The best basis problem, compaction problem and PCFB design problems [filter banks]

In a companion paper, we have considered the problem of optimization of filter banks (FBs) for given input statistics. We have pointed out a strong connection between FB optimality and the principal component property. We have shown that principal component filter banks (PCFBs) are optimal for various signal processing schemes such as coding and denoising. In the present paper, we examine the nature of the FB optimization problems for these schemes in situations where a PCFB does not exist. We describe an algorithm involving a sequential design of compaction filters, which is known to produce PCFBs if they exist. We then demonstrate in an insightful manner how this algorithm can be suboptimal when PCFBs do not exist. This was earlier shown only by numerical examples.