Resolution of the wavefront set using continuous shearlets

Abstract:

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 $\mathcal {SH}_\psi f(a,s,t) = \langle {f}{\psi _{ast}}\rangle$, where the analyzing elements $\psi _{ast}$ are dilated and translated copies of a single generating function $\psi$. The dilation matrices form a two-parameter matrix group consisting of products of parabolic scaling and shear matrices. We show that the elements $\{\psi _{ast}\}$ form a system of smooth functions at continuous scales $a>0$, locations $t \in \mathbb {R}^2$, and oriented along lines of slope $s \in \mathbb {R}$ in the frequency domain. We then prove that the Continuous Shearlet Transform does exactly resolve the wavefront set of a distribution $f$.