project.md

---
title: Final Project
---

# Computed Tomography


## Background

- [Radon Transform](http://www-math.mit.edu/~helgason/Radonbook.pdf) by Sigurdur Helgason


## Introduction

Computed Tomography (CT) is used to turn 2d projections of a 3d shape like these (TODO insert image)

into a 3d model like this (TODO insert image).

This is pretty cool. In particular, voxels in the computed model can tell you the density at
specific locations *inside* the scanned object, whereas pixels in the projections can only tell you
the average density along lines that pass *through* the object. So we're getting a lot more out of
those 2d images than meets the eye.

How does it work? That's what I'd like to understand. In particular, I hope to clearly explain the
basic principles of CT, use them to implement a basic reconstruction from scratch, and experiment
with existing solutions to get a sense of the state of the art.

I'm going to describe the algorithms in 2d, since it will make everything simpler. In this case we
have an image like this (TODO insert image), and we'd like to reconstruct it only knowing its 1d
projections at different angles.


## Fourier Reconstruction

### Fourier Slice Theorem

Let $$f : \mathbb{R}^2 \rightarrow \mathbb{R}$$ be a density function. If we map density to
brightness we can view $$f$$ as describing an image. We'll assume that this function is defined
everywhere, but is always zero outside some finite neighborhood of the origin (say, the bounds of
the image).

The projection of $$f$$ to the x axis is obtained by integrating along y:

$$
p(x) = \int_\mathbb{R} f(x, y) dy
$$

Meanwhile, the [Fourier transform](../notes/fourier_transform.html) of $$f$$ is

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

Now comes the key insight. The slice along the $$u$$ axis in frequency space is

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

So the Fourier transform of the 1d projection is a 1d slice through the 2d Fourier transform of the
image. Since the x axis is arbitrary (we can rotate the image however we want), this works for other
angles as well. That is, if you project $$f$$ onto the line that makes an angle $$\theta$$ with the
x axis, its Fourier transform is equal to the slice along the corresponding line in frequency space.
This result is known as the Fourier Slice Theorem, and is the foundation of most reconstruction
techniques.

Conceptually this tells us everything we need to know about the reconstruction. First we take the 1D
Fourier transform of each projection. Then we combine them by arranging them radially. Finally we
take the inverse Fourier transform of the resulting 2d function. We'll end up with the reconstructed
image.

It also tells us how to generate the projections, given that we don't have a 1d x-ray machine. First
we take the Fourier transform of the image. Then we extract radial slices from it. Finally we take
the inverse Fourier transform of each slice. These are the projections. This will come in handy for
generating testing data.


### Discretization

Naturally the clean math of the theory has be modified a bit to make room for reality. In
particular, we only have discrete samples of $$f$$ (i.e. pixel values) rather than the full
continuous function (which in our current formalism may contain infinite information). This has two
important implications.

First, we'll want our Fourier transforms to be discrete Fourier transforms (DFTs). Luckily the
continuous and discrete Fourier transforms are effectively interchangeable, as long as the functions
we work with are mostly spatially and bandwidth limited, and we take an appropriate number of
appropriately spaced samples. You can read more about these requirements
[here](../notes/fourier_series.html).

Second, since we combine the DFTs of the projections radially, we'll end up with samples (of the 2D
Fourier transform of our image) on a polar grid rather than a cartesian one. So we'll have to
interpolate. This step is tricky and tends to introduce a lot of error. I'll rather arbitrarily use
[Catmull-Rom interpolation](http://entropymine.com/imageworsener/bicubic/), but there are better
algorithms out there that come closer to the theoretically ideal
[sinc interpolation](https://en.wikipedia.org/wiki/Whittaker%E2%80%93Shannon_interpolation_formula).
The popular [gridrec](https://www.ncbi.nlm.nih.gov/pubmed/23093766) method is one.