[proofplan] We choose a nonnegative cutoff $\chi$ supported in the measured set $V$ and nonzero at a data point realizing the visibility of $(x_0,\xi_0)$. The Radon transform is a Fourier integral operator whose canonical relation records the incidence relation $s=x\cdot\omega$ together with the normal covector direction $\xi=\lambda\omega$. Composing this canonical relation with its transpose gives the diagonal relation near the visible branch, so $R^*\chi R$ is microlocally a pseudodifferential operator of order $-1$. Its principal symbol is a nonzero constant times $|\xi|^{-1}$ multiplied by the sum of the two antipodal cutoff values, hence is elliptic near $(x_0,\xi_0)$. The elliptic parametrix theorem then gives an order $1$ inverse modulo smoothing terms, and the Sobolev wave front equivalence follows from the mapping and elliptic regularity properties of pseudodifferential operators. [/proofplan]

[step:Choose a measured cutoff that is nonzero on a visible branch] Because $(x_0,\xi_0)$ is visible from $V$, there exist $\omega_0\in S^1$ and $\lambda_0\in\mathbb{R}\setminus\{0\}$ such that $s_0:=x_0\cdot\omega_0$ satisfies \begin{align*} (\omega_0,s_0)\in V \end{align*} and \begin{align*} \xi_0=\lambda_0\omega_0. \end{align*} Since $V$ is open and $(\omega_0,s_0)\ne(-\omega_0,-s_0)$, choose an open neighbourhood $W\subset V$ of $(\omega_0,s_0)$ whose closure in $S^1\times\mathbb{R}$ is disjoint from a neighbourhood of the antipodal data point $(-\omega_0,-s_0)$. Choose a function \begin{align*} \chi:S^1\times\mathbb{R}\to[0,\infty) \end{align*} with $\chi\in C_c^\infty(V)$, $\operatorname{supp}\chi\subset W$, and $\chi(\omega_0,s_0)>0$. This is possible by the smooth bump function construction on the smooth manifold $S^1\times\mathbb{R}$. The nonnegativity is used to prevent cancellation between the two oriented parametrizations of the same unoriented line. Since $\operatorname{supp}\chi\subset V$, the distribution $\chi\,Rf$ depends only on the known restriction of $Rf$ to $V$, and hence so does $N_\chi f=R^*(\chi\,Rf)$. [/step]

[step:Record the canonical relation of the Radon transform]Let $\mathcal{L}^1$ denote one-dimensional Lebesgue measure on $\mathbb{R}$, and let $\mathcal{L}^2$ denote two-dimensional Lebesgue measure on $\mathbb{R}^2$. For $f\in C_c^\infty(\mathbb{R}^2)$, the Radon transform can be represented microlocally by the oscillatory integral \begin{align*} Rf(\omega,s)=\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}^2}e^{i\sigma(s-y\cdot\omega)}f(y)\,d\mathcal{L}^2(y)\,d\mathcal{L}^1(\sigma). \end{align*} Here the phase function \begin{align*} \phi:S^1\times\mathbb{R}\times\mathbb{R}^2\times(\mathbb{R}\setminus\{0\})\to\mathbb{R} \end{align*} is defined by \begin{align*} \phi(\omega,s,y,\sigma)=\sigma(s-y\cdot\omega). \end{align*} The critical equation in the frequency variable is \begin{align*} \partial_\sigma\phi(\omega,s,y,\sigma)=s-y\cdot\omega=0. \end{align*} Thus the canonical relation $C_R\subset T^*(S^1\times\mathbb{R})\setminus 0\times T^*\mathbb{R}^2\setminus 0$ is \begin{align*} C_R=\{((\omega,s;\eta,\sigma),(y,\xi)):s=y\cdot\omega,\ \xi=-\sigma\omega,\ \eta(v)=-\sigma\,y\cdot v\text{ for every }v\in T_\omega S^1,\ \sigma\ne0\}. \end{align*} Changing the parameter $\sigma$ to $-\sigma$ gives the equivalent convention $\xi=\sigma\omega$. We use this latter sign convention below; the sign has no effect on ellipticity because $\sigma$ ranges over $\mathbb{R}\setminus\{0\}$. By the standard microlocal calculus of the Radon transform, the preceding phase representation is a Fourier integral representation of $R$. The phase is nondegenerate on the incidence hypersurface because $\partial_\sigma\phi=s-y\cdot\omega$ and the differential of $s-y\cdot\omega$ in the $s$ variable is $1$. The single nonzero oscillatory parameter $\sigma$ parametrizes the conormal direction to this hypersurface, and the standard order convention for this Radon kernel gives order $-1/2$. Hence $R$ is a Fourier integral operator of order $-1/2$ associated to $C_R$, and its adjoint $R^*$ is a Fourier integral operator of order $-1/2$ associated to the transposed canonical relation $C_R^t$.[/step]

