6 research outputs found
Sparse signal reconstruction from polychromatic X-ray CT measurements via mass attenuation discretization
We propose a method for reconstructing sparse images from polychromatic x-ray computed tomography (ct) measurements via mass attenuation coefficient discretization. The material of the inspected object and the incident spectrum are assumed to be unknown. We rewrite the Lambert-Beer’s law in terms of integral expressions of mass attenuation and discretize the resulting integrals. We then present a penalized constrained least-squares optimization approach forreconstructing the underlying object from log-domain measurements, where an active set approach is employed to estimate incident energy density parameters and the nonnegativity and sparsity of the image density map are imposed using negative-energy and smooth ℓ1-norm penalty terms. We propose a two-step scheme for refining the mass attenuation discretization grid by using higher sampling rate over the range with higher photon energy, and eliminating the discretization points that have little effect on accuracy of the forward projection model. This refinement allows us to successfully handle the characteristic lines (Dirac impulses) in the incident energy density spectrum. We compare the proposed method with the standard filtered backprojection, which ignores the polychromatic nature of the measurements and sparsity of theimage density map. Numerical simulations using both realistic simulated and real x-ray ct data are presented
A Max-Product EM Algorithm for Reconstructing Markov-tree Sparse Signals from Compressive Samples
We propose a Bayesian expectation-maximization (EM) algorithm for
reconstructing Markov-tree sparse signals via belief propagation. The
measurements follow an underdetermined linear model where the
regression-coefficient vector is the sum of an unknown approximately sparse
signal and a zero-mean white Gaussian noise with an unknown variance. The
signal is composed of large- and small-magnitude components identified by
binary state variables whose probabilistic dependence structure is described by
a Markov tree. Gaussian priors are assigned to the signal coefficients given
their state variables and the Jeffreys' noninformative prior is assigned to the
noise variance. Our signal reconstruction scheme is based on an EM iteration
that aims at maximizing the posterior distribution of the signal and its state
variables given the noise variance. We construct the missing data for the EM
iteration so that the complete-data posterior distribution corresponds to a
hidden Markov tree (HMT) probabilistic graphical model that contains no loops
and implement its maximization (M) step via a max-product algorithm. This EM
algorithm estimates the vector of state variables as well as solves iteratively
a linear system of equations to obtain the corresponding signal estimate. We
select the noise variance so that the corresponding estimated signal and state
variables obtained upon convergence of the EM iteration have the largest
marginal posterior distribution. We compare the proposed and existing
state-of-the-art reconstruction methods via signal and image reconstruction
experiments.Comment: To appear in IEEE Transactions on Signal Processin
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 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
Sparse Signal Reconstruction via ECME Hard Thresholding
We propose a probabilistic model for sparse signal reconstruction and develop several novel algorithms for computing the maximum likelihood (ML) parameter estimates under this model. The measurements follow an underdetermined linear model where the regression-coefficient vector is the sum of an unknown deterministic sparse signal component and a zero-mean white Gaussian component with an unknown variance. Our reconstruction schemes are based on an expectation-conditional maximization either (ECME) iteration that aims at maximizing the likelihood function with respect to the unknown parameters for a given signal sparsity level. Compared with the existing iterative hard thresholding (IHT) method, the ECME algorithm contains an additional multiplicative term and guarantees monotonic convergence for a wide range of sensing (regression) matrices. We propose a double overrelaxation (DORE) thresholding scheme for accelerating the ECME iteration. We prove that, under certain mild conditions, the ECME and DORE iterations converge to local maxima of the likelihood function. The ECME and DORE iterations can be implemented exactly in small-scale applications and for the important class of large-scale sensing operators with orthonormal rows used e.g., partial fast Fourier transform (FFT). If the signal sparsity level is unknown, we introduce an unconstrained sparsity selection (USS) criterion and a tuning-free automatic double overrelaxation (ADORE) thresholding method that employs USS to estimate the sparsity level. We compare the proposed and existing sparse signal reconstruction methods via one-dimensional simulation and two-dimensional image reconstruction experiments using simulated and real X-ray CT data
Convex-Set–Constrained Sparse Signal Recovery: Theory and Applications
Convex-set constrained sparse signal reconstruction facilitates flexible measurement model and accurate recovery. The objective function that we wish to minimize is a sum of a convex differentiable data-fidelity (negative log-likelihood (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. Signal sparsity is imposed using the l1-norm penalty on the signal\u27s linear transform coefficients.
First, we present a projected Nesterov’s proximal-gradient (PNPG) approach that employs a projected Nesterov\u27s acceleration step with restart and a duality-based inner iteration to compute the proximal mapping. We propose an adaptive step-size selection scheme to obtain a good local majorizing function of the NLL and reduce the time spent backtracking. 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 convex-set constraint. The tuning of PNPG is largely application independent. Tomographic and compressed-sensing reconstruction experiments with Poisson generalized linear and Gaussian linear measurement models demonstrate the performance of the proposed approach.
We then address the problem of upper-bounding the regularization constant for the convex-set--constrained sparse signal recovery problem behind the PNPG framework. This bound defines the maximum influence the regularization term has to the signal recovery. We formulate an optimization problem for finding these bounds when the regularization term can be globally minimized and develop an alternating direction method of multipliers (ADMM) type method for their computation. Simulation examples show that the derived and empirical bounds match.
Finally, we show application of the PNPG framework to X-ray computed tomography (CT) and outline a method for sparse image reconstruction from Poisson-distributed polychromatic X-ray CT measurements under the blind scenario where the material of the inspected object and the incident energy spectrum are unknown. To obtain a parsimonious mean measurement-model parameterization, we first rewrite the measurement equation 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. We apply a block coordinate-descent algorithm that alternates between an NPG image reconstruction step and a limited-memory BFGS with box constraints (L-BFGS-B) iteration for updating mass-attenuation spectrum parameters. Our NPG-BFGS algorithm is the first physical-model based image reconstruction method for simultaneous blind sparse image reconstruction and mass-attenuation spectrum estimation from polychromatic measurements. Real X-ray CT reconstruction examples demonstrate the performance of the proposed blind scheme