Sinogram

Description

Let $f$ be a function on $\R^2$ with Radon transform $R f$. Parametrize directions by an angle, $\theta(\varphi) = (\cos\varphi, \sin\varphi)$. The sinogram of $f$ is the function $(\varphi, s) \mapsto R f(\theta(\varphi), s)$ on $[0, \pi) \times \R$. It is drawn as an image with the angle $\varphi$ on one axis and the offset $s$ on the other. The range $[0, \pi)$ suffices, because $R f(-\theta, -s) = R f(\theta, s)$.

Check of the name: let $f$ be concentrated near a single point $p = (\abs{p}\cos\alpha, \abs{p}\sin\alpha)$, a narrow bump of total mass $1$. A line $x \cdot \theta(\varphi) = s$ meets the bump only when $s$ is close to $p \cdot \theta(\varphi) = \abs{p} \cos(\varphi - \alpha)$. So the sinogram is concentrated near the curve $s = \abs{p} \cos(\varphi - \alpha)$, a sinusoid in $\varphi$.

Second check: take the indicator of a disk of radius $a$ centered at $0$. Its sinogram is $2\sqrt{a^2 - s^2}$ for $\abs{s} \le a$ and $0$ otherwise, independent of $\varphi$: a band of constant profile.