Visible covectors for limited-angle data

Let $\mathrm{\Theta }\subset S^1$ be a nonempty proper open set of line normals, assume $\theta \in \mathrm{\Theta }$ if and only if $-\theta \in \mathrm{\Theta }$, and assume $S^1\setminus \bar{\mathrm{\Theta }}$ contains a nonempty open arc. The measured line with normal $\theta$ and signed offset $s$ is ${\mathrm{\ell }}_{\theta ,s}=\{y\in {\mathbb{R}}^2:y\cdot \theta =s\}$. Identify the nonzero cotangent vectors of the Euclidean plane with pairs $(x,\xi )\in {\mathbb{R}}^2\times ({\mathbb{R}}^2\setminus \{0\})$. Such a covector is visible from $\mathrm{\Theta }$ when $\xi \mathrm{/}\mathrm{\mid }\xi \mathrm{\mid }\in \mathrm{\Theta }$, and invisible when $\xi \mathrm{/}\mathrm{\mid }\xi \mathrm{\mid }\mathrm{\notin }\mathrm{\Theta }$. The associated measured Fourier wedge is

$W_{\mathrm{\Theta }}=\{\sigma \theta :\theta \in \mathrm{\Theta },\sigma \in \mathbb{R}\setminus \{0\}\}\mathrm{.}$

IP-36 line-normal convention; current body inspected

Mikko Salo, On geometric inverse problems and microlocal analysis, §2.1

Venkataswaran P. Krishnan and Eric Todd Quinto, Microlocal Analysis in Tomography, §1.5.1

This schematic chooses one open measured arc $J$ and its antipode on $S^1$. Their open radial sectors form the corresponding part of $W_{\mathrm{\Theta }}$; the dashed blue boundary rays are excluded, the orange horizontal sectors illustrate unmeasured arcs, and the origin is excluded from the measured wedge. For general $\mathrm{\Theta }$, take the union of the open sectors generated by all its arcs. The picture fixes the angle convention only; every claim above is stated and proved algebraically.