Let $\mathcal{L}^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure) on $\mathbb{R}$, let $\mathcal{L}^2$ denote two-dimensional Lebesgue measure on $\mathbb{R}^2$, and let $\mathcal{H}^1$ denote one-dimensional [Hausdorff measure](/page/Hausdorff%20Measure). Let $R:\mathcal{E}'(\mathbb{R}^2)\to \mathcal{D}'(S^1\times\mathbb{R})$ be the two-dimensional Radon transform, where

\begin{align*} Rf(\omega,s)=\int_{\{y\in\mathbb{R}^2:y\cdot\omega=s\}}f(y)\,d\mathcal{H}^1(y) \end{align*}

for $f\in C_c^\infty(\mathbb{R}^2)$ and by distributional extension for $f\in\mathcal{E}'(\mathbb{R}^2)$. Let $R^*:\mathcal{E}'(S^1\times\mathbb{R})\to\mathcal{D}'(\mathbb{R}^2)$ denote the distributional adjoint of $R$. Let $V\subset S^1\times\mathbb{R}$ be open, let $f\in\mathcal{E}'(\mathbb{R}^2)$, and suppose that the restriction of $Rf$ to $V$ is known.

A nonzero covector $(x,\xi)\in T^*\mathbb{R}^2\setminus 0$ is called visible from $V$ if there exist $\omega\in S^1$ and $\lambda\in\mathbb{R}\setminus\{0\}$ such that $(\omega,x\cdot\omega)\in V$ and $\xi=\lambda\omega$.

Assume the standard microlocal calculus for the two-dimensional Radon transform with the following objects and hypotheses. The transform $R$ is a Fourier integral operator of order $-1/2$ associated to the incidence canonical relation of oriented lines. For each open conic set $\Omega\subset T^*\mathbb{R}^2\setminus 0$, define the oriented-normal maps $\omega_+:\Omega\to S^1$ and $\omega_-:\Omega\to S^1$ by $\omega_+(x,\xi)=\xi/|\xi|$ and $\omega_-(x,\xi)=-\xi/|\xi|$, and define the oriented line-parameter maps $\ell_+:\Omega\to S^1\times\mathbb{R}$ and $\ell_-:\Omega\to S^1\times\mathbb{R}$ by $\ell_+(x,\xi)=(\omega_+(x,\xi),x\cdot\omega_+(x,\xi))$ and $\ell_-(x,\xi)=(\omega_-(x,\xi),x\cdot\omega_-(x,\xi))$. Multiplication by any $\chi\in C_c^\infty(S^1\times\mathbb{R})$ is a pseudodifferential operator of order $0$ on $S^1\times\mathbb{R}$. For every nonnegative $\chi\in C_c^\infty(S^1\times\mathbb{R})$ and every open conic set $\Omega\subset T^*\mathbb{R}^2\setminus 0$ such that each of the sets $\ell_+(\Omega)\cap\operatorname{supp}\chi$ and $\ell_-(\Omega)\cap\operatorname{supp}\chi$ is either empty or contained in a single isolated oriented branch of the incidence relation, the clean composition $R^*\chi R$ is pseudodifferential of order $-1$ microlocally on $\Omega$, and its principal symbol is the positive Radon normal symbol

\begin{align*} \sigma_{-1}(R^*\chi R)(x,\xi)=c_R|\xi|^{-1}\left[\chi\left(\frac{\xi}{|\xi|},x\cdot\frac{\xi}{|\xi|}\right)+\chi\left(-\frac{\xi}{|\xi|},-x\cdot\frac{\xi}{|\xi|}\right)\right] \end{align*}

for a constant $c_R>0$ fixed by the chosen Fourier normalization. Also assume the standard elliptic parametrix theorem and microlocal Sobolev mapping property for classical pseudodifferential operators on $\mathbb{R}^2$: an [elliptic operator](/page/Elliptic%20Operator) of order $m$ has a microlocal parametrix of order $-m$ modulo microlocally smoothing remainders, and a pseudodifferential operator of order $m$ maps microlocal $H^t$ regularity to microlocal $H^{t-m}$ regularity.

If $(x_0,\xi_0)\in T^*\mathbb{R}^2\setminus 0$ is visible from $V$, then there exist $\omega_0\in S^1$, $\lambda_0\in\mathbb{R}\setminus\{0\}$, and $\chi\in C_c^\infty(V)$ such that $\xi_0=\lambda_0\omega_0$, $\chi(\omega_0,x_0\cdot\omega_0)\ne 0$, and the localized normal operator $N_\chi:\mathcal{E}'(\mathbb{R}^2)\to\mathcal{D}'(\mathbb{R}^2)$ defined by

\begin{align*} N_\chi f=R^*(\chi\,Rf) \end{align*}

is, microlocally near $(x_0,\xi_0)$, a classical pseudodifferential operator of order $-1$ that is elliptic at $(x_0,\xi_0)$. Consequently, there are conic neighbourhoods $U\subset T^*\mathbb{R}^2\setminus 0$ of $(x_0,\xi_0)$ and a classical pseudodifferential operator $Q$ of order $1$ such that

\begin{align*} QN_\chi f-f \end{align*}

is smooth microlocally on $U$. Equivalently, for every $t\in\mathbb{R}$ and every $(x,\xi)\in U$,

\begin{align*} (x,\xi)\notin WF^t(f)\quad\Longleftrightarrow\quad (x,\xi)\notin WF^{t+1}(N_\chi f). \end{align*}

Thus the measured data on $V$ determine the Sobolev wave front presence and Sobolev order of $f$ at visible covectors, modulo microlocally smooth remainders, by elliptic inversion of $N_\chi$.