The ability to efficiently and sparsely represent seismic data is becoming an increasingly important problem in geophysics. Over the last thirty years many transforms such as wavelets, curvelets, contourlets, surfacelets, shearlets, and many other types of ‘x-lets’ have been developed. Such transform were leveraged to resolve this issue of sparse representations. In this work we compare the properties of four of these commonly used transforms, namely the shift-invariant wavelets, complex wavelets, curvelets and surfacelets. We also explore the performance of these transforms for the problem of recovering seismic wavefields from incomplete measurements.

In this talk, directional frames, known as curvelets, are used to recover seismic data and images from noisy and incomplete data. Sparsity and invariance properties of curvelets are exploited to formulate the recovery by a `1-norm promoting program. It is shown that our data recovery approach is closely linked to the recent theory of “compressive sensing” and can be seen as a first step towards a nonlinear sampling theory for wavefields. The second problem that will be discussed concerns the recovery of the amplitudes of seismic images in clutter. There, the invariance of curvelets is used to approximately invert the Gramm operator of seismic imaging. In the high-frequency limit, this Gramm matrix corresponds to a pseudo-differential operator, which is near diagonal in the curvelet domain.

The ability to efficiently and sparsely represent seismic data is becoming an increasingly important problem in geophysics. Over the last thirty years many transforms such as wavelets, curvelets, contourlets, surfacelets, shearlets, and many other types of ‘x-lets’ have been developed. Such transform were leveraged to resolve this issue of sparse representations. In this work we compare the properties of four of these commonly used transforms, namely the shift-invariant wavelets, complex wavelets, curvelets and surfacelets. We also explore the performance of these transforms for the problem of recovering seismic wavefields from incomplete measurements.

The ability to efficiently and sparsely represent seismic data is becoming an increasingly important problem in geophysics. Over the last thirty years many transforms such as wavelets, curvelets, contourlets, surfacelets, shearlets, and many other types of ‘x-lets’ have been developed. Such transform were leveraged to resolve this issue of sparse representations. In this work we compare the properties of four of these commonly used transforms, namely the shift-invariant wavelets, complex wavelets, curvelets and surfacelets. We also explore the performance of these transforms for the problem of recovering seismic wavefields from incomplete measurements. / Science, Faculty of / Mathematics, Department of / Graduate

Humans perceive the world in three dimensions, but many sensing capabilities only display two-dimensional information to users by way of images. In this work we develop two novel reconstruction techniques utilizing synthetic aperture radar (SAR) data in three dimensions given sparse amounts of available data. We additionally leverage a hybrid joint-sparsity and sparsity approach to remove a-priori influences on the environment and instead explore general imaging properties in our reconstructions. We evaluate the required sampling rates for our techniques and a thorough analysis of the accuracy of our methods. The results presented in this thesis suggest a solution to sparse three-dimensional object reconstruction that effectively uses a substantially less amount of phase history data (PHD) while still extracting critical features off an object of interest.

One of the fundamental tasks in science is to find explainable relationships between observed phenomena. Recent work has addressed this problem by attempting to learn the structure of graphical models - especially Gaussian models - by the imposition of sparsity constraints. The graphical lasso is a popular method for learning the structure of a Gaussian model. It uses regularisation to impose sparsity. In real-world problems, there may be latent variables that confound the relationships between the observed variables. Ignoring these latents, and imposing sparsity in the space of the visibles, may lead to the pruning of important structural relationships. We address this problem by introducing an expectation maximisation (EM) method for learning a Gaussian model that is sparse in the joint space of visible and latent variables. By extending this to a conditional mixture, we introduce multiple structures, and allow side information to be used to predict which structure is most appropriate for each data point. Finally, we handle non-Gaussian data by extending each sparse latent Gaussian to a Gaussian copula. We train these models on a financial data set; we find the structures to be interpretable, and the new models to perform better than their existing competitors. A potential problem with the mixture model is that it does not require the structure to persist in time, whereas this may be expected in practice. So we construct an input-output HMM with sparse Gaussian emissions. But the main result is that, provided the side information is rich enough, the temporal component of the model provides little benefit, and reduces efficiency considerably. The GWishart distribution may be used as the basis for a Bayesian approach to learning a sparse Gaussian. However, sampling from this distribution often limits the efficiency of inference in these models. We make a small change to the state-of-the-art block Gibbs sampler to improve its efficiency. We then introduce a Hamiltonian Monte Carlo sampler that is much more efficient than block Gibbs, especially in high dimensions. We use these samplers to compare a Bayesian approach to learning a sparse Gaussian with the (non-Bayesian) graphical lasso. We find that, even when limited to the same time budget, the Bayesian method can perform better. In summary, this thesis introduces practically useful advances in structure learning for Gaussian graphical models and their extensions. The contributions include the addition of latent variables, a non-Gaussian extension, (temporal) conditional mixtures, and methods for efficient inference in a Bayesian formulation.

