Discussion

The iterative/recursive algorithm at all but the lowest-level recurrence interrupts each iteration of the BID algorithm, defines a new, related deblurring problem, and then executes the algorithm on the related problem. The power of the algorithm lies in the way the new problem is defined.

Repeated blurring by the PSF produces a progressively simpler image, eventually blurring the image to a constant. Only a Gaussian blur is guaranteed not to create spurious local structure during progressive blurring (Koenderink 1992), but any realistic PSF will eventually blur an image to a constant.

Each difference image estimates the derivative of image intensity with respect to blurring by the PSF. This kind of derivative is central to the multiscale geometric image decompositions used in recent computer vision research; Coggins (1992) has called the computer vision approach based on progressive blurring spatial spectroscopy. Each recursion level implements the next higher order of this differentiation. The revised problem defined in each recursion, then, is to deblur a higher-order derivative of the image with respect to blurring by the PSF.

Blom (1993) has recently shown that the computation of higher-order derivatives is more robust against noise than the computation of lower-order derivatives in the presence of (white) noise. This result is surprising because the ill-posedness of differentiation has become an informal axiom of image and signal processing. Blom's result can be partly explained by the larger spatial extent of higher-order derivative kernels. The larger spatial extent, required to fit in all of the ripples of the high-order derivative operator, regularizes the data by smoothing out small scale noise. In the present application, the higher-order derivatives are simpler, more regular images, so the iterative deblurring algorithm in the last recursion level is executed on a simpler deblurring problem having less noise contamination. The deblurred result is then used to correct the estimates of the true image at the previous recurrence.

Some reblurring algorithms (Kawata 1980b) that blur the observed image before iterating in order to suppress noise appear to be roughly equivalent to performing one recursion level of the iterative/recursive algorithm.

The iterative/recursive restoration algorithm will still encounter numerical problems if the noise is severe enough to persist in the lowest recursion level. However, the recursions make the iterative/recursive algorithm more robust than the BID algorithm, linear and flux-conserving, faster to converge toward the inverse filter than the BID algorithm, and faster to run than other deblurring algorithms (Busko 1993). The power of this algorithm suggests that research into restoration of chemical spectra, medical images, and other restoration problems should be revisited using this new algorithm.