A Smoothed Dual Approach for Variational Wasserstein Problems

Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by Carlier et al. (2014) can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals

Compressive sensing using the modified entropy functional

Cataloged from PDF version of article.In most compressive sensing problems, 1 norm is used during the signal reconstruction process. In this article, a modified version of the entropy functional is proposed to approximate the 1 norm. The proposed modified version of the entropy functional is continuous, differentiable and convex. Therefore, it is possible to construct globally convergent iterative algorithms using Bregman’s row-action method for compressive sensing applications. Simulation examples with both 1D signals and images are presented. © 2013 Elsevier Inc. All rights reserved

Entropy-Functional-Based Online Adaptive Decision Fusion Framework with Application to Wildfire Detection in Video

Cataloged from PDF version of article.In this paper, an entropy-functional-based online adaptive decision fusion (EADF) framework is developed for image analysis and computer vision applications. In this framework, it is assumed that the compound algorithm consists of several subalgorithms, each of which yields its own decision as a real number centered around zero, representing the confidence level of that particular subalgorithm. Decision values are linearly combined with weights that are updated online according to an active fusion method based on performing entropic projections onto convex sets describing subalgorithms. It is assumed that there is an oracle, who is usually a human operator, providing feedback to the decision fusion method. A video-based wildfire detection system was developed to evaluate the performance of the decision fusion algorithm. In this case, image data arrive sequentially, and the oracle is the security guard of the forest lookout tower, verifying the decision of the combined algorithm. The simulation results are presented

Image Restoration

This book represents a sample of recent contributions of researchers all around the world in the field of image restoration. The book consists of 15 chapters organized in three main sections (Theory, Applications, Interdisciplinarity). Topics cover some different aspects of the theory of image restoration, but this book is also an occasion to highlight some new topics of research related to the emergence of some original imaging devices. From this arise some real challenging problems related to image reconstruction/restoration that open the way to some new fundamental scientific questions closely related with the world we interact with

