Wavelet-Packet Content for Positive Operators

Abstract: We give a simple way to attach ``content" to the nodes of a wavelet packet tree when a positive operator is given. At a fixed packet depth, the packet projections split the operator into positive pieces, and this decomposition induces a boundary measure on the packet path space, together with vector-dependent densities that show how energy is distributed across the tree. We then study a sequential extraction procedure and two depth-fixed greedy rules for choosing packet blocks, one based on trace weights and one based on Hilbert-Schmidt weights. The main results are explicit geometric decay estimates for the remainder under these greedy removals. In the Hilbert-Schmidt case we also isolate a coherence quantity that measures how close the operator is to being block-diagonal in the packet partition. We close with a concrete patch-based denoising procedure for images, where packet blocks are selected by these content weights computed from an empirical second-moment operator; the construction ensures that both the approximants and the remainders stay positive at every step.