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1. Constrained one‐step material decomposition reconstruction of head CT data from a silicon photon‐counting prototype

   https://doi.org/10.1002/mp.16649  

   Schmidt, Taly Gilat ; Sidky, Emil Y. ; Pan, Xiaochuan ; Barber, Rina Foygel ; Grönberg, Fredrik ; Sjölin, Martin ; Danielsson, Mats ( July 2023 , Medical Physics)

   Spectral CT material decomposition provides quantitative information but is challenged by the instability of the inversion into basis materials. We have previously proposed the constrained One‐Step Spectral CT Image Reconstruction (cOSSCIR) algorithm to stabilize the material decomposition inversion by directly estimating basis material images from spectral CT data. cOSSCIR was previously investigated on phantom data.

   This study investigates the performance of cOSSCIR using head CT datasets acquired on a clinical photon‐counting CT (PCCT) prototype. This is the first investigation of cOSSCIR for large‐scale, anatomically complex, clinical PCCT data. The cOSSCIR decomposition is preceded by a spectrum estimation and nonlinear counts correction calibration step to address nonideal detector effects.

   Head CT data were acquired on an early prototype clinical PCCT system using an edge‐on silicon detector with eight energy bins. Calibration data of a step wedge phantom were also acquired and used to train a spectral model to account for the source spectrum and detector spectral response, and also to train a nonlinear counts correction model to account for pulse pileup effects. The cOSSCIR algorithm optimized the bone and adipose basis images directly from the photon counts data, while placing a grouped total variation (TV) constraint on the basis images. For comparison, basis images were also reconstructed by a two‐step projection‐domain approach of Maximum Likelihood Estimation (MLE) for decomposing basis sinograms, followed by filtered backprojection (MLE + FBP) or a TV minimization algorithm (MLE + TV_min) to reconstruct basis images. We hypothesize that the cOSSCIR approach will provide a more stable inversion into basis images compared to two‐step approaches. To investigate this hypothesis, the noise standard deviation in bone and soft‐tissue regions of interest (ROIs) in the reconstructed images were compared between cOSSCIR and the two‐step methods for a range of regularization constraint settings.

   cOSSCIR reduced the noise standard deviation in the basis images by a factor of two to six compared to that of MLE + TV_min, when both algorithms were constrained to produce images with the same TV. The cOSSCIR images demonstrated qualitatively improved spatial resolution and depiction of fine anatomical detail. The MLE + TV_minalgorithm resulted in lower noise standard deviation than cOSSCIR for the virtual monoenergetic images (VMIs) at higher energy levels and constraint settings, while the cOSSCIR VMIs resulted in lower noise standard deviation at lower energy levels and overall higher qualitative spatial resolution. There were no statistically significant differences in the mean values within the bone region of images reconstructed by the studied algorithms. There were statistically significant differences in the mean values within the soft‐tissue region of the reconstructed images, with cOSSCIR producing mean values closer to the expected values.

   The cOSSCIR algorithm, combined with our previously proposed spectral model estimation and nonlinear counts correction method, successfully estimated bone and adipose basis images from high resolution, large‐scale patient data from a clinical PCCT prototype. The cOSSCIR basis images were able to depict fine anatomical details with a factor of two to six reduction in noise standard deviation compared to that of the MLE + TV_mintwo‐step approach.

    

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2. Exact gram filtering and efficient backprojection for iterative CT reconstruction

   https://doi.org/10.1002/mp.15547  

   Shu, Ziyu ; Entezari, Alireza ( March 2022 , Medical Physics)

   Forward and backprojections are the basis of all model‐based iterative reconstruction (MBIR) methods. However, computing these accurately is time‐consuming. In this paper, we present a method for MBIR in parallel X‐ray beam geometry that utilizes a Gram filter to efficiently implement forward and backprojection.

   We propose using voxel‐basis and modeling its footprint in a box spline framework to calculate the Gram filter exactly and improve the performance of backprojection. In the special case of parallel X‐ray beam geometry, the forward and backprojection can be implemented by an estimated Gram filter efficiently if the sinogram signal is bandlimited. In this paper, a specialized sinogram interpolation method is proposed to eliminate the bandlimited prerequisite and thus improve the reconstruction accuracy. We build on this idea by utilizing the continuity of the voxel‐basis' footprint, which provides a more accurate sinogram interpolation and further improves the efficiency and quality of backprojection. In addition, the detector blur effect can be efficiently accounted for in our method to better handle realistic scenarios.

