TY - JOUR
T1 - Sample Distortion for Compressed Imaging
AU - Guo, Chunli
AU - E. Davies, Mike
N1 - 11 pages, 11 figures
PY - 2013
Y1 - 2013
N2 - 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 certainencoder-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. Wesubsequently 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 Schniterâ€™s turbo message passingapproach 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 allocationprofile based on average image statistics not only illustrates preferable performance but also makes the scheme practical.
AB - 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 certainencoder-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. Wesubsequently 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 Schniterâ€™s turbo message passingapproach 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 allocationprofile based on average image statistics not only illustrates preferable performance but also makes the scheme practical.
KW - sample distortion function
KW - bandwise sampling
KW - compressed sensing
U2 - 10.1109/TSP.2013.2286775
DO - 10.1109/TSP.2013.2286775
M3 - Article
VL - 61
SP - 6431
EP - 6442
JO - IEEE Transactions on Signal Processing
JF - IEEE Transactions on Signal Processing
SN - 1053-587X
IS - 24
ER -