Group-Invariant Colour Morphology Based on Frames

Mathematical morphology is a very popular framework for processing binary or grayscale images. One of the key problems in applying this framework to color images is the notorious false color problem. We discuss the nature of this problem and its origins. In doing so, it becomes apparent that the lack of invariance of operators to certain transformations (forming a group) plays an important role. The main culprits are the basic join and meet operations, and the associated lattice structure that forms the theoretical basis for mathematical morphology. We show how a lattice that is not group invariant can be related to another lattice that is. When all transformations in a group are linear, these lattices can be related to one another via the theory of frames. This provides all the machinery to let us transform any (grayscale or color) morphological filter into a group-invariant filter on grayscale or color images. We then demonstrate the potential for both subjective and objective improvement in selected tasks.