Analysis and Identification of Multidimensional Singularities using the Continuous Shearlet Transform

In this chapter, we illustrate the properties of the continuous shearlet transform with respect to its ability to describe the set of singularities of multidimensional functions and distributions. This is of particular interest since singularities and other irregular structures typically carry the most essential information in multidimensional phenomena. Consider, for example, the edges of natural images or the moving fronts in the solutions of transport equations. In the following, we show that the continuous shearlet transform provides a precise geometrical characterization of the singularity sets of multidimensional functions and precisely characterizes the boundaries of 2D and 3D regions through its asymptotic decay at fine scales. These properties go far beyond the continuous wavelet transform and other classical methods, and set the groundwork for very competitive algorithms for edge detection and feature extraction of 2D and 3D data.