Reconstruction of Nonnegative Sparse Signals Using Accelerated Proximal-Gradient Algorithms

We develop an accelerated proximal-gradient scheme for reconstructing nonnegative signals that are sparse in a transform domain from underdetermined measurements. This signal model is motivated by tomographic applications where the signal of interest is known to be nonnegative because it represents a tissue or material density. It is also applicable to optical and hyperspectral imaging, where energy within certain spectral band is nonnegative. We adopt the unconstrained regularization framework where the objective function to be minimized is a sum of a convex data fidelity (negative log-likelihood (NLL)) term and a convex regularization term that imposes signal nonnegativity and sparsity by using indicator-function and l1-norm constraints on the signal and its transform coefficients, respectively. We apply the Nesterov's proximal-gradient (NPG) method with function restart to minimize this objective function and the alternating direction method of multipliers (ADMM) to compute the proximal mapping. To accelerate convergence of the NPG iteration, we apply a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. We also apply adaptive continuation, which provides numerical stability and can accelerate the convergence of the NPG iteration. We construct compressed-sensing and tomographic reconstruction experiments with Gaussian linear and Poisson generalized linear measurement models, where we compare the proposed reconstruction approach with existing signal reconstruction methods. By exploiting both the nonnegativity of the underlying signal and sparsity of its wavelet coefficients, we can achieve significantly better reconstruction performance than the existing methods.