We introduce bendlets, a shearlet-like system that is based on anisotropic scaling, translation, shearing, and bending of a compactly supported generator. With shearing being linear and bending quadratic in spatial coordinates, bendlets provide what we term a second-order shearlet system. As we show in this article, the decay rates of the associated transform enable the precise characterization of location, orientation and curvature of discontinuities in piecewise constant images. These results yield an improvement over existing directional representation systems where curvature only controls the constant of the decay rate of the transform. We also detail the construction of shearlet systems of arbitrary order. A practical implementation of bendlets is provided as an extension of the ShearLab toolbox, which we use to verify our theoretical classification results.
Numerous generalizations of the just described method have been developed, including a generalization to 3D domains in @cite_28 , with a particular emphasis on detecting line singularities within the 2D manifolds given by the boundaries @cite_20 . In another line of research, the region @math is no longer required to be constant but can be smooth, see @cite_29 . All the results above deal with band-limited generators. @cite_32 the decay of a continuous shearlet transform with a compactly supported generator was analyzed. It was demonstrated that a similar classification of edges as with band-limited generators could be achieved in both 2D and 3D, with the additional improvement that the decay rates are uniform, meaning they can be analyzed in a pre-asymptotic regime. Furthermore, it was also established that information on the curvature of @math could be extracted from the shearlet transform in a weak form. This is, however, very different from the results in the present paper that provide an exact description of the asymptotic behavior of a higher-order shearlet transform for different curvatures.
@cite_28 @cite_29 @cite_32 @cite_20
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Abstract Directional multiscale transforms such as the shearlet transform have emerged in recent years for their ability to capture the geometrical information associated with the singularity sets of bivariate functions and distributions. One of the most striking features of the continuous shearlet transform is that it provides a very simple and precise geometrical characterization for the boundary curves of general planar regions. However, no specific results were known so far in higher dimensions, since the arguments used in dimension n = 2 do not directly carry over to the higher dimensional setting. In this paper, we extend this framework for the analysis of singularities to the 3-dimensional setting, and show that the 3-dimensional continuous shearlet transform precisely characterizes the boundary set of solid regions in R 3 by identifying both its location and local orientation. Abstract The analysis and detection of edges is a central problem in applied mathematics and image processing. A number of results in recent years have shown that directional multiscale methods such as continuous curvelet and shearlet transforms offer a powerful theoretical framework to capture the geometry of edge singularities, going far beyond the capabilities of the conventional wavelet transform. The continuous shearlet transform, in particular, provides a precise geometric characterization of edges in piecewise constant functions in R 2 and R 3 , including corner points. However, a question has been raised frequently: What happens if the function is piecewise smooth and not just piecewise constant? Clearly, a piecewise smooth function is a much more realistic model of images with edges. In this paper, we extend the characterization results previously known and show that, also in the case of piecewise smooth functions, the continuous shearlet transform can detect the location and orientation of edge points, including corner points, through its asymptotic decay at fine scales. The new proof introduces innovative technical constructions to deal with the more challenging problem. The new results set the theoretical groundwork for the application of the shearlet framework to a wider class of problems from image processing. Suppose that is a three dimensional solid with boundary surface S = S1[ [ Sq, where each Sr is a smooth surface with boundary curve r. Multiscale directional representation systems (e.g., shearlets) are able to capture the essential geometry of by precisely identifying the boundary set