Inverse problem of image processing

Abstract

A Gel’fand–Levitan inverse problem of image processing is used to determine an optimum trade-off potential U(f) or V(f) from the knowledge of the restored object Q(f) or the output transfer function Ψ(f) and the design knowledge of the optical transfer function (OTF) S(f). Analytic examples and exact solutions are given for the coherent and incoherent image restorations. The minimum mean-square image spread (MMIS) filter is found to yield an identical minimum mean square estimation-error (MMSE) filter of Wiener in the special case, similar to the visual MTF or contrast sensitivity function, peaked at the middle band of spatial frequency channels. The trade-off potential for the linear and noisy motion-blurred image restoration is obtained. By means of the Gel’fand–Levitan inverse transform technique for the reference potential, the design of the energy constrained MMIS filter follows. Lastly, both the direct and inverse problems of incoherent imaging are simultaneously solved with a specific example of the dc incision, namely the central dark field method of Abbe and Zernike. It illustrates the main concept that a useful and efficient new approach of image processing requires a systematical design of both the imaging OTF and image restoration filters under noise.