Browsing by Subject "Sparse Signal Processing"
Now showing 1 - 2 of 2
Results Per Page
Sort Options
Item Open Access Signal and image processing algorithms using interval convex programming and sparsity(2012) Köse, KıvançIn this thesis, signal and image processing algorithms based on sparsity and interval convex programming are developed for inverse problems. Inverse signal processing problems are solved by minimizing the ℓ1 norm or the Total Variation (TV) based cost functions in the literature. A modified entropy functional approximating the absolute value function is defined. This functional is also used to approximate the ℓ1 norm, which is the most widely used cost function in sparse signal processing problems. The modified entropy functional is continuously differentiable, and convex. As a result, it is possible to develop iterative, globally convergent algorithms for compressive sensing, denoising and restoration problems using the modified entropy functional. Iterative interval convex programming algorithms are constructed using Bregman’s D-Projection operator. In sparse signal processing, it is assumed that the signal can be represented using a sparse set of coefficients in some transform domain. Therefore, by minimizing the total variation of the signal, it is expected to realize sparse representations of signals. Another cost function that is introduced for inverse problems is the Filtered Variation (FV) function, which is the generalized version of the Total Variation (VR) function. The TV function uses the differences between the pixels of an image or samples of a signal. This is essentially simple Haar filtering. In FV, high-pass filter outputs are used instead of differences. This leads to flexibility in algorithm design adapting to the local variations of the signal. Extensive simulation studies using the new cost functions are carried out. Better experimental restoration, and reconstructions results are obtained compared to the algorithms in the literatureItem Open Access Uncertain linear equations(2010) Pilancı, MertIn this thesis, new theoretical and practical results on linear equations with various types of uncertainties and their applications are presented. In the first part, the case in which there are more equations than unknowns (overdetermined case) is considered. A novel approach is proposed to provide robust and accurate estimates of the solution of the linear equations when both the measurement vector and the coefficient matrix are subject to uncertainty. A new analytic formulation is developed in terms of the gradient flow to analyze and provide estimates to the solution. The presented analysis enables us to study and compare existing methods in literature. We derive theoretical bounds for the performance of our estimator and show that if the signal-to-noise ratio is low than a treshold, a significant improvement is made compared to the conventional estimator. Numerical results in applications such as blind identification, multiple frequency estimation and deconvolution show that the proposed technique outperforms alternative methods in mean-squared error for a significant range of signal-to-noise ratio values. The second type of uncertainty analyzed in the overdetermined case is where uncertainty is sparse in some basis. We show that this type of uncertainty on the coefficient matrix can be recovered exactly for a large class of structures, if we have sufficiently many equations. We propose and solve an optimization criterion and its convex relaxation to recover the uncertainty and the solution to the linear system. We derive sufficiency conditions for exact and stable recovery. Then we demonstrate with numerical examples that the proposed method is able to recover unknowns exactly with high probability. The performance of the proposed technique is compared in estimation and tracking of sparse multipath wireless channels. The second part of the thesis deals with the case where there are more unknowns than equations (underdetermined case). We extend the theory of polarization of Arikan for random variables with continuous distributions. We show that the Hadamard Transform and the Discrete Fourier Transform, polarizes the information content of independent identically distributed copies of compressible random variables, where compressibility is measured by Shannon’s differential entropy. Using these results we show that, the solution of the linear system can be recovered even if there are more unknowns than equations if the number of equations is sufficient to capture the entropy of the uncertainty. This approach is applied to sampling compressible signals below the Nyquist rate and coined ”Polar Sampling”. This result generalizes and unifies the sparse recovery theory of Compressed Sensing by extending it to general low entropy signals with an information theoretical analysis. We demonstrate the effectiveness of Polar Sampling approach on a numerical sub-Nyquist sampling example.