Extraction of Digital Wavefront Sets Using Applied Harmonic Analysis and Deep Neural Networks

Abstract

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.

Edge, Ridge, and Blob Detection with Symmetric Molecules

“…On one hand, this could lead to new ideas to improve the proposed measures in order to to better handle difficulties such as corner points, intersections of ridges, or densely grouped blobs. On the other hand, neural network architectures such as [6], which have already been trained with great success for similar tasks, could potentially also be further improved by considering the computational principles behind the proposed measures. Ideally, such work would lead to hybrid approaches that combine the best of both worlds in the sense that they yield state-of-the-art performance results for specific applications that are nowadays often only achievable with learning-based methods while also being robust in applications where pure learning is not possible due to a lack of training data.…”

Shearlets as feature extractor for semantic edge detection: the model-based and data-driven realm

“…. , F (1) , F (2) , F (3) , A At the end, a K-grouped 1 × 1 convolutional layer is applied to A f , generating a semantic edge map with K channels, where the k-th channel represents the edge map for the k-th category. Summarizing, the first four stages of CASENet produce category-agnostic edge feature maps with different levels of refinement.…”

“…To show this in the context of digitized images and data, we make use of the digital wavefront set extraction applied to the head-phantom dataset introduced in §5a and the microlocal canonical relation of the ray transform ( §2c), in order to obtain the digital wavefront set an image from the digital wavefront set of its sinogram without a previous inversion. This approach is based on the work presented by the authors in [2]. We simulate tomographic data (sinograms) using the Python implementation of the digital ray transform in the operator discretization library (http://github.com/odlgroup/odl).…”

“…In computer vision, semantic edge detection aims to detect edges and object boundaries in natural images and to classify the points in those objects, which is of significant importance in various applications such as object recognition, semantic segmentation and image reconstruction. Examples of such classification could be to label each point of an edge according to the object this edge belongs to [ 1 ] or the orientation of the edge at that particular point [ 2 ]. Thus, this problem combines two different classification tasks.…”

“…For the special case of wavefront set extraction (see § 1 a), where the class an edge point belongs to represents the orientation of the edge in that particular point, assuming a finite set of orientations, a hybrid approach was suggested in [ 2 ]. This method is based on a careful combination of the shearlet transform and a CNN.…”