Add outtakes section

project.md

@@ -143,6 +143,46 @@ specifically for polar resampling for CT reconstruction, but there are better al
 anyway so I'll move on.
 
 
+### Outtakes
+
+I got a number of interesting failures before getting my code to work correctly.
+
+![outtake](../assets/img/project_cool_1.png)
+![outtake](../assets/img/project_cool_2.png)
+![outtake](../assets/img/project_cool_3.png)
+![outtake](../assets/img/project_cool_4.png)
+
+These all started as attempts to project and reconstruct a disk, though I did experiment once
+interesting mistakes started happening. They mostly result from indexing errors and an issue with my
+integration with FFTW. The latter problem relates to the periodicity of discrete Fourier transforms
+and the resulting ambiguity in frequency interpretation. In a nutshell, the DFT doesn't give you
+samples of the continuous Fourier transform; it gives you samples of the periodic summation of the
+continuous Fourier transform. So each sample isn't representative of one frequency, it's
+representative of a whole equivalence class of frequencies.
+
+For this application it's very important that the center of the polar coordinate system used for
+resampling is right at the DC sample. So though the raw Fourier transform of the image looks like
+this (real and imaginary parts shown separately),
+
+![transform](../assets/img/project_transform_swapped_re.png)
+![transform](../assets/img/project_transform_swapped_im.png)
+
+we want to permute the quadrants so that it looks like this:
+
+![transform](../assets/img/project_transform_re.png)
+![transform](../assets/img/project_transform_im.png)
+
+Then the center of the image can be the center of the polar coordinate system. (Note: to make these
+I linearly mapped the full range of each image to [1, e], then applied the natural logarithm. So
+though they aren't the same color, both tend toward zero away from the center.) This is akin to
+viewing the sample frequencies not as $$0$$, $$1/N$$, $$\ldots$$, $$(N - 1)/N$$, but as $$0$$,
+$$1/N$$, $$\ldots$$, $$(N/2 - 1)/N$$, $$-1/2$$, $$\ldots$$, $$-1/N$$.
+
+Interestingly, you can do this by literally swapping quadrants of the image, or by multiplying the
+results element-wise by a checker board of 1s and -1s. This seems like magic until you just write
+out the math.
+
+
 ## Filtered Back Projection
 
 It would be nice if we could avoid the interpolation required for Fourier reconstruction. The