Linear and Nonlinear Image Deblurring: A Documented Study

Nonlinear image deblurring procedures based on probabilistic considerations are widely believed to outperform conventional linear methods. This paper is exclusively concerned with nonsmooth images such as those that occur in biomedical imaging, where reconstruction of high frequency detail is of prime interest, and where avoidance of a priori smoothness constraints is a major concern. The theoretical basis behind each of the following nonlinear procedures is examined: the Lucy--Richardson method, the maximum likelihood E-M algorithm, the Poisson maximum a posteriori method, and the Nuñez--Llacer version of the maximum entropy method. A linear iterative method, VanCittert's iteration, is also studied. It is shown thateach of the first three methods, as well as VanCittert's method, lack a necessary ingredient for successful solution of the ill-posed deblurring problem, while in the maximum entropy method, the enforced smoothness may have adverse consequences in medical imaging. A direct linear method, the slow evolution from the continuation boundary (SECB) method, designed specifically for nonsmooth images, is also considered. That method is stabilized by constraining the blurring operator as well as the solution and does not require smoothness constraints. It is shown that useful error estimates can be obtained in the SECB method while this is impossible in Tikhonov's method without a priori bounds on derivatives of the unknown solution. Reconstruction experiments on low noise synthetic MRI data show that thousands of iterations are necessary to achieve sufficient resolution in the iterative procedures. However, the SECB method provides higher resolution at considerable savings in computer time. At high noise levels, the iterative algorithms are shown to diverge. At these same noise levels, the SECB method produces reconstructions comparable in quality to those that would be obtained in the iterative methods, were one able to terminate the divergent algorithm at that iteration which best approximates the true solution in the L1 norm.