A fast proximal gradient algorithm for reconstructing nonnegative signals with sparse transform coefficients

We develop a fast 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. 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 regularization term that imposes signal nonnegativity and sparsity via an ℓ₁-norm constraint on the signal´s transform coefficients. This objective function is minimized via Nesterov´s proximal-gradient method with function restart, where the proximal mapping is computed via alternating direction method of multipliers (ADMM). To accelerate the convergence, we develop an adaptive continuation scheme and a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. In the numerical examples, we consider Gaussian linear and Poisson generalized linear measurement models. We compare the proposed penalized NLL minimization approach and existing signal reconstruction methods via compressed sensing and tomographic reconstruction experiments and demonstrate that, 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.