project.md

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}(\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*}