Analytic tomography by Andrew Markoe

By Andrew Markoe

This ebook is a finished examine of the Radon remodel, which operates on a functionality by means of integrating it over hyperplanes. The e-book starts off with an hassle-free and graphical creation to the Radon remodel, tomography and CT scanners, through a rigorous improvement of the fundamental homes of the Radon remodel. subsequent the writer introduces Grassmann manifolds within the examine of the k-plane rework (a model of the Radon remodel) which integrates over k-dimensional planes instead of hyperplanes. the rest chapters are inquisitive about extra complex themes.

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Sample text

Therefore, the number of photons that exit the last plate on the right is less than the number that entered on the left. This experiment was repeated by using many different materials for the plates. In each case it was observed that the number of photons that did not survive the trip over a small distance d is a fraction of the number beginning the trip. However, over a large distance, the relationship between the number of photons entering the material and the number exiting is more intricate.

From the definition, µd = fraction of surviving photons, provided the distance d is small. 003 = 1,000 , so 3 of every 1,000 photons will be blocked while traversing the distance d. 7% of the photons will survive the passage of the small distance d. Let N0 denote the number of photons entering the material and let N1 denote the number of photons exiting. 01 before exiting, we had N0 = 1,000 and N1 = 997. 14 This result holds only for monoenergetic x-ray beams. The x-ray beams used in CT scanners actually consist of photons of many different energies.

The resulting quantity, µ, is the attenuation coefficient of the material. 05 cm. 5 10,000 and we see that we get the same attenuation coefficient. 34 leads to the Lambert-Beer law. One aspect of the Lambert-Beer law is that the attenuation coefficient is constant for homogeneous materials. Of course each material has a different attenuation coefficient. To obtain the Lambert-Beer law, we need to use the idea of the attenuation coefficient in the following way. From the definition, µd = fraction of surviving photons, provided the distance d is small.

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