CT problem set for undergraduate Medical Physics course.
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\section{Radon Transform of a Gaussian}
Calculate the Radon transform of $p(\xi, \phi)$ of $f(x,y) = e^{-x^2 - y^2}$. (Hint: there is symmetry you can exploit to simplify this problem).
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\section{Radon Transform of Shifted Function}
Show that if the Radon transform of $f(x,y)$ is $p(\xi,\phi)$, then the Radon transform of $f(x-x_0,y-y_0)$ is $p(\xi-x_0\cos{\phi} - y_0\sin{\phi})$. Also give a graphical explanation of this result. [Hint: this is somewhat easier to prove if you use the delta function form of the Radon transform given in the book: $p(\xi,\phi) = \int_{-\infty}^\infty \int_{-\infty}^{\infty} f(x,y)\delta(x\cos{\phi} + y\sin{\phi} - \xi)\,dxdy$.]
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\section{Radon transform consistency conditions}
Let $p(\xi,\phi)$ be a parallel-beam sinogram and $P_\phi(\nu)$ be its 1D Fourier transform with respect to $\xi$ for fixed $\phi$, as defined in the lecture. Show that
$$
P_{\phi + \pi}(\nu) = P_{\phi}(-\nu)
$$
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\section{Problem 6.12 from Prince book}
Consider an object comprising two small metal pellets located at $(x,y) = (2,0)$ and $(2,2)$ and a pice of wire stretched straight between $(0,-2)$ and $(0,0)$.\\
(a) Sketch this object. Assume $N$ photons are fired at each lateral position $\ell$ in a parallel-ray configuration. For simplicity, assumes that each metal object stops 1/2 the photons that are incident upon it no matter what angle it is hit.\\
(b) Sketch the number of photons you would expect to see as a function of $\ell$ for $\theta = 0^\circ$ and $\theta = 90^\circ$. \\
(c) Draw the projections you would see at $\theta = 0^\circ$ and $\theta = 90^\circ$.\\
(d) Sketh the backprojection image you would get at $\theta = 0^\circ$ (without filtering).
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