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title: Final Project
---
# CT Imaging from Scratch
## Background
- [Radon Transform](http://www-math.mit.edu/~helgason/Radonbook.pdf) 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
\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)