Nonlinear approximation based image recovery using adaptive sparse reconstructions

We study the robust estimation of missing regions in images and video using adaptive, sparse reconstructions. We assume that we are given a 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. We adaptively determine these small magnitude coefficients through thresholding, establish sparsity constraints, and estimate missing regions in images using information surrounding these regions. We show that the region types we can effectively estimate in a mean squared error sense are those for which the given transform provides a close approximation using nonlinear approximation. We show the nature of the constructed estimators and how these estimators relate to the utilized transform and its sparsity over regions of interest. For images the developed algorithms are applicable over locally uniform (smooth, high frequency, texture, etc.) regions separated by edges or edge-like singularities. However, the developed estimation framework is general, and can readily be applied to other nonstationary signals with a suitable choice of linear transforms. Equations are derived and extensive simulation examples included.