Two-stage decomposition of the DCT

Abstract

The DCT kernel matrix is first decomposed into a block diagonal structure (BDS) with diagonal skew-circular correlated (SCCR) sub-matrices of length 2, 4, ..., N/2 by coset decomposition, and then each of these independent SCCR sub-matrices is further split into two stages by decomposing its elements into a linear combination of other simple basis functions. The preprocessing stage can be treated as a new transform approximated to the DCT, and is suitable for image compression. Various preprocessing stages are obtained by choosing various basis functions. The postprocessing stage is used for converting the preprocessing stage back to the DCT. Both the preprocessing stage and the postprocessing stage are BDSs containing independent diagonal SCCR sub-matrices, thus the fast and parallel computation of both the preprocessing and the postprocessing stages is feasible using methods such as a semisystolic array or distributed arithmetic implementation.