Iterative deconvolution begins by guessing what the true image, ,
might be. This initial guess is denoted
. If this guess is
correct, then the convolution
will produce the observed
image,
. If the guess is wrong, it can be corrected based on the
residual between the observed image and the blurred guess:
. In fact, the correction might be simply to add that
difference to
.
The observed image will serve as the initial guess. The first step in an iterative deblurring method, then, is to blur the observed image again. While this might seem surprising, it is correct because the observed image is the closest data we have to the true image. A flat field could be used as the initial guess, but then the correction factor would be the observed image, so the estimate used for the second iteration would be the observed image anyway.
Formally, the Basic Iterative Deconvolution (BID) procedure is defined as follows:
This approach is essentially the Jacobi method for solving
simultaneous linear systems as applied to signal processing by Van
Cittert (1931), extended by Jansson (1968, 1970a, 1970b) and
independently developed by Iinuma (1967a, 1967b). It
converges to the correct inverse filter if one exists; otherwise it
can be terminated after a finite number of iterations to obtain an
approximation to the true image. The iterative method can also be
understood as computing a power series expansion of the inverse filter
based on the identity
Convergence is slow after the first few iterations, with rapidly diminishing returns. The approach also is very sensitive to noise in the signal or error in the estimate of the PSF.
Mathematically, the effect of the BID algorithm is most easily understood via the frequency domain. This discussion follows Metz (1969) and Kawata (1980a, 1980b). Transforming equations 5 and 6 into the frequency domain gives
Reordering the terms and factoring yields
This equation shows how each iteration is obtained from the previous one. Combining this form of the iteration with the initial condition in Eq. 4, one can construct successive iterates
to find the general form for the successive estimates,
which contains the power series expansion for the inverse filter.
This expression for the th estimate of the true image can be reduced
to a more compact form using the series identity
to yield
When , the estimate of the true image's spectrum is
. As
, the filter approaches
and the estimate
approaches the correct answer.
Define
is the effective filter after
iterations of the BID
algorithm. The estimate of the true image's spectrum after
iterations
of the BID algorithm can be written