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 L-2 norm of the Tikhonov-Miller inverse operator by almost a factor of ten.