The Mathematics of Computerized Tomography

Computerized Tomography (CT) is a medical imaging technique that uses X-rays and computer algorithms to produce detailed images of the body. The mathematics behind CT involves the use of mathematical transforms such as the Radon transform and the Fourier transform.

The Radon transform is used to convert a 2D image into a set of line integrals. This transform is the basis for the creation of CT images. A CT scanner rotates around the patient and takes multiple X-ray images from different angles. The Radon transform is applied to each of these images to obtain a set of line integrals that represent the attenuation of X-rays through the patient's body.

The Fourier transform is used to convert the line integrals obtained from the Radon transform into a 3D image. The Fourier transform is applied to the set of line integrals to obtain a set of frequency components. These frequency components represent the different spatial frequencies present in the image. By inverting the Fourier transform, a 3D image can be reconstructed from the frequency components.

The mathematics of CT also involves the use of algorithms to process the data obtained from the CT scanner. One of the most commonly used algorithms is the filtered back projection algorithm. This algorithm involves filtering the line integrals obtained from the Radon transform to remove noise and other artifacts. The filtered line integrals are then back-projected to obtain a 3D image.

In summary, the mathematics of CT involves the use of mathematical transforms such as the Radon transform and the Fourier transform, as well as algorithms to process the data obtained from the CT scanner. These mathematical techniques are essential for the creation of detailed and accurate CT images, which are used in the diagnosis and treatment of a wide range of medical conditions.