Nonlinear approximation based image recovery using adaptive sparse reconstructions and iterated denoising - Part II: Adaptive algorithms

We combine the main ideas introduced in Part I with adaptive techniques to arrive at a powerful algorithm that estimates missing data in nonstationary signals. The proposed approach operates automatically based on a chosen linear transform that is expected to provide sparse decompositions over missing regions such that a portion of the transform coefficients over missing regions are zero or close to zero. Unlike prevalent algorithms, our method does not necessitate any complex preconditioning, segmentation, or edge detection steps, and it can be written as a progression of denoising operations. We show that constructing estimates based on nonlinear approximants is fundamentally a nonconvex problem and we propose a progressive algorithm that is designed to deal with this issue directly. The algorithm is applied to images through an extensive set of simulation examples, primarily on missing regions containing textures, edges, and other image features that are not readily handled by established estimation and recovery methods. We discuss the properties required of good transforms, and in conjunction, show the types of regions over which well-known transforms provide good predictors. We further discuss extensions of the algorithm where the utilized transforms are also chosen adaptively, where unpredictable signal components in the progressions are identified and not predicted, and where the prediction scenario is more general.