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...quired on the average about 2–5 min on a 3.20-GHz Xeon PC for 256 256 images. Note that the running time of the algorithms can be improved by utilizing preconditioning methods (see, for example, [38]–=-=[41]-=-). VI. CONCLUSION We have presented two new methods for the simultaneous estimation of the image and the unknown hyperparameters in TV-based image restoration problems. We adopt a variational approach...

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...rmed spatially (as in the case of single image restoration), temporal consistency cannot be guaranteed if these methods are applied to a video in a frame-by-frame basis. Because of these unique features in video, we seek a video restoration algorithm that utilizes motion information, exploits the spatially invariant properties and enforces spatial and temporal consistency. B. Related Work Majority of existing video restoration methods recover a video by solving a sequence of individual image restorations. To improve the temporal consistency among the frames, various approaches can be adopted: [19] modified Equation (1) to incorporate the geometric warp caused by motion; [20] utilized the motion vector field as a prior to the restoration; [21] considered a regularization function of the residue between the current solution and the motion compensated version of the previous solution. Another class of methods are based on the concept of “space-time volume”, which is first introduced in the early 90’s by Jahne [22], and rediscovered by Wexler, Shechtman, 2 Caspi and Irani [23], [24]. The idea of space-time volume is to stack the frames of a video to form a three-dimensional data structure...

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...d topic. Recently, motivated by its success in image recovery problems, the use of the total variation (TV) function and its variants has become popular in super resolution. Both regularization-based =-=[2,3]-=- and Bayesian [4] formulations have been proposed which utilize TV functions to characterize the HR images. However, both of these approaches involve certain model parameters to be set by the user, wh...

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...rporation of accurate image models. The TV function is utilized successfully in a number of image recovery methods including denoising [29], blind deconvolution [30], inpainting, and super resolution =-=[31]-=-. TV priors are very effective in preserving edges while imposing smoothness. In this work, as the HR image prior, we utilize the quadratic approximation of the TV prior, that is where is a constant a...

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...[24]–[26] and Bayesian [27] total variation (TV) regularization techniques. As far as the imaging models are concerned, many techniques are incorporating the motion field (MF) information [22], [24], =-=[25]-=-, [28], whereas others do not use this information at all. The term multichannel [29] in the context of video recovery implies the use of the between frames correlation. Multichannel approaches have b...

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...al variation (TV) norm. Total variation minimization was originally introduced for noise reduction [25] [7], and has also been used for image deblurring [14] and super-resolution image reconstruction =-=[17]-=-. The TV regularization (ℓ2-TV) is obtained from (1.2) by setting p = 1, q = 2, and A(u) the gradient magnitude of u. If we let ∇ui := (Gx,iu, Gy,iu) T , with Gx,i, Gy,i representing the ith rows of t...

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...s that Tikhonov regularization cannot preserve sharp edges. Fig. 6 shows some comparisons between the proposed spatial regularization and Tikhonov regularization. Detailed discussions can be found in =-=[47]-=- and [48]. C. Temporal Regularization Although the spatial regularization function can be applied to each frame of a video individually, the temporal consistency of the video is not guaranteed. Tempor...

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...ient number of LR images, ill-conditioned registration and unknown blurring operators. Therefore one can add various regularization items to further stabilize the inversion of this ill-posed problem (=-=Ng et al., 2007-=-; Chan et al., 2007), which are named as regularization-based methods. However, their performance degrades rapidly when the magnification factor becomes large (Baker and Kanade, 2002). Another catalog...

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...unts for occlusion effects at defocused boundaries. We remark that in all the above works the observations are not affected by parallax and their formulations do not account for it. In a recent paper =-=[25]-=-, joint super-resolution of video and inpainting has been demonstrated. The inverse problem is solved using a total variation based regularization approach thereby tackling the inpainting problem also...