Iterative algorithms for deblurring and deconvolution with constraints

The deblurring problem is that of recovering the function c = c(t) from (noisy) values integral h(s, t)c(t) dr = d(s). The discrete finite version of the problem is to solve the system of linear equations He = d for c, where H is a matrix and d and c are vectors. When the kernel Iz(s, t) is a function of the difference (s - t), the deblurring problem becomes a deconvolution problem. The use of iterative algorithms to effect deblurring subject to non-negativity constraints on c has been presented by Snyder er al for the case of non-negative kernel function h. In this paper we extend these algorithms to include upper and lower bounds on the entries of the desired solution. We show that any linear deblurring problem involving a real kernel h can be transformed into a linear deblurring problem involving a non-negative kernel. Therefore our algorithms apply to general deblurring and deconvolution problems. These algorithms converge to a solution of the system of equations y = Pr, with P = [P-ij] P-ij greater than or equal to 0 for i = 1,..., I, j = 1,..., J, satisfying the vector inequalities a less than or equal to x less than or equal to b, whenever such a solution exists. When there is no solution satisfying the constraints the simultaneous versions converge to an approximate solution that minimizes a cost function related to the Kullback-Leibler cross-entropy and the Fermi-Dirac generalized entropy.