Results on Principal Component Filter Banks : Colored Noise Suppression and Existence Issues 1

We have recently made explicit the precise connection between the optimization of orthonor-mal lter banks (FB's) and the principal component property: The principal component lter bank (PCFB) is optimal whenever the minimization objective is a concave function of the subband variances of the FB. This explains PCFB optimality for compression, progressive transmission and various hitherto unnoticed white{noise suppression applications such as subband Wiener ltering. The present work examines the nature of the FB optimization problems for such schemes when PCFB's do not exist. Using the geometry of the optimization search{spaces, we explain exactly why these problems are usually analytically intractable. We show the relation between compaction lter design (i.e., variance maximization) and optimum FB's: A sequential maximization of subband variances produces a PCFB if one exists, but is otherwise suboptimal for several concave objectives. We then study PCFB optimality for colored noise suppression. Unlike case when the noise is white, here the minimization objective is a function of both the signal and the noise subband variances. We show that for the transform coder class, if a common signal and noise PCFB (KLT) exists, it is optimal for a large class of concave objectives. Common PCFB's for general FB classes have a considerably more restricted optimality, as we show using the class of unconstrained orthonormal FB's. For this class, we also show how to nd an optimum FB when the signal and noise spectra are both piecewise constant with all discontinuities at rational multiples of .