Start explaining the projection slice theorem

project.md

@@ -7,3 +7,32 @@ title: Final Project
 ## 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
+
+$$
+\begin{align*}
+\hat{f}(\hat{x}, 0)
+&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i \hat{x} x} dx dy \\
+&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i \hat{x} x} dx \\
+&= \int_\mathbb{R} p(x) e^{-2 \pi i \hat{x} x} dx \\
+&= \hat{p}(\hat{x})
+\end{align*}
+$$