Restoration methods for biomedical images in confocal microscopy

Diese Arbeit stellt neue Loesungen zum Problem Bildrestauration im biomedizinischen Bereich vor. Das Konfokal-Mikroskop ist eine verhaeltnismaessig neue Bildungstechnik, die als Standardwerkzeug in biomedizinischen Studien eingesetzt wird. Diese Technik dient zum Sammeln einer Reihe von 2D Bildern der einzelnen Abschnitte innerhalb eines Probestuecks, um eine 3D Darstellung des Gegenstandes zu erzeugen. Trotz seiner verbesserten Belichtungseigenschaften unterliegen die beobachteten Bilder Stoerungen augrund der begrenzten Groesse der Punktantwort (PSF) und das Poisson-Rauschens. Bildrestaurationstechniken versuchen diese Stoerungen herauszurechnen und das Originalbild zu rekonstruieren. Diese Doktorarbeit beginnt mit der Beschreibung des Konfokal-Mikroskops und den Quellen von Artefakten. Dann werden die vorhandenen Bildwiederherstellungsmethoden vorgestellt und verglichen. Die Arbeit ist in drei Teile gegliedert: Im ersten Teil wird eine neue begrenzte blinde Dekonvolutionsmethode eingefuehrt. Durch eine passende Re-Parametrisierung wird dabei a priori Wissen eingebaut. Fuer die PSF wird ein parametrisches Modell, mit einem begrenzten Satz von Basisunktionen benutzt, um Nicht-Negativitaet, zirkulaaere Symmetrie und Limitierung der Frequenzbandbreite sicher zu stellen. Fuer das Bild stellt die quadratische Re-Parametrisierung die Nicht-Negativitaet sicher. Die Entfaltungsmethode wird anhand von simulierten und realen Konfokal-Mikroskopie Daten ausgewertet. Der Vergleich mit einem nicht-parametrisierten Algorithmus zeigt, dass die vorgeschlagene Methode verbesserte Leistung und schnellere Konvergenz erreicht. Im zweiten Teil der Arbeit wird eine neue Methode eingefuehrt, die versucht die anisotrope tiefabhaengige Unschaerfe zu beheben. Wenn roehrenfoermige Gegenstaende -wie Neuronen- abgebildet werden, sind die aufgenommenen Bilder degradiert und die Extraktion der genauen Morphologie der Neuronen wird erschwert. Es wird eine neue Methode vorgeschlagen, mit der sich die PSF ohne irgendein Vorwissen ueber das Belichtungssystem aus dem augenommenen Bild schaetzen laesst. Diese Methode, die auf der Schaetzung des urspruenglichen Gegenstandes basiert ist fuer Faelle verwendbar, in denen der abgebildete Gegenstand eine bekannte Geometrie hat. Mit der vorgeschlagenen Dekonvolutionsmethode werden geometrische Verzerrungen beseitigt und die wiederhergestellten Bilder sind fuer weitere Analysen besser verwendbar. Im dritten Teil wird eine neue Methode zur adaptiven Regularisierung vorgeschlagen. Diese vorgeschlagene Technik passt ihr Verhalten abhaengig von den lokalen Intensitaetsgradient im Bild an. Die neue Technik wird getestet und mit der ''total variation'' und der Tikhonov Regularisierungtechnik verglichen. Die Experimente zeigen, dass mit dem adaptiven Verahren, die Qualitaet der rekonstruierten Bilder verbessert wird.This thesis introduces new solutions to the problem of image restoration in biomedical fields. The confocal microscope is a relatively new imaging technique that is emerging as a standard tool in biomedical studies. This technique is capable of collecting a series of 2D images of single sections inside a specimen to form a 3D image of the object. Despite of its improved imaging properties, the observed images are blurred due to the finite size of the the point spread function and corrupted by Poisson noise due to the counting nature of image detection. Image restoration techniques aim at reversing the degradation and recovering an estimate of the true image. This thesis starts with the description of the confocal microscope and the sources of degradation. Then, the existing image restoration methods are studied and compared. The work done in this thesis is divided into three parts: In the first part, a new constrained blind deconvolution method is introduced. Re-parameterization is used to strictly enforce prior knowledge. A parametric model based on a set of constrained basis functions is used for the PSF to ensure non-negativity, circular symmetry, and band-limitedness. For the image, quadratic re-parameterization ensures non-negativity. The deconvolution method is evaluated on both simulated and real confocal microscopy data sets. The comparison with non-parameteric algorithms shows that the proposed method exhibits improved performance and faster convergence. In the second part, a new method to correct the anisotropic, depth-variant blur is introduced. When objects of tubular-like structure, like neurons, are imaged, the acquired images are degraded and the extraction of accurate morphology of neurons is hampered. A new method to estimate the PSF from the acquired image, without any prior knowledge about the imaging system, is proposed. This method which is based on the estimation of the original object and is suitable for cases in which, the object being imaged has a known geometry. Using the proposed restoration method, geometric distortions are eliminated and the restored images are more suitable for further analysis. In the third part, a new method for adaptive regularization is proposed. The proposed technique adapts its behavior depending on the local activities in the image, as reflected in the magnitude of the intensity gradient. The new technique is tested and compared to both the total variation and the Tikhonov regularization techniques. Experiments show that, using the adaptive technique, the quality of the restored images is improved

Proximity Operators of Discrete Information Divergences

Information divergences allow one to assess how close two distributions are from each other. Among the large panel of available measures, a special attention has been paid to convex $\varphi$-divergences, such as Kullback-Leibler, Jeffreys-Kullback, Hellinger, Chi-Square, Renyi, and I$_{\alpha}$ divergences. While $\varphi$-divergences have been extensively studied in convex analysis, their use in optimization problems often remains challenging. In this regard, one of the main shortcomings of existing methods is that the minimization of $\varphi$-divergences is usually performed with respect to one of their arguments, possibly within alternating optimization techniques. In this paper, we overcome this limitation by deriving new closed-form expressions for the proximity operator of such two-variable functions. This makes it possible to employ standard proximal methods for efficiently solving a wide range of convex optimization problems involving $\varphi$-divergences. In addition, we show that these proximity operators are useful to compute the epigraphical projection of several functions of practical interest. The proposed proximal tools are numerically validated in the context of optimal query execution within database management systems, where the problem of selectivity estimation plays a central role. Experiments are carried out on small to large scale scenarios