Edge analysis and identification using the continuous shearlet transform (Q1026461)

The authors explore and improve results on how one with the aid of the continuous shearlet transform of a function of the form \(f=\sum_{n=1}^N f_n\chi_{\Omega_n}\) can characterize the locate and orientation of the boundaries \(\partial \Omega_n\). The shearlet transform is the mapping \(f\in L^2(\mathbb{R})\to \langle f,\psi_{ast}\rangle\) where \(\psi_{ast}(x)=|\text{ det}(M_{as})|^{-\frac 12}\psi(M_{as}^{-1}(x-t))\) with \(M_{as}=\begin{pmatrix} a & -a^\alpha s \