that paper this conjecture was proved only for certain special cases—for example, if S and T are arcs. This paper proves the conjecture for arbitrary simply connected sets.