Efficient analysis and detection of edges through directional multiscale representations


The analysis and detection of edges and interface boundaries is a fundamental problem in applied mathematics and image processing. In the study of the wave equation, for example, one is interested in the evolution of moving fronts; in image processing and computer vision, the detection and analysis of edges is an essential task for applications such as shape recognition, image enhancement, and classification. Multiscale methods and wavelets have been very successful in this area, due to a combination of useful micro-analytical properties and fast numerical implementations. The continuous wavelet transform in particular has the ability to signal the location of the singularities of functions and distributions through its asymptotic decay at fine scales. However, this approach is unable to provide additional information about the geometry of the singularity set, such as the edge orientation. This limitation can be overcome by using the continuous shearlet transform, an approach combining the analytical power of multiscale analysis and high directional sensitivity. This chapter gives an overview of the microlocal properties of the shearlet transform and illustrates its ability to provide a precise geometric characterization of edges and interface boundaries in images and other multidimensional data. These results provide the theoretical groundwork for innovative applications in problems of edge detection, feature extraction, and geometric separation.



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Applied and Numerical Harmonic Analysis