Dual constrained TV-based regularization

International audienceAlgorithms based on the minimization of the Total Variation are prevalent in computer vision. They are used in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa-Osher's projection algorithms for TV minimization in a flexible graph data representation by generalizing the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of a fast parallel proximal algorithm, which performs better than the classical TV approach for denoising, and is also applicable to inverse problems such as image deblurring

A tight frame algorithm in image inpainting.

Cheng, Kei Tsi Daniel.Thesis (M.Phil.)--Chinese University of Hong Kong, 2007.Includes bibliographical references (leaves 45-49).Abstracts in English and Chinese.Abstract --- p.iAcknowledgement --- p.iiiChapter 1 --- Introduction --- p.1Chapter 2 --- Background Knowledge --- p.6Chapter 2.1 --- Image Restoration using Total Variation Norm --- p.6Chapter 2.2 --- An Example of Tight Frame system --- p.10Chapter 2.3 --- Sparse and compressed representation --- p.13Chapter 2.4 --- Existence of minimizer in convex analysis --- p.16Chapter 3 --- Tight Frame Based Minimization --- p.18Chapter 3.1 --- Tight Frames --- p.18Chapter 3.2 --- Minimization Problems and Algorithms --- p.19Chapter 3.3 --- Other Minimization Problems --- p.22Chapter 4 --- Algorithm from minimization problem 3 --- p.24Chapter 5 --- Algorithm from minimization problem 4 --- p.28Chapter 6 --- Convergence of Algorithm 2 --- p.31Chapter 6.1 --- Inner Iteration --- p.31Chapter 6.2 --- Outer Iteration --- p.33Chapter 6.2.1 --- Existence of minimizer --- p.33Chapter 7 --- Numerical Results --- p.37Chapter 8 --- Conclusion --- p.4

An efficient algorithm for total variation regularization with applications to the single pixel camera and compressive sensing

In this thesis, I propose and study an efficient algorithm for solving a class of compressive sensing problems with total variation regularization. This research is motivated by the need for efficient solvers capable of restoring images to a high quality captured by the single pixel camera developed in the ECE department of Rice University. Based on the ideas of the augmented Lagrangian method and alternating minimization to solve subproblems, I develop an efficient and robust algorithm called TVAL3. TVAL3 is compared favorably with other widely used algorithms in terms of reconstruction speed and quality. Convincing numerical results are presented to show that TVAL3 is suitable for the single pixel camera as well as many other applications

