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

Method and apparatus to reduce artifacts in a computed-tomography (CT) image by iterative reconstruction (IR) using a cost function with a de-emphasis operator

An apparatus and method are provided for computed tomography (CT) imaging to reduce artifacts due to objects outside the field of view (FOV) of a reconstructed image. The artifacts are suppressed by using an iterative reconstruction method to minimize a cost function that includes a de-emphasis operator. The de-emphasis operator operates in the data domain, and minimizes the contributions of data inconsistencies arising from attenuation due to objects outside the FOV. This can be achieved by penalizing images that manifest indicia of artifacts due to outside objects especially those outside objects have high-attenuation densities and minimizing components of the data inconsistency likely attributable to the outside object

Low-rank and sparse reconstruction in dynamic magnetic resonance imaging via proximal splitting methods

Dynamic magnetic resonance imaging (MRI) consists of collecting multiple MR images in time, resulting in a spatio-temporal signal. However, MRI intrinsically suffers from long acquisition times due to various constraints. This limits the full potential of dynamic MR imaging, such as obtaining high spatial and temporal resolutions which are crucial to observe dynamic phenomena. This dissertation addresses the problem of the reconstruction of dynamic MR images from a limited amount of samples arising from a nuclear magnetic resonance experiment. The term limited can be explained by the approach taken in this thesis to speed up scan time, which is based on violating the Nyquist criterion by skipping measurements that would be normally acquired in a standard MRI procedure. The resulting problem can be classified in the general framework of linear ill-posed inverse problems. This thesis shows how low-dimensional signal models, specifically lowrank and sparsity, can help in the reconstruction of dynamic images from partial measurements. The use of these models are justified by significant developments in signal recovery techniques from partial data that have emerged in recent years in signal processing. The major contributions of this thesis are the development and characterisation of fast and efficient computational tools using convex low-rank and sparse constraints via proximal gradient methods, the development and characterisation of a novel joint reconstruction–separation method via the low-rank plus sparse matrix decomposition technique, and the development and characterisation of low-rank based recovery methods in the context of dynamic parallel MRI. Finally, an additional contribution of this thesis is to formulate the various MR image reconstruction problems in the context of convex optimisation to develop algorithms based on proximal splitting methods

Compressed sensing on terahertz imaging

Most terahertz (THz) time-domain (pulsed) imaging experiments that have been performed by raster scanning the object relative to a focused THz beam require minutes or even hours to acquire a complete image. This slow image acquisition is a major limiting factor for real-time applications. Other systems using focal plane detector arrays can acquire images in real-time, but they are too expensive or are limited by low sensitivity in the THz range. More importantly, such systems cannot provide spectroscopic information of the sample. To develop faster and more efficient THz time-domain (pulsed) imaging systems, this research used random projection approach to reconstruct THz images from the synthetic and real-world THz data based on the concept of compressed/compressive sensing/sampling (CS). Compared with conventional THz time-domain (pulsed) imaging, no raster scanning of the object is required. The simulation results demonstrated that CS has great potential for real-time THz imaging systems because its use can dramatically reduce the number of measurements in such systems. We then implemented two different CS-THz systems based on the random projection method. One is a compressive THz time-domain (pulsed) spectroscopic imaging system using a set of independent optimized masks. A single-point THz detector, together with a set of 40 optimized two-dimensional binary masks, was used to measure the THz waveforms transmitted through a sample. THz time- and frequency-domain images of the sample comprising 20×20 pixels were subsequently reconstructed. This demonstrated that both the spatial distribution and the spectral characteristics of a sample can be obtained by this means. Compared with conventional THz time-domain (pulsed) imaging, ten times fewer THz spectra need to be taken. In order to further speed up the image acquisition and reconstruction process, another hardware implementation - a single rotating mask (i.e., the spinning disk) with random binary patterns - was utilized to spatially modulate a collimated THz. After propagating through the sample, the THz beam was measured using a single detector, and a THz image was subsequently reconstructed using the CS approach. This demonstrated that a 32×32 pixel image could be obtained from 160 to 240 measurements. This spinning disk configuration allows the use of an electric motor to rotate the spinning disk, thus enabling the experiment to be performed automatically and continuously. To the best of our knowledge, this is the first experimental implementation of a spinning disk configuration for high speed compressive image acquisition. A three-dimensional (3D) joint reconstruction approach was developed to reconstruct THz images from random/incomplete subsets of THz data. Such a random sampling method provides a fast THz imaging acquisition and also simplifies the current THz imaging hardware implementation. The core idea is extended in image inpainting to the case of 3D data. Our main objective is to exploit both spatial and spectral/temporal information for recovering the missing samples. It has been shown that this approach has superiority over the case where the spectral/temporal images are treated independently. We first proposed to learn a spatio-spectral/temporal dictionary from a subset of available training data. Using this dictionary, the THz images can then be jointly recovered from an incomplete set of observations. The simulation results using the measured THz image data confirm that this 3D joint reconstruction approach also provides a significant improvement over the existing THz imaging methods

Large Covariance Matrix Estimation by Composite Minimization

The present thesis concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure. Existing methods like POET (Principal Orthogonal complEment Thresholding) perform estimation by extracting principal components and then applying a soft thresholding algorithm. In contrast, our method recovers the low rank plus sparse decomposition of the covariance matrix by least squares minimization under nuclear norm plus $l_1$ norm penalization. This non-smooth convex minimization procedure is based on semidefinite programming and subdifferential methods, resulting in two separable problems solved by a singular value thresholding plus soft thresholding algorithm. The most recent estimator in literature is called LOREC (Low Rank and sparsE Covariance estimator) and provides non-asymptotic error rates as well as identifiability conditions in the context of algebraic geometry. Our work shows that the unshrinkage of the estimated eigenvalues of the low rank component improves the performance of LOREC considerably. The same method also recovers covariance structures with very spiked latent eigenvalues like in the POET setting, thus overcoming the necessary condition $p\leq n$. In addition, it is proved that our method recovers structures with intermediate degrees of spikiness, obtaining a loss which is bounded accordingly. Then, an ad hoc model selection criterion which detects the optimal point in terms of composite penalty is proposed. Empirical results coming from a wide original simulation study where various low rank plus sparse settings are simulated according to different parameter values are described outlining in detail the improvements upon existing methods. Two real data-sets are finally explored highlighting the usefulness of our method in practical applications

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