Sample Distortion for Compressed Imaging

TL;DR

This paper introduces a sample distortion framework for compressed sensing, analyzing achievable reconstruction performance, optimizing sample allocation, and validating results with natural image simulations.

Contribution

It develops a novel SD function concept for compressed sensing, derives bounds, and applies it to optimize sampling strategies in compressed imaging.

Findings

SD functions accurately predict compressed sensing gains with oracle statistics.

Optimized sample allocation improves reconstruction performance.

Practical sample allocation profiles based on average statistics are effective.

Abstract

We propose the notion of a sample distortion (SD) function for independent and identically distributed (i.i.d) compressive distributions to fundamentally quantify the achievable reconstruction performance of compressed sensing for certain encoder-decoder pairs at a given sampling ratio. Two lower bounds on the achievable performance and the intrinsic convexity property is derived. A zeroing procedure is then introduced to improve non convex SD functions. The SD framework is then applied to analyse compressed imaging with a multi-resolution statistical image model using both the generalized Gaussian distribution and the two-state Gaussian mixture distribution. We subsequently focus on the Gaussian encoder-Bayesian optimal approximate message passing (AMP) decoder pair, whose theoretical SD function is provided by the rigorous analysis of the AMP algorithm. Given the image statistics, analytic bandwise sample allocation for bandwise independent model is derived as a reverse water-filling scheme. Som and Schniter's turbo message passing approach is further deployed to integrate the bandwise sampling with the exploitation of the hidden Markov tree structure of wavelet coefficients. Natural image simulations confirm that with oracle image statistics, the SD function associated with the optimized sample allocation can accurately predict the possible compressed sensing gains. Finally, a general sample allocation profile based on average image statistics not only illustrates preferable performance but also makes the scheme practical.