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I need to calculate the Radon transform of $f(x,y)=cos(2\pi y)$ using the projection-slice theorem. After doing the 2-D polar Fourier transform, I get two 2-D delta functions which then need to go through a 1-D inverse Fourier transform. Those delta functions have functions of $\rho$ and $\theta$(my polar coordinates in frequency domain) as their arguments, so I am stuck calculating the integral of the 1-D inverse Fourier transform.

I know that for a 1-D delta fuction of a fuction we have the following relation:

$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$

What would this be for a 2-D delta function?

That is to say, what would $\delta\big(f(\rho,\theta),g(\rho,\theta)\big)$ be equal to?

TinyRick
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  • The general formula from here becomes $$\delta(f(x, y)) , \delta(g(x, y)) = \sum_{(x_0, y_0): \hspace {1.5px} f(x_0, y_0) = g(x_0, y_0) = 0} \frac {\delta(x - x_0) , \delta(y - y_0)} {\left| \nabla f(x_0, y_0) \right| \left| \nabla g(x_0, y_0) \right| \sin \phi},$$ where $\phi$ is the angle at which the curves $f = 0$ and $g = 0$ intersect at the point $(x_0, y_0)$.

    The Fourier transform can be obtained by applying this functional to $e^{i (x, y) \cdot (\xi, \zeta)}$ as if the latter were a test function.

     – Maxim Jan 18 '20 at 22:46
  • Thanks! Do you know where I can find the proof for this? – TinyRick Jan 18 '20 at 23:31
  • Essentially it's just a transformation to a coordinate system in which $f$ and $g$ are coordinate curves. The general theory is given in Gelfand and Shilov, Generalized functions, Vol. 1, Ch. 3. – Maxim Jan 18 '20 at 23:55

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