Next: Test Results
Up: Iterative/Recursive Deconvolution with Application
Previous: A New Enhancement
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.
Next: Test Results
Up: Iterative/Recursive Deconvolution with Application
Previous: A New Enhancement