Constrained Iterations for Blind Deconvolution and Convexity Issues

The need for image restoration arises in many applications of various scientific disciplines, such as medicine and astronomy and, in general, whenever an unknown image must be recovered from blurred and noisy data. The algorithm studied in this work restores the image without the knowledge of the blur, using little a priori information and a 'blind inverse filter' iteration. It represents a variation of the methods proposed by Kundur and Hatzinakos (1998) and by Ng, Plemmons and Qiao (2000). The problem of interest here is an 'inverse' one, that cannot be solved by simple filtering since it is ill--posed. The imaging system is assumed to be linear and space--invariant: this allows a simplified relationship between unknown and observed images, described by a 'point spread function' modeling the distortion. The blurring, though, makes the restoration ill--conditioned: 'regularization' is therefore also needed, obtained by adding constraints to the formulation of the estimated solution. The problem is modeled as a constrained minimization: particular attention is given here to the analysis of the objective function and on establishing whether or not it is a convex function, whose minima can be located by classic optimization techniques and descent methods. Numerical examples are applied to simulated data and to real data derived from various applications. Comparison with the behavior of the two other methods, mentioned above, show the effectiveness of our variant.