Miniature cameras play a key role in numerous imaging applications ranging from endoscopy and metrology inspection devices to smartphones and head-mount acquisition systems. However, due to the physical constraints, the imaging conditions, and the low quality of small optics, their imaging capabilities are limited in terms of the delivered resolution, the acquired depth of field, and the captured dynamic range. Computational imaging jointly addresses the imaging system and the reconstructing algorithms to bypass the traditional limits of optical systems and deliver better restorations for various applications. The scene is encoded into a set of efficient measurements which could then be computationally decoded to output a richer estimate of the scene as compared with the raw images captured by conventional imagers. In this dissertation, three task-based computational imaging techniques are developed to make low-quality miniature cameras capable of delivering realistic high-resolution reconstructions, providing full-focus imaging, and acquiring depth information for high dynamic range objects. For the superresolution task, a non-regularized direct superresolution algorithm is developed to achieve realistic restorations without being penalized by improper assumptions (e.g., optimizers, priors, and regularizers) made in the inverse problem. An adaptive frequency-based filtering scheme is introduced to upper bound the reconstruction errors while still producing more fine details as compared with previous methods under realistic imaging conditions. For the full-focus imaging task, a computational depth-based deconvolution technique is proposed to bring a scene captured by an ordinary fixed-focus camera to a full-focus based on a depth-variant point spread function prior. The ringing artifacts are suppressed on three levels: block tiling to eliminate boundary artifacts, adaptive reference maps to reduce ringing initiated by sharp edges, and block-wise deconvolution or depth-based masking to suppress artifacts initiated by neighboring depth-transition surfaces. Finally for the depth acquisition task, a multi-polarization fringe projection imaging technique is introduced to eliminate saturated points and enhance the fringe contrast by selecting the proper polarized channel measurements. The developed technique can be easily extended to include measurements captured under different exposure times to obtain more accurate shape rendering for very high dynamic range objects.

Two complementary solution strategies to the least-squares migration problem with sparseness- & continuity constraints are proposed. The applied formalism explores the sparseness of curvelets on the reflectivity and their invariance under the demigration migration operator. Sparseness is enhanced by (approximately) minimizing a (weighted) l1-norm on the curvelet coefficients. Continuity along imaged reflectors is brought out by minimizing the anisotropic difussion or total variation norm which penalizes variations along and in between reflectors. A brief sketch of the theory is provided as well as a number of synthetic examples. Technical details on the implementation of the optimization strategies are deferred to an accompanying paper: implementation.

Commonly the thermal maturity of the Late Devonian to Early Mississippian Woodford shale found on the flanks of the Arbuckle Mountains of Oklahoma is determined by vitrinite reflectance, values ranging from 0.3-1.5%. Using phyllosilicate minerals, specifically diagenetic mixed layer illite/smectite and diagenetic illite, an understanding of the extent and processes leading to the thermal maturation may be developed. Analysis by XRD of the clay mineralogy of the Woodford shale found kaolinite and mixed layer illite/smectite with <5% smectite and R≥3 stacking order. Modeling of the Woodford shale also suggests the percentage of smectite present in mixed layer illite/smectite to be <5% and commonly <2.5%. Deconvolution of the illite (001) peak supports the low smectite content and high illite crystallinity. The long range ordered illite, R≥3, and high illite crystallinity values are indicative of diagenesis to anchizone conditions suggesting a higher thermal maturity relative to previously measured values of vitrinite reflectance.

Axicons have been proposed for imaging applications since they offer the advantage of extended depth of field (DOF). This enhanced DOF comes at the cost of degraded image quality. Image processing has been proposed to improve the image quality. Initial efforts were focused on the use of an axicon in a borescope thereby extending depth of focus and eliminating the need for a focusing mechanism. Though promising, it is clear that image processing would lead to improved image quality. This would also eliminate the need, in certain applications, for a fiber optic imaging bundle as many modern day video borescopes use an imaging sensor coupled directly to the front end optics. In the present work, three types of refractive axicons are examined: a linear axicon, a logarithmic axicon and a Fresnel axicon. The linear axicon offers the advantage of simplicity and a significant amount of scientific literature including the application of image restoration techniques. The Fresnel axicon has the advantage of compactness and potential low cost of production. As no physical prior examples of the Fresnel axicons were available for experimentation until recently, very little literature exists. The logarithmic axicon has the advantage of nearly constant longitudinal intensity distribution and an aspheric design producing superior pre-processed images over the aforementioned elements. Point Spread Functions (PSFs) for each of these axicons have been measured. These PSFs form the basis for the design of digital image restoration filters. The performance of these three optical elements and a number of restoration techniques are demonstrated and compared.

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We present the modes identified for frequencies found in spectroscopic observations of the Doradus stars HD 189631, QW Puppis, and IR Draconis. A cross-correlation tech- nique was used to create mean line profiles for HD 189631. Four frequencies and modes were identified for this star: 1.6774±0.0002 d⁻¹, 1.4174±0.0002 d⁻¹, 0.0714±0.0002 d⁻¹, and 1.8228 ± 0.0002 d⁻¹ which were identified with the modes (l,m) = (1, +1), (1, +1), (2,−2), and (1, +1) respectively. A least-squares deconvolution method was implemented for line profile generation in the study of QW Puppis and IR Draconis. Three frequen- cies were identified for QW Puppis: 0.055972 ± 0.000004 d⁻¹, 0.064846 ± 0.000004, and 5.219398±0.000002 d⁻¹. These frequencies were identified with the modes (l,m) = (1,−1), (4,−1), (4, +1). Two frequencies were identified in spectra of the rapidly rotating star IR Draconis: 0.00515 ± 0.00003 d⁻¹ and 2.35538 ± 0.00004 d⁻¹; which were identified with (l,m) = (1,−1), and (1, +1) modes respectively. These mode identifications will assist in modelling the structure and interior conditions of these main sequence, non-radially pulsating stars.

The connectivity and structural integrity of the white matter of the brain is known to be implicated in a wide range of brain-related diseases and injuries. However, it is only since the advent of diffusion magnetic resonance imaging (dMRI) that researchers have been able to probe the miscrostructure of white matter in vivo. Presently, among a range of methods of dMRI, high angular resolution diffusion imaging (HARDI) is known to excel in its ability to provide reliable information about the local orientations of neural fasciculi (aka fibre tracts). It preserves the high angular resolution property of diffusion spectrum imaging (DSI) but requires less measurements. Meanwhile, as opposed to the more traditional diffusion tensor imaging (DTI), HARDI is capable of distinguishing the orientations of multiple fibres passing through a given spatial voxel. Unfortunately, the ability of HARDI to discriminate neural fibres that cross each other at acute angles is always limited. The limitation becomes the motivation to develop numerous post-processing tools, aiming at the improvement of the angular resolution of HARDI. Among such methods, spherical deconvolution (SD) is the one which attracts the most attentions. Due to its ill-posed nature, however, standard SD relies on a number of a priori assumptions needed to render its results unique and stable. In the present thesis, we introduce a novel approach to the problem of non-blind SD of HARDI signals, which does not only consider the existence of anisotropic diffusion component of HARDI signal but also explicitly take the isotropic diffusion component into account. As a result of that, in addition to reconstruction of fODFs, our algorithm can also yield a useful estimation of its related IDM, which quantifies a relative contribution of the isotropic diffusion component as well as its spatial pattern. Moreover, one of the principal contributions is to demonstrate the effectiveness of exploiting different prior models for regularization of the spatial-domain behaviours of the reconstructed fODFs and IDMs. Specifically, the fibre continuity model has been used to force the local maxima of the fODFs to vary consistently throughout the brain, whereas the bounded variation model has helped us to achieve piecewise smooth reconstruction of the IDMs. The proposed algorithm is formulated as a convex minimization problem, which admits a unique and stable minimizer. Moreover, using ADMM, we have been able to find the optimal solution via a sequence of simpler optimization problems, which are both computationally efficient and amenable to parallel computations. In a series of both in silico and in vivo experiments, we demonstrate how the proposed solution can be used to successfully overcome the effect of partial voluming, while preserving the spatial coherency of cerebral diffusion at moderate to severe noise levels. The performance of the proposed method is compared with that of several available alternatives, with the comparative results clearly supporting the viability and usefulness of our approach. Moreover, the results illustrate the power of applied spatial regularization terms.