Magnetic resonance imaging (MRI) is a widely employed imaging modality that allows observation of the interior of human body. Compared to other imaging modalities such as the computed tomography (CT), MRI features a relatively long scan time that gives rise to many potential issues. The advent of parallel MRI, which employs multiple receiver coils, has started a new era in speeding up the scan of MRI by reducing the number of data acquisitions. However, the finally recovered images from under-sampled data sets often suffer degraded image quality. This thesis explores methods that incorporate prior knowledge of the image to be reconstructed to achieve improved image recovery in parallel MRI, following the philosophy that ‘if some prior knowledge of the image to be recovered is known, the image could be recovered better than without’. Specifically, the prior knowledge of image sparsity is utilized. Image sparsity exists in different domains. Image sparsity in the image domain refers to the fact that the imaged object only occupies a portion of the imaging field of view; image sparsity may also exist in a transform domain for which there is a high level of energy concentration in the image transform. The use of both types of sparsity is considered in this thesis. There are three major contributions in this thesis. The first contribution is the development of ‘GUISE’. GUISE employs an adaptive sampling design method that achieves better exploitation of image domain sparsity in parallel MRI. Secondly, the development of ‘PBCS’ and ‘SENSECS’. PBCS achieves better exploitation of transform domain sparsity by incorporating a prior estimate of the image to be recovered. SENSECS is an application of PBCS that achieves better exploitation of transform domain sparsity in parallel MRI. The third contribution is the implementation of GUISE and PBCS in contrast enhanced MR angiography (CE MRA). In their applications in CE MRA, GUISE and PBCS share the common ground of exploiting the high sparsity of the contrast enhanced angiogram. The above developments are assessed in various ways using both simulated and experimental data. The potential extensions of these methods are also suggested.

Image super-resolution technique mainly aims at restoring high-resolution image with satisfactory novel details. In recent years, leaning-based single-image super-resolution has been developed and proved to produce satisfactory results. With one or some dictionaries trained from a training set, learning-based super-resolution is able to establish a mapping relationship between low-resolution images and their corresponding high-resolution ones. Among all these algorithms, sparsity-based super-resolution has been proved with outstanding performance from extensive experiments. By utilizing compact dictionaries, this class of super-resolution algorithms can be efficient with lower computation complexity and has shown great potential for the practical applications. Our proposed model, which is known as Joint Dictionary-based Super-Resolution (JDSR) algorithm, is a new sparsity-based super-resolution approach. Based on the observation that the initial values of Non-locally Centralized Sparse Representation (NCSR) model will affect the final reconstruction, we change its initial values by using results of Zeyde's model. Besides, with the purpose of further improvement, we also add a gradient histogram preservation term in the sparse model of NCSR, and modify the reference histogram estimation by a simple edge detection based enhancement so that the estimated histogram will be closer to the ground truth. The experimental results illustrate that our method outperforms the state-of-the-art methods in terms of sharper edges, clearer textures and better novel details.

