Unifying regularization and Bayesian estimation methods for enhanced imaging with remotely sensed Data-part I: theory

The problem of estimating, from one sampled realization of the remotely sensed data signal, the power spatial spectrum pattern (SSP) of the wave field scattered from the probing surface is treated as it is required for enhanced radar imaging of the remotely sensed scenes. Specifically, we propose to unify the Bayesian estimation strategy with the maximum-entropy (ME) information-theoretic principle for incorporating the prior knowledge through developing the fused Bayesian-regularization (FBR) technique for SSP estimation. The first aspect of the proposed approach concerns the ME-based incorporating the a priori information about the geometrical properties of an image to tailor the metrics structure in the solution space to the problem at hand. The second aspect alleviates the problem ill-posedness associated with preserving the boundary values, calibration, and spectral a priori fixed model properties of an image through the regularizing projection constraints imposed on the solution. When applied to SSP estimation without incorporating the metrics and regularization considerations, the procedure leads to the previously derived maximum-likelihood method. When such considerations are incorporated, the optimal FBR technique leads to a new nonlinear imaging algorithm that implies adaptive formation of the second-order sufficient statistics of the data, their smoothing, and projection applying the composite regularizing window operator. We provide analytical techniques to find these statistics and windows, and the optimal FBR estimator itself. Numerical recipes, performance issues, and simulation examples are treated in a companion paper.