We develop a projected Nesterov’s proximalgradient (PNPG) scheme for reconstructing sparse signals from compressive Poissondistributed 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 loglikelihood (NLL)) and regularization terms. We apply sparse signal regularization where the signal belongs to a nonempty closed convex set within the domain of the NLL and signal sparsity is imposed using totalvariation (TV) penalty. We present analytical upper bounds on the regularization tuning constant. The proposed PNPG method employs projected Nesterov’s acceleration step, function restart, andmore »
Upperbounding the regularization constant for convex sparse signal reconstruction
Consider reconstructing a signal x by minimizing a weighted sum of a convex differentiable negative loglikelihood (NLL) (datafidelity) term and a convex regularization term that imposes a convexset constraint on x and enforces its sparsity using ℓ1norm analysis regularization.We compute upper bounds on the regularization tuning constant beyond which the regularization term overwhelmingly dominates the NLL term so that the set of minimum points of the objective function does not change. Necessary and sufficient conditions for irrelevance of sparse signal regularization and a condition for the existence of finite
upper bounds are established. We formulate an optimization problem for finding these bounds when the regularization term can be globally minimized by a feasible x and also develop an alternating direction method of multipliers (ADMM) type method for their computation. Simulation examples show that the derived and empirical bounds match.
 Award ID(s):
 1421480
 Publication Date:
 NSFPAR ID:
 10023710
 Journal Name:
 arXiv.org
 ISSN:
 23318422
 Sponsoring Org:
 National Science Foundation
More Like this


We develop a projected Nesterov’s proximalgradient (PNPG) approach for sparse signal reconstruction that combines adaptive step size with Nesterov’s momentum acceleration. The objective function that we wish to minimize is the sum of a convex differentiable datafidelity (negative loglikelihood (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; the convexset constraint facilitates flexible NLL domains and accurate signal recovery. Signal sparsity is imposed using the ℓ₁norm penalty on the signal’s linear transform coefficients. The PNPG approach employs a projectedmore »

We develop a sparse image reconstruction method for Poissondistributed polychromatic Xray computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. We employ our massattenuation spectrum parameterization of the noiseless measurements for singlematerial objects and express the massattenuation spectrum as a linear combination of Bspline basis functions of order one. A block coordinatedescent algorithm is developed for constrained minimization of a penalized Poisson negative loglikelihood (NLL) cost function, where constraints and penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the densitymap image; themore »

We give nearly matching upper and lower bounds on the oracle complexity of finding ϵstationary points (∥∇F(x)∥≤ϵ in stochastic convex optimization. We jointly analyze the oracle complexity in both the local stochastic oracle model and the global oracle (or, statistical learning) model. This allows us to decompose the complexity of finding nearstationary points into optimization complexity and sample complexity, and reveals some surprising differences between the complexity of stochastic optimization versus learning. Notably, we show that in the global oracle/statistical learning model, only logarithmic dependence on smoothness is required to find a nearstationary point, whereas polynomial dependence on smoothness ismore »

We develop a framework for reconstructing images that are sparse in an appropriate transform domain from polychromatic computed tomography (CT) measurements under the blind scenario where the material of the inspected object and incidentenergy spectrum are unknown. Assuming that the object that we wish to reconstruct consists of a single material, we obtain a parsimonious measurementmodel parameterization by changing the integral variable from photon energy to mass attenuation, which allows us to combine the variations brought by the unknown incident spectrum and mass attenuation into a single unknown massattenuation spectrum function; the resulting measurement equation has the Laplaceintegral form. Themore »