  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.
