  Microlocal analysis provides deep insight into singularity structures and is
often crucial for solving inverse problems, predominately, in imaging sciences.
Of particular importance is the analysis of wavefront sets and the correct
extraction of those. In this paper, we introduce the first algorithmic approach
to extract the wavefront set of images, which combines data-based and
model-based methods. Based on a celebrated property of the shearlet transform
to unravel information on the wavefront set, we extract the wavefront set of an
image by first applying a discrete shearlet transform and then feeding local
patches of this transform to a deep convolutional neural network trained on
labeled data. The resulting algorithm outperforms all competing algorithms in
edge-orientation and ramp-orientation detection.
