I have trouble following Deans's derivation of the inverse Radon transform formula for $n=2$ on this page of his book "The Radon Transform and Some of its Applications" (see snapshot)
Formulas (3.9) and (3.10) make sense, but I can't quite see the change of variables used to get (3.11) and (3.12)?
Any insight appreciated, thanks!
p.
Let us represent points on the plane using polar coordinates $(p,\phi)$, where the radial coordinate $p$ is allowed to be negative. Then, for any function $F(p,\phi)$, it follows that $F(p,\phi+\pi) = F(-p,\phi)$. Using this property,
\begin{align} \int_0^{2\pi} \int_{-\infty}^{\infty} F(p,\phi) \, dp \, d\phi &= \int_0^{\pi} \int_{-\infty}^{\infty} F(p,\phi) \, dp \, d\phi + \int_0^{\pi} \int_{-\infty}^{\infty} F(p,\phi + \pi) \, dp \, d\phi \\ &= \int_0^{\pi} \int_{-\infty}^{\infty} F(p,\phi) \, dp \, d\phi + \int_0^{\pi} \int_{-\infty}^{\infty} F(-p,\phi) \, dp \, d\phi \\ &= \int_0^{\pi} \left[\int_{-\infty}^{\infty} F(p,\phi) \, dp + \int_{-\infty}^{\infty} F(-p,\phi) \, dp \right] d\phi \\ &= \int_0^{\pi} \left[2 \int_{-\infty}^{\infty} F(p,\phi) \, dp \right] d\phi \\ &= 2 \int_0^{\pi} \int_{-\infty}^{\infty} F(p,\phi) \, dp \, d\phi. \end{align}
Then (3.11) follows immediately if you set the integrand of (3.10) as $F(p,\phi)$ in the equation above. Equation (3.12) can be derived similarly using the relation above, except you will need to define the $\phi$-integral over $[-\pi/2,3\pi/2]$ and then split it at $\phi = \pi/2$.