Non-line-of-sight (NLOS) imaging is the inference of the properties of objects or scenes outside of the direct line-of-sight of the observer. Such inferences can range from a 2D photograph-like image of a hidden area, to determining the position, motion or number of hidden objects, to 3D reconstructions of a hidden volume. NLOS imaging has many enticing potential applications, such as leveraging the existing hardware in many automobiles to identify hidden pedestrians, vehicles or other hazards and hence plan safer trajectories. Other potential application areas include improving navigation for robots or drones by anticipating occluded hazards, peering past obstructions in medical settings, or in surveying unreachable areas in search-and-rescue operations. Most modern NLOS imaging methods fall into one of two categories: active imaging methods that have some control of the illumination of the hidden area, and passive methods that simply measure light that already exists. This thesis introduces two NLOS imaging methods, one of each category, along with modeling and data processing techniques that are more broadly applicable. The methods are linked by their use of objects (‘occluders’) that reside somewhere between the observer and the hidden scene and block some possible light paths. Computational periscopy, a passive method, can recover the unknown position of an occluding object in the hidden area and then recover an image of the hidden scene behind it. It does so using only a single photograph of a blank relay wall taken by an ordinary digital camera. We develop also a framework using an optimized preconditioning matrix to improve the speed at which these reconstructions can be made and greatly improve the robustness to ambient light. Lastly, we develop tools necessary to demonstrate recovery of scenes at multiple unknown depths – paving the way towards three-dimensional reconstructions. Edge-resolved transient imaging, an active method, enables the formation of 2.5D representations – a plan view plus heights – of large-scale scenes. A pulsed laser illuminates spots along a small semi-circle on the floor, centered on the edge of a vertical wall such as in a doorway. The wall edge occludes some light paths, only allowing the laser light reflecting off of the floor to illuminate certain portions of the hidden area beyond the wall, depending on where along the semi-circle it is illuminating. The time at which photons return following a laser pulse is recorded. The occluding wall edge provides angular resolution, and time-resolved sensing provides radial resolution. This novel acquisition strategy, along with a scene response model and reconstruction algorithm, allow for 180° field of view reconstructions of large-scale scenes unlike other active imaging methods. Lastly, we introduce a sparsity penalty named mutually exclusive group sparsity (MEGS), that can be used as a constraint or regularization in optimization problems to promote solutions in which certain components are mutually exclusive. We explore how this penalty relates to other similar penalties, develop fast algorithms to solve MEGS-regularized problems, and demonstrate how enforcing mutual exclusivity structure can provide great utility in NLOS imaging problems.

<p> The focus of this thesis is on the utilization of the sparsity concept in solving some challenging problems, e.g., finding a unique solution to the under-determined linear system of equations in which the number of equations is less than the number of unknowns. This concept is extended to the problem of solving the under-determined blind source separation (BSS) problem in which the number of source signals is greater than the number of sensors and both the mixing matrix and the source signals are unknowns. In this respect we study three problems: </p> <p> 1. Developing some algorithms for solving the under-determined linear system of equations for the case of a sparse solution vector. In this thesis we develop a new methodology for minimizing a class of non-convex (concave on the non-negative orthant) functions for solving the aforementioned problem. The proposed technique is based on locally replacing the original objective function by a quadratic convex function which is easily minimized. For a certain selection of the convex objective function, the existing class of algorithms called Iterative Re-weighted Least Squares (IRLS) can be derived from the proposed methodology. Thus the proposed algorithms are a generalization and unification of the previous methods. In this thesis we also propose a convex objective function that produces an algorithm that can converge to a sparse solution vector in significantly fewer iterations than the IRLS algorithms.</p> <p> 2. Solving the under-determined BSS problem by developing new clustering algorithms for estimating the mixing matrix. The under-determined BSS problem is usually solved by following a two-step approach, in which the mixing matrix is estimated in the first step, then the sources are estimated in the second step. For the case of sparse sources, the mixing matrix is usually estimated by clustering the columns of the observation matrix. In this thesis we develop three novel clustering algorithms that can efficiently estimate the mixing matrix, as well as the number of sources, which is usually unknown. Numerical simulations verify the efficiency of the proposed algorithms compared to some well known algorithms that are usually used for solving the same problem.</p> <p> 3. Extraction of a desired source signal from a linear mixture of hidden sources when prior information is available about the desired source signal. There are many situations in which one is interested in extracting a specific source signal. The a priori available information about the desired source signal could be temporal, spatial, or both. In this thesis we develop new algorithms for extracting a desired sparse source signal from a linear mixture of hidden sources. The information available about the desired source signal, as well as its sparsity, are incorporated in an optimization problem for extracting this source signal. Four different algorithms have been developed for solving this problem. Numerical simulations show that the proposed algorithms can be used successfully for removing different kind of artifacts from real electroencephalographic (EEG) data and for estimating the event related potential (ERP) signal from synthesized EEG data.</p> / Thesis / Doctor of Philosophy (PhD)