Statistical imaging and complexity regularization

We apply the complexity regularization principle to statistical ill-posed inverse problems in imaging. The class of problems studied includes restoration of images corrupted by Gaussian or Poisson noise and nonlinear transformations. We formulate a natural distortion measure in image space and develop nonasymptotic bounds on estimation performance in terms of an index of resolvability that characterizes the compressibility of the true image. These bounds extend previous results that were obtained in the literature under simpler observational models. The notion of an asymptotic imaging experiment is clarified and used to characterize consistency and convergence rates of the estimator. We present a connection between complexity-regularized estimation and rate-distortion theory, which suggests a method for constructing optimal codebooks. However, the design of computationally tractable complexity regularized image estimators is quite challenging; we present some of the issues involved and illustrate them with a Poisson-imaging application.