Morphological Diversity and Sparse Image Denoising

Overcomplete representations are attracting interest in image processing theory, particularly due to their potential to generate sparse representations of data based on their morphological diversity. We here consider a scenario of image denoising using an overcomplete dictionary of sparse linear transforms. Rather than using the basic approach where the denoised image is obtained by simple averaging of denoised estimates provided by each sparse transform, we here develop an elegant Bayesian framework to optimally combine the individual estimates. Our derivation of the optimally combined denoiser relies on a scale mixture of Gaussian (SMG) prior on the coefficients in each representation transform. Exploiting this prior, we design a Bayesian ℓ₂-risk (mean field) nonlinear estimator and we derive a closed-form for its expression when the SMG specializes to the Bessel K form prior. Experimental results are carried out to show the striking profits gained from exploiting sparsity of data and their morphological diversity.