  Transcranial ultrasound imaging is currently limited by attenuation and
aberration induced by the skull. First used in contrast-enhanced ultrasound
(CEUS), highly echoic microbubbles allowed for the development of novel imaging
modalities such as ultrasound localization microscopy (ULM). Herein, we develop
an inverse problem approach to aberration correction (IPAC) that leverages the
sparsity of microbubble signals. We propose to use the \textit{a priori}
knowledge of the medium based upon microbubble localization and wave
propagation to build a forward model to link the measured signals directly to
the aberration function. A standard least-squares inversion is then used to
retrieve the aberration function. We first validated IPAC on simulated data of
a vascular network using plane wave as well as divergent wave emissions. We
then evaluated the reproducibility of IPAC \textit{in vivo} in 5 mouse brains.
We showed that aberration correction improved the contrast of CEUS images by
4.6 dB. For ULM images, IPAC yielded sharper vessels, reduced vessel
duplications, and improved the resolution from 21.1 $\mu$m to 18.3 $\mu$m.
Aberration correction also improved hemodynamic quantification for velocity
magnitude and flow direction.
