Convergence of Fourier-wavelet models for Gaussian random processes

Mean square convergence and convergence in probability of Fourier-Wavelet Models (FWM) of stationary Gaussian Random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolski\u{i}-Besov classes cannot be improved

Convergence of Fourier-Wavelet models for Gaussian random processes

Mean square convergence and convergence in probability of Fourier-Wavelet Models (FWM) of stationary Gaussian Random processes in the metric of Banach space of continuously differentiable functions and in Sobolev space are studied. Sufficient conditions for the convergence formulated in the frame of spectral functions are given. It is shown that the given rates of convergence of FWM in the mean square obtained in the Nikolskiui-Besov classes cannot be improved

Graph Signal Processing: Overview, Challenges and Applications

Research in Graph Signal Processing (GSP) aims to develop tools for processing data defined on irregular graph domains. In this paper we first provide an overview of core ideas in GSP and their connection to conventional digital signal processing. We then summarize recent developments in developing basic GSP tools, including methods for sampling, filtering or graph learning. Next, we review progress in several application areas using GSP, including processing and analysis of sensor network data, biological data, and applications to image processing and machine learning. We finish by providing a brief historical perspective to highlight how concepts recently developed in GSP build on top of prior research in other areas.Comment: To appear, Proceedings of the IEE

Structured Compressed Sensing: From Theory to Applications

Compressed sensing (CS) is an emerging field that has attracted considerable research interest over the past few years. Previous review articles in CS limit their scope to standard discrete-to-discrete measurement architectures using matrices of randomized nature and signal models based on standard sparsity. In recent years, CS has worked its way into several new application areas. This, in turn, necessitates a fresh look on many of the basics of CS. The random matrix measurement operator must be replaced by more structured sensing architectures that correspond to the characteristics of feasible acquisition hardware. The standard sparsity prior has to be extended to include a much richer class of signals and to encode broader data models, including continuous-time signals. In our overview, the theme is exploiting signal and measurement structure in compressive sensing. The prime focus is bridging theory and practice; that is, to pinpoint the potential of structured CS strategies to emerge from the math to the hardware. Our summary highlights new directions as well as relations to more traditional CS, with the hope of serving both as a review to practitioners wanting to join this emerging field, and as a reference for researchers that attempts to put some of the existing ideas in perspective of practical applications.Comment: To appear as an overview paper in IEEE Transactions on Signal Processin

A stochastic fractal model of the universe related to the fractional Laplacian

A new stochastic fractal model based on a fractional Laplace equation is developed. Exact representation for the spectral and correlation functions under random boundary excitation are obtained. Randomized spectral expansion is constructed for simulation of the solution of the fractional Laplace equation. We present calculations for 2D and 3D spaces for a series of fractional parameters showing a strong memory effect: the decay of correlations is several order of magnitudes less compared to the conventional Laplace equation model

Beyond the noise : high fidelity MR signal processing

This thesis describes a variety of methods developed to increase the sensitivity and resolution of liquid state nuclear magnetic resonance (NMR) experiments. NMR is known as one of the most versatile non-invasive analytical techniques yet often suffers from low sensitivity. The main contribution to this low sensitivity issue is a presence of noise and level of noise in the spectrum is expressed numerically as “signal-to-noise ratio”. NMR signal processing involves sensitivity and resolution enhancement achieved by noise reduction using mathematical algorithms. A singular value decomposition based reduced rank matrix method, composite property mapping, in particular is studied extensively in this thesis to present its advantages, limitations, and applications. In theory, when the sum of k noiseless sinusoidal decays is formatted into a specific matrix form (i.e., Toeplitz), the matrix is known to possess k linearly independent columns. This information becomes apparent only after a singular value decomposition of the matrix. Singular value decomposition factorises the large matrix into three smaller submatrices: right and left singular vector matrices, and one diagonal matrix containing singular values. Were k noiseless sinusoidal decays involved, there would be only k nonzero singular values appearing in the diagonal matrix in descending order providing the information of the amplitude of each sinusoidal decay. The number of non-zero singular values or the number of linearly independent columns is known as the rank of the matrix. With real NMR data none of the singular values equals zero and the matrix has full rank. The reduction of the rank of the matrix and thus the noise in the reconstructed NMR data can be achieved by replacing all the singular values except the first k values with zeroes. This noise reduction process becomes difficult when biomolecular NMR data is to be processed due to the number of resonances being unknown and the presence of a large solvent peak