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

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.