Range conditions for the Radon transform

Remarks

Necessity is a direct computation. Condition 1 holds because $(\theta, s)$ and $(-\theta, -s)$ give the same hyperplane. For condition 2, Fubini's theorem gives $\int_\R R f(\theta, s)\, s^k\, ds = \int_{\R^n} f(x)\, (x \cdot \theta)^k\, dx$, and expanding $(x \cdot \theta)^k$ shows a homogeneous polynomial of degree $k$ in $\theta$. Sufficiency is the deep part. Following the projection-slice theorem, one defines $\hat f(\sigma\theta)$ as the Fourier transform of $g(\theta, \cdot)$. The moment conditions are what make this function smooth at the origin $\xi = 0$.

Example: for $n = 2$, $g(\theta, s) = \sqrt{\pi}\, e^{-s^2}$ is the Radon transform of $e^{-\abs{x}^2}$. Its moments vanish for odd $k$ and are constants $c_k$ for even $k$. On the circle, $c_k = c_k (\theta_1^2 + \theta_2^2)^{k/2}$, a homogeneous polynomial of degree $k$.

Non-example: for $n = 2$, $g(\theta, s) = \theta_1^2 e^{-s^2}$ satisfies condition 1. But for $k = 0$ its integral in $s$ is $\sqrt{\pi}\, \theta_1^2$, which is not constant on the circle. So condition 2 fails, and $g$ is not a Radon transform.

These are often called the Helgason–Ludwig consistency conditions.

Going deeper

For a student

In tomography, measured data are noisy and usually violate these conditions. Projecting the data onto the range, or checking the conditions, helps to detect errors, to estimate unknown calibration parameters and to fill in missing data. The case $k = 0$ says that the mass is the same in every direction, and $k = 1$ says that the center of mass is a single point of $\R^n$. The theorem describes the range of $R$ on $\mathcal{S}(\R^n)$; for other function spaces the range is described by similar conditions.