CAT Scan Imaging
OverviewTomography means reconstruction of an image from its sections. The word comes from the greek ‘‘tomos’’ (‘‘slice’’) and ‘‘graph’’ (‘‘description’’). The problem arises in many fields, ranging from astronomy to medical imaging. Computerized Axial Tomography (CAT) is a medical imaging method that processes large amounts of two-dimensional X-ray images in order to produce a three-dimensional image. The goal is to picture for example the tissue density of the different parts of the brain, in order to detect anomalies (such as brain tumors). Typically, the X-ray images represent ‘‘slices’’ of the part of the body (such as the brain) that is examined. Those slices are indirectly obtained via axial measurements of X ray attenuation, as explained below. Thus, in CAT for medical imaging, we use axial (line) measurements to get two-dimensional images (slices), and from that scan of images, we may proceed to digitally reconstruct a three-dimensional view. Here, we focus on the process that produces a single two-dimensional image from axial measurements.
From 1D to 2D: axial tomographyIn CAT-based medical imaging, a number of X rays are sent through the tissues to be examined along different directions, and their intensity after they have traversed the tissues is captured by a camera. For each direction, we record the attenuation of the X ray, by comparing the intensity of the X ray at the source, , to the intensity after the X ray has traversed the tissues, at the receiver's end, . Linear equations for a single sliceSimilar to the Beer-Lambert law of optics, it turns out that, to a reasonable degree of approximation, the log-ratio of the intensities at the source and at the receiver is linear in the densities of the tissues traversed. With the discretization, the linear relationship between intensities log-ratios and densities can be expressed as where denotes the indices of pixel areas traversed by the X ray, the density in the area, and the proportion of the area within the pixel that is traversed by the ray. Thus, we can relate the vector to the observed intensity log-ratio vector in terms of a linear equation where , with . Note that depending on the number of pixels used, and the number of measurements, the matrix can be quite large. In general, the matrix is fat, in the sense that it has (many) more columns than rows (). Thus, the above system of equations is usually undetermined. In the example pictured above, we have |