Notes for ECEN 3400 Edward

The word tomography comes from the Greek words tomo (tomos: a slice or cross section) and grafein (graphein: to write). It means a technique for producing cross-sectional images, usually of the human body for medical diagnostics, but it is also used in many other applications. The data acquired by X-ray scans of biological tissues and other materials is essentially a set of projections—the shadows cast by obstructions to the propagation of the X-rays. Such projections provide some information, but important objects can be hidden behind others and remain undetected in the traditional X-ray diagnostics. The data from projections can be enhanced in its usefulness by doing supplemental computations with it. We describe below a (vastly oversimplified) method for doing this Computed Axial Tomography or Computer-Aided Tomography. The acronym (CAT) is used for both, more familiarly they are called CAT-scans. The method outlined here for computer-processing of the data is called the Algebraic Reconstruction Technique (ART). We divide the region to be scanned into N pixels as shown in the figure below. Each pixel is assumed to be small enough that it is approximately uniform in its material properties. At X-ray frequencies, a plane wave is reflected very little from most materials (the atoms and molecules cannot respond fast enough to have an ǫ or μ significantly different from their free space values). However, because typical samples are huge in size compared to a wavelength (at X-ray frequencies, the wavelength ranges from 0.25 to 100 Å; 1 Å = 10 m), the attenuation of a plane wave in a material is important, and we denote by αi the attenuation constant of the material in the ith pixel. An X-ray emitter sends waves through this region along various paths called rays, as indicated. Even though the waves are in the form of narrow beams, they still behave locally like plane waves. The intensity (i. e., power) carried by the wave is measured by a detector after it has passed through the region to the opposite side from the emitter. The ratio of power measured to power emitted will be due to the cumulative effect of attenuation through each of the pixels the ray passes through: