A process for the use of multihypothesis prediction in the reconstruction of images is proposed for use in both compressed-sensing reconstruction as well as single-image super-resolution. Specifically, for compressed-sensing reconstruction of a single still image, multiple predictions for an image block are drawn from spatially surrounding blocks within an initial non-predicted reconstruction. The predictions are used to generate a residual in the domain of the compressed-sensing random projections. This residual being typically more compressible than the original signal leads to improved compressed-sensing reconstruction quality. To appropriately weight the hypothesis predictions, a Tikhonov regularization to an ill-posed least-squares optimization is proposed. An extension of this framework is applied to the compressed-sensing reconstruction of hyperspectral imagery is also studied. Finally, the multihypothesis paradigm is employed for single-image superresolution wherein each patch of a low-resolution image is represented as a linear combination of spatially surrounding hypothesis patches.

In this dissertation, the general problem of the dimensionality reduction of hyperspectral imagery is considered. Data dimension can be reduced through compression, in which an original image is encoded into bitstream of greatly reduced size; through application of a transformation, in which a high-dimensional space is mapped into a low-dimensional space; and through a simple process of subsampling, wherein the number of pixels is reduced spatially during image acquisition. All three techniques are investigated in the course of the dissertation. For data compression, an approach to calculate an operational bitrate for JPEG2000 in conjunction with principal component analysis is proposed. It is shown that an optimal bitrate for such a lossy compression method can be estimated while maintaining both class separability as well as anomalous pixels in the original data. On the other hand, the transformation paradigm is studied for spectral dimensionality reduction; specifically, dataindependent random spectral projections are considered, while the compressive projection principal component analysis algorithm is adopted for data reconstruction. It is shown that, by incorporating both spectral and spatial partitioning of the original data, reconstruction accuracy can be improved. Additionally, a new supervised spectral dimensionality reduction approach using a sparsity-preserving graph is developed. The resulting sparse graph-based discriminant analysis is seen to yield superior classification performance at low dimensionality. Finally, for spatial dimensionality reduction, a simple spatial subsampling scheme is considered for a multitemporal hyperspectral image sequence, such that the original image is reconstructed using a sparse dictionary learned from a prior image in the sequence.

Previous approach for narrowband interference cancellation based on compressed sensing (CS) in wideband receivers uses orthogonal projections to project away from the interference. This is not effective in the presence of nonlinear LNA (low noise amplifier) and finite bit ADCs (analog-to-digital converters) due to the fact that the nonidealities present will result in irresolvable intermodulation components and corrupt the signal reconstruction. Cancelling out the interferer before reaching the LNA thus becomes very important. A CS measurement matrix with randomly placed zeros in the frequency domain helps in this regard by removing the effect of interference when the signal measurements are performed before the LNA. Using this idea, under much idealized hardware assumptions impressive performance is obtained. The use of binary sequences which makes the hardware implementation simplistic is investigated in this thesis. Searching sequences with many spectral nulls turns out to be nontrivial. A theoretical approach for estimating probability of nulls is provided to reduce significant computational effort in the search and is shown to be close to actual search iterations. The use of real binary sequences (generated using ideal switches) obtained through the search does not do better compared to the orthogonal projection method in the presence of nonlinear LNA.

