In this paper we describe how to use the direct product of the dihedral group D(4) and the symmetric group S(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 will 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.

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In applications of principal component analysis (PCA) it has often been observed that the eigenvector with the largest eigenvalue has only nonnegative entries when the vectors of the underlying stochastic process have only nonnegative values. This has been used to show that the coordinate vectors in PCA are all located in a cone. We prove that the nonnegativity of the first eigenvector follows from the Perron-Frobenius (and Krein-Rutman theory). Experiments show also that for stochastic processes with nonnegative signals the mean vector is often very similar to the first eigenvector. This is not true in general, but we first give a heuristical explanation why we can expect such a similarity. We then derive a connection between the dominance of the first eigenvalue and the similarity between the mean and the first eigenvector and show how to check the relative size of the first eigenvalue without actually computing it. In the last part of the paper we discuss the implication of theoretical results for multispectral color processing.

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Understanding the properties of time-varying illumination spectra is of importance in all applications where dynamical color changes due to changes in illumination characteristics have to be analyzed or synthesized. Examples are (dynamical) color constancy and the creation of realistic animations. In this article we show how group theoretical methods can be used to describe sequences of time changing illumination spectra with only few parameters. From the description we can also derive a differential equation that describes the illumination changes. We illustrate the method with investigations of black-body radiation and measured sequences of daylight spectra.