[guided]We first need to know which singularities of $f$ can contribute to singularities of $Rf$. The oscillatory representation gives this information directly. For $f\in C_c^\infty(\mathbb{R}^2)$, write \begin{align*} Rf(\omega,s)=\frac{1}{2\pi}\int_{\mathbb{R}}\int_{\mathbb{R}^2}e^{i\sigma(s-y\cdot\omega)}f(y)\,d\mathcal{L}^2(y)\,d\mathcal{L}^1(\sigma). \end{align*} This is just the Fourier representation of the delta distribution imposing the equation $s=y\cdot\omega$. The phase function is the smooth map \begin{align*} \phi:S^1\times\mathbb{R}\times\mathbb{R}^2\times(\mathbb{R}\setminus\{0\})\to\mathbb{R} \end{align*} defined by \begin{align*} \phi(\omega,s,y,\sigma)=\sigma(s-y\cdot\omega). \end{align*} The critical equation in the auxiliary variable $\sigma$ is \begin{align*} \partial_\sigma\phi(\omega,s,y,\sigma)=s-y\cdot\omega=0. \end{align*} Therefore the incidence relation is exactly the condition that $y$ lies on the line with normal $\omega$ and signed distance $s$. The covectors are obtained by differentiating the phase in the base variables. The derivative in $s$ is $\sigma$. If $v\in T_\omega S^1$ is a tangent vector at $\omega$, the derivative in the $\omega$ variable is $-\sigma\,y\cdot v$. The derivative in the spatial variable $y$ is $-\sigma\omega$. Hence the canonical relation is \begin{align*} C_R=\{((\omega,s;\eta,\sigma),(y,\xi)):s=y\cdot\omega,\ \xi=-\sigma\omega,\ \eta(v)=-\sigma\,y\cdot v\text{ for every }v\in T_\omega S^1,\ \sigma\ne0\}. \end{align*} Because $\sigma$ runs through all nonzero real numbers, replacing $\sigma$ by $-\sigma$ gives the equivalent convention $\xi=\sigma\omega$. This convention matches the visibility condition $\xi=\lambda\omega$. The standard Radon-transform Fourier integral operator theorem applies to this phase. Its nondegeneracy is visible directly: the defining incidence function $s-y\cdot\omega$ has derivative $1$ in the $s$ variable, so the incidence hypersurface is smooth. The nonzero parameter $\sigma$ parametrizes the conormal line to that hypersurface, and the standard Radon kernel order convention gives order $-1/2$. Therefore this oscillatory representation makes $R$ a Fourier integral operator of order $-1/2$ associated to $C_R$, and $R^*$ is a Fourier integral operator of order $-1/2$ associated to $C_R^t$. This is the microlocal mechanism behind visibility: spatial covectors normal to measured lines are precisely the covectors that enter the canonical relation.[/guided]

