A fast and portable Re-Implementation of Piskunov and Valenti's Optimal-Extraction Algorithm with improved Cosmic-Ray Removal and Optimal Sky Subtraction

We present a fast and portable re-implementation of Piskunov and Valenti's optimal-extraction algorithm (Piskunov & Valenti, 2002} in C/C++ together with full uncertainty propagation, improved cosmic-ray removal, and an optimal background-subtraction algorithm. This re-implementation can be used with IRAF and most existing data-reduction packages and leads to signal-to-noise ratios close to the Poisson limit. The algorithm is very stable, operates on spectra from a wide range of instruments (slit spectra and fibre feeds), and has been extensively tested for VLT/UVES, ESO/CES, ESO/FEROS, NTT/EMMI, NOT/ALFOSC, STELLA/SES, SSO/WiFeS, and finally, P60/SEDM-IFU data.Comment: 23 pages, 12 figure

Performance Analysis of Sparse Recovery Based on Constrained Minimal Singular Values

The stability of sparse signal reconstruction is investigated in this paper. We design efficient algorithms to verify the sufficient condition for unique $\ell_1$ sparse recovery. One of our algorithm produces comparable results with the state-of-the-art technique and performs orders of magnitude faster. We show that the $\ell_1$-constrained minimal singular value ($\ell_1$-CMSV) of the measurement matrix determines, in a very concise manner, the recovery performance of $\ell_1$-based algorithms such as the Basis Pursuit, the Dantzig selector, and the LASSO estimator. Compared with performance analysis involving the Restricted Isometry Constant, the arguments in this paper are much less complicated and provide more intuition on the stability of sparse signal recovery. We show also that, with high probability, the subgaussian ensemble generates measurement matrices with $\ell_1$-CMSVs bounded away from zero, as long as the number of measurements is relatively large. To compute the $\ell_1$-CMSV and its lower bound, we design two algorithms based on the interior point algorithm and the semi-definite relaxation