Estimating the Spectrum in Computed Tomography Via Kullback–Leibler Divergence Constrained Optimization

Purpose We study the problem of spectrum estimation from transmission data of a known phantom. The goal is to reconstruct an x‐ray spectrum that can accurately model the x‐ray transmission curves and reflects a realistic shape of the typical energy spectra of the CT system. Methods Spectrum estimation is posed as an optimization problem with x‐ray spectrum as unknown variables, and a Kullback–Leibler (KL)‐divergence constraint is employed to incorporate prior knowledge of the spectrum and enhance numerical stability of the estimation process. The formulated constrained optimization problem is convex and can be solved efficiently by use of the exponentiated‐gradient (EG) algorithm. We demonstrate the effectiveness of the proposed approach on the simulated and experimental data. The comparison to the expectation–maximization (EM) method is also discussed. Results In simulations, the proposed algorithm is seen to yield x‐ray spectra that closely match the ground truth and represent the attenuation process of x‐ray photons in materials, both included and not included in the estimation process. In experiments, the calculated transmission curve is in good agreement with the measured transmission curve, and the estimated spectra exhibits physically realistic looking shapes. The results further show the comparable performance between the proposed optimization‐based approach and EM. Conclusions Our formulation of a constrained optimization provides an interpretable and flexible framework for spectrum estimation. Moreover, a KL‐divergence constraint can include a prior spectrum and appears to capture important features of x‐ray spectrum, allowing accurate and robust estimation of x‐ray spectrum in CT imaging

Blind polychromatic X-ray CT reconstruction from Poisson measurements

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 (LBFGS- 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

System Characterizations and Optimized Reconstruction Methods for Novel X-ray Imaging

In the past decade there have been many new emerging X-ray based imaging technologies developed for different diagnostic purposes or imaging tasks. However, there exist one or more specific problems that prevent them from being effectively or efficiently employed. In this dissertation, four different novel X-ray based imaging technologies are discussed, including propagation-based phase-contrast (PB-XPC) tomosynthesis, differential X-ray phase-contrast tomography (D-XPCT), projection-based dual-energy computed radiography (DECR), and tetrahedron beam computed tomography (TBCT). System characteristics are analyzed or optimized reconstruction methods are proposed for these imaging modalities. In the first part, we investigated the unique properties of propagation-based phase-contrast imaging technique when combined with the X-ray tomosynthesis. Fourier slice theorem implies that the high frequency components collected in the tomosynthesis data can be more reliably reconstructed. It is observed that the fringes or boundary enhancement introduced by the phase-contrast effects can serve as an accurate indicator of the true depth position in the tomosynthesis in-plane image. In the second part, we derived a sub-space framework to reconstruct images from few-view D-XPCT data set. By introducing a proper mask, the high frequency contents of the image can be theoretically preserved in a certain region of interest. A two-step reconstruction strategy is developed to mitigate the risk of subtle structures being oversmoothed when the commonly used total-variation regularization is employed in the conventional iterative framework. In the thirt part, we proposed a practical method to improve the quantitative accuracy of the projection-based dual-energy material decomposition. It is demonstrated that applying a total-projection-length constraint along with the dual-energy measurements can achieve a stabilized numerical solution of the decomposition problem, thus overcoming the disadvantages of the conventional approach that was extremely sensitive to noise corruption. In the final part, we described the modified filtered backprojection and iterative image reconstruction algorithms specifically developed for TBCT. Special parallelization strategies are designed to facilitate the use of GPU computing, showing demonstrated capability of producing high quality reconstructed volumetric images with a super fast computational speed. For all the investigations mentioned above, both simulation and experimental studies have been conducted to demonstrate the feasibility and effectiveness of the proposed methodologies

Basis Vector Model Method for Proton Stopping Power Estimation using Dual-Energy Computed Tomography

Accurate estimation of the proton stopping power ratio (SPR) is important for treatment planning and dose prediction for proton beam therapy. The state-of-the-art clinical practice for estimating patient-specific SPR distributions is the stoichiometric calibration method using single-energy computed tomography (SECT) images, which in principle may introduce large intrinsic uncertainties into estimation results. One major factor that limits the performance of SECT-based methods is the Hounsfield unit (HU) degeneracy in the presence of tissue composition variations. Dual-energy computed tomography (DECT) has shown the potential of reducing uncertainties in proton SPR prediction via scanning the patient with two different source energy spectra. Numerous methods have been studied to estimate the SPR by dual-energy CT DECT techniques using either image-domain or sinogram-domain decomposition approaches. In this work, we implement and evaluate a novel DECT approach for proton SPR mapping, which integrates image reconstruction and material characterization using a joint statistical image reconstruction (JSIR) method based on a linear basis vector model (BVM). This method reconstructs two images of material parameters simultaneously from the DECT measurement data and then uses them to predict the electron densities and the mean excitation energies, which are required by the Bethe equation for computing proton SPR. The proposed JSIR-BVM method is first compared with image-domain and sinogram-domain decomposition approaches based on three available SPR models including the BVM in a well controlled simulation framework that is representative of major uncertainty sources existing in practice. The intrinsic SPR modeling accuracy of the three DECT-SPR models is validated via theoretical computed radiological quantities for various reference human tissues. The achievable performances of the investigated methods in the presence of image formation uncertainties are evaluated using synthetic DECT transmission sinograms of virtual cylindrical phantoms and virtual patients, which consist of reference human tissues with known densities and compositions. The JSIR-BVM method is then experimentally commissioned using the DECT measurement data acquired on a Philips Brilliance Big Bore CT scanner at 90 kVp and 140 kVp for two phantoms of different sizes, each of which contains 12 different soft and bony tissue surrogates. An image-domain decomposition method that utilizes the two HU images reconstructed via the scanner\u27s software is implemented for comparison The JSIR-BVM method outperforms the other investigated methods in both the simulation and experimental settings. Although all investigated DECT-SPR models support low intrinsic modeling errors (i.e., less than 0.2% RMS errors for reference human tissues), the achievable accuracy of the image- and sinogram-domain methods is limited by the image formation uncertainties introduced by the reconstruction and decomposition processes. In contrast, by taking advantage of an accurate polychromatic CT data model and a joint DECT statistical reconstruction algorithm, the JSIR-BVM method accounts for both systematic bias and random noise in the acquired DECT measurement data. Therefore, the JSIR-BVM method achieves much better accuracy and precision on proton SPR estimation compared to the image- and sinogram-domain methods for various materials and object sizes, with an overall RMS-of-mean error of 0.4% and a maximum absolute-mean error of 0.7% for test samples in the experimental setting. The JSIR-BVM method also reduces the pixel-wise random variation by 4-fold to 6-fold within homogeneous regions compared to the image- and sinogram-domain methods while exhibiting relatively higher spatial resolution. The results suggest that the JSIR-BVM method has the potential for better SPR prediction in clinical settings