A multi-scale imaging approach to understand osteoarthritis development

X-ray phase-contrast imaging is an innovative and advanced imaging method. Contrary to conventional radiology, where the image contrast is primarily determined by X-ray attenuation, phase-contrast images contain additional information generated by the phase shifts or refraction of the X-rays passing through matter. The refractive effect on tissue samples is orders of magnitude higher than the absorption effect in the X-ray energy range used in biomedical imaging. This technique makes it possible to produce excellent and enhanced image contrast, particularly when examining soft biological tissues or features with similar X-ray attenuation properties. In combination with high spatial resolution detector technology and computer tomography, X-ray phase-contrast imaging has been proved to be a powerful method to examine tissue morphology and the evolution of pathologies three-dimensionally, with great detail and without the need of contrast agents. This Thesis work has focused on developing an accurate, multi-scale X-ray-based methodology for imaging and characterizing the early stages of osteoarthritis. X-ray phase-contrast images acquired at different spatial resolutions provide unprecedented insights into cartilage and the development of its degeneration, i.e., osteoarthritis. Other types of X-ray phase-contrast imaging techniques and setups using spatial resolutions ranging from micrometer down to nanometer were applied. Lower spatial resolutions allow large sample coverage and comprehensive representations, while the nanoscale analysis provides a precise depiction of anatomical details and pathological signs. X-ray phase-contrast results are correlated to data obtained, on the same specimens, by standard laboratory methods, such as histology and transmission electron microscopy. Furthermore, X-ray phase-contrast images of cartilage were acquired using different X-ray sources and results were compared in terms of image quality. It was shown that with the use of synchrotron radiation, more detailed images and much faster data acquisitions could be achieved. A second focus in this Thesis work has been the investigation of the reaction of healthy and degenerated cartilage under different physical pressures, simulating the different levels of stress to which the tissue is subject during daily movements. A specifically designed setup was used to dynamically study cartilage response to varying pressures with X-ray phase-contrast micro-computed tomography, and a fully volumetric and quantitative methodology to accurately describe the tissue morphological variations. This study revealed changes in the behavior of the cartilage cell structure, which differ between normal and osteoarthritic cartilage tissues. The third focus of this Thesis is the realization of an automated evaluation procedure for the discrimination of healthy and cartilage images with osteoarthritis. In recent years, developments in neural networks have shown that they are excellently suited for image classification tasks. The transfer learning method was applied, in which a pre-trained neural network with cartilage images is further trained and then used for classification. This enables a fast, robust and automated grouping of images with pathological findings. A neural network constructed in this way could be used as a supporting instrument in pathology. X-ray phase-contrast imaging computed tomography can provide a powerful tool for a fully 3D, highly accurate and quantitative depiction and characterization of healthy and early stage-osteoarthritic cartilage, supporting the understanding of the development of osteoarthritis.Röntgen-Phasenkontrast-Bildgebung ist eine innovative und weiterführende Bildgebungsmethode. Im Gegensatz zu herkömlichen Absorptions-Röntgenaufnahmen, wie sie in der Radiologie verwendet werden, wird der Kontrast bei dieser Methode aus dem Effekt der Phasenverschiebung oder auch Brechung der Röngtenstrahlen gebildet. Der Brechungseffekt bei Gewebeproben ist um ein Vielfaches höher als der Absorptionseffekt des elektromagnetischen Spektrums der Röntgenstrahlen. Diese Methode ermöglicht die Darstellung von großen Kontraste im Gewebe. Unter Verwendung eines hochauflösenden Detektors und in Kombination mit der Computer-Tomographie, ist Phasenkontrast-Bildgebung eine sehr gute Methode um Knorpelgewebe und Arthrose im Knorpel zu untersuchen. Diese Arbeit beschreibt primär ein Verfahren zur Darstellung arthrotischen Knorpels im Anfangsstadium. Die mit verschiedenen Auflösungen und 3D-Phasen-Kontrast-Methoden produzierten Aufnahmen ermöglichen einen noch nie dagewesenen Einblick in den Knorpel und die Entwicklung von Arthrose im Anfangsstadium. Hierbei kam die propagationsbasierte Phasenkontrastmethode mit einer Auflösung im mikrometer Bereich und die (Nano)-Holotomographie-Methode mit einer Auflösung im Submicrometer Bereich zum Einsatz. Durch Auflösung im mikrometer Bereich kann ein großes Volumen im Knorpel gescannt werden, während die Nano-Holotomographie Methode eine sehr große Detailauflösung aufweißt. Die Phasenkontrast-Aufnahmen werden mit zwei anderen wissenschaftlichen Methoden verglichen: mikroskopische Abbildungen histologisch aufgearbeiteter Knorpelproben und Aufnahmen eines Transmissionselektroskop zeigen sehr große Übereinstimmungen zur Röntgen-Phasenkontrast-Bildgebung. Desweiteren wurden Phasenkontrast-Aufnahmen von Knorpel aus unterschiedlichen Röntgenquellen verglichen. Hierbei zeigte sich, dass mit Hilfe des Teilchenbeschleunigers (Synchrotron) detailreichere und schnellere Aufnahmen erzielt werden können. Bilder aus Flüssig-Metall-Quellen zeigen sich durchaus von guter Qualität, erfordern jedoch sehr lange Aufnahmezeiten. In dieser Arbeit wird zudem das Verhalten von Knorpelgewebe, welches ein Anfangsstadium von Arthrose aufweist, unter physikalischem Druck untersucht. Hierfür wurden 3D-Computertomographie-Aufnahmen von komprimiertem Knorpelgewebe angefertig und mit Aufnahmen ohne Komprimierung verglichen. Ein quantitativer Vergleich machte Veränderungen des Verhaltens der Knorpelzellstruktur (Chondronen) sichtbar. Es konnte gezeigt werden, dass Chondrone bei arthrotischem Knorpel ein verändertes Kompressionsverhalten haben. Der dritte Fokus dieser Arbeit liegt auf der automatisierten Auswertung von Aufnahmen gesunden und arthrotischen Knorpelgewebes. Die Entwicklungen im Bereich der Neuronale Netze zeigten in den letzten Jahren, dass diese sich hervoragend für Bildklassifizierungsaufgaben eignen. Es wurde die Methode des transferierenden Lernens angewandt, bei der ein vortrainiertes Neuronales Netz mit Knorpelbildern weitertrainiert und anschließend zur Klassifizierung eingesetzt wird. Dadurch ist eine schnelle, robuste und automatisierte Gruppierung von Bildern mit pathologischen Befunden möglich. Ein derart konstruiertes Neuronales Netz könnte als unterstützendes Instrument in der Pathologie angewandt werden. Röntgen-Phasenkontrast-CT kann ein leistungsstarkes Werkzeug für eine umfassende, hochpräzise und quantitative 3D-Darstellung und Charakterisierung von gesundem Knorpel und athrotischem Knorpel im Frühstadium bieten, um das Verständnis der Entwicklung von Osteoarthritis zu erweitern

