44 research outputs found
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 -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 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
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
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