Error Bounds In Nonsmooth Image Deblurring

This paper deals with image deblurring when the unknown original image is not smooth and a priori bounds on its derivatives cannot be prescribed in the inversion algorithm. A significant class of such deblurring problems occurring in medical, industrial, military, astronomical, and environmental applications is shown to be equivalent to backwards-in-time continuation in a generalized diffusion equation that may involve fractional Laplacians. The slow-evolution-from-the-continuation-boundary (SECB) constraint, introduced by the author in [SIAM J. Numer. Anal., 31 (1994), pp. 1535--1557], is applicable to such nonsmooth image deblurring. A new analytical approach based on Fourier analysis provides sharp error estimates for SECB deblurring explicitly in terms of the constants entering the a priori constraints. It also leads to an explicit formula that expresses SECB's improvement over the classical Tikhonov--Miller method. An example from positron emission tomography (PET) imaging is used to illustrate the meaning of the SECB constraint. In this application, use of the SECB constraint reduces the L2 norm of the Tikhonov--Miller inverse operator by almost a factor of ten.