Resolution of the Wavefront Set using Continuous Shearlets

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It is known that the continuous wavelet transform of a function f decays very rapidly near the points where f is smooth, while it decays slowly near the irregular points. This property allows one to precisely identify the singular support of f. However, the continuous wavelet transform is unable to provide additional information about the geometry of the singular points. In this paper, we introduce a new transform for functions and distributions on ², called the Continuous Shearlet Transform. This is defined by SHf(a,s,t) = fψast, where the analyzing elements ψast are dilated and translated copies of a single generating function ψ and, thus, they form an affine system. The resulting continuous shearlets ψast are smooth functions at continuous scales a >0, locations t ∈ ² and oriented along lines of slope s ∈ in the frequency domain. The Continuous Shearlet Transform transform is able to identify not only the location of the singular points of a distribution f, but also the orientation of their distributed singularities. As a result, we can use this transform to exactly characterize the wavefront set of f.