System of linear equations in Computed Tomography (CT)

Abstract

In X-rays the internal structures often overlap, thereby it reduces the chances of detecting any anomalies (if any) and hence the patient may be given an improper diagnosis or they will have to suffer due to participating in multiple scans, which is inevitable if their illness continues and the radiologists are unable to find a cause. In that aspect Computed Tomography (CT), uses multiple X-rays at variable angles directed at a section of the patient’s body, to acquire images (single X-ray shot gives image from a specific angle) of the entire cross section. These images are then taken by the scanner and the data (all the images) is then processed to create a detailed image of the cross section. This image is very detailed and so chances of detecting anomaly (if any) goes up. This thesis focuses on representing the scans of CT using a field of linear algebra, namely the System of Linear Equations to be more specific. That is to say, in this thesis we acquire a system of linear equations using the arbitrary data collected after sending and detecting X-ray beams at a patient’s cross section (slice), then solving it using Gaussian Elimination and the Matrix Inversion method to acquire roots or solution values which represent the substances present in the slice. These values are then cross referenced with a table that shows the range of values (linearly attenuated for this arbitrary scan) that represent various substances like bones, tissues (healthy or tumorous) and metallic objects that may or may not be present in the slice. Thus by using this table we can identify as to what the roots actually are within the slice.