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<p>Localized Magnetic Resonance Spectroscopy is a non-invasive tool that offers insight into physiological status via signals arising from biological compounds. Unambiguous evaluation of said signals, however; is intrinsically limited by self interference through signal overlap. J Resolved Spectroscopy introduces an additional dimension to the measured signal which reduces overlap at the cost of increasing the scan duration. Compressed Sensing is a growing mathematical framework that asserts that under certain conditions, if a signal admits a sparse representation then it can be recovered from fewer measurements than required by classical signal theory. This framework has been successfully applied in high resolution Nuclear Magnetic Resonance experiments, justifying the investigation into its applicability in the realm of localized Magnetic Resonance Spectroscopy. The problem is addressed by optimizing the Compressed Sensing recovery on model spectra and evaluated in vitro through a parametric approach.</p> / <p>Localized Magnetic Resonance Spectroscopy is a non-invasive tool that o↵ers insight into physiological status via signals arising from biological compounds. Unambiguous evaluation of said signals, however; is intrinsically limited by self interference through signal overlap. J Resolved Spectroscopy introduces an additional dimension to the measured signal which reduces overlap at the cost of increasing the scan duration. Compressed Sensing is a growing mathematical framework that asserts that under certain conditions, if a signal admits a sparse representation then it can be recovered from fewer measurements than required by classical signal theory. This framework has been successfully applied in high resolution Nuclear Magnetic Resonance experiments, justifying the investigation into its applicability in the realm of localized Magnetic Resonance Spectroscopy. The problem is addressed by optimizing the Compressed Sensing recovery on model spectra and evaluated in vitro through a parametric approach.</p> / Master of Applied Science (MASc)

This dissertation focuses on problems in design, optimization and control of complex, large-scale dynamical systems from different viewpoints. The goal is to develop new algorithms and methods, that solve real problems more efficiently, together with providing mathematical insight into the success of those methods. There are three main contributions in this dissertation. In Chapter 3, we provide a new method to solve large-scale algebraic Riccati equations, which arise in optimal control, filtering and model reduction. We present a projection based algorithm utilizing proper orthogonal decomposition, which is demonstrated to produce highly accurate solutions at low rank. The method is parallelizable, easy to implement for practitioners, and is a first step towards a matrix free approach to solve AREs. Numerical examples for n >= 100,000 unknowns are presented. In Chapter 4, we develop a system identification method which is motivated by tangential interpolation. This addresses the challenge of fitting linear time invariant systems to input-output responses of complex dynamics, where the number of inputs and outputs is relatively large. The method reduces the computational burden imposed by a full singular value decomposition, by carefully choosing directions on which to project the impulse response prior to assembly of the Hankel matrix. The identification and model reduction step follows from the eigensystem realization algorithm. We present three numerical examples, a mass spring damper system, a heat transfer problem, and a fluid dynamics system. We obtain error bounds and stability results for this method. Chapter 5 deals with control and observation design for parameter dependent dynamical systems. We address this by using local parametric reduced order models, which can be used online. Data available from simulations of the system at various configurations (parameters, boundary conditions) is used to extract a sparse basis to represent the dynamics (via dynamic mode decomposition). Subsequently, a new compressed sensing based classification algorithm is developed which incorporates the extracted dynamic information into the sensing basis. We show that this augmented classification basis makes the method more robust to noise, and results in superior identification of the correct parameter. Numerical examples consist of a Navier-Stokes, as well as a Boussinesq flow application. / Ph. D.

The research presented in this dissertation pertains to the development of a real time SLAM solution that can be performed by a low cost autonomous underwater vehicle equipped with low cost and memory constrained computing resources. The design of a custom rangefinder for underwater applications is presented. The rangefinder makes use of two laser line generators and a camera to measure the unknown distance to objects in an underwater environment. A visual odometry algorithm is introduced that makes use of a downward facing camera to provide our underwater vehicle with localization information. The sensor suite composed of the laser rangefinder, downward facing camera, and a digital compass are verified, using the Extended Kalman Filter based solution to the SLAM problem along with the particle filter based solution known as FastSLAM, to ensure that they provide in- formation that is accurate enough to solve the SLAM problem for out low cost underwater vehicle. Next, an extension of the FastSLAM algorithm is presented that stores the map of the environment using an occupancy grid is introduced. The use of occupancy grids greatly increases the amount of memory required to perform the algorithm so a version of the Fast- SLAM algorithm that stores the occupancy grids using the Haar wavelet representation is presented. Finally, a form of the FastSLAM algorithm is presented that stores the occupancy grid in compressed form to reduce the amount memory required to perform the algorithm. It is shown in experimental results that the same result can be achieved, as that produced by the algorithm that stores the complete occupancy grid, using only 40% of the memory required to store the complete occupancy grid. / Ph. D.