Mathematical Methods of Imaging

Imaging techniques have a wide array of applications in various fields of technology. One such application is medical imaging, such as the use of Computerized Assisted Tomography (CAT) or Magnetic Resonance Imaging (MRI). The mathematical techniques behind such imaging technologies will be discussed. In particular, the data measured by the CAT scanners correspond to the Radon transform of the image function, which one would like to recover. However, the Radon transform presents the data in a sinusoidal form, which must then be modified into the image of the object by the Fourier transform, certain filtering, and the inverse Fourier transform. The back-projection will be applied on this modified set of data. This final data set should reflect the original object provided by the original measurements. The Radon transforms of several elementary shapes will be presented. Some variations on the Radon Transform will also be discussed, such as the effect of having limited data from the Radon Transform on the final reconstructed image.