Nonuniform principal component filter banks: definitions, existence, and optimality

Abstract

The optimality of principal component filter banks (PCFBs) for data compression has been observed in many works to varying extents. Recent work by the authors has made explicit the precise connection between the optimality of uniform orthonormal filter banks (FBs) and the principal component property: The PCFB is optimal whenever the minimization objective is a concave function of the subband variances of the FB. This gives a unified explanation of PCFB optimality for compression, denoising and progressive transmission. However not much is known for the case when the optimization is over a class of nonuniform Fbs. In this paper we first define the notion of a PCFB for a class of nonuniform orthonormal Fbs. We then show how it generalizes the uniform PCFBs by being optimal for a certain family of concave objectives. Lastly, we show that existence of nonuniform PCFBs could imply severe restrictions on the input power spectrum. For example, for the class of unconstrained orthonormal nonuniform Fbs with any given set of decimators that are not all equal, there is no PCFB if the input spectrum is strictly monotone.

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