Space-Variant Image Restoration with Running Sinusoidal Transforms

Abstract

Various restoration methods (linear, nonlinear, iterative, noniterative, deterministic, stochastic, etc.) optimized with respect to different criteria have been introduced (Bertero & Boccacci, 1998; Biemond et al., 1990; Kundur, & Hatzinakos, 1996; Banham & Katsaggelos, 1997; Jain, 1989; Bovik, 2005; Gonzalez & Woods, 2008). These techniques may be broadly divided in two classes: (i) fundamental algorithms and (ii) specialized algorithms. One of the most popular fundamental techniques is a linear minimum mean square error (LMMSE) method. It finds the linear estimate of the ideal image for which the mean square error between the estimate and the ideal image is minimum. The linear operator acting on the observed image to determine the estimate is obtained on the basis of a priori second-order statistical information about the image and noise processes. In the case of stationary processes and space-invariant blurs, the LMMSE estimator takes the form of the Wiener filter (Jain, 1989). The Kalman filter determines the causal LMMSE estimate recursively. Specialized algorithms can be viewed as extensions of the fundamental algorithms to specific restoration problems. It is based on a state-space representation of the imaging system, and image data are used to define the state vectors. Specialized algorithms can be viewed as extensions of the fundamental algorithms to specific restoration problems. In this paper we deal with restoration of images degraded by space-variant blurs. Basically, all fundamental algorithms apply to the restoration of images degraded by space-variant blurs. However, because Fourier transforms cannot be utilized when the blur is space-variant, space-domain implementations of these algorithms may be computationally formidable due to large matrix operations. Several specialized methods were developed to attack the spacevariant restoration problem. The first class referred to as sectioning is based on assumption that the blur is approximately space-invariant within local regions of the image. Therefore, the entire image can be restored by applying well-known space-invariant techniques to the local image regions. A drawback of sectioning methods is the generation of artifacts at the region boundaries. The second class is based on a coordinate transformation (Sawchuk, 1974), which is applied to the observed image so that the blur in the transformed coordinates becomes space-invariant. Therefore, the transformed image can be restored by a space-invariant filter and then transformed back to obtain the final restored image. However, the statistical properties of the image and noise processes are affected by the