From 3ad32bbd063fcc6a825bcbe554d8ff00c08e1d01 Mon Sep 17 00:00:00 2001
From: Erik Strand <erik.strand@cba.mit.edu>
Date: Wed, 22 May 2019 13:23:39 -0400
Subject: [PATCH] Add Fourier reconstruction results

---
 assets/img/project_brain.png                  | Bin 0 -> 279175 bytes
 assets/img/project_fourier_reconstruction.png | Bin 0 -> 225271 bytes
 assets/img/project_sinogram.png               | Bin 0 -> 368037 bytes
 project.md                                    |  34 ++++++++++++++++--
 4 files changed, 32 insertions(+), 2 deletions(-)
 create mode 100644 assets/img/project_brain.png
 create mode 100644 assets/img/project_fourier_reconstruction.png
 create mode 100644 assets/img/project_sinogram.png

diff --git a/assets/img/project_brain.png b/assets/img/project_brain.png
new file mode 100644
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diff --git a/project.md b/project.md
index e77cf28..79bebb2 100644
--- a/project.md
+++ b/project.md
@@ -105,13 +105,43 @@ appropriately spaced samples. You can read more about these requirements
 
 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
+interpolate. This step is tricky and tends to introduce a lot of error. , 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.
 
 
+### Implementation
+
+I used [FFTW](http://www.fftw.org/) to compute Fourier transforms. It's written in C and is very
+[fast](http://www.fftw.org/benchfft/). I implemented my own polar resampling routine. It uses a
+[Catmull-Rom interpolation](http://entropymine.com/imageworsener/bicubic/) kernel.
+
+I started with this image of a
+[brain](https://commons.wikimedia.org/wiki/File:FMRI_coronal_scan.jpg) from Wikimedia Commons.
+
+![brain](../assets/img/project_brain.png)
+
+Fourier reconstruction produces a nice [sinogram](https://en.wikipedia.org/wiki/Radon_transform).
+Each row is one projection, with angle going from 0 at the top to $$\pi$$ at the bottom, and $$r =
+0$$ down the middle column. You can clearly see the skull (a roughly circular feature) unwrapped to
+a line on the left side of the sinogram.
+
+![sinogram](../assets/img/project_sinogram.png)
+
+The reconstruction, however, isn't so clean.
+
+![brain](../assets/img/project_fourier_reconstruction.png)
+
+The Fourier library I'm using is solid (and indeed it reproduces images very well even after
+repeated applications), so the error must be coming from my interpolation code. Indeed, the high
+frequency content looks ok, but there's a lot of error in low frequency content. This is encoded in
+the middle of the Fourier transform, which is what is most distorted by the polar resampling. I
+could implement a resampling routine specifically designed for polar resampling, or indeed
+specifically for polar resampling for CT reconstruction, but there are better algorithms out there
+anyway so I'll move on.
+
+
 ## Filtered Back Projection
 
 It would be nice if we could avoid the interpolation required for Fourier reconstruction. The
-- 
GitLab