MCF-homomorphisms of cost functions for minimum cost flow InSAR phase unwrapping

Branchcut methods represent an important class of phase unwrapping (PU) approaches in SAR interferometry (InSAR). Their main idea is to circumvent the problem of path dependency for the integration of the wrapped gradient of the wrapped interferometric phase by placing suitable barriers (branchcuts) to the possible integration paths in the image. In 1996 M. Costantini proposed to transform the problem of InSAR PU to a minimum cost flow (MCF) problem. The critical point of this new approach is to generate cost functions which represent the a priori knowledge necessary for phase unwrapping. Since many algorithms for generating cost functions from a priori knowledge have been proposed in the last years, it has become important to decide whether two given methods of computing cost functions lead to the same branchcut system on every interferogram. A formal tool to treat this problem mathematically are universal MCF-homomorphisms. Any mapping a from the space of cost functions into itself is called an universal MCF-homomorphism if every cost function c and its image α(c) have the same MCF solution and if α can be defined separately for each arc. This paper presents the proof of the following mathematical theorem: The only existing universal MCF-homomorphisms are the componentwise multiplications with a fixed positive scalar