[step:Compose the Radon canonical relation with its transpose near the visible covector] Let $\Omega\subset T^*\mathbb{R}^2\setminus0$ be a sufficiently small conic neighbourhood of $(x_0,\xi_0)$. Define the oriented-normal maps \begin{align*} \omega_+:\Omega\to S^1 \end{align*} and \begin{align*} \omega_-:\Omega\to S^1 \end{align*} by \begin{align*} \omega_+(x,\xi)=\frac{\xi}{|\xi|} \end{align*} and \begin{align*} \omega_-(x,\xi)=-\frac{\xi}{|\xi|}. \end{align*} The corresponding oriented line-parameter maps are \begin{align*} \ell_+:\Omega\to S^1\times\mathbb{R} \end{align*} and \begin{align*} \ell_-:\Omega\to S^1\times\mathbb{R} \end{align*} defined by \begin{align*} \ell_+(x,\xi)=(\omega_+(x,\xi),x\cdot\omega_+(x,\xi)) \end{align*} and \begin{align*} \ell_-(x,\xi)=(\omega_-(x,\xi),x\cdot\omega_-(x,\xi)). \end{align*} Shrink $\Omega$ so that the branch realizing visibility stays in $W$ and the nonselected antipodal branch avoids $\operatorname{supp}\chi$. More explicitly, if $\lambda_0>0$, shrink $\Omega$ so that $\ell_+(\Omega)\subset W$ and $\ell_-(\Omega)\cap\operatorname{supp}\chi=\varnothing$; if $\lambda_0<0$, shrink $\Omega$ so that $\ell_-(\Omega)\subset W$ and $\ell_+(\Omega)\cap\operatorname{supp}\chi=\varnothing$. This shrinking is possible because $\ell_+$ and $\ell_-$ are continuous, $\operatorname{supp}\chi$ is a closed subset of $W$, and $W$ was chosen away from the antipodal data point $(-\omega_0,-s_0)$. Consequently each set $\ell_+(\Omega)\cap\operatorname{supp}\chi$ and $\ell_-(\Omega)\cap\operatorname{supp}\chi$ is either empty or contained in a single localized oriented branch. This is exactly the branch-isolation condition needed for the standard Radon normal-operator composition theorem. Suppose two nearby spatial covectors $(x,\xi)$ and $(y,\eta)$ are connected through the same oriented Radon covector in the localized composition. Then the canonical relation gives a common $\omega\in S^1$ and nonzero parameters $\sigma,\tau\in\mathbb{R}\setminus\{0\}$ such that $\xi=\sigma\omega$, $\eta=\tau\omega$, $x\cdot\omega=y\cdot\omega$, and the full data covectors agree. Equality of the data covectors includes equality of the $s$-covector components, so $\sigma=\tau$. Writing $v\in T_\omega S^1$ for the unit tangent vector, equality of the $S^1$-covector components gives $\sigma x\cdot v=\sigma y\cdot v$. Since $\sigma\ne0$, this gives $x\cdot v=y\cdot v$. Since $\{\omega,v\}$ is a basis of $\mathbb{R}^2$, the two scalar equalities $x\cdot\omega=y\cdot\omega$ and $x\cdot v=y\cdot v$ imply $x=y$, and then $\xi=\eta$. Thus the localized composed relation has diagonal output near the selected visible branch. We now use the standard Radon normal-operator composition theorem. Its hypotheses are met on the chosen conic neighbourhood: multiplication by $\chi$ is a pseudodifferential operator of order $0$ on $S^1\times\mathbb{R}$, each supported oriented branch is isolated in the explicit sense that $\ell_+(\Omega)\cap\operatorname{supp}\chi$ and $\ell_-(\Omega)\cap\operatorname{supp}\chi$ are separately localized or empty, and the preceding calculation identifies the localized composed relation with the diagonal in $T^*\mathbb{R}^2\setminus0$. Therefore clean composition gives that $N_\chi=R^*\chi R$ is a classical pseudodifferential operator of order $-1$ microlocally near $(x_0,\xi_0)$. [/step]

