diff --git a/project.md b/project.md index 2e878471c552009003bcb1e3f4ddab70179a22fe..ccd904f4dc98d20e9c74e5fd2e0195fd90926631 100644 --- a/project.md +++ b/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 + +$$ +\begin{align*} +\hat{f}(\hat{x}, 0) +&= \int_\mathbb{R} \int_\mathbb{R} f(x, y) e^{-2 \pi i \hat{x} x} dx dy \\ +&= \int_\mathbb{R} \left( \int_\mathbb{R} f(x, y) dy \right) e^{-2 \pi i \hat{x} x} dx \\ +&= \int_\mathbb{R} p(x) e^{-2 \pi i \hat{x} x} dx \\ +&= \hat{p}(\hat{x}) +\end{align*} +$$