Approximation of the Karhunen-Loeve transformation and its application to colour images

Analysis of colour images in the Red, Green and Blue acquisition space and in the intensity and chrominance spaces shows that colour components are closely correlated (Carron, Ph.D. Thesis, Univ. Savoie. France, 1995; Ocadis, Ph.D. Thesis, Univ. Grenoble, France, 1985). These have to be decorrelated so that each component of the colour image can be studied separately. The Karhunen-Loeve transformation provides optimal decorrelation of these colour data. However, this transformation is related to the colour distribution in the image, i.e. to the statistical properties of the colour image and is therefore dependent on the image under analysis. In order to enjoy the advantages of direct, independent and rapid transformation and the advantages of the Karhunen-Loeve properties, this paper presents the study of the approximation of the Karhunen-Loeve transformation. The approximation is arrived at through exploitation of the properties of Toeplitz matrices. The search for eigenvectors of a Toeplitz matrix shows that complex or real orthogonal mappings such as the discrete Fourier transform and its decompositions approximate the Karhunen-Loeve transformation in the case of first-order Markov processes.