Nonlinear image deblurring procedures are widely believed to outperform
conventional linear methods. Such nonlinear methods need * iterative*
solutions that may require upwards of 20 hours of computing time on current
desk-top workstations. Well-designed * non-iterative* methods that can
produce results of comparable quality, * a thousand times faster*, would
offer considerable advantages in medical diagnostic environments, in industrial
process control systems, or in military and surveillance applications. Through
its long-term research interest in ill-posed inverse problems, and its recent
patented invention of the SECB deblurring method, ACMD has accumulated
considerable expertise that enables it to undertake systematic evaluation of
various image deblurring procedures.

The best-known nonlinear methods are based on probabilistic ideas centered on Bayes's theorem. Such a formulation offers the advantage of positivity and radiant flux conservation. In addition, optimal performance with respect to Poisson noise can often be designed into the algorithm. This is the case for the Lucy-Richardson method, the Maximum Likelihood method, the Expectation-Maximization algorithm, Hunt's Maximum A Posteriori method, and the Nunez-Llacer version of the Maximum Entropy method. These techniques are applied to the ill-posed linear imaging problem,

where is the known system point spread function, is the blurred noisy image, and is the desired deblurred image. As one example, Hunt's MAP procedure is based on the iteration,

where is the adjoint of the
integral operator **H**.

Carasso has shown mathematically that except for the Maximum Entropy method,
all of the above iterative methods necessarily * converge to the wrong
solution* of **Hf=g**, in which severe noise amplification overwhelms the true
image. The Maximum Entropy method converges to a solution whose smoothness is
determined by an input parameter . No successful theory exists for
choosing in terms of a-priori information. Too small a value for
leads to * oversmooth* solutions that may be dangerous in medical diagnostic
contexts. Thus, small lesions, tumors, microcalcifications, and other
singularities in the true image may be * erased* by the algorithm as it
maximizes the image entropy. Too large a value for brings out noise
which obscures important features in the image. A trial and error process for
choosing the optimal must be used. This becomes very time consuming as
one must iterate to convergence, and evaluate the result, prior to selecting a
new value for .

Generated by boisvert@nist.gov on Mon Aug 19 10:08:42 EDT 1996