A unified framework for image modeling and estimation using measurement constraints

This thesis studies the problem of modeling “natural” images by integrating available prior information about diverse image attributes. To this end, a general estimation framework based on the maximum entropy (maxent) principle is proposed. An equivalence between two conceptually different methods of signal estimation under modeling uncertainty viz. set-theoretic estimation and maximum entropy MAP estimation is established. Broad conditions under which the two aforementioned estimation paradigms produce the same signal estimate are provided. It is shown how entropy can be used to provide solutions to the following three important problems: how to select sizes of constraint sets in set-theoretic estimation; how to choose the values of parameters in regularized restoration when using multiple regularization functions; and how to trade-off model complexity and goodness of fit in the problem of model selection.; As a technical offshoot of this work, new sufficient conditions that guarantee the existence and uniqueness of the maxent distribution consistent with specified bounds on certain generalized moments are provided. Unlike earlier results, which are in terms of conditions on the collection of distributions satisfying moment constraints with equality , these results hold for moment inequality constraints, and the technical conditions are explicitly on the individual moment functions. An analytical characterization of the maxent distribution is also provided. It is shown how the infinite-dimensional constrained entropy maximization problem can be converted to a finite-dimensional convex minimization by invoking Lagrange duality theory. As a consequence, maxent models consistent with specified moment measurements are shown to be asymptotically equivalent to maximum likelihood models that use a certain parametric exponential family of signal priors.; A new, rich class of maxent priors for natural images from imprecise subband statistics in multiple orthonormal wavelet bases is developed. Experimental results for the problem of image restoration in additive white Gaussian noise are presented. Denoising and restoration of natural images using algorithms based on these maxent priors demonstrate significant improvements in terms of both perceptual quality as well as mean-squared error over classical approaches such as adaptive Wiener filtering. Under appropriate conditions, a variety of classical wavelet-domain image models and denoising algorithms are shown to be subsumed by the proposed multiple-domain maxent modeling and estimation framework.