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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 projected Nesterov’s acceleration step with restart and a dualitybased inner iteration to compute the proximal mapping. We propose an adaptive stepsize selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. Thanks to stepsize adaptation, PNPG converges faster than the methods that do not adjust to the local curvature of the NLL. We present an integrated derivation of the momentum acceleration and proofs of O(k⁻²) objective function convergence rate and convergence of the iterates, which account for adaptive step size, inexactness of the iterative proximal mapping, and the convexset constraint. The tuning of PNPG is largely application independent. Tomographic and compressedsensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach.more » « less

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.more » « less

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; the image sparsity is imposed using a convex totalvariation (TV) norm penalty term. This algorithm alternates between a Nesterov’s proximalgradient (NPG) step for estimating the densitymap image and a limitedmemory BroydenFletcherGoldfarbShanno with box constraints (LBFGSB) step for estimating the incidentspectrum parameters. We establish conditions for biconvexity of the penalized NLL objective function, which, if satisfied, ensures monotonicity of the NPGBFGS iteration. We also show that the penalized NLL objective satisfies the KurdykaŁojasiewicz property, which is important for establishing local convergence of blockcoordinate descent schemes in biconvex optimization problems. Simulation examples demonstrate the performance of the proposed scheme.more » « less

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. The massattenuation spectrum is then expanded into basis functions using B splines of order one. We consider a Poisson noise model and establish conditions for biconvexity of the corresponding negative loglikelihood (NLL) function with respect to the densitymap and massattenuation spectrum parameters. We derive a blockcoordinate descent algorithm for constrained minimization of a penalized NLL objective function, where penalty terms ensure nonnegativity of the massattenuation spline coefficients and nonnegativity and gradientmap sparsity of the densitymap image, imposed using a convex totalvariation (TV) norm; the resulting objective function is biconvex. This algorithm alternates between a Nesterov’s proximalgradient (NPG) step and a limitedmemory BroydenFletcherGoldfarbShanno with box constraints (LBFGSB) iteration for updating the image and massattenuation spectrum parameters, respectively. We prove the KurdykaŁojasiewicz property of the objective function, which is important for establishing local convergence of blockcoordinate descent schemes in biconvex optimization problems. Our framework applies to other NLLs and signalsparsity penalties, such as lognormal NLL and ℓ₁ norm of 2D discrete wavelet transform (DWT) image coefficients. Numerical experiments with simulated and real Xray CT data demonstrate the performance of the proposed scheme.more » « less

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, and an adaptive stepsize selection scheme that aims at obtaining a good local majorizing function of the NLL and reducing the time spent backtracking. We establish O(k⁻²) convergence of the PNPG method with stepsize backtracking only and no restart. Numerical examples demonstrate the performance of the PNPG method.more » « less

We develop a sparse image reconstruction method for polychromatic tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. To obtain a parsimonious measurement model parameterization, we first rewrite the measurement equation using our mass attenuation parameterization, which has the Laplace integral form. The unknown massattenuation spectrum is expanded into basis functions using a Bspline basis of order one. We develop a block coordinatedescent algorithm for constrained minimization of a penalized negative loglikelihood function, where constraints and penalty terms ensure nonnegativity of the spline coefficients and sparsity of the density map image in the wavelet domain. This algorithm alternates between a Nesterov’s proximalgradient step for estimating the density map image and an activeset step for estimating the incident spectrum parameters. Numerical simulations demonstrate the performance of the proposed scheme.more » « less