A group theoretical toolbox for color image operators

In this paper we describe how to use the direct product of the dihedral group D(4) and the symmetric group 5(3) to automatically derive low-level image processing filter systems for RGB images. For important classes of stochastic processes it can be shown that the resulting operators lead to a block-diagonalization of the correlation matrix. We show that the group theoretical derivation of the operators leads to a very fast implementation consisting mainly of additions and subtractions. They can therefore be implemented in fast graphics computation hardware such as a GPU. We then illustrate the block-diagonalization property in an experiment where we used 20,000 subimage patches collected from 10,000 random images in a large image database. The very short execution times make these operators suitable for applications where many images have to be processed. Typical examples are video processing and content-based image database search. We describe one example where we use the operators to compute content based descriptors of images. These descriptors are currently used in one search mode in an image database browser operating on a database with more than 750,000 images. The group theoretical tools used to derive these filters are very general and can directly be applied for other types of image data. Examples are the following generalizations of the methodology: filter systems for multiband images with more than the ordinary three RGB-channels or images with other grid geometries such as hexagonal sampling.