In general, image restoration problems are ill posed and need to be regularized. For applications such as realtime video, fast restorations are also needed to keep up with the frame rate. Restoration based on 2D FFT's provides a fast implementation assuming a constant regularization term over the image. Unfortunately, this assumption creates significant ringing artifacts on edges as well as blurrier edges in the restored image. On the other hand, shift-variant regularization will reduce edge artifacts and provide better quality but it destroys the structure that makes use of the 2D FFT possible, thus no longer have the computational efficiency of the FFT. In this paper, we use a Bayesian approach-maximum a posteriori (MAP) estimation to compute an estimate of the original image given the blurred image. To avoid the smoothing of edges, shift-variant regularization must be used. The Huber-Markov random field model is applied to preserve the discontinuities on edges. For fast minimization of the above model, a new algorithm involving the Sherman-Morrison matrix inversion lemma is
This results in a restored image with good edge preservation and less computation. Experiments show restored images with sharper edges. Convergence is fast, and the computational speed can be improved considerably by breaking the image into subimages.
In applications of PMMW imaging such as real-time video, fast restorations are needed to keep up with the frame rate. FFT-based restoration provides a fast implemention but at the expense of assuming that the regularization term is constant over the image. Unfortunately, this assumption can create significant ringing artifacts in the presence of edges as well as edges that are blurrier than necessary. Furthermore,shift-invariant regularization does not allow for the possibility of superresolution.
Shift-variant regularization provides a way to vary the roughness penalty as a function of spatial coordinates to reduce edge artifacts and provide a degree of superresolution. Virtually all edge-preserving regularization approaches exploit this concept. However, this approach destroys the structure that makes the use of the FFT possible, since the deblurring operation is no longer shift-invariant. Thus, the restoration methods available for this problem no longer have the computational efficiency of the FFT.
We propose a new restoration method for the shift-variant regularization approach that can be implemented in a fast and flexible manner. We decompose the restoration into a sum of two independent restorations. One restoration yields an image that comes directly from an FFT-based approach. This image is a shift-invariant restoration containing the usual artifacts. The other restoration involves a set of unknowns whose number equals the number of pixels with a local smoothing penalty significantly different from the typical value in the image. This restoration represents the artifact correction image. By summing the two, the artifacts are canceled. Because the second restoration has a significantly reduced set of unknowns, it can be calculated very efficiently even though no circular convolution structure exists.