Clean up discussion of Fourier reconstruction

_notes/fourier_series.md

@@ -81,7 +81,7 @@ $$
 ## Derivation of the Discrete Time Fourier Transform
 
 We can apply the above results to $$\hat{f}$$ as well. Recall that the Fourier transform of
-$$\hat{f}$$ is $$f(-x)$$. Thus the periodic summation of the Fourier transform of $$f$$ is
+$$\hat{f}$$ is $$f(-x)$$. Thus the periodic summation of $$\hat{f}$$ is
 
 $$
 \begin{align*}

project.md

@@ -12,37 +12,40 @@ title: Final Project
 
 ## Introduction
 
-Computed Tomography (CT) can turn 2d projections of a 3d shape like these (TODO insert image)
+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 images than meets the eye.
+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.
 
-## Theory
 
-Let's start 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
 
-We can describe the image as a density function $$f : \mathbb{R}^2 \rightarrow \mathbb{R}$$. Think
-of $$f(x, y)$$ as the brightness of the image at $$(x, y)$$. We'll assume that this function is
-defined everywhere, but is always zero outside our image.
+### Fourier Slice Theorem
 
-The projection of this density function to the x axis is obtained by integrating along y:
+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 of $$f$$ is
+Meanwhile, the [Fourier transform](../notes/fourier_transform.html) of $$f$$ is
 
 $$
 \hat{f}(u, v)
@@ -63,12 +66,14 @@ $$
 
 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 take the projection onto the line that makes an angle $$\theta$$
-with the x axis, its Fourier transform is equal to the slice along the same line in frequency space.
-
-Conceptually this tells us everything we need to know about the reconstruction. First we take the
-Fourier transform of each projection individually. 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
+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
@@ -77,10 +82,23 @@ the inverse Fourier transform of each slice. These are the projections. This wil
 generating testing data.
 
 
-## Discretization
+### 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).
 
-Of course the clean math of the theory has be modified a bit to make room for reality. First of all,
-our projections will be represented as discrete pixels, not perfect functions. So we'll want our
-Fourier transforms to be DFTs. And when we combine the transforms of the projections, we won't
-naturally end up with a nice 2d grid of values on which to perform an inverse DFT. So we'll have to
-interpolate.
+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.