Image Restoration for Shift Invariant Blurs-A Survey <s> Alternating Minimization <s> Shock filters for image enhancement are developed. The filters use new nonlinear time dependent partial differential equations and their discretizations. The evolution of the initial image $u_0 (x,y)$ as $t \to \infty $ into a steady state solution $u_\infty (x,y)$ through $u(x,y,t)$, $t > 0$, is the filtering process. The partial differential equations have solutions which satisfy a maximum principle. Moreover the total variation of the solution for any fixed $t > 0$ is the same as that of the initial data. The processed image is piecewise smooth, nonoscillatory, and the jumps occur across zeros of an elliptic operator (edge detector). The algorithm is relatively fast and easy to program. <s> BIB001 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Alternating Minimization <s> In this chapter, ensemble learning is applied to the problem of blind source separation and deconvolution of images. It is assumed that the observed images were constructed by mixing a set of images (consisting of independent, identically distributed pixels), convolving the mixtures with unknown blurring filters and then adding Gaussian noise. <s> BIB002 </s> Image Restoration for Shift Invariant Blurs-A Survey <s> Alternating Minimization <s> Camera shake during exposure leads to objectionable image blur and ruins many photographs. Conventional blind deconvolution methods typically assume frequency-domain constraints on images, or overly simplified parametric forms for the motion path during camera shake. Real camera motions can follow convoluted paths, and a spatial domain prior can better maintain visually salient image characteristics. We introduce a method to remove the effects of camera shake from seriously blurred images. The method assumes a uniform camera blur over the image and negligible in-plane camera rotation. In order to estimate the blur from the camera shake, the user must specify an image region without saturation effects. We show results for a variety of digital photographs taken from personal photo collections. <s> BIB003
In most of real life situation both the PSF and the true image must be estimated & this estimation process could be separate or joint. Estimation process uses various constraints such as nonnegativity, symmetry, and smoothness about the PSF and the true image. To find both the PSF h and the true image f that maximize the following posterior probability (39) where is the likelihood of observations is the image prior, and is the PSF prior. In optimization probability is dropped as it is constant with respect to f & h. If parameterization is used for the prior distributions and introduced parameters are considered as random variables called hyperparameters than the posterior probability becomes; unknown image could be obtained as; (41) Instead of determining the optimal f and h simultaneously alternating minimization is used which alternatively updates the one variable while keeping the other fixed  BIB003 . Equations for next estimate of f & h are as follows; (42) (43) and the corresponding alternating minimization iterations are Initial true image estimation obtained by applying a shock filter BIB001 or a bilateral filter BIB002 results in highly successful blind deconvolution algorithms with sharp edges in image.