Rekonstrukcija signala iz nepotpunih merenja sa primenom u ubrzanju algoritama za rekonstrukciju slike magnetne rezonance

In dissertation a problem of reconstruction of images from undersampled measurements is considered which has direct application in creation of magnetic resonance images. The topic of the research is proposition of new regularization based methods for image reconstruction which are based on statistical Markov random field models and theory of compressive sensing. With the proposed signal model which follows the statistics of images, a new regularization functions are defined and four methods for reconstruction of magnetic resonance images are derived.У докторској дисертацији разматран је проблем реконструкције сигнала слике из непотпуних мерења који има директну примену у креирању слика магнетне резнонаце. Предмет истраживања је везан за предлог нових регуларизационих метода реконструкције коришћењем статистичких модела Марковљевог случајног поља и теорије ретке репрезентације сигнала. На основу предложеног модела који на веродостојан начин репрезентује статистику сигнала слике предложене су регуларизационе функције и креирана четири алгоритма за реконструкцију слике магнетне резонанце.U doktorskoj disertaciji razmatran je problem rekonstrukcije signala slike iz nepotpunih merenja koji ima direktnu primenu u kreiranju slika magnetne reznonace. Predmet istraživanja je vezan za predlog novih regularizacionih metoda rekonstrukcije korišćenjem statističkih modela Markovljevog slučajnog polja i teorije retke reprezentacije signala. Na osnovu predloženog modela koji na verodostojan način reprezentuje statistiku signala slike predložene su regularizacione funkcije i kreirana četiri algoritma za rekonstrukciju slike magnetne rezonance

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Chambolle's Projection Algorithm for Total Variation Denoising

Denoising is the problem of removing the inherent noise from an image. The standard noise model is additive white Gaussian noise, where the observed image f is related to the underlying true image u by the degradation model f=u+n, and n is supposed to be at each pixel independently and identically distributed as a zero-mean Gaussian random variable. Since this is an ill-posed problem, Rudin, Osher and Fatemi introduced the total variation as a regularizing term. It has proved to be quite efficient for regularizing images without smoothing the boundaries of the objects. This paper focuses on the simple description of the theory and on the implementation of Chambolle's projection algorithm for minimizing the total variation of a grayscale image. Furthermore, we adapt the algorithm to the vectorial total variation for color images. The implementation is described in detail and its parameters are analyzed and varied to come up with a reliable implementation