Mathematical morphology for angle-valued images

Mathematical Morphology (MM) can be defined in terms of complete lattices. Thus, MM is useful for the processing of binary images or of single-valued intensity images – images for which a partial ordering, hence a lattice structure, is apparent. The lattice structure of an intensity image is manifest through set inclusion with ordering on intensity. It is always possible to define majorants and minorants for collections of sets that are intensities with spatial support. Not all the components of a color image can be ordered trivially. In particular, hue is angle-valued. Consequently, MM has not been as useful for color image processing (as for grayscale or binary image processing) because it has not been not clear how to define set inclusion for angle-valued images. This paper contains definitions for erosion, dilation, opening, and closing for angle-valued images using hue as the exemplar. The fundamental idea is to define a structuring element (SE) with a given hue or hues. From each image neighborhood of the SE, the erosion operation returns the hue value that is closest to the hue of the corresponding SE member. Examples of the effects of the operators on a color noise field are shown. Histograms demonstrate the effects of the operators on the hue distributions.