Fast Exact Reconstruction Algorithms For Multisource Cone-Beam CT

The pursuance for higher temporal resolution has largely driven the development of computed tomography, and recently led to the emergence of Siemens dual-source CT scanner in 2005. Given the impact and limitation of dual-source CT, multi-source geometry seems a promising mode for next generation of CT scanner. On the other side, exact algorithms are developing very fast these years after the great breakthrough made by Katsevich in 2002.Inspired by the great success of dual-source CT and Katsevich's algorithm, in this work we are investigating the fast exact algorithms for multi-source cone-beam CT. The main contributions of this dissertation are as follows.(1) We have proposed an exact shift-invariant filtered backprojection (FBP) algorithm for triple-source saddle-curve cone-beam CT. In this imaging geometry, the X-ray sources are symmetrically positioned along a circle, and the trajectory of each source is a saddle curve. Then, we extend Yang's formula from the single-source case to the triple-source case. The saddle curves can be divided into four parts to yield four datasets. Each of them contains three data segments associated with different saddle curves, respectively. Images can be reconstructed on the planes orthogonal to the z-axis. Each plane intersects the trajectories at six points (or three points at the two ends) which can be used to define the filtering directions. Then, we discuss the properties of these curves and study the case of 2N + 1 sources (N≥2). A new concept of efficient curves is defined. And a necessary condition and a sufficient condition are given to find efficient curves. Finally, we perform numerical simulations to demonstrate the feasibility of our triplesource saddle-curve approach. The results show that the triple-source geometry is advantageous for high temporal resolution imaging, especially important for cardiac imaging and small animal imaging.(2) We have proposed two filtered-backprojection algorithms for triple-source helical cone-beam CT. Based on Prof. Zhao's work on triple-helixes geometry, we find and prove the relationship among three inter-PI lines. A-curve, T -curve and Bs -curve are introduced and extended. A new auxiliary named L -curve is also defined. We analyze in detail the distribution of the intersections of the Radon planes and the triple-helixes trajectories, and design two weight functions for two fast FBP algorithms. The first algorithm utilizes two families of filtering lines. These lines are parallel to the tangent of the scanning trajectory and the so-called L lines. The second algorithm utilizes two families of filtering lines tangent to the boundaries of the Zhao window and L lines, respectively, but it eliminates the filtering paths along the tangent of the scanning trajectory, thus reducing the required detector size greatly. The first algorithm is theoretically exact for r < 0.265R and quasi-exact for 0.265 R≤r < 0.495R, and the second algorithm is quasi-exact for r < 0.495R, where r and R denote the object radius and the trajectory radius, respectively. Both algorithms are computationally efficient. Numerical results are presented to verify and showcase the proposed algorithms.(3) We have proposed an aglortihm to reconstruct an image from overlapped projections so that the data acquisition process can be shortened while the image quality remains essentially uncompromised. To perform image reconstruction from overlapped projections, the conventional reconstruction approach (e.g., FBP) cannot be directly used because of two problems. First, overlapped projections represent an imaging system in terms of summed exponentials, which cannot be transformed into a linear form. Second, the overlapped measurement carries less information than the traditional line integrals. To meet these challenges, we propose a compressive sensing-(CS-) inspired iterative algorithm for reconstruction from overlapped data. This algorithm starts with a good initial guess, relies on adaptive linearization, and minimizes the total variation (TV). Then, we demonstrated the feasibility of this algorithm in numerical tests.The contents of this dissertation cover the crucial issues in recent CT research, including helical scanning trajectory, saddle-curve scanning trajectory, multi-source geometry and compressive sensing technique, and pay special attentions to the algorithms' computational efficinecy. The results of our work are the foundation for the development of future CT systems.