A necessary and sufficient condition for transform optimality in source coding

Abstract

Abstract-It is well-known for transform coding of multivariate Gaussian sources, that the Karhunen Loeve transform (KLT) minimizes the mean square error distortion. However, finding the optimal transform for general non-Gaussian sources has been an open problem for decades, despite several important advances that provide some partial answers regarding KLT optimality. In this paper, we present a necessary and sufficient condition for optimality of a transform when high resolution, variable rate quantizers are employed. We present not only a complete characterization of when KLT is optimal, but also a determining condition for optimality of a general (non-KLT) transform. This necessary and sufficient condition is shown to have direct connections to the well studied source separation problem. This observation can impact source separation itself, as illustrated with a new optimality result. Finally, we combine the transform optimality condition with algorithmic tools from source separation, to derive a practical numerical method to search for the optimal transform in source coding.

Towards Optimality in Multiterminal Transform Coding

“…Single Terminal Transform Optimality Theorem [5]: Unitary transform U * is optimal if and only if the following is satisfied:…”

“…In our recent work, [5], we derived a necessary and sufficient condition for optimality of a transform in conjunction with variable rate coding at high resolution. Specifically, we showed that the optimal transform is the one that minimizes the divergence between the joint distribution of the resulting transform coefficients and the product of their marginals.…”