We develop a sparse image reconstruction method for Poisson-distributed polychromatic X-ray 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 mass-attenuation spectrum parameterization of the noiseless measurements for single-material objects and express the mass-attenuation spectrum as a linear combination of B-spline basis functions of order one. A block coordinate-descent algorithm is developed for constrained minimization of a penalized Poisson negative log-likelihood (NLL) cost function, where constraints and penalty terms ensure nonnegativity of the spline coefficients and nonnegativity and sparsity of the
density-map image; the image sparsity is imposed using a convex total-variation (TV) norm penalty term. This algorithm alternates between a Nesterov’s proximal-gradient (NPG) step for estimating the density-map image and a limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) step for estimating the incident-spectrum parameters. We establish conditions for biconvexity of the penalized NLL
objective function, which, if satisfied, ensures monotonicity of the NPG-BFGS iteration. We also show that the penalized NLL objective satisfies the Kurdyka-Łojasiewicz property, which is important for establishing local convergence of block-coordinate descent schemes in biconvex optimization problems. Simulation examples demonstrate the performance of the proposed scheme.
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Blind X-ray CT Image Reconstruction from Polychromatic Poisson Measurements
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 incident-energy spectrum are unknown. Assuming that the object that we wish to reconstruct consists of a single material, 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 basis functions using B splines of order one. We consider a Poisson noise model and establish conditions for biconvexity of the corresponding negative log-likelihood (NLL) function with respect to the density-map and mass-attenuation spectrum parameters. We derive a block-coordinate descent algorithm for constrained minimization of a penalized NLL objective function, where penalty terms ensure nonnegativity of the mass-attenuation spline coefficients and nonnegativity and gradient-map sparsity of the density-map image, imposed using a convex total-variation (TV) norm; the resulting objective function is biconvex. This algorithm alternates between a Nesterov’s proximal-gradient (NPG) step and a limited-memory Broyden-Fletcher-Goldfarb-Shanno with box constraints (L-BFGS-B) iteration for updating the image and mass-attenuation spectrum parameters, respectively. We prove the Kurdyka-Łojasiewicz property of the objective function, which is important for establishing local convergence of block-coordinate descent schemes in biconvex optimization problems. Our framework applies to other NLLs and signal-sparsity penalties, such as lognormal NLL and ℓ₁ norm of 2D discrete wavelet transform (DWT) image coefficients. Numerical experiments
with simulated and real X-ray CT data demonstrate the performance of the proposed scheme.
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- Award ID(s):
- 1421480
- NSF-PAR ID:
- 10013990
- Date Published:
- Journal Name:
- IEEE Transactions on Computational Imaging
- ISSN:
- 2334-0118
- Page Range / eLocation ID:
- 1 to 1
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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 mass-attenuation spectrum is expanded into basis functions using a B-spline basis of order one. We develop a block coordinate-descent algorithm for constrained minimization of a penalized negative log-likelihood 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 proximal-gradient step for estimating the density map image and an active-set step for estimating the incident spectrum parameters. Numerical simulations demonstrate the performance of the proposed scheme.more » « less
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Summary Variable selection for recovering sparsity in nonadditive and nonparametric models with high-dimensional variables has been challenging. This problem becomes even more difficult due to complications in modeling unknown interaction terms among high-dimensional variables. There is currently no variable selection method to overcome these limitations. Hence, in this article we propose a variable selection approach that is developed by connecting a kernel machine with the nonparametric regression model. The advantages of our approach are that it can: (i) recover the sparsity; (ii) automatically model unknown and complicated interactions; (iii) connect with several existing approaches including linear nonnegative garrote and multiple kernel learning; and (iv) provide flexibility for both additive and nonadditive nonparametric models. Our approach can be viewed as a nonlinear version of a nonnegative garrote method. We model the smoothing function by a Least Squares Kernel Machine (LSKM) and construct the nonnegative garrote objective function as the function of the sparse scale parameters of kernel machine to recover sparsity of input variables whose relevances to the response are measured by the scale parameters. We also provide the asymptotic properties of our approach. We show that sparsistency is satisfied with consistent initial kernel function coefficients under certain conditions. An efficient coordinate descent/backfitting algorithm is developed. A resampling procedure for our variable selection methodology is also proposed to improve the power.
- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1109/TCI.2016.2523431
