Four-dimensional Cone Beam CT Reconstruction and Enhancement using a Temporal Non-Local Means Method

Four-dimensional Cone Beam Computed Tomography (4D-CBCT) has been developed to provide respiratory phase resolved volumetric imaging in image guided radiation therapy (IGRT). Inadequate number of projections in each phase bin results in low quality 4D-CBCT images with obvious streaking artifacts. In this work, we propose two novel 4D-CBCT algorithms: an iterative reconstruction algorithm and an enhancement algorithm, utilizing a temporal nonlocal means (TNLM) method. We define a TNLM energy term for a given set of 4D-CBCT images. Minimization of this term favors those 4D-CBCT images such that any anatomical features at one spatial point at one phase can be found in a nearby spatial point at neighboring phases. 4D-CBCT reconstruction is achieved by minimizing a total energy containing a data fidelity term and the TNLM energy term. As for the image enhancement, 4D-CBCT images generated by the FDK algorithm are enhanced by minimizing the TNLM function while keeping the enhanced images close to the FDK results. A forward-backward splitting algorithm and a Gauss-Jacobi iteration method are employed to solve the problems. The algorithms are implemented on GPU to achieve a high computational efficiency. The reconstruction algorithm and the enhancement algorithm generate visually similar 4D-CBCT images, both better than the FDK results. Quantitative evaluations indicate that, compared with the FDK results, our reconstruction method improves contrast-to-noise-ratio (CNR) by a factor of 2.56~3.13 and our enhancement method increases the CNR by 2.75~3.33 times. The enhancement method also removes over 80% of the streak artifacts from the FDK results. The total computation time is ~460 sec for the reconstruction algorithm and ~610 sec for the enhancement algorithm on an NVIDIA Tesla C1060 GPU card.Comment: 20 pages, 3 figures, 2 table

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

Inversion of the star transform

We define the star transform as a generalization of the broken ray transform introduced by us in previous work. The advantages of using the star transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium separately and simultaneously (from the same data) and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is discarded. In this paper, we derive the star transform from physical principles, discuss its mathematical properties and analyze numerical stability of inversion. In particular, it is shown that stable inversion of the star transform can be obtained only for configurations involving odd number of rays. Several computationally-efficient inversion algorithms are derived and tested numerically.Comment: Accepted to Inverse Problems in this for

Artefact Reduction Methods for Iterative Reconstruction in Full-fan Cone Beam CT Radiotherapy Applications

A cone beam CT (CBCT) system acquires two-dimensional projection images of an imaging object from multiple angles in one single rotation and reconstructs the object geometry in three dimensions for volumetric visualization. It is mounted on most modern linear accelerators and is routinely used in radiotherapy to verify patient positioning, monitor patient contour changes throughout the course of treatment, and enable adaptive radiotherapy planning. Iterative image reconstruction algorithms use mathematical methods to iteratively solve the reconstruction problem. Iterative algorithms have demonstrated improvement in image quality and / or reduction in imaging dose over traditional filtered back-projection (FBP) methods. However, despite the advancement in computer technology and growing availability of open-source iterative algorithms, clinical implementation of iterative CBCT has been limited. This thesis does not report development of codes for new iterative image reconstruction algorithms. It focuses on bridging the gap between the algorithm and its implementation by addressing artefacts that are the results of imperfections from the raw projections and from the imaging geometry. Such artefacts can severely degrade image quality and cannot be removed by iterative algorithms alone. Practical solutions to solving these artefacts will be presented and this in turn will better enable clinical implementation of iterative CBCT reconstruction

Automatic alignment for three-dimensional tomographic reconstruction

In tomographic reconstruction, the goal is to reconstruct an unknown object from a collection of line integrals. Given a complete sampling of such line integrals for various angles and directions, explicit inverse formulas exist to reconstruct the object. Given noisy and incomplete measurements, the inverse problem is typically solved through a regularized least-squares approach. A challenge for both approaches is that in practice the exact directions and offsets of the x-rays are only known approximately due to, e.g. calibration errors. Such errors lead to artifacts in the reconstructed image. In the case of sufficient sampling and geometrically simple misalignment, the measurements can be corrected by exploiting so-called consistency conditions. In other cases, such conditions may not apply and we have to solve an additional inverse problem to retrieve the angles and shifts. In this paper we propose a general algorithmic framework for retrieving these parameters in conjunction with an algebraic reconstruction technique. The proposed approach is illustrated by numerical examples for both simulated data and an electron tomography dataset

Preasymptotic Convergence of Randomized Kaczmarz Method

Kaczmarz method is one popular iterative method for solving inverse problems, especially in computed tomography. Recently, it was established that a randomized version of the method enjoys an exponential convergence for well-posed problems, and the convergence rate is determined by a variant of the condition number. In this work, we analyze the preasymptotic convergence behavior of the randomized Kaczmarz method, and show that the low-frequency error (with respect to the right singular vectors) decays faster during first iterations than the high-frequency error. Under the assumption that the inverse solution is smooth (e.g., sourcewise representation), the result explains the fast empirical convergence behavior, thereby shedding new insights into the excellent performance of the randomized Kaczmarz method in practice. Further, we propose a simple strategy to stabilize the asymptotic convergence of the iteration by means of variance reduction. We provide extensive numerical experiments to confirm the analysis and to elucidate the behavior of the algorithms.Comment: 20 page

Development and Implementation of Fully 3D Statistical Image Reconstruction Algorithms for Helical CT and Half-Ring PET Insert System

X-ray computed tomography: CT) and positron emission tomography: PET) have become widely used imaging modalities for screening, diagnosis, and image-guided treatment planning. Along with the increased clinical use are increased demands for high image quality with reduced ionizing radiation dose to the patient. Despite their significantly high computational cost, statistical iterative reconstruction algorithms are known to reconstruct high-quality images from noisy tomographic datasets. The overall goal of this work is to design statistical reconstruction software for clinical x-ray CT scanners, and for a novel PET system that utilizes high-resolution detectors within the field of view of a whole-body PET scanner. The complex choices involved in the development and implementation of image reconstruction algorithms are fundamentally linked to the ways in which the data is acquired, and they require detailed knowledge of the various sources of signal degradation. Both of the imaging modalities investigated in this work have their own set of challenges. However, by utilizing an underlying statistical model for the measured data, we are able to use a common framework for this class of tomographic problems. We first present the details of a new fully 3D regularized statistical reconstruction algorithm for multislice helical CT. To reduce the computation time, the algorithm was carefully parallelized by identifying and taking advantage of the specific symmetry found in helical CT. Some basic image quality measures were evaluated using measured phantom and clinical datasets, and they indicate that our algorithm achieves comparable or superior performance over the fast analytical methods considered in this work. Next, we present our fully 3D reconstruction efforts for a high-resolution half-ring PET insert. We found that this unusual geometry requires extensive redevelopment of existing reconstruction methods in PET. We redesigned the major components of the data modeling process and incorporated them into our reconstruction algorithms. The algorithms were tested using simulated Monte Carlo data and phantom data acquired by a PET insert prototype system. Overall, we have developed new, computationally efficient methods to perform fully 3D statistical reconstructions on clinically-sized datasets