This monograph is about a class of optimization algorithms called proximal algorithms. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here, we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that commonly arise in practice.

Proximal Algorithms

From the Publisher: 
The accessible presentation of this book gives both a general view of the entire computer vision enterprise and also offers sufficient detail to be able to build useful applications. Users learn techniques that have proven to be useful by first-hand experience and a wide range of mathematical methods. A CD-ROM with every copy of the text contains source code for programming practice, color images, and illustrative movies. Comprehensive and up-to-date, this book includes essential topics that either reflect practical significance or are of theoretical importance. Topics are discussed in substantial and increasing depth. Application surveys describe numerous important application areas such as image based rendering and digital libraries. Many important algorithms broken down and illustrated in pseudo code. Appropriate for use by engineers as a comprehensive reference to the computer vision enterprise.

Computer vision : a modern approach = 计算机视觉 : 一种现代的方法

International Joint Conference on Neural Networks

Three-Way Multidimensional Scaling

Forensic analyses of digital images rely heavily on the traces of in-camera and out-camera processes left on the acquired images. Such traces represent a sort of camera fingerprint. If one is able to recover them, by suppressing the high-level scene content and other disturbances, a number of forensic tasks can be easily accomplished. A notable example is the PRNU pattern, which can be regarded as a device fingerprint, and has received great attention in multimedia forensics. In this paper, we propose a method to extract a camera model fingerprint, called noiseprint, where the scene content is largely suppressed and model-related artifacts are enhanced. This is obtained by means of a Siamese network, which is trained with pairs of image patches coming from the same (label +1) or different (label −1) cameras. Although the noiseprints can be used for a large variety of forensic tasks, in this paper we focus on image forgery localization. Experiments on several datasets widespread in the forensic community show noiseprint-based methods to provide state-of-the-art performance.

Noiseprint: A CNN-Based Camera Model Fingerprint

In this paper we investigate the use of discriminative model learning through Convolutional Neural Networks (CNNs) for SAR image despeckling. The network uses a residual learning strategy, hence it does not recover the filtered image, but the speckle component, which is then subtracted from the noisy one. Training is carried out by considering a large multitemporal SAR image and its multilook version, in order to approximate a clean image. Experimental results, both on synthetic and real SAR data, show the method to achieve better performance with respect to state-of-the-art techniques.

https://hal.archives-ouvertes.fr/hal-01710036/document

SAR image despeckling through convolutional neural networks

Graphics editing programs of the last generation provide ever more powerful tools, which allow for the retouching of digital images leaving little or no traces of tampering. The reliable detection of image forgeries requires, therefore, a battery of complementary tools that exploit different image properties. Techniques based on the photo-response non-uniformity (PRNU) noise are among the most valuable such tools, since they do not detect the inserted object but rather the absence of the camera PRNU, a sort of camera fingerprint, dealing successfully with forgeries that elude most other detection strategies. In this paper, we propose a new approach to detect image forgeries using sensor pattern noise. Casting the problem in terms of Bayesian estimation, we use a suitable Markov random field prior to model the strong spatial dependences of the source, and take decisions jointly on the whole image rather than individually for each pixel. Modern convex optimization techniques are then adopted to achieve a globally optimal solution and the PRNU estimation is improved by resorting to nonlocal denoising. Large-scale experiments on simulated and real forgeries show that the proposed technique largely improves upon the current state of the art, and that it can be applied with success to a wide range of practical situations.

A Bayesian-MRF Approach for PRNU-Based Image Forgery Detection

We propose a proximal approach to deal with a class of convex variational problems involving nonlinear constraints. A large family of constraints, proven to be effective in the solution of inverse problems, can be expressed as the lower-level set of a sum of convex functions evaluated over different blocks of the linearly transformed signal. For such constraints, the associated projection operator generally does not have a simple form. We circumvent this difficulty by splitting the lower-level set into as many epigraphs as functions involved in the sum. In particular, we focus on constraints involving \(\varvec{\ell }_q\)-norms with \(q\ge 1\), distance functions to a convex set, and \(\varvec{\ell }_{1,p}\)-norms with \(p\in \{2,{+\infty }\}\). The proposed approach is validated in the context of image restoration by making use of constraints based on Non-Local Total Variation. Experiments show that our method leads to significant improvements in term of convergence speed over existing algorithms for solving similar constrained problems.

/pdf/epigraphical-projection-and-proximal-tools-for-solving-5bxl18aq2m.pdf

Epigraphical projection and proximal tools for solving constrained convex optimization problems

Nonlocal total variation (NLTV) has emerged as a useful tool in variational methods for image recovery problems. In this paper, we extend the NLTV-based regularization to multicomponent images by taking advantage of the structure tensor (ST) resulting from the gradient of a multicomponent image. The proposed approach allows us to penalize the nonlocal variations, jointly for the different components, through various l(1, p)-matrix-norms with p ≥ 1. To facilitate the choice of the hyperparameters, we adopt a constrained convex optimization approach in which we minimize the data fidelity term subject to a constraint involving the ST-NLTV regularization. The resulting convex optimization problem is solved with a novel epigraphical projection method. This formulation can be efficiently implemented because of the flexibility offered by recent primal-dual proximal algorithms. Experiments are carried out for color, multispectral, and hyperspectral images. The results demonstrate the interest of introducing a nonlocal ST regularization and show that the proposed approach leads to significant improvements in terms of convergence speed over current state-of-the-art methods, such as the alternating direction method of multipliers.

A nonlocal structure tensor-based approach for multicomponent image recovery problems

To detect some image forgeries one can rely on the Photo-Response Non-Uniformity (PRNU), a deterministic pattern associated with each individual camera, which can be loosely modeled as low-intensity multiplicative noise. A very promising algorithm for PRNU-based forgery detection has been recently proposed by Chen et al. Image denoising is a key step of the algorithm, since it allows to single out and remove most of the signal components and reveal the PRNU pattern. In this work we analyze the influence of denoising on the overall performance of the method and show that the use of a suitable state-of-the art denoising technique improves performance appreciably w.r.t. the original algorithm.

Giovanni Chierchia

Papers

SAR image despeckling through convolutional neural networks

A Bayesian-MRF Approach for PRNU-Based Image Forgery Detection

Epigraphical projection and proximal tools for solving constrained convex optimization problems

A nonlocal structure tensor-based approach for multicomponent image recovery problems

On the influence of denoising in PRNU based forgery detection