Group theoretical transforms and their applications in image coding

Uses the representation theory of finite groups to simplify the Karhunen-Loeve transform (KLT) of systems with group theoretically defined symmetries. In the paper the authors focus on applications of the dihedral groups D(n) which consist of all isometries that map the n-sided regular polygon into itself. The group D(4) is of special importance for all problems on square grids. Connected to each group is a type of Fourier transform. This transform block-diagonalizes all operators that commute with the group operations. As a result all correlation matrices of processes with group theoretically defined symmetries are block-diagonalized. This simplifies the computation of the KLT considerably. For real world data the symmetry assumptions leading to the simplification of the KLT are never exactly fulfilled and the KLT based on the block-diagonal correlation matrix is only an approximation to the correct KLT. In the second part of the paper the authors compare several approximations to the KLT for a large data base consisting of blocks collected from a standard TV-channel. Finally they discuss some of the consequences for image coding applications