A biorthogonal wavelet design technique using Karhunen–Loéve transform approximation

Abstract

The lifting style biorthogonal wavelet implementation has a nice property of enabling flexible design; it is immediately reversible and has a simple relation to subband filters. In this work, we present a method to design prediction (P) and update (U) filters of two-channel lifting structures by minimising the difference between Block Wavelet Transform (BWT) matrix of the wavelet and the Karhunen–Loéve Transform (KLT) of a stochastic process with certain autocorrelation. Here, BWTs are transform matrices that are generated by constructing columns through balanced wavelet trees fed by shifted impulse trains. Although wavelets already have fast implementations through subband filtering or lifting, parametric optimisation of the filter coefficients is still possible through indirectly mimicking the corresponding wavelet and a class of signals with certain KLT. This paper describes the above optimisation by putting constraints on the P and U filters for regularity and constructing the filter coefficients in a least-squares sense. In this part of the paper, the iterated orthogonality constraint over the BWT is resolved with a generalised 12k+1-tap/6k+1-tap P / U construction with numerical results for the 4×4 and 8×8 BWT cases. Experimental approximation to several KLT cases with numerical results in terms of filter coefficients, their spectral behaviour, compaction gain, etc. are provided.