Add filtered back projection results

project.md

@@ -32,6 +32,7 @@ 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
@@ -111,7 +112,7 @@ algorithms out there that come closer to the theoretically ideal
 The popular [gridrec](https://www.ncbi.nlm.nih.gov/pubmed/23093766) method is one.
 
 
-### Implementation
+### Results
 
 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
@@ -148,6 +149,7 @@ It would be nice if we could avoid the interpolation required for Fourier recons
 simplest way of doing so, and still the most popular method of performing image reconstruction, is
 called filtered back projection.
 
+
 ### Theory
 
 Let's hop back to the continuous theory for a moment. The Fourier reconstruction technique is based
@@ -189,6 +191,11 @@ $$
 f(x, y) = \int_0^\pi \mathcal{F}^{-1}[q_\theta](x \cos \theta + y \sin \theta) \mathrm{d} \theta
 $$
 
+Since multiplication in the frequency domain is equivalent to convolution in the spatial domain,
+$$\mathcal{F}^{-1}[q_\theta]$$ is simply a filtered version of $$p_\theta$$. Hence the name filtered
+back projection.
+
+
 ### Discretization
 
 We'll replace the continuous Fourier transforms with DFTs as before.
@@ -210,6 +217,13 @@ them directly for quadrature. For $$r$$, on the other hand, the regular samples
 line up with the values $$x \cos \theta + y \sin \theta$$ that we want. So we'll still have to do
 some interpolation. But now it's a simple 1D interpolation problem that's much easier to do without
 introducing as much error. In particular, we only have to do interpolation in the spatial domain, so
-errors will only accumulate locally. [Lanczos
-resampling](https://en.wikipedia.org/wiki/Lanczos_resampling) would probably be the best, but I'll
-just use Catmull-Rom again since I already have it implemented.
+errors will only accumulate locally.
+
+
+### Results
+
+I again use [FFTW](http://www.fftw.org/) for Fourier transforms. For interpolation I just take the
+sample to the left of the desired location, and for integration I just use a left Riemann sum. Even
+with these lazy techniques, the reconstruction looks quite good.
+
+![brain](../assets/img/project_fbp_reconstruction.png)