Extraction of digital wavefront sets using applied harmonic analysis and deep neural networks
This addresses a key challenge in imaging sciences for researchers and practitioners by providing a more accurate method for wavefront set extraction, though it appears incremental as it builds on existing shearlet transform properties.
The paper tackles the problem of extracting wavefront sets from images, which is crucial for understanding singularities in inverse problems like imaging, by introducing an algorithmic approach that combines shearlet transforms and deep neural networks, resulting in outperforming competing algorithms in edge-orientation and ramp-orientation detection.
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.