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.
Besides approximation, shearlets also provide a powerful tool for feature analysis of functions. It was first demonstrated in @cite_2 that, analogous to the curvelet transform @cite_4 , the shearlet transform provides a precise characterization of the wavefront set, which corresponds to the aforementioned boundary curves between image regions. In particular, the wavefront set is characterized as those positions and orientations in parameter space where the transform decays slowly for increasing scales. By now, the analysis of edges and singularities of functions utilizing a continuous shearlet transform is a well-established area of research @cite_1 .
@cite_1 @cite_4 @cite_2
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In recent years directional multiscale transformations like the curvelet- or shearlet transformation have gained considerable attention. The reason for this is that these transforms are—unlike more traditional transforms like wavelets—able to efficiently handle data with features along edges. The main result in Kutyniok and Labate (Trans. Am. Math. Soc. 361:2719–2754, 2009) confirming this property for shearlets is due to Kutyniok and Labate where it is shown that for very special functions ψ with frequency support in a compact conical wegde the decay rate of the shearlet coefficients of a tempered distribution f with respect to the shearlet ψ can resolve the wavefront set of f. We demonstrate that the same result can be verified under much weaker assumptions on ψ, namely to possess sufficiently many anisotropic vanishing moments. We also show how to build frames for ( L^2( R ^2) ) from any such function. To prove our statements we develop a new approach based on an adaption of the Radon transform to the shearlet structure. It is well known that the wavelet transform provides a very effective framework for analysis of multiscale edges. In this paper, we propose a novel approach based on the shearlet transform: a multiscale directional transform with a greater ability to localize distributed discontinuities such as edges. Indeed, unlike traditional wavelets, shearlets are theoretically optimal in representing images with edges and, in particular, have the ability to fully capture directional and other geometrical features. Numerical examples demonstrate that the shearlet approach is highly effective at detecting both the location and orientation of edges, and outperforms methods based on wavelets as well as other standard methods. Furthermore, the shearlet approach is useful to design simple and effective algorithms for the detection of corners and junctions. 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. This paper shows that the continuous shearlet transform, a novel directional multiscale transform recently introduced by the authors and their collaborators, provides a precise geometrical characterization for the boundary curves of very general planar regions. This study is motivated by imaging applications, where such boundary curves represent edges of images. The shearlet approach is able to characterize both locations and orientations of the edge points, including corner points and junctions, where the edge curves exhibit abrupt changes in tangent or curvature. Our results encompass and greatly extend previous results based on the shearlet and curvelet transforms which were limited to very special cases such as polygons and smooth boundary curves with nonvanishing curvature.