Our modern society has transformed to an information-demanding system, seeking voice, video, and data in quantities that could not be imagined even a decade ago. The mobility of communicators has added more challenges. One of the new challenges is to conceive highly reliable and fast communication system unaffected by the problems caused in the multipath fading wireless channels. Our quest is to remove one of the obstacles in the way of achieving ultimately fast and reliable wireless digital communication, namely Inter-Symbol Interference (ISI), the intensity of which makes the channel noise inconsequential. The theoretical background for wireless channels modeling and adaptive signal processing are covered in first two chapters of dissertation. The approach of this thesis is not based on one methodology but several algorithms and configurations that are proposed and examined to fight the ISI problem. There are two main categories of channel equalization techniques, supervised (training) and blind unsupervised (blind) modes. We have studied the application of a new and specially modified neural network requiring very short training period for the proper channel equalization in supervised mode. The promising performance in the graphs for this network is presented in chapter 4. For blind modes two distinctive methodologies are presented and studied. Chapter 3 covers the concept of multiple "cooperative" algorithms for the cases of two and three cooperative algorithms. The "select absolutely larger equalized signal" and "majority vote" methods have been used in 2-and 3-algoirithm systems respectively. Many of the demonstrated results are encouraging for further research. Chapter 5 involves the application of general concept of simulated annealing in blind mode equalization. A limited strategy of constant annealing noise is experimented for testing the simple algorithms used in multiple systems. Convergence to local stationary points of the cost function in parameter space is clearly demonstrated and that justifies the use of additional noise. The capability of the adding the random noise to release the algorithm from the local traps is established in several cases.

Low-dimensional signal structures naturally arise in a large set of applications in various fields such as medical imaging, machine learning, signal, and array processing. A ubiquitous low-dimensional structure in signals and images is sparsity, and a new sampling theory; namely, compressive sensing, proves that the sparse signals and images can be reconstructed from incomplete measurements. The signal recovery is achieved using efficient algorithms such as \ell_1-minimization. Recently, the research focus has spun-off to encompass other interesting low-dimensional signal structures such as group-sparsity and low-rank structure. This thesis considers low-rank matrix recovery (LRMR) from various structured-random measurement ensembles. These results are then employed for the in depth investigation of the classical blind-deconvolution problem from a new perspective, and for the development of a framework for the efficient sampling of correlated signals (the signals lying in a subspace). In the first part, we study the blind deconvolution; separation of two unknown signals by observing their convolution. We recast the deconvolution of discrete signals w and x as a rank-1 matrix wx* recovery problem from a structured random measurement ensemble. The convex relaxation of the problem leads to a tractable semidefinite program. We show, using some of the mathematical tools developed recently for LRMR, that if we assume the signals convolved with one another live in known subspaces, then this semidefinite relaxation is provably effective. In the second part, we design various efficient sampling architectures for signals acquired using large arrays. The sampling architectures exploit the correlation in the signals to acquire them at a sub-Nyquist rate. The sampling devices are designed using analog components with clear implementation potential. For each of the sampling scheme, we show that the signal reconstruction can be framed as an LRMR problem from a structured-random measurement ensemble. The signals can be reconstructed using the familiar nuclear-norm minimization. The sampling theorems derived for each of the sampling architecture show that the LRMR framework produces the Shannon-Nyquist performance for the sub-Nyquist acquisition of correlated signals. In the final part, we study low-rank matrix factorizations using randomized linear algebra. This specific method allows us to use a least-squares program for the reconstruction of the unknown low-rank matrix from the samples of its row and column space. Based on the principles of this method, we then design sampling architectures that not only acquire correlated signals efficiently but also require a simple least-squares program for the signal reconstruction. A theoretical analysis of all of the LRMR problems above is presented in this thesis, which provides the sufficient measurements required for the successful reconstruction of the unknown low-rank matrix, and the upper bound on the recovery error in both noiseless and noisy cases. For each of the LRMR problem, we also provide a discussion of a computationally feasible algorithm, which includes a least-squares-based algorithm, and some of the fastest algorithms for solving nuclear-norm minimization.