Globally optimal smoothing functional for edge-enhancing regularized image restoration

In regularized image restoration a solution is sought which preserves the fidelity to the noisy and blurred image data and also satisfies some constraints which represent our prior knowledge about the original image. A standard expression of this prior knowledge is that the original image is smooth. The regularization parameter balances these two requirements, i.e., fidelity to the data and smoothness of the solution. The smoothness requirement on the solution, however, results in a globally smooth image, i.e., no attention is paid to the preservation of the high spatial frequency information (edges). One approach towards the solution of this problem is the introduction of spatial adaptivity. A different approach is presented in this paper. According to this approach besides the constraint which bounds from above the energy of the restored image at high frequencies, a second constraint is used. With this constraint the high frequency energy of the restored image is also bounded from below. This means that very smooth solutions are not allowed, thus preserving edges and fine details in the restored image. Extending our previous work, we propose a nonlinear formulation of the regularization functional and derive an iterative algorithm for obtaining the unique minimum of this functional. The regularization parameters are evaluated simultaneously with the restored image, in an iterative fashion based on the partially restored image.