Mathematical Analysis of the Iterative/Recursive Algorithm

This section presents the results of a mathematical analysis of the Iterative/Recursive Algorithm. (For details of the analysis procedure, see Coggins 1993.) This analysis is based on an iterative/recursive deblurring procedure with iterations at each of recursion levels. The number of iterations could be different at different recursion levels, but keeping them equal simplifies this analysis. An expression will be derived for the effective linear filter applied by the entire iterative/recursive deblurring process for several values of and . The examples will be developed in the frequency domain, with capitals representing the Fourier spectra of the image given by the corresponding lower case letter.

The deblurring algorithm at the lowest recursion level is the BID algorithm, the effect of which is given in Eq. 14.

The key to understanding the iterative/recursive algorithm is to expand the restoration function for various values of and . We have worked out these effects for various pairs. The method involves algebraically bootstrapping the expressions for from the expressions for and . This analysis unrolls the iterations and unfolds the recursions, resulting in a complicated expression in terms of BID iterations. The contributions of iteration and recursion to this algorithm can be compared by converting the BID iterations into truncated power series form and then algebraically simplifying into a sum of powers of . That is, we can express the restoration functions in the form

where the limit of the summation, , is a function of and . We have worked out the summation relationships shown in Table 1. Line 1 describes the BID algorithm. Line 2 describes an algorithm consisting of recursions alone. Subsequent lines explore the interaction of recursions and iterations. Clearly, the iterative/recursive algorithm is producing a closer approximation to , as indicated by the rapidly growing summation limit, than either the recursions or the iterations alone.

This analysis demonstrates that the iterative/recursive algorithm does converge toward the inverse filter and that it provides a better approximation to the inverse filter for a given amount of computation than the BID algorithm.