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

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