The Convergence of Two Algorithms for Compressed Sensing Based Tomography

The constrained total variation minimization has been developed successfully for image reconstruction in computed tomography. In this paper, the block component averaging and diagonally-relaxed orthogonal projection methods are proposed to incorporate with the total variation minimization in the compressed sensing framework. The convergence of the algorithms under a certain condition is derived. Examples are given to illustrate their convergence behavior and noise performance

Selection of Step Size for Total Variation Minimization in CT

Medical image reconstruction by total variation minimization is a newly developed area in computed tomography (CT). In compressed sensing literature, it hasbeen shown that signals with sparse representations in an orthonormal basis may be reconstructed via l1-minimization. Furthermore, if an image can be approximately modeled to be piecewise constant, then its gradient is sparse. The application of l1-minimization to a sparse gradient, known as total variation minimization, may then be used to recover the image. In this paper, the steepest descent method is employed to update the approximation of the image. We propose a way to estimate an optimal step size so that the total variation is minimized. A new minimization problem is also proposed. Numerical tests are included to illustrate the improvement

Precise Phase Transition of Total Variation Minimization

Characterizing the phase transitions of convex optimizations in recovering structured signals or data is of central importance in compressed sensing, machine learning and statistics. The phase transitions of many convex optimization signal recovery methods such as $\ell_1$ minimization and nuclear norm minimization are well understood through recent years' research. However, rigorously characterizing the phase transition of total variation (TV) minimization in recovering sparse-gradient signal is still open. In this paper, we fully characterize the phase transition curve of the TV minimization. Our proof builds on Donoho, Johnstone and Montanari's conjectured phase transition curve for the TV approximate message passing algorithm (AMP), together with the linkage between the minmax Mean Square Error of a denoising problem and the high-dimensional convex geometry for TV minimization.Comment: 6 page

Domain decomposition methods for compressed sensing

We present several domain decomposition algorithms for sequential and parallel minimization of functionals formed by a discrepancy term with respect to data and total variation constraints. The convergence properties of the algorithms are analyzed. We provide several numerical experiments, showing the successful application of the algorithms for the restoration 1D and 2D signals in interpolation/inpainting problems respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles.Comment: 4 page

Cardiac CT Image Reconstruction Based on Compressed Sensing

AbstractIn this paper, the authors review the compressed sensing concept and analyze the basics for CT image reconstruction. The application of compressed sensing is investigated in cardiac CT image reconstruction and a comprehensive sparse reconstruction approach is presented. This approach incorporates the total variation norm, l1 norm and subtraction of different images norm minimization to reconstruct image from Sheep-Logan and Cardiac phantom. The reconstruction results show that our algorithm can achieve high quality cardiac image

Autonomous Electron Tomography Reconstruction with Machine Learning

Modern electron tomography has progressed to higher resolution at lower doses by leveraging compressed sensing methods that minimize total variation (TV). However, these sparsity-emphasized reconstruction algorithms introduce tunable parameters that greatly influence the reconstruction quality. Here, Pareto front analysis shows that high-quality tomograms are reproducibly achieved when TV minimization is heavily weighted. However, in excess, compressed sensing tomography creates overly smoothed 3D reconstructions. Adding momentum into the gradient descent during reconstruction reduces the risk of over-smoothing and better ensures that compressed sensing is well behaved. For simulated data, the tedious process of tomography parameter selection is efficiently solved using Bayesian optimization with Gaussian processes. In combination, Bayesian optimization with momentum-based compressed sensing greatly reduces the required compute time$-$an 80% reduction was observed for the 3D reconstruction of SrTiO$_3$ nanocubes. Automated parameter selection is necessary for large scale tomographic simulations that enable the 3D characterization of a wider range of inorganic and biological materials.Comment: 8 pages, 4 figure

A General Total Variation Minimization Theorem for Compressed Sensing Based Interior Tomography

Recently, in the compressed sensing framework we found that a two-dimensional interior region-of-interest (ROI) can be exactly reconstructed via the total variation minimization if the ROI is piecewise constant (Yu and Wang, 2009). Here we present a general theorem charactering a minimization property for a piecewise constant function defined on a domain in any dimension. Our major mathematical tool to prove this result is functional analysis without involving the Dirac delta function, which was heuristically used by Yu and Wang (2009)

A Convergent Overlapping Domain Decomposition Method for Total Variation Minimization

