Computerized Tomography

The word "tomography" comes from the Greek word τoμoσ, meaning slice. Computerized tomography is a medical imaging technique that deals with reconstruction of a bivariate f(x, y) or trivariate f(x, y, z) function from its line integrals.

The idea in computerized tomography is to construct an image of some part of the human body using a sufficient number of X-ray beams in many directions by measuring the intensity difference between the incoming and outgoing rays. In this manner, we create an image whose gray scale values are consistent with the measurements. Using Beer's Law, changes in intensity of an X-ray beam provide the line integral (along the line connecting the initial position and final position of the beam) of the attenuation coefficient function. The attenuation coefficient measures the ability of a medium to absorb energy from the beam. The attenuation coefficient functions for the 2D and 3D cases are represented as A(x,y) and A(x,y,z), respectively. After the reconstruction process for a particular point (x,y) we should obtain a value for A(x,y) which is a number from 0 to 1 that represents the gray scale value of that point.

Using the Radon transform, each of these line integrals represents the Radon transform of the line. What we have at the beginning of the reconstruction process is the Radon transform of the object being reconstructed. In practice, there are several ways to reconstruct the Radon transform once you have its values for a finite number of lines. The most widely used method is the filtered-back projection algorithm, which includes filtering of the Radon transform for the purpose of removing high frequency components followed by a back-projection process. The back projection of a point gives the average value of all the line integrals averaged over all lines passing through that point.

Beam characteristics depend on dimensionality of the imaging. The 2D scanning process can use parallel beam or fan beam tomography, whereas 3D scanning utilizes cone beam tomography.

Don't be afraid of the mathematics; it is nothing more than applied calculus. A good place to start would be the free ebook The Mathematics of Medical Imaging: A Beginner's Guide. Chapter 3 from the book Principles of Computerized Tomographic Imaging illustrates parallel beam, fan beam and cone beam tomography reconstruction processes.

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