Super-resolution: a comprehensive survey <s> Maximum a posteriori <s> This paper describes a Bayesian method for constructing a super-resolved surface model by combining information from a set of images of the given surface. We develop the theory and algorithms in detail for the 2-D reconstruction problem, appropriate for the case where all images are taken from roughly the same direction and under similar lighting conditions. We show the results of this 2-D reconstruction on Viking Martian data. These results show dramatic improvements in both spatial and gray-scale resolution. The Bayesian approach uses a neighbor correlation model as well as pixel data from the image set. Some extensions of this method are discussed, including 3-D surface reconstruction and the resolution of diffraction blurred images. <s> BIB001 </s> Super-resolution: a comprehensive survey <s> Maximum a posteriori <s> Subsequent to the work of Kim, Bose, and Valenzuela in 1990 on the simultaneous filtering and interpolation of a registered sequence of undersampled noisy and shift-invariant blur degraded images, Bose and Boo tackled in 1998 the problem of reconstructing a high-resolution image from multiple undersampled, shifted, degraded frames with subpixel displacement errors. This led to a formulation involving a periodically shift-variant system model. Lertrattanapanich and Bose advanced in 1999 a procedure for construction of a high-resolution video mosaic following the estimation of motion parameters between successive frames in a video sequence generated from a video camera. Current research is focused on simultaneous blur identification and robust superresolution. The blur is not restricted to be linear shift-invariant and could not only be of the linear shift-variant type but also some nonlinear blurs could be accommodated. The optimal tuning parameter may, if desired, be calculated analytically and not by trial-and-error. <s> BIB002 </s> Super-resolution: a comprehensive survey <s> Maximum a posteriori <s> Superresolution reconstruction produces a high-resolution image from a set of low-resolution images. Previous iterative methods for superresolution had not adequately addressed the computational and numerical issues for this ill-conditioned and typically underdetermined large scale problem. We propose efficient block circulant preconditioners for solving the Tikhonov-regularized superresolution problem by the conjugate gradient method. We also extend to underdetermined systems the derivation of the generalized cross-validation method for automatic calculation of regularization parameters. The effectiveness of our preconditioners and regularization techniques is demonstrated with superresolution results for a simulated sequence and a forward looking infrared (FLIR) camera image sequence. <s> BIB003 </s> Super-resolution: a comprehensive survey <s> Maximum a posteriori <s> In this chapter, we present a parametric method for generating a super-resolution image from a sequence consisting of blurred and noisy observations. The high resolution image is modeled as a Markov random field (MRF) and a maximum a posteriori (MAP) estimation technique is used for super-resolution restoration. Unlike other super-resolution imaging methods, the proposed technique does not require sub-pixel registration of given observations. A simple gradient descent method is used to optimize the cost. The discontinuities in the intensity process can be preserved by introducing suitable line processes. Superiority of this technique to standard methods of image interpolation is illustrated. The motivation for using blur as a cue is also explained. <s> BIB004 </s> Super-resolution: a comprehensive survey <s> Maximum a posteriori <s> In many surveillance video applications, it is of interest to recognize a region of interest (ROI), which often occupies a small portion of a low-resolution, noisy video. This paper proposes an edge-preserving maximum a posteriori (MAP) estimation based super-resolution algorithm using a weighted directional Markov image prior model for a ROI from more than one low-resolution surveillance image. Conjugate gradient (CG) optimization based on standard operations on images is then developed to improve the computational efficiency of the algorithm. The proposed algorithm is tested on different series of surveillance images. The experimental results indicate that the proposed algorithm has considerable effectiveness in terms of both objective measurements and visual evaluation. <s> BIB005 </s> Super-resolution: a comprehensive survey <s> Maximum a posteriori <s> Image super-resolution (SR) reconstruction has been a hot research topic in recent years. This technique allows the recovery of a high-resolution (HR) image from several low-resolution (LR) images that are noisy, blurred and down-sampled. Among the available reconstruction frameworks, the maximum a posteriori (MAP) model is widely used. In this model, the regularization parameter plays an important role. If the parameter is too small, the noise will not be effectively restrained; conversely, the reconstruction result will become blurry. Therefore, how to adaptively select the optimal regularization parameter has been widely discussed. In this paper, we propose an adaptive MAP reconstruction method based upon a U-curve. To determine the regularization parameter, a U-curve function is first constructed using the data fidelity term and prior term, and then the left maximum curvature point of the curve is regarded as the optimal parameter. The proposed algorithm is tested on both simulated and actual data. Experimental results show the effectiveness and robustness of this method, both in its visual effects and in quantitative terms. <s> BIB006 </s> Super-resolution: a comprehensive survey <s> Maximum a posteriori <s> Stochastic regularized methods are quite advantageous in super-resolution (SR) image reconstruction problems. In the particular techniques, the SR problem is formulated by means of two terms, the data-fidelity term and the regularization term. The present work examines the effect of each one of these terms on the SR reconstruction result with respect to the presence or absence of noise in the low-resolution (LR) frames. Experimentation is carried out with the widely employed L"2, L"1, Huber and Lorentzian estimators for the data-fidelity term. The Tikhonov and Bilateral (B) Total Variation (TV) techniques are employed for the regularization term. The extracted conclusions can, in practice, help to select an effective SR method for a given sequence of LR frames. Thus, in case that the potential methods present common data-fidelity or regularization term, and frames are noiseless, the method which employs the most robust regularization or data-fidelity term should be used. Otherwise, experimental conclusions regarding performance ranking vary with the presence of noise in frames, the noise model as well as the difference in robustness of efficiency between the rival terms. Estimators employed for the data-fidelity term or regularizations stand for the rival terms. <s> BIB007
Given one or more LR images, g k , MAP methods BIB001 ( Table 7) find an estimate of the HR image,f , using Bayes's rules: By deleting the known denominator of the above equation and taking logarithms, the estimated response of the superresolved imagef using the MAP method is: (g 1 , g 2 , . . . , Since the LR images are independent of each other, the above equation becomes: which can be rewritten aŝ Using the same notations as used for ML, and assuming a zero mean white Gaussian noise, the likelihood distribution p(g k |f ) can be rewritten as: where C 1 is constant, and σ 2 k is the error variance. Then, the a priori can be written in Gibbs form as: where C 2 is constant and Γ ( f ) is the a priori energy function. Putting Eqs. (40) and (41) into Eq. (39), the MAP solution can be written as: where λ is the regularization parameter BIB004  BIB005 . For finding the best possible value of λ, different approaches can be used BIB007 . For example, in BIB002 the relationship between the residual term, ||A f − g|| 2 , and the regularization term, ||ρ( f )|| 2 , is studied for different values of λ to generate an L-curve. Then, it is decided that the desired value for λ is located at the corner of this curve. In BIB003 , GCV has been used to find the best possible value for λ. In BIB006 , it is suggested to use U -curves for this purpose and also discussed that these curves not only perform better than L-curves, but also can provide an interval in which the best value of the regularization term can be found. In this method first a Ucurve function is defined based on the data fidelity and a priori terms of the SR algorithm, and then the left maximum curvature point of the function is chosen as the best value of the regularization term BIB006 . Based on the nature of the terms of Eq. (42), different optimization methods can be followed, which are discussed in Sect. 6.2. The first term of Eq. (42) is already given in Eq. (32); there are however many different possibilities for the second term, the regularization term, which are discussed in the next subsections. The regularization terms are mostly, but not always, used when the solution of an inverse problem is under-determined (they might be used when the system is determined or over-determined as well: in this case they are used mainly for removing any artifacts which might appear after reconstruction). Therefore, there might be many solutions for a given set of observations. The regularization terms, in the form of a priori knowledge, are used to identify one of these available solutions which best fits some predefined desired conditions, and also to help the convergence of the problem. Such conditions can be, e.g., the smoothness of the solution, discontinuity preserving, etc. These can be achieved by, e.g., penalizing the high gradients of the solution or by defining some specific relationships between the neighboring pixels in the solution. Regularization terms have been used along with different SR methods: from iterative methods, direct methods, POCS and ML, to MAP methods.