A Fast and Accurate Basis Pursuit Denoising Algorithm With Application to Super-Resolving Tomographic SAR

<inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula> regularization is used for finding sparse solutions to an underdetermined linear system. As sparse signals are widely expected in remote sensing, this type of regularization scheme and its extensions have been widely employed in many remote sensing problems, such as image fusion, target detection, image super-resolution, and others, and have led to promising results. However, solving such sparse reconstruction problems is computationally expensive and has limitations in its practical use. In this paper, we proposed a novel efficient algorithm for solving the complex-valued <inline-formula> <tex-math notation="LaTeX">$L_{1}$ </tex-math></inline-formula> regularized least squares problem. Taking the high-dimensional tomographic synthetic aperture radar (TomoSAR) as a practical example, we carried out extensive experiments, both with the simulation data and the real data, to demonstrate that the proposed approach can retain the accuracy of the second-order methods while dramatically speeding up the processing by one or two orders. Although we have chosen TomoSAR as the example, the proposed method can be generally applied to any spectral estimation problems.