| dc.description.abstract | Compressed sensing has roused great interest in research and many industries over the last few decades. This is because we can recover signals from vastly undersampled measurements, under certain assumptions: sparsity, incoherence and uniform random subsampling. However, recent research has shown that the traditional theory yields poor recovery results in many practical cases. This has lead to the development of a new compressed sensing theory, based on local structure in the signals. The new theory defines asymptotic sparsity, asymptotic incoherence and multilevel random subsampling. With these new principles, we see much better recovery results. In order to apply CS in practice, we need to be able to solve the main optimization problem basis pursuit efficiently for large data sets. The spectral projected gradient ℓ₁ (SPGL1) algorithm serves this purpose. It restates the optimization problem as a root finding problem of a single-variable non-linear equation, and utilizes an inexact Newton method to find this root. The purpose of this text is to give an introduction to the field of compressed sensing, provide the mathematical motivation for the SPGL1 algorithm and highlight some recent advances in compressed sensing. | eng |