EPFL | Biomedical Imaging Group | Karhunen-Loève Transform

An Extension of the Karhunen-Loève Transform for Wavelets and Perfect Reconstruction Filterbanks

M. Unser

Proceedings of the SPIE Conference on Mathematical Imaging: Wavelet Applications in Signal and Image Processing, San Diego CA, USA, July 15-16, 1993, vol. 2034, pp. 45–56.

Most orthogonal signal decompositions, including block transforms, wavelet transforms, wavelet packets, and perfect reconstruction filterbanks in general, can be represented by a paraunitary system matrix. The author considers the general problem of finding the optimal P × P paraunitary transform that minimizes the approximation error when a signal is reconstructed from a reduced number of components Q < P. This constitutes a direct extension of the Karhunen-Loève transform which provides the optimal solution for block transforms (unitary system matrix). General solutions are presented for the optimal representation of arbitrary wide sense stationary processes. The author also investigates a variety of suboptimal schemes using FIR filterbanks. In particular, it is shown that low-order Daubechies wavelets and wavelet packets (D2 and D3) are near optimal for the representation of Markov-1 processes.

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