Characterization of Piecewise-Smooth Surfaces Using the 3D Continuous Shearlet Transform

• Published in: The Journal of fourier analysis and applications 2012-06, Vol.18 (3), p.488-516
• Main Authors: Guo, Kanghui, Labate, Demetrio
• Format: Article
• Language: English
• Subjects: Abstract Harmonic Analysis; Approximations and Expansions; Computer programs; Fourier Analysis; Image processing; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Partial Differential Equations; Signal,Image and Speech Processing
• ISSN: 1069-5869; 1531-5851
• DOI: 10.1007/s00041-011-9209-y

One of the most striking features of the Continuous Shearlet Transform is its ability to precisely characterize the set of singularities of multivariable functions through its decay at fine scales. In dimension n =2, it was previously shown that the continuous shearlet transform provides a precise geometrical characterization for the boundary curves of very general planar regions, and this property sets the groundwork for several successful image processing applications. The generalization of this result to dimension n =3 is highly nontrivial, and so far it was known only for the special case of 3D bounded regions where the boundary set is a smooth 2-dimensional manifold with everywhere positive Gaussian curvature. In this paper, we extend this result to the general case of 3D bounded regions with piecewise-smooth boundaries, and show that also in this general situation the continuous shearlet transform precisely characterizes the geometry of the boundary set.