When studying the Radon transform, we saw that we could reconstruct the Fourier transform of a function from its Radon Transform:
- The Fourier Transform integrates the product of a function with “waves” that are constant on hyperplanes.
- The Radon Transform computes the integral of a function over these hyperplanes.
- So the Fourier Transform of a function is a sort of phase-weighted sum of integrals over hyperplanes, i.e. a phase-weighted sum of the Radon transform.
When evaluating the Fourier Transform of a function at a point , the hyperplane appears with “weight” . So we can write the Fourier transform of in terms of the Radon Transform as follows:
The same arguments apply to the X-Ray transform. In fact, we can compute the Radon transform from the X-Ray transform at a hyperplane by integrating over a set of parallel lines that covers the hyperplane (there is more than one way to do this!).
To compute the Fourier Transform directly from the X-Ray transform, consider the following. Say we want to compute the Fourier Transform of at a frequency vector . We can pick any orthogonal , and have . Now the “wave” with frequency will be constant in direction , so to compute the Fourier Transform of at we just need to add up the integrals of along the lines in direction — — weighted it by the value of the wave on these lines. In other words
for . Notice that our choice of was arbitrary, so this really gives us a continuum of formulas, one for each .