   The proposed method is tested on both phantom and real computed tomography (CT) images under different resolutions, sinogram sampling steps, and noise levels. The proposed method consistently outperforms other state‐of‐the‐art projection models in terms of speed and accuracy for both backprojection and reconstruction.

   We proposed a iterative reconstruction methodology for 3D parallel‐beam X‐ray CT reconstruction. Our experimental results demonstrate that the proposed methodology is accurate, fast, and reproducible, and outperforms alternative state‐of‐the‐art projection models on both backprojection and reconstruction results significantly.

    

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3. A content-adaptive unstructured grid based regularized CT reconstruction method with a SART-type preconditioned fixed-point proximity algorithm

   https://doi.org/10.1088/1361-6420/ac490f  

   Chen, Yun ; Lu, Yao ; Ma, Xiangyuan ; Xu, Yuesheng ( January 2022 , Inverse Problems)

   Abstract The goal of this study is to develop a new computed tomography (CT) image reconstruction method, aiming at improving the quality of the reconstructed images of existing methods while reducing computational costs. Existing CT reconstruction is modeled by pixel-based piecewise constant approximations of the integral equation that describes the CT projection data acquisition process. Using these approximations imposes a bottleneck model error and results in a discrete system of a large size. We propose to develop a content-adaptive unstructured grid (CAUG) based regularized CT reconstruction method to address these issues. Specifically, we design a CAUG of the image domain to sparsely represent the underlying image, and introduce a CAUG-based piecewise linear approximation of the integral equation by employing a collocation method. We further apply a regularization defined on the CAUG for the resulting ill-posed linear system, which may lead to a sparse linear representation for the underlying solution. The regularized CT reconstruction is formulated as a convex optimization problem, whose objective function consists of a weighted least square norm based fidelity term, a regularization term and a constraint term. Here, the corresponding weighted matrix is derived from the simultaneous algebraic reconstruction technique (SART). We then develop a SART-type preconditioned fixed-point proximity algorithm to solve the optimization problem. Convergence analysis is provided for the resulting iterative algorithm. Numerical experiments demonstrate the superiority of the proposed method over several existing methods in terms of both suppressing noise and reducing computational costs. These methods include the SART without regularization and with the quadratic regularization, the traditional total variation (TV) regularized reconstruction method and the TV superiorized conjugate gradient method on the pixel grid. 

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4. Blind X-ray CT Image Reconstruction from Polychromatic Poisson Measurements

   https://doi.org/10.1109/TCI.2016.2523431  

   Gu, Renliang ; Dogandzic, Aleksandar ( February 2016 , IEEE Transactions on Computational Imaging)

   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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5. Nonlinear reconstruction of coded spectral X-ray CT based on material decomposition

   https://doi.org/10.1364/OE.426732  

   Zhang, Tong ; Zhao, Shengjie ; Ma, Xu ; Cuadros, Angela P. ; Zhao, Qile ; Arce, Gonzalo R. ( June 2021 , Optics Express)

   Coded spectral X-ray computed tomography (CT) based on K-edge filtered illumination is a cost-effective approach to acquire both 3-dimensional structure of objects and their material composition. This approach allows sets of incomplete rays from sparse views or sparse rays with both spatial and spectral encoding to effectively reduce the inspection duration or radiation dose, which is of significance in biological imaging and medical diagnostics. However, reconstruction of spectral CT images from compressed measurements is a nonlinear and ill-posed problem. This paper proposes a material-decomposition-based approach to directly solve the reconstruction problem, without estimating the energy-binned sinograms. This approach assumes that the linear attenuation coefficient map of objects can be decomposed into a few basis materials that are separable in the spectral and space domains. The nonlinear problem is then converted to the reconstruction of the mass density maps of the basis materials. The dimensionality of the optimization variables is thus effectively reduced to overcome the ill-posedness. An alternating minimization scheme is used to solve the reconstruction with regularizations of weighted nuclear norm and total variation. Compared to the state-of-the-art reconstruction method for coded spectral CT, the proposed method can significantly improve the reconstruction quality. It is also capable of reconstructing the spectral CT images at two additional energy bins from the same set of measurements, thus providing more spectral information of the object.

    

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