Image reconstruction and geometric modeling in computed tomography

Computed tomography (CT) is an important area in the modern science and technology. This dissertation focuses on the development of innovative CT theory, reconstruction algorithms, and geometric modeling techniques. Our main results can be summarized into the following five aspects. (1) When the spectrum of an underlying image is not absolutely integrable, we use the method of limited bandwidth to analyze filtered backprojection-based image reconstruction. Our analytic findings improve the understanding on the limit behavior of the filtered backprojection algorithms. (2) Nonstandard spiral cone-beam scanning trajectories are needed for cutting edge CT research, such as bolus-chasing CT angiography and electron-beam micro-CT which are pioneered by our CT/Micro-CT laboratory. We generalize the Tam-Danielsson window into the case of nonstandard spiral cone beam scanning, show that PI-line exists, and find the sufficient and necessary condition for the uniqueness of the PI-line. These results are a prerequisite for exact cone-beam reconstruction with general scanning curves. (3) Feldkamp-type and Katsevich-type algorithms are popular methods for approximate and exact image reconstruction, respectively. Numerical studies are performed on these two types of algorithms for cone-beam reconstruction with variable radius spiral loci. It is observed that they produce similar image quality if the cone angle is not large and/or there is no sharp density change along the z-direction. The Katsevich-type algorithm is generally preferred due to its exactness. (4) Geometric modeling plays an indispensable role in the evaluation of CT reconstruction algorithms. We formulate the X-ray transform and 3D Radon transform for arbitrarily positioned ellipsoids and tetrahedra. These formulas are used in our projects for development of various cone-beam algorithms. (5) It is highly desirable in the CT simulation to have anatomically realistic mathematical phantoms. Superquadrics are a family of three-dimensional objects, which can be used to model a variety of anatomical structures. We propose an algorithm for computation of X-ray transforms for superellipsoids and tori. Their usefulness and efficiency are demonstrated by projection generation and image reconstruction of a superquadric-based thorax phantom. Our data indicate that superquadric modeling is more realistic than the quadratic counterpart, and faster than the spline methods.