We describe two broad classes of useful and physically meaningful image models that can be used to construct novel smoothing constraints for use in the regularized image restoration problem. The two classes, termed piecewise image models (PIMs) and focal image models (LIMs), respectively, capture unique image properties that can be adapted to the image and that reflect structurally significant surface characteristics. Members of the PIM and LIM classes are easily formed into regularization operators that replace differential-type constraints. We also develop an adaptive strategy for selecting the best PIM or LIM for a given problem (from among the defined class), and we explain the construction of the corresponding regularization operators. Considerable attention is also given to determining the regularization parameter via a cross-validation technique, and also to the selection of an optimization strategy for solving the problem. Several results are provided that illustrate the processes of model selection, parameter selection, and image restoration. The overall approach provides a new viewpoint on the restoration problem through the use of new image models that capture salient image features that are not well represented through traditional approaches.
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