  Ultrasound Localization Microscopy (ULM) can map microvessels at a resolution
of a few micrometers (\mu m). Transcranial ULM remains challenging in presence
of aberrations caused by the skull, which lead to localization errors. Herein,
we propose a deep learning approach based on complex-valued convolutional
neural networks (CV-CNNs) to retrieve the aberration function, which can then
be used to form enhanced images using standard delay-and-sum beamforming.
CV-CNNs were selected as they can apply time delays through multiplication with
in-phase quadrature input data. Predicting the aberration function rather than
corrected images also confers enhanced explainability to the network. In
addition, 3D spatiotemporal convolutions were used for the network to leverage
entire microbubble tracks. For training and validation, we used an anatomically
and hemodynamically realistic mouse brain microvascular network model to
simulate the flow of microbubbles in presence of aberration. The proposed
CV-CNN performance was compared to the coherence-based method by using
microbubble tracks. We then confirmed the capability of the proposed network to
generalize to transcranial \textit{in vivo} data in the mouse brain (n=3).
Vascular reconstructions using a locally predicted aberration function included
additional and sharper vessels. The CV-CNN was more robust than the
coherence-based method and could perform aberration correction in a 6-month-old
mouse. After correction, we measured a resolution of 15.6 \mu m for younger
mice, representing an improvement of 25.8 $\%$, while the resolution was
improved by 13.9 $\%$ for the 6-month-old mouse. This work leads to different
applications for complex-valued convolutions in biomedical imaging and
strategies to perform transcranial ULM.
