Improving the Performance of the Prony Method Using a Wavelet Domain Filter for MRI Denoising

The Prony methods are used for exponential fitting. We use a variant of the Prony method for abnormal brain tissue detection in sequences of T2 weighted magnetic resonance images. Here, MR images are considered to be affected only by Rician noise, and a new wavelet domain bilateral filtering process is implemented to reduce the noise in the images. This filter is a modification of Kazubek’s algorithm and we use synthetic images to show the ability of the new procedure to suppress noise and compare its performance with respect to the original filter, using quantitative and qualitative criteria. The tissue classification process is illustrated using a real sequence of T2 MR images, and the filter is applied to each image before using the variant of the Prony method

Resolution Enhancement in Magnetic Resonance Imaging by Frequency Extrapolation

This thesis focuses on spatial resolution enhancement of magnetic resonance imaging (MRI). In particular, it addresses methods of performing such enhancement in the Fourier domain. After a brief review of Fourier theory, the thesis reviews the physics of the MRI acquisition process in order to introduce a mathematical model of the measured data. This model is later used to develop and analyze methods for resolution enhancement, or "super-resolution'', in MRI. We then examine strategies of performing super-resolution MRI (SRMRI). We begin by exploring strategies that use multiple data sets produced by spatial translations of the object being imaged, to add new information to the reconstruction process. This represents a more detailed mathematical examination of the author's Master's work at the University of Calgary. Using our model of the measured data developed earlier in the thesis, we describe how the acquisition strategy determines the efficacy of the SRMRI process that employs multiple data sets. The author then explores the self-similarity properties of MRI data in the Fourier domain as a means of performing spatial resolution enhancement. To this end, a fractal-based method over (complex-valued) Fourier Transforms of functions with compact spatial support, derived from a fractal transform in the spatial domain, is explored. It is shown that this method of "Iterated Fourier Transform Systems" (IFTS) can be tailored to perform frequency extrapolation, hence spatial resolution enhancement. The IFTS method, however, is limited in scope, as it assumes that a spatial function f(x) may be approximated by linear combinations of spatially-contracted and range-modified copies of the entire function. In order to improve the approximation, we borrow from traditional fractal image coding in the spatial domain, where subblocks of an image are approximated by other subblocks, and employ such a block-based strategy in the Fourier domain. An examination of the statistical properties of subblock approximation errors shows that, in general, Fourier data can be locally self-similar. Furthermore, we show that such a block-based self-similarity method is actually equivalent to a special case of the auto-regressive moving average (ARMA) modeling method. The thesis concludes with a chapter on possible future research directions in SRMRI

Variable Splitting as a Key to Efficient Image Reconstruction

The problem of reconstruction of digital images from their degraded measurements has always been a problem of central importance in numerous applications of imaging sciences. In real life, acquired imaging data is typically contaminated by various types of degradation phenomena which are usually related to the imperfections of image acquisition devices and/or environmental effects. Accordingly, given the degraded measurements of an image of interest, the fundamental goal of image reconstruction is to recover its close approximation, thereby "reversing" the effect of image degradation. Moreover, the massive production and proliferation of digital data across different fields of applied sciences creates the need for methods of image restoration which would be both accurate and computationally efficient. Developing such methods, however, has never been a trivial task, as improving the accuracy of image reconstruction is generally achieved at the expense of an elevated computational burden. Accordingly, the main goal of this thesis has been to develop an analytical framework which allows one to tackle a wide scope of image reconstruction problems in a computationally efficient manner. To this end, we generalize the concept of variable splitting, as a tool for simplifying complex reconstruction problems through their replacement by a sequence of simpler and therefore easily solvable ones. Moreover, we consider two different types of variable splitting and demonstrate their connection to a number of existing approaches which are currently used to solve various inverse problems. In particular, we refer to the first type of variable splitting as Bregman Type Splitting (BTS) and demonstrate its applicability to the solution of complex reconstruction problems with composite, cross-domain constraints. As specific applications of practical importance, we consider the problem of reconstruction of diffusion MRI signals from sub-critically sampled, incomplete data as well as the problem of blind deconvolution of medical ultrasound images. Further, we refer to the second type of variable splitting as Fuzzy Clustering Splitting (FCS) and show its application to the problem of image denoising. Specifically, we demonstrate how this splitting technique allows us to generalize the concept of neighbourhood operation as well as to derive a unifying approach to denoising of imaging data under a variety of different noise scenarios

Non-linear Recovery of Sparse Signal Representations with Applications to Temporal and Spatial Localization

