First-order Convex Optimization Methods for Signal and Image Processing

In this thesis we investigate the use of first-order convex optimization methods applied to problems in signal and image processing. First we make a general introduction to convex optimization, first-order methods and their iteration com-plexity. Then we look at different techniques, which can be used with first-order methods such as smoothing, Lagrange multipliers and proximal gradient meth-ods. We continue by presenting different applications of convex optimization and notable convex formulations with an emphasis on inverse problems and sparse signal processing. We also describe the multiple-description problem. We finally present the contributions of the thesis. The remaining parts of the thesis consist of five research papers. The first paper addresses non-smooth first-order convex optimization and the trade-off between accuracy and smoothness of the approximating smooth function. The second and third papers concern discrete linear inverse problems and reliable numerical reconstruction software. The last two papers present a convex opti-mization formulation of the multiple-description problem and a method to solve it in the case of large-scale instances. i i

Compressive Sensing for Spread Spectrum Receivers

With the advent of ubiquitous computing there are two design parameters of wireless communication devices that become very important power: efficiency and production cost. Compressive sensing enables the receiver in such devices to sample below the Shannon-Nyquist sampling rate, which may lead to a decrease in the two design parameters. This paper investigates the use of Compressive Sensing (CS) in a general Code Division Multiple Access (CDMA) receiver. We show that when using spread spectrum codes in the signal domain, the CS measurement matrix may be simplified. This measurement scheme, named Compressive Spread Spectrum (CSS), allows for a simple, effective receiver design. Furthermore, we numerically evaluate the proposed receiver in terms of bit error rate under different signal to noise ratio conditions and compare it with other receiver structures. These numerical experiments show that though the bit error rate performance is degraded by the subsampling in the CS-enabled receivers, this may be remedied by including quantization in the receiver model. We also study the computational complexity of the proposed receiver design under different sparsity and measurement ratios. Our work shows that it is possible to subsample a CDMA signal using CSS and that in one example the CSS receiver outperforms the classical receiver.Comment: 11 pages, 11 figures, 1 table, accepted for publication in IEEE Transactions on Wireless Communication

A Fast Interior Point Method for Atomic Norm Soft Thresholding

The atomic norm provides a generalization of the $\ell_1$-norm to continuous parameter spaces. When applied as a sparse regularizer for line spectral estimation the solution can be obtained by solving a convex optimization problem. This problem is known as atomic norm soft thresholding (AST). It can be cast as a semidefinite program and solved by standard methods. In the semidefinite formulation there are $O(N^2)$ dual variables which complicates the implementation of a standard primal-dual interior-point method based on symmetric cones. That has lead researcher to consider alternating direction method of multipliers (ADMM) for the solution of AST, but this method is still somewhat slow for large problem sizes. To obtain a faster algorithm we reformulate AST as a non-symmetric conic program. That has two properties of key importance to its numerical solution: the conic formulation has only $O(N)$ dual variables and the Toeplitz structure inherent to AST is preserved. Based on it we derive FastAST which is a primal-dual interior point method for solving AST. Two variants are considered with the fastest one requiring only $O(N^2)$ flops per iteration. Extensive numerical experiments demonstrate that FastAST solves AST significantly faster than a state-of-the-art solver based on ADMM.Comment: 31 pages, accepted for publication in Elsevier Signal Processin