X-ray Computed Tomography for Medical Imaging

Themathematical principles of computed tomography (CT) were first investigated by J. Radon [28] as early as 1917 and were later extended to complex fields by Kirillov [22]. The first application of tomography was, surprisingly, in radio astronomy [2], and today, it finds usage in diverse fields such as medical imaging, seismology, and underwater acoustic imaging. In medical imaging, CT has had a tremendous impact in noninvasive diagnostics, surgical planning, etc., as a diagnostic tool. Newer scanning techniques such as the spiral CT are being used that have extended the traditional CT technology. There is vigorous ongoing research in cone-beam CT, in which the mathematical principles are being understood arid extended for practical scanner implementations. In this article we will exclusively deal with nondiffractive-transmission CT. Tomography literally means "slice" or cross-sectional imaging. The central idea is to reconstruct an object from planar integrals of the data. In an n-dimensional world, the object is reconstructed from data obtained by integration along hyper-planes intersecting it. In 2-D, the hyper-plane integrals degenerate to line integrals. In this article, we shall limit our discussion to 2-D and 3-D objects. A 3-D object can be examined in two ways. o The object can be visualized as a stack of 2-D slices. • The object is examined in its natural 3-D representation. Various scanning geometries have evolved in 2-D CT. The original CT theory was developed for parallel beam geometries. In the second-generation scanners, fan-beam geometries were used. In this generation of scanners, the detectors were placed on an arc of a circle or a straight line, and the source-detector assembly rotated around the object. In the third-generation scanners, the detectors were placed on a complete circular ring and the x-ray source rotated around the object. The fourth and the current generation of scanners use the spiral or helical CT technology. Practical 3-D scanners are yet to be built. Various prototypes such as Mayo Clinic's Dynamic Spatial Reconstructor (DSR) and Imatron Inc.'s Electron Beam CT have been developed and show promise, but they are not true 3-D scanners. In this article we will examine the physical and mathematical concepts of the Radon transform, and the basic parallel beam reconstruction algorithms are discussed. We also develop the algorithms for fan-beam CT, and we briefly discuss the mathematical principles of cone-beam CT. 1. (a)A 3-D object within a sphere of radius R. One octant of the sphere is depicted. (b) The same object is shown as a stack of slices along the z direction. f(x,y) = w(x,y,zk) is one such slice at z