[step:Compute the principal symbol and prove ellipticity at the visible covector] By the positive Radon normal-symbol formula, the principal symbol of the localized Radon normal operator has the form \begin{align*} \sigma_{-1}(N_\chi)(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*} where $c_R>0$ is the positive density constant obtained from the stationary-phase evaluation of the one-dimensional clean fiber in the Radon normal composition. The positivity follows because the normal operator is formed as $R^*\chi R$ and the principal density contribution is the positive fiber density multiplied by the nonnegative cutoff values. Evaluate this formula at $(x_0,\xi_0)$. If $\lambda_0>0$, then \begin{align*} \frac{\xi_0}{|\xi_0|}=\omega_0, \end{align*} and the first term equals $\chi(\omega_0,s_0)>0$. If $\lambda_0<0$, then \begin{align*} -\frac{\xi_0}{|\xi_0|}=\omega_0, \end{align*} and the second term equals $\chi(\omega_0,s_0)>0$. Since $\chi\ge0$, the sum cannot cancel. Hence \begin{align*} \sigma_{-1}(N_\chi)(x_0,\xi_0)\ne0. \end{align*} By continuity and homogeneity of the principal symbol, there exists a conic neighbourhood $U_0\subset T^*\mathbb{R}^2\setminus0$ of $(x_0,\xi_0)$ and a constant $a_0>0$ such that \begin{align*} |\sigma_{-1}(N_\chi)(x,\xi)|\ge a_0|\xi|^{-1} \end{align*} for every $(x,\xi)\in U_0$. Thus $N_\chi$ is elliptic of order $-1$ on $U_0$. [/step]

[step:Construct a microlocal parametrix for the elliptic normal operator] By the elliptic parametrix theorem for classical pseudodifferential operators, applied to the order $-1$ operator $N_\chi$ on the elliptic conic neighbourhood $U_0$, there exist a conic neighbourhood $U\subset U_0$ of $(x_0,\xi_0)$ and a classical pseudodifferential operator \begin{align*} Q:\mathcal{D}'(\mathbb{R}^2)\to\mathcal{D}'(\mathbb{R}^2) \end{align*} of order $1$ such that, with $I:\mathcal{D}'(\mathbb{R}^2)\to\mathcal{D}'(\mathbb{R}^2)$ denoting the identity operator, \begin{align*} QN_\chi-I=S \end{align*} microlocally on $U$, where $S$ is microlocally smoothing on $U$. Therefore, for every $f\in\mathcal{E}'(\mathbb{R}^2)$, \begin{align*} QN_\chi f-f=Sf \end{align*} is smooth microlocally on $U$. [/step]

[step:Translate elliptic inversion into Sobolev wave front equivalence] Fix $t\in\mathbb{R}$ and $(x,\xi)\in U$. Since $N_\chi$ is a pseudodifferential operator of order $-1$, the Sobolev mapping property for pseudodifferential operators gives \begin{align*} (x,\xi)\notin WF^t(f)\quad\Longrightarrow\quad (x,\xi)\notin WF^{t+1}(N_\chi f). \end{align*} Indeed, an operator of order $-1$ raises local Sobolev regularity by one. Conversely, suppose $(x,\xi)\notin WF^{t+1}(N_\chi f)$. Since $Q$ has order $1$, the same mapping property gives \begin{align*} (x,\xi)\notin WF^t(QN_\chi f). \end{align*} The parametrix identity gives $f=QN_\chi f-Sf$ microlocally on $U$, and $Sf$ is smooth microlocally on $U$. Smooth microlocal regularity implies absence from every Sobolev wave front set, so \begin{align*} (x,\xi)\notin WF^t(f). \end{align*} This proves \begin{align*} (x,\xi)\notin WF^t(f)\quad\Longleftrightarrow\quad (x,\xi)\notin WF^{t+1}(N_\chi f) \end{align*} for every $t\in\mathbb{R}$ and every $(x,\xi)\in U$. The only data used to form $N_\chi f$ are $\chi Rf$ with $\operatorname{supp}\chi\subset V$. Hence the measured Radon data on $V$ determine the Sobolev wave front presence and Sobolev order of $f$ at the visible covectors in $U$, modulo the microlocally smooth remainder produced by the elliptic parametrix. [/step]