Minimum cross-entropy formulations in image super-resolution

Abstract

Super-resolution is defined as the ability to algorithmically or physically form an image with meaningful spatial frequency content at spatial frequencies for which the optical instrument has an optical transfer function equal to zero. Historically, the method of least-squares has played a significant role in numerous estimation problems including the super-resolution problem. A viable alternative for the recovery of non-negative signals is the minimum cross-entropy principle. This principle is a generalization of minimum discrimination information in statistics and information theory. In the first part of the dissertation, various minimum cross-entropy methods for super-resolution are presented. Alternating Projections, a special case of which is the class of Expectation-Maximization (EM) algorithms, and Picard-type iterations are employed in our investigations. A cross-entropic Projection-Onto-Convex-Sets (POCS) formulation is developed to provide an alternate interpretation of the minimum cross-entropy based EM-type algorithms. This interpretation provides a theoretical basis for including some a priori object constraints in iterative super-resolution algorithms. The performance of signal recovery algorithms is dependent on the sparsity of the signal. This fact has been observed empirically and theoretically by several researchers. Indeed, the Gerchberg-Papoulis (GP) algorithm achieves bandwidth extrapolation primarily from the finite spatial extent a priori knowledge, a special form of signal sparsity. Unfortunately, in real-world applications, objects are rarely sparse. In the second part of the dissertation, some approximately sparse representations of signals, viz., background-foreground, trend-fluctuations and wavelet representations are proposed to circumvent the sparsity requirement. Multigrid methods and wavelet decompositions are two closely related concepts. Multigrid methods were proposed to improve the convergence rates of iterative smoothers by appending corrections from coarse grids to an approximate estimate at the fine grid. Wavelet representation schemes, on the other hand, show great promise in alternatively representing an object as sparse components. A wavelet-subspace based multigrid formulation for recovery of nonsparse objects is proposed. A unified space-decomposition formulation that ties related concepts found in varied application areas, viz., Grenander's method of sieves in statistical inference, intrinsic correlation functions in astronomy, method of resolution kernels, wavelet-based space-decompositions, space-decompositions in multigrid methods etc., is presented.