Recovery of singularities for the weighted cone transform appearing in the Compton camera imaging

We study the weighted cone transform $I_\kappa$ of distributions with compact support in a domain $M$ of $\mathbb{R}^3$, over cone surfaces whose vertexes are located on a smooth surface away from $M$ and opening angles are limited to an open interval of $(0,\pi/2)$. We show that when the weight function has compact support and satisfies certain nonvanishing assumptions, the normal operator $I^*_\kappa I_\kappa$ is an elliptic $\Psi$DO at the accessible singularities. Then the accessible singularities are stably recoverable from local data. We prove a microlocal stability estimate for $I_\kappa$. Moreover, we show the same analysis can be applied to the restricted cone transform.