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Time series datasets are useful in a wide range of diverse real world applications. Retrieving or querying from a collection of time series is a fundamental task, with a key example being the similarity query. A similarity query returns all time series from the collection that are similar to a given reference time series. This type of query is particularly useful in prediction and forecasting applications. / A key challenge for similarity queries is efficiency and for large datasets, it is important to develop efficient indexing techniques. Existing approaches in this area are mainly based on the Generic Multimedia Indexing Method (GEMINI), which is a framework that uses spatial indexes such as the R-tree to index reduced time series. For processing a similarity query, the index is first used to prune candidate time series using a lower bounding distance. Then, all remaining time series are compared using the original similarity measure, to derive the query result. Performance within this framework depends on the tightness of the lower bounding distance with respect to the similarity measure. Indeed much work has been focused on representation and dimensionality reduction, in order to provide a tighter lower bounding distance. / Existing work, however, has not used employed dimensionality reduction in a flexible way, requiring all time series to be reduced to have the same dimension. In contrast, in this thesis, we investigate the possibility of allowing a variable dimension reduction. To this end, we develop a new and more flexible tree based indexing structure called the Multi-Resolution Index (MR-Index), which allows dimensionality to vary across different levels of the tree. We provide efficient algorithms for querying, building and maintaining this structure. Through an experimental analysis, we show that the MR-Index can deliver improved query efficiency compared to the traditional R-tree index, using both the Euclidean and dynamic time warping similarity measures.

Time series datasets are useful in a wide range of diverse real world applications. Retrieving or querying from a collection of time series is a fundamental task, with a key example being the similarity query. A similarity query returns all time series from the collection that are similar to a given reference time series. This type of query is particularly useful in prediction and forecasting applications. / A key challenge for similarity queries is efficiency and for large datasets, it is important to develop efficient indexing techniques. Existing approaches in this area are mainly based on the Generic Multimedia Indexing Method (GEMINI), which is a framework that uses spatial indexes such as the R-tree to index reduced time series. For processing a similarity query, the index is first used to prune candidate time series using a lower bounding distance. Then, all remaining time series are compared using the original similarity measure, to derive the query result. Performance within this framework depends on the tightness of the lower bounding distance with respect to the similarity measure. Indeed much work has been focused on representation and dimensionality reduction, in order to provide a tighter lower bounding distance. / Existing work, however, has not used employed dimensionality reduction in a flexible way, requiring all time series to be reduced to have the same dimension. In contrast, in this thesis, we investigate the possibility of allowing a variable dimension reduction. To this end, we develop a new and more flexible tree based indexing structure called the Multi-Resolution Index (MR-Index), which allows dimensionality to vary across different levels of the tree. We provide efficient algorithms for querying, building and maintaining this structure. Through an experimental analysis, we show that the MR-Index can deliver improved query efficiency compared to the traditional R-tree index, using both the Euclidean and dynamic time warping similarity measures.

The main contribution of this thesis is a new method of image compression based on a recently developed adaptive transform called Mixtures of Principal Components (MPC). Our multi-resolution extension of MPC-called Multi-Resolution Mixtures of Principal Components (MR-MPC) compresses and decompresses images in stages. The first stage processes the original images at very low resolution and is followed by stages that process the encoding errors of the previous stages at incrementally higher resolutions. To evaluate our multi-resolution extension of MPC we compared it with MPC and with the excellent performing wavelet based scheme called SPIHT. Fifty chest radiographs were compressed and compared to originals in two ways. First, Peak Signal to Noise Ratio (PSNR) and five distortion factors from a perceptual distortion measure called PQS were used to demonstrate that our multi-resolution extension of MPC can achieve rate distortion performance that is 220% to 720% better than MPC and much closer to that of SPIHT. And second, in a study involving 724 radiologists' evaluations of compressed chest radiographs, we found that the impact of MR-MPC and SPIHT at 25:1, 50:1, 75:1 on subjective image quality scores was less than the difference of opinion between four radiologists. / Thesis / Master of Science (MS)

The ability to predict and control laminar-to-turbulent transition in high-speed flow has a substantial effect on heat transfer and skin friction, thus improving the design and operation of hypersonic vehicles. The control of transition on blunt bodies is essential to improve the performance of lifting and control surfaces. The objective of this Ph.D. research is to develop efficient and accurate algorithms for the detection of high-frequency heat flux fluctuations supported by hypersonic flow in transitional boundary layers. The focus of this research is on understanding the mathematical properties of the reconstruction such as regularity, sensitivity to noise, multi-resolution, and accuracy. This research is part of an effort to develop small-footprint heat flux sensors able to measure high-frequency fluctuations on test articles in a hypersonic wind tunnel with a small curvature radius. In the present theoretical/numerical study a multi-resolution formulation for the direct and inverse reconstruction of the heat flux from temperature sensors distributed over a multidimensional solid in a hypersonic flow was developed and validated. The solution method determines the thermal response by approximating the system Green's function with the Galerkin method and optimizes the heat flux distribution by fitting the distributed surface temperature data. Coating and glue layers are treated as separate domains for which the Green's function is obtained independently. Connection conditions for the system Green's function are derived by imposing continuity of heat flux and temperature concurrently at all interfaces. The solution heat flux is decomposed on a space-time basis with the temporal basis a multi-resolution wavelet with arbitrary scaling function. Quadrature formulas for the convolution of wavelets and the Green's function, a reconstruction approach based on isoparametric mapping of three-dimensional geometries, and a boundary wavelet approach for inverse problems were developed and verified. This approach was validated against turbulent conjugate heat transfer simulations at Mach 6 on a blunted wedge at 0 angle of attack and wind tunnel experiments of round impinging jet at Mach 0.7 It was found that multidimensional effects were important near the wedge shoulder in the short time scale, that the L-curve regularization needed to be locally corrected to analyze transitional flows and that proper regularization led to sub-cell resolution of the inverse problem. While the L2 regularization techniques are accurate they are also computationally inefficient and lack mathematical rigor. Optimal non-linear estimators were researched both as means to promote sparsity in the regularization and to pre-threshold the inverse heat conduction problem. A novel class of nonlinear estimators is presented and validated against wind tunnel experiments for a flat-faced cylinder also at Mach 6. The new approach to hypersonic heat flux reconstruction from discrete temperature data developed in this thesis is more efficient and accurate than existing techniques. / Doctor of Philosophy / The harsh environment supported by hypersonic flows is characterized by high-frequency turbulent bursts, acoustic noise, and vibrations that pollute the signals of the sensors that probe at high frequencies the state of the boundary layers developing on the walls. This research describes the search for optimal estimators of the noisy signal, i.e., those that lead to the maximum attenuation of the risk of error pollution by non-coherent scales. This research shows that linear estimators perform poorly at high-frequency and non-linear estimators can be optimized over a sparse projection of the signal in a discrete wavelet basis. Optimal non-linear estimators are developed and validated for wind tunnel experiments conducted at Mach 6 in the Advanced Propulsion and Power Laboratory at Virginia Tech.

Image descriptors play an important role in image representation and analysis. Multi-resolution image descriptors can effectively characterize complex images and extract their hidden information. Wavelets descriptors have been widely used in multi-resolution image analysis. However, making the wavelets transform shift and rotation invariant produces redundancy and requires complex matching processes. As to other multi-resolution descriptors, they usually depend on other theories or information, such as filtering function, prior-domain knowledge, etc.; that not only increases the computation complexity, but also generates errors. We propose a novel multi-resolution scheme that is capable of transforming any kind of image descriptor into its multi-resolution structure with high computation accuracy and efficiency. Our multi-resolution scheme is based on sub-sampling an image into an odd-even image tree. Through applying image descriptors to the odd-even image tree, we get the relative multi-resolution image descriptors. Multi-resolution analysis is based on downsampling expansion with maximum energy extraction followed by upsampling reconstruction. Since the maximum energy usually retained in the lowest frequency coefficients; we do maximum energy extraction through keeping the lowest coefficients from each resolution level. Our multi-resolution scheme can analyze images recursively and effectively without introducing artifacts or changes to the original images, produce multi-resolution representations, obtain higher resolution images only using information from lower resolutions, compress data, filter noise, extract effective image features and be implemented in parallel processing.

Image descriptors play an important role in image representation and analysis. Multi-resolution image descriptors can effectively characterize complex images and extract their hidden information. Wavelet descriptors have been widely used in multi-resolution image analysis. However, making the wavelet transform shift and rotation invariant produces redundancy and requires complex matching processes. As to other multi-resolution descriptors, they usually depend on other methods, such as filtering function, prior-domain knowledge, etc.; that not only increases the computation complexity, but also generates errors. We propose a novel multi-resolution scheme that is capable of transforming any kind of image descriptor into its multi-resolution structure with high computation accuracy and efficiency. Our multi-resolution scheme is based on sub-sampling each image into an odd-even image tree. Through applying image descriptors to the odd-even image tree, we get the relative multi-resolution image descriptors. Multi-resolution analysis is based on downsampling expansion with maximum energy extraction followed by upsampling reconstruction. Since the maximum energy usually retained in the lowest frequency coefficients; we do maximum energy extraction through keeping the lowest coefficients from each resolution level. Our multi-resolution scheme can analyze images recursively and effectively without introducing artifacts or changes to the original images, produce multi-resolution representations, obtain higher resolution images only using information from lower resolutions, compress data, filter noise, extract effective image features and be implemented in parallel processing.

Image segmentation is an important step in image processing, with many applications such as pattern recognition, object detection, and medical image analysis. It is a technique that separates objects of interests from the background in an image. Geometric active contour is a recent image segmentation method that overcomes previous problems with snakes. It is an attractive method for medical image segmentation as it is able to capture the object of interest in one continuous curve. The theory and implementation details of geometric active contours are discussed in this work. The robustness of the algorithm is tested through a series of tests, involving both synthetic images and medical images. Curve leaking past boundaries is a common problem in cases of non-ideal edges. Noise is also problematic for the advancement of the curve. Smoothing and parameters selection are discussed as ways to help solve these problems. This work also explores the incorporation of the multi-resolution method of Gaussian pyramids into the algorithm. Multi-resolution methods, used extensively in the areas of denoising and edge-selection, can help capture the spatial structure of an image. Results show that similar to the multi-resolution methods applied to parametric active contours, the multi-resolution can greatly increase the computation without sacrificing performance. In fact, results show that with successive smoothing and sub-sampling, performance often improves. Although smoothing and parameter adjustment help improve the performance of geometric active contours, the edge-based approach is still localized and the improvement is limited. Region-based approaches are recommended for further work on active contours.

Image segmentation is an important step in image processing, with many applications such as pattern recognition, object detection, and medical image analysis. It is a technique that separates objects of interests from the background in an image. Geometric active contour is a recent image segmentation method that overcomes previous problems with snakes. It is an attractive method for medical image segmentation as it is able to capture the object of interest in one continuous curve. The theory and implementation details of geometric active contours are discussed in this work. The robustness of the algorithm is tested through a series of tests, involving both synthetic images and medical images. Curve leaking past boundaries is a common problem in cases of non-ideal edges. Noise is also problematic for the advancement of the curve. Smoothing and parameters selection are discussed as ways to help solve these problems. This work also explores the incorporation of the multi-resolution method of Gaussian pyramids into the algorithm. Multi-resolution methods, used extensively in the areas of denoising and edge-selection, can help capture the spatial structure of an image. Results show that similar to the multi-resolution methods applied to parametric active contours, the multi-resolution can greatly increase the computation without sacrificing performance. In fact, results show that with successive smoothing and sub-sampling, performance often improves. Although smoothing and parameter adjustment help improve the performance of geometric active contours, the edge-based approach is still localized and the improvement is limited. Region-based approaches are recommended for further work on active contours.

There are a number of techniques that are used to solve reinforcement learning problems, but very few that have been developed for and tested on highly reconfigurable systems cast as reinforcement learning problems. Reconfigurable systems refers to a vehicle (air, ground, or water) or collection of vehicles that can change its geometrical features, i.e. shape or formation, to perform tasks that the vehicle could not otherwise accomplish. These systems tend to be optimized for several operating conditions, and then controllers are designed to reconfigure the system from one operating condition to another. Q-learning, an unsupervised episodic learning technique that solves the reinforcement learning problem, is an attractive control methodology for reconfigurable systems. It has been successfully applied to a myriad of control problems, and there are a number of variations that were developed to avoid or alleviate some limitations in earlier version of this approach. This dissertation describes the development of three modular enhancements to the Q-learning algorithm that solve some of the unique problems that arise when working with this class of systems, such as the complex interaction of reconfigurable parameters and computationally intensive models of the systems. A multi-resolution state-space discretization method is developed that adaptively rediscretizes the state-space by progressively finer grids around one or more distinct Regions Of Interest within the state or learning space. A genetic algorithm that autonomously selects the basis functions to be used in the approximation of the action-value function is applied periodically throughout the learning process. Policy comparison is added to monitor the state of the policy encoded in the action-value function to prevent unnecessary episodes at each level of discretization. This approach is validated on several problems including an inverted pendulum, reconfigurable airfoil, and reconfigurable wing. Results show that the multi-resolution state-space discretization method reduces the number of state-action pairs, often by an order of magnitude, required to achieve a specific goal and the policy comparison prevents unnecessary episodes once the policy has converged to a usable policy. Results also show that the genetic algorithm is a promising candidate for the selection of basis functions for function approximation of the action-value function.