project.md

---
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

$$
\hat{f}(\hat{x}, \hat{y})
= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i (\hat{x} x + \hat{y} y)} dx dy
$$

Note that the slice along the $$\hat{x}$$ axis in frequency space is described by

$$
\begin{align*}
\hat{f}(\hat{x}, 0)
&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i \hat{x} x} dx dy \\
&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i \hat{x} x} dx \\
&= \int_\mathbb{R} p(x) e^{-2 \pi i \hat{x} x} dx \\
&= \hat{p}(\hat{x})
\end{align*}
$$