This paper is concerned with the analysis of convergent sequential and parallel overlapping domain decomposition methods for the minimization of functionals formed by a discrepancy term with respect to data and a total variation constraint. To our knowledge, this is the first successful attempt of addressing such strategy for the nonlinear, nonadditive, and nonsmooth problem of total variation minimization. We provide several numerical experiments, showing the successful application of the algorithm for the restoration of 1D signals and 2D images in interpolation/inpainting problems respectively, and in a compressed sensing problem, for recovering piecewise constant medical-type images from partial Fourier ensembles.Comment: Matlab code and numerical experiments of the methods provided in this paper can be downloaded at the web-page: http://homepage.univie.ac.at/carola.schoenlieb/webpage_tvdode/tv_dode_numerics.ht

Geometrische Interpretationen und Algorithmische Verifikation von exakten Lösungen in Compressed Sensing

In an era dominated by the topic big data, in which everyone is confronted with spying scandals, personalized advertising, and retention of data, it is not surprising that a topic as compressed sensing is of such a great interest. Further the field of compressed sensing is very interesting for problems in signal- and image processing. Similarly, the question arises how many measurements are necessarily required to capture and represent high-resolution signal or objects. In the thesis at hand, the applicability of three of the most applied optimization problems with linear restrictions in compressed sensing is studied. These are basis pursuit, analysis l1-minimization und isotropic total variation minimization. Unique solutions of basis pursuit and analysis l1-minimization are considered and, on the basis of their characterizations, methods are designed which verify whether a given vector can be reconstructed exactly by basis pursuit or analysis l1-minimization. Further, a method is developed which guarantees that a given vector is the unique solution of isotropic total variation minimization. In addition, results on experiments for all three methods are presented where the linear restrictions are given as a random matrix and as a matrix which models the measurement process in computed tomography. Furthermore, in the present thesis geometrical interpretations are presented. By considering the theory of convex polytopes, three geometrical objects are examined and placed within the context of compressed sensing. The result is a comprehensive study of the geometry of basis pursuit which contains many new insights to necessary geometrical conditions for unique solutions and an explicit number of equivalence classes of unique solutions. The number of these equivalence classes itself is strongly related to the number of unique solutions of basis pursuit for an arbitrary matrix. In addition, the question is addressed for which linear restrictions do exist the most unique solutions of basis pursuit. For this purpose, upper bounds are developed and explicit restrictions are given under which the most vectors can be reconstructed via basis pursuit.In Zeiten von Big Data, in denen man nahezu täglich mit Überwachungsskandalen, personalisierter Werbung und Vorratsdatenspeicherung konfrontiert wird, ist es kein Wunder dass ein Forschungsgebiet wie Compressed Sensing von so grossem Interesse ist. Es stellt sich die Frage, wie viele Messungen tatsächlich nötig sind, um ein Signal oder ein Objekt hochaufgelöst darstellen zu können. In der vorliegenden Arbeit wird die Anwendungsmöglichkeit von drei in Compressed Sensing verwendeten Optimierungsprobleme mit linearen Nebenbedingungen untersucht. Hierbei handelt es sich namentlich um Basis Pursuit, Analysis l1-Minimierung und Isotropic Total Variation. Es werden eindeutige Lösungen von Basis Pursuit und der Analysis l1-Minimierung betrachtet, um auf der Grundlage ihrer Charakterisierungen Methoden vorzustellen, die Verifizieren ob ein gegebener Vektor exakt durch Basis Pursuit oder der Analysis l1-Minimierung rekonstruiert werden kann. Für Isotropic Total Variation werden hinreichende Bedingungen aufgestellt, die garantieren, dass ein gegebener Vektor die eindeutige Lösung von Isotropic Total Variation ist. Darüber hinaus werden Ergebnisse zu Experimenten mit Zufallsmatrizen als linearen Nebenbedingungen sowie Ergebnisse zu Experimenten mit Matrizen vorgestellt, die den Aufnahmeprozess bei Computertomographie simulieren. Weiterhin werden in der vorliegenden Arbeit verschiedene geometrische Interpretationen von Basis Pursuit vorgestellt. Unter Verwendung der konvexen Polytop-Theorie werden drei unterschiedliche geometrische Objekte untersucht und in den Zusammenhang mit Compressed Sensing gestellt. Das Ergebnis ist eine umfangreiche Studie der Geometrie von Basis Pursuit mit vielen neuen Einblicken in notwendige geometrische Bedingungen für eindeutige Lösungen und in die explizite Anzahl von Äquivalenzklassen eindeutiger Lösungen. Darüber hinaus wird der Frage nachgegangen, unter welchen linearen Nebenbedingungen die meisten eindeutigen Lösungen existieren. Zu diesem Zweck werden obere Schranken entwickelt, sowie explizite Nebenbedingungen genannt unter denen die meisten Vektoren exakt rekonstruiert werden können