Polychromatic X-ray CT Image Reconstruction and Mass-Attenuation Spectrum Estimation

We develop a method for sparse image reconstruction from polychromatic computed tomography (CT) measurements under the blind scenario where the material of the inspected object and the incident-energy spectrum are unknown. We obtain a parsimonious measurement-model 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 mass-attenuation spectrum function; the resulting measurement equation has the Laplace integral form. The mass-attenuation spectrum is then expanded into first order B-spline basis functions. We derive a block coordinate-descent algorithm for constrained minimization of a penalized negative log-likelihood (NLL) cost function, where penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the density map. The image sparsity is imposed using total-variation (TV) and $\ell_1$ norms, applied to the density-map image and its discrete wavelet transform (DWT) coefficients, respectively. This algorithm alternates between Nesterov's proximal-gradient (NPG) and limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) steps for updating the image and mass-attenuation spectrum parameters. To accelerate convergence of the density-map NPG step, we apply a step-size selection scheme that accounts for varying local Lipschitz constant of the NLL. We consider lognormal and Poisson noise models and establish conditions for biconvexity of the corresponding NLLs. We also prove the Kurdyka-{\L}ojasiewicz property of the objective function, which is important for establishing local convergence of the algorithm. Numerical experiments with simulated and real X-ray CT data demonstrate the performance of the proposed scheme

Chernoff bounds on pairwise error probabilities of space-time codes

We derive Chernoff bounds on pairwise error probabilities of coherent and noncoherent space-time signaling schemes. First, general Chernoff bound expressions are derived for a correlated Ricean fading channel and correlated additive Gaussian noise. Then, we specialize the obtained results to the cases of space-time-separable noise, white noise, and uncorrelated fading. We derive approximate Chernoff bounds for high and low signal-to-noise ratios (SNRs) and propose optimal signaling schemes. We also compute the optimal number of transmitter antennas for noncoherent signaling with unitary mutually orthogonal space-time codes

Projected Nesterov’s Proximal-Gradient Signal Recovery from Compressive Poisson Measurements

We develop a projected Nesterov’s proximalgradient (PNPG) scheme for reconstructing sparse signals from compressive Poisson-distributed 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 log-likelihood (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 total-variation (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 accounts for varying local Lipschitz constant of the NLL.We establish O k2 convergence of the PNPG method with step-size backtracking only and no restart. Numerical examples compare PNPG with the state-of-the-art sparse Poisson-intensity reconstruction algorithm (SPIRAL)

ECME Hard Thresholding Methods for Image Reconstruction from Compressive Samples

We propose two hard thresholding schemes for image reconstruction from compressive samples. The measurements follow an underdetermined linear model, where the regression-coefficient vector is a sum of an unknown deterministic sparse signal component and a zero-mean white Gaussian component with an unknown variance. We derived an expectation-conditional maximization either (ECME) iteration that converges to a local maximum of the likelihood function of the unknown parameters for a given image sparsity level. Here, we present and analyze a double overrelaxation (DORE) algorithm that applies two successive overrelaxation steps after one ECME iteration step, with the goal to accelerate the ECME iteration. To analyze the reconstruction accuracy, we introduce minimum sparse subspace quotient (minimum SSQ), a more flexible measure of the sampling operator than the well-established restricted isometry property (RIP). We prove that, if the minimum SSQ is sufficiently large, the DORE algorithm achieves perfect or near-optimal recovery of the true image, provided that its transform coefficients are sparse or nearly sparse, respectively. We then describe a multiple-initialization DORE algorithm (DOREMI) that can significantly improve DORE’s reconstruction performance. We present numerical examples where we compare our methods with existing compressive sampling image reconstruction approaches