We analyze the detection and classification of singularities of functions @math , where @math and @math . It will be shown how the set @math can be extracted by a continuous shearlet transform associated with compactly supported shearlets. Furthermore, if @math is a @math dimensional piecewise smooth manifold with @math or @math , we will classify smooth and non-smooth components of @math . This improves previous results given for shearlet systems with a certain band-limited generator, since the estimates we derive are uniform. Moreover, we will show that our bounds are optimal. Along the way, we also obtain novel results on the characterization of wavefront sets in @math dimensions by compactly supported shearlets. Finally, geometric properties of @math such as curvature are described in terms of the continuous shearlet transform of @math .
It is known that the Continuous Wavelet Transform of a distribution f decays rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows the identification of the singular support of f. However, the Continuous Wavelet Transform is unable to describe the geometry of the set of singularities of f and, in particular, identify the wavefront set of a distribution. In this paper, we employ the same framework of affine systems which is at the core of the construction of the wavelet transform to introduce the Continuous Shearlet Transform. This is defined by SH ψ f(a,s,t) = (fψ ast ), where the analyzing elements ψ ast are dilated and translated copies of a single generating function ψ. The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements ψ ast form a system of smooth functions at continuous scales a > 0, locations t ∈ R 2 , and oriented along lines of slope s ∈ R in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution f. In many x-ray images, we find crossing structures, for example the rib bones of a thorax image. Hence, many edges in such images intersect with other edges. We propose a new algorithm, which simultaneously detects and separates such crossing edges. We prove the functionality of the algorithm under the geometric assumption that the edges possess different orientations at the points of intersection. To this end, we model edges as singularities of distributions with a submanifold structure and define the edge orientation to be the orientation of the corresponding singularity in the sense of microlocal analysis. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 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.