Foundations of signal processing are heavily based on Shannon's sampling theorem for acquisition, representation and reconstruction. This theorem states that signals should not contain frequency components higher than the Nyquist rate, which is half of the sampling rate. Then, the signal can be perfectly reconstructed from its samples. Increasing evidence shows that the requirements imposed by Shannon's sampling theorem are too conservative for many naturally-occurring signals, which can be accurately characterized by sparse representations that require lower sampling rates closer to the signal's intrinsic information rates. Finite rate of innovation (FRI) is a new theory that allows to extract underlying sparse signal representations while operating at a reduced sampling rate. The goal of this PhD work is to advance reconstruction techniques for sparse signal representations from both theoretical and practical points of view. Specifically, the FRI framework is extended to deal with applications that involve temporal and spatial localization of events, including inverse source problems from radiating fields. We propose a novel reconstruction method using a model-fitting approach that is based on minimizing the fitting error subject to an underlying annihilation system given by the Prony's method. First, we showed that this is related to the problem known as structured low-rank matrix approximation as in structured total least squares problem. Then, we proposed to solve our problem under three different constraints using the iterative quadratic maximum likelihood algorithm. Our analysis and simulation results indicate that the proposed algorithms improve the robustness of the results with respect to common FRI reconstruction schemes. We have further developed the model-fitting approach to analyze spontaneous brain activity as measured by functional magnetic resonance imaging (fMRI). For this, we considered the noisy fMRI time course for every voxel as a convolution between an underlying activity inducing signal (i.e., a stream of Diracs) and the hemodynamic response function (HRF). We then validated this method using experimental fMRI data acquired during an event-related study. The results showed for the first time evidence for the practical usage of FRI for fMRI data analysis. We also addressed the problem of retrieving a sparse source distribution from the boundary measurements of a radiating field. First, based on Green's theorem, we proposed a sensing principle that allows to relate the boundary measurements to the source distribution. We focused on characterizing these sensing functions with particular attention for those that can be derived from holomorphic functions as they allow to control spatial decay of the sensing functions. With this selection, we developed an FRI-inspired non-iterative reconstruction algorithm. Finally, we developed an extension to the sensing principle (termed eigensensing) where we choose the spatial eigenfunctions of the Laplace operator as the sensing functions. With this extension, we showed that eigensensing principle allows to extract partial Fourier measurements of the source functions from boundary measurements. We considered photoacoustic tomography as a potential application of these theoretical developments

ИНТЕЛЛЕКТУАЛЬНЫЙ числовым программным ДЛЯ MIMD-компьютер

For most scientific and engineering problems simulated on computers the solving of problems of the computational mathematics with approximately given initial data constitutes an intermediate or a final stage. Basic problems of the computational mathematics include the investigating and solving of linear algebraic systems, evaluating of eigenvalues and eigenvectors of matrices, the solving of systems of non-linear equations, numerical integration of initial- value problems for systems of ordinary differential equations.Для більшості наукових та інженерних задач моделювання на ЕОМ рішення задач обчислювальної математики з наближено заданими вихідними даними складає проміжний або остаточний етап. Основні проблеми обчислювальної математики відносяться дослідження і рішення лінійних алгебраїчних систем оцінки власних значень і власних векторів матриць, рішення систем нелінійних рівнянь, чисельного інтегрування початково задач для систем звичайних диференціальних рівнянь.Для большинства научных и инженерных задач моделирования на ЭВМ решение задач вычислительной математики с приближенно заданным исходным данным составляет промежуточный или окончательный этап. Основные проблемы вычислительной математики относятся исследования и решения линейных алгебраических систем оценки собственных значений и собственных векторов матриц, решение систем нелинейных уравнений, численного интегрирования начально задач для систем обыкновенных дифференциальных уравнений

Random observations on random observations: Sparse signal acquisition and processing

In recent years, signal processing has come under mounting pressure to accommodate the increasingly high-dimensional raw data generated by modern sensing systems. Despite extraordinary advances in computational power, processing the signals produced in application areas such as imaging, video, remote surveillance, spectroscopy, and genomic data analysis continues to pose a tremendous challenge. Fortunately, in many cases these high-dimensional signals contain relatively little information compared to their ambient dimensionality. For example, signals can often be well-approximated as a sparse linear combination of elements from a known basis or dictionary. Traditionally, sparse models have been exploited only after acquisition, typically for tasks such as compression. Recently, however, the applications of sparsity have greatly expanded with the emergence of compressive sensing, a new approach to data acquisition that directly exploits sparsity in order to acquire analog signals more efficiently via a small set of more general, often randomized, linear measurements. If properly chosen, the number of measurements can be much smaller than the number of Nyquist-rate samples. A common theme in this research is the use of randomness in signal acquisition, inspiring the design of hardware systems that directly implement random measurement protocols. This thesis builds on the field of compressive sensing and illustrates how sparsity can be exploited to design efficient signal processing algorithms at all stages of the information processing pipeline, with a particular focus on the manner in which randomness can be exploited to design new kinds of acquisition systems for sparse signals. Our key contributions include: (i) exploration and analysis of the appropriate properties for a sparse signal acquisition system; (ii) insight into the useful properties of random measurement schemes; (iii) analysis of an important family of algorithms for recovering sparse signals from random measurements; (iv) exploration of the impact of noise, both structured and unstructured, in the context of random measurements; and (v) algorithms that process random measurements to directly extract higher-level information or solve inference problems without resorting to full-scale signal recovery, reducing both the cost of signal acquisition and the complexity of the post-acquisition processing

Adaptive techniques for the detection and localization of event related potentials from EEGs using reference signals

In this thesis we show the methods we developed for the detection and localisation of P300 signals from the electroencephalogram. We utilised signal processing theory in order to enhance the current methodology. The work done can be applied both to EEG averages and single trial EEG data. We developed a variety of methods dealing with the extraction of the P300 and its subcomponents using independent component analysis and least squares. Moreover, we developed novel localisation methods that localise the desired P300 subcomponent from EEG data. Throughout the thesis the main idea was the use of reference signals, which describe the prior information we have about the sources of interest. The main objective of this thesis is to utilize adaptive techniques, namely blind source separation (BSS), least squares (LS) and spatial filtering, in order to extract the P300 subcomponents from the electroencephalogram (EEG) with greater accuracy than the traditional methods. The first topic of research, is the development of constrained BSS and blind signal extraction (BSE) algorithms, to enhance the estimation of the conventional BSS and BSE algorithms. In these methods we use reference signals as prior information, obtained from real EEG data, to aid BSS and BSE in the extraction of the P300 subcomponents. Although, this method exhibits very good behaviour in terms of EEG averaged data, its performance degrades when applied to single trial data, which is the response of the brain after one single stimulus. The second topic deals with single trial EEG data and is based on least squares. Again, we use reference signals to describe the prior knowledge of the P300 subcomponents. In contrast to the first method, the reference signals are Gaussian spike templates with variable latency and width. The target of this algorithm is to measure the properties of the extracted P300 subcomponents and obtain features that can be used in the classification of schizophrenic patients and healthy subjects. Finally, the idea of spatial filtering combined with the use of a reference signal for localisation is introduced for the first time. The designed algorithm localises our desired source from within a mixture of sources where the propagation model of the sources is available. It performs well in the presence of noise and correlated sources. The research presented in this thesis paves the path in introducing adaptive techniques based on reference signals into ERP estimation. The results have been very promising and provide a big step in establishing a foundation for future research

New approaches for EEG signal processing: artifact EOG removal by ICA-RLS scheme and tracks extraction method

Localizing the bioelectric phenomena originating from the cerebral cortex and evoked by auditory and somatosensory stimuli are clear objectives to both understand how the brain works and to recognize different pathologies. Diseases such as Parkinson’s, Alzheimer’s, schizophrenia and epilepsy are intensively studied to find a cure or accurate diagnosis. Epilepsy is considered the disease with major prevalence within disorders with neurological origin. The recurrent and sudden incidence of seizures can lead to dangerous and possibly life-threatening situations. Since disturbance of consciousness and sudden loss of motor control often occur without any warning, the ability to predict epileptic seizures would reduce patients’ anxiety, thus considerably improving quality of life and safety. The common procedure for epilepsy seizure detection is based on brain activity monitorization via electroencephalogram (EEG) data. This process consumes a lot of time, especially in the case of long recordings, but the major problem is the subjective nature of the analysis among specialists when analyzing the same record. From this perspective, the identification of hidden dynamical patterns is necessary because they could provide insight into the underlying physiological mechanisms that occur in the brain. Time-frequency distributions (TFDs) and adaptive methods have demonstrated to be good alternatives in designing systems for detecting neurodegenerative diseases. TFDs are appropriate transformations because they offer the possibility of analyzing relatively long continuous segments of EEG data even when the dynamics of the signal are rapidly changing. On the other hand, most of the detection methods proposed in the literature assume a clean EEG signal free of artifacts or noise, leaving the preprocessing problem opened to any denoising algorithm. In this thesis we have developed two proposals for EEG signal processing: the first approach consists in electrooculogram (EOG) removal method based on a combination of ICA and RLS algorithms which automatically cancels the artifacts produced by eyes movement without the use of external “ad hoc” electrode. This method, called ICA-RLS has been compared with other techniques that are in the state of the art and has shown to be a good alternative for artifacts rejection. The second approach is a novel method in EEG features extraction called tracks extraction (LFE features). This method is based on the TFDs and partial tracking. Our results in pattern extractions related to epileptic seizures have shown that tracks extraction is appropriate in EEG detection and classification tasks, being practical, easily applicable in medical environment and has acceptable computational cost