Super-resolution (SR) is a technique that allows increasing the resolution of a given image. Having applications in many areas, from medical imaging to consumer electronics, several SR methods have been proposed. Currently, the best performing methods are based on convolutional neural networks (CNNs) and require extensive datasets for training. However, at test time, they fail to impose consistency between the super-resolved image and the given low-resolution image, a property that classic reconstruction-based algorithms naturally enforce in spite of having poorer performance. Motivated by this observation, we propose a new framework that joins both approaches and produces images with superior quality than any of the prior methods. Although our framework requires additional computation, our experiments on Set5, Set14, and BSD100 show that it systematically produces images with better peak signal to noise ratio (PSNR) and structural similarity (SSIM) than the current state-of-the-art CNN architectures for SR.
Total variation methods A different line of work directly encodes prior knowledge into the optimization problem. For instance, @cite_6 introduced the concept of TV to capture the number of edges in an image and observed that natural images have small TV. Since then, several algorithms have been proposed to super-resolve images by minimizing TV. For example, @cite_16 addresses the problem by discretizing a differential equation that relates changes in the values of individual pixels to the curvature of level sets. More recently, with the development of nondifferentiable convex optimization methods, TV minimization has been addressed in the discrete setting by a wide range of algorithms @cite_35 @cite_4 @cite_38 @cite_26 @cite_23 . There are essentially two versions of discrete TV, according to the norm that is applied to the discretized gradient at each pixel: isotropic ( @math -norm) and anisotropic ( @math -norm); see, e.g., @cite_4 @cite_23 . Since both versions are convex, discrete TV minimization algorithms not only have small computational complexity (as matrix-vector operations can be performed via the FFT algorithm), but are also guaranteed to find a global minimizer.
@cite_35 @cite_38 @cite_26 @cite_4 @cite_6 @cite_23 @cite_16
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We present a new method for solving total variation (TV) minimization problems in image restoration. The main idea is to remove some of the singularity caused by the nondifferentiability of the quantity @math in the definition of the TV-norm before we apply a linearization technique such as Newton's method. This is accomplished by introducing an additional variable for the flux quantity appearing in the gradient of the objective function, which can be interpreted as the normal vector to the level sets of the image u. Our method can be viewed as a primal-dual method as proposed by Conn and Overton [ A Primal-Dual Interior Point Method for Minimizing a Sum of Euclidean Norms, preprint, 1994] and Andersen [Ph. D. thesis, Odense University, Denmark, 1995] for the minimization of a sum of Euclidean norms. In addition to possessing local quadratic convergence, experimental results show that the new method seems to be globally convergent. Image restoration models based on total variation (TV) have become popular since their introduction by Rudin, Osher, and Fatemi (ROF) in 1992. The dual formulation of this model has a quadratic objective with separable constraints, making projections onto the feasible set easy to compute. This paper proposes application of gradient projection (GP) algorithms to the dual formulation. We test variants of GP with different step length selection and line search strategies, including techniques based on the Barzilai-Borwein method. Global convergence can in some cases be proved by appealing to existing theory. We also propose a sequential quadratic programming (SQP) approach that takes account of the curvature of the boundary of the dual feasible set. Computational experiments show that the proposed approaches perform well in a wide range of applications and that some are significantly faster than previously proposed methods, particularly when only modest accuracy in the solution is required. We present a blind deconvolution algorithm based on the total variational (TV) minimization method proposed by Acar and Vogel (1994). The motivation for regularizing with the TV norm is that it is extremely effective for recovering edges of images as well as some blurring functions, e.g., motion blur and out-of-focus blur. An alternating minimization (AM) implicit iterative scheme is devised to recover the image and simultaneously identify the point spread function (PSF). Numerical results indicate that the iterative scheme is quite robust, converges very fast (especially for discontinuous blur), and both the image and the PSF can be recovered under the presence of high noise level. Finally, we remark that PSFs without sharp edges, e.g., Gaussian blur, can also be identified through the TV approach. TV regularization is an L1 penalization of the flow gradient magnitudes, and due to the tendency of the L1 norm to favor sparse solutions (i.e. lots of ‘zeros’), the fill-in effect caused by the regularizer leads to piecewise constant solutions in weakly textured areas. This effect, known as ‘staircasing’ in a 1D setting, can be reduced significantly by using a quadratic penalization for small gradient magnitudes while sticking to linear penalization for larger magnitudes to maintain the discontinuity preserving properties known from TV. A comparison of isotropic TV and isotropic Huber regularity is shown in Fig. 1 by means of rendering the disparities u1 of the Dimetrodon dataset. The color coded flow (cf. Fig. 1(a)) is superimposed as texture. Based on the two observations that motion discontinuities often occur along object boundaries and that in turn object boundaries often coincide