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Erik Strand authoredErik Strand authored
project.md 949 B
title: Final Project
CT Imaging from Scratch
Background
- Radon Transform by Sigurdur Helgason
2d Reconstruction
In two dimensions, the theory of image reconstruction from projections is pretty simple. Assume some density function
f : \mathbb{R}^2 \rightarrow \mathbb{R}
(with compact support). The projection
of this density function to the x axis is
p(x) = \int_\mathbb{R} f(x, y) dy
Meanwhile, the Fourier transform of
f
is
\hat{f}(u, v)
= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i (u x + v y)} dx dy
Note that the slice along the
u
axis in frequency space is described by
\begin{align*}
\hat{f}(u, 0)
&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i u x} dx dy \\
&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i u x} dx \\
&= \int_\mathbb{R} p(x) e^{-2 \pi i u x} dx \\
&= \hat{p}(u)
\end{align*}