| We develop a projected Nesterov’s proximal-gradient (PNPG) approach for sparse signal reconstruction. The objective function to be minimized is the sum of a convex differentiable data-fidelity (negative log-likelihood (NLL)) term and a convex regularization term. We apply sparse signal regularization where the signal belongs to a closed convex set within the closure of the domain of the NLL. Signal sparsity is imposed using the ℓ₁-norm or total-variation penalties on the signal’s transform coefficients or gradient map, respectively. The PNPG approach employs projected Nesterov’s acceleration step with restart and an inner iteration to compute the proximal mapping. We propose an adaptive step-size selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. We establish O(k⁻²) convergence of the PNPG scheme; our convergence-rate analysis accounts for inexactness of the iterative proximal mapping. Tomographic and compressed-sensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach. If time permits, the speaker will also outline a method for sparse image reconstruction from Poisson-distributed polychromatic X-ray computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. |
