Structure-Constrained Basis Pursuit for Compressively Sensing Speech
Compressed Sensing (CS) exploits the sparsity of many signals to enable sampling below the Nyquist rate. If the original signal is sufficiently sparse, the Basis Pursuit (BP) algorithm will perfectly reconstruct the original signal. Unfortunately many signals that intuitively appear sparse do not meet the threshold for "sufficient sparsity". These signals require so many CS samples for accurate reconstruction that the advantages of CS disappear. This is because Basis Pursuit/Basis Pursuit Denoising only models sparsity. We developed a "Structure-Constrained Basis Pursuit" that models the structure of somewhat sparse signals as upper and lower bound constraints on the Basis Pursuit Denoising solution. We applied it to speech, which seems sparse but does not compress well with CS, and gained improved quality over Basis Pursuit Denoising. When a single parameter (i.e. the phone) is encoded, Normalized Mean Squared Error (NMSE) decreases by between 16.2% and 1.00% when sampling with CS between 1/10 and 1/2 the Nyquist rate, respectively. When bounds are coded as a sum of Gaussians, NMSE decreases between 28.5% and 21.6% in the same range. SCBP can be applied to any somewhat sparse signal with a predictable structure to enable improved reconstruction quality with the same number of samples.