Radon transform

Remarks

$R f(\theta, s)$ is the integral of $f$ over the hyperplane $x \cdot \theta = s$. Since $(-\theta, -s)$ gives the same hyperplane, $R f(-\theta, -s) = R f(\theta, s)$. The transform is also defined for $f \in L^1(\R^n)$, at almost every $(\theta, s)$, by Fubini's theorem. One often writes $R_\theta f(s) = R f(\theta, s)$.

Example: $f(x) = e^{-\abs{x}^2}$. For $y \in \theta^\perp$, $\abs{s\theta + y}^2 = s^2 + \abs{y}^2$, so $R f(\theta, s) = e^{-s^2} \int_{\R^{n-1}} e^{-\abs{y}^2}\, dy = \pi^{(n-1)/2} e^{-s^2}$.

Example: in $\R^2$, the indicator of the unit disk has $R f(\theta, s) = 2\sqrt{1 - s^2}$ for $\abs{s} \le 1$ and $0$ otherwise.

Non-example: $f = 1$ is not integrable on any hyperplane, so it has no Radon transform.

For $n = 2$ the Radon transform coincides with the X-ray transform up to the parametrization of lines.