Projected Nesterov´s proximal-gradient signal recovery from compressive poisson measurements

We develop a projected Nesterov´s proximal-gradient (PNPG) scheme for reconstructing sparse signals from compressive Poisson-distributed measurements with the mean signal intensity that follows an affine model with known intercept. The objective function to be minimized is a sum of convex data fidelity (negative log-likelihood (NLL)) and regularization terms. We apply sparse signal regularization where the signal belongs to a closed convex set within the domain of the NLL and signal sparsity is imposed using total-variation (TV) penalty. We present analytical upper bounds on the regularization tuning constant. The proposed PNPG method employs projected Nesterov´s acceleration step, function restart, and an adaptive step-size selection scheme that aims at obtaining a good local majorizing function of the N LL and reducing the time spent backtracking. We establish O (k^-2) convergence of the PNPG method with step-size backtracking only and no restart. Numerical examples demonstrate the performance of the PNPG method.