13611 - Tomography Larry Shepp Prof of Statistics Rutgers...

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Tomography Larry Shepp Prof of Statistics Rutgers University
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Tomography The Radon transform is the key technology in CAT scanning, now used in every hospital since 1972. Nowadays the research frontier has shifted to MRI= magnetic resonance imaging. I will discuss both. An X-ray moves through an object of density f(x,y) at the point (x,y). It is absorbed or deflected with probability f(x,y) ds where ds is the element of length along a line, L, through (x,y). The chance it goes all the way along L is
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Why line integrals? Beer’s law says that the log of the ratio of input to detected X-ray photons is proportional to the line integral of the density along the straight line path of the X-ray beam. If an X-ray passes through an object of density f(s) at the point s, then the probability that it gets to s+ds given that it gets to s is 1-f(s) ds +o(ds). Multiplying all these probabilities proves Beer’s law. So the Radon transform, Pf(L), the line integral of a object with density (gm/cc), f(x,y) can be measured. Radon’s theorem does the rest.
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Rando phantom He has no neck. He is used to calibrate scanners. Note that X-ray images are better for finding cavities than to study brain tumors. Why is this?
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Interior tissue density Fat = .9 Bone = 2 Water = 1 Blood = 1.05 Tumor = 1.03 Gray matter = 1.02
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First commercial CAT scanner EMI 1972 Godfrey Hounsfield It measured one line integral at a time. The X-ray source is visible at the bottom, there is a detector at the top. It measured 100 line integrals and then rotated 1 degree and went back.
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Anatomical “phantom” model Hounsfield invented tomography but didn’t think of using an anatomical model. This idea turned out useful. The line integrals can be calculated exactly. Errors in algorithm can be separated from errors due to noise in data.
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The density values are chosen Note the “skull”, “ventricles”, “tumors” Seems pretty silly, but I got very lucky with this idea as we’ll see.
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The line integrals of the phantom If the line misses the head the integral is zero. The small “tumors” contribute only to the 4 th decimal place. Need many projections.
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transform, ie “reconstruct” The Fourier transform of the projection is equal to the two-dimensional Fourier transform of the object.
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13611 - Tomography Larry Shepp Prof of Statistics Rutgers...

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