Resolution of the Wavefront Set using Continuous Shearlets

Abstract

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 $\R^2$, called the Continuous Shearlet Transform. This is defined by $\mathcal{S}\mathcal{H}_f(a,s,t) = \ip{f}{ψ_{ast}}$, where the analyzing elements $ψ_{ast}$ are dilated and translated copies of a single generating function $ψ$ and, thus, they form an affine system. The resulting continuous shearlets $ψ_{ast}$ are smooth functions at continuous scales $a >0$, locations $t \in \R^2$ and oriented along lines of slope $s \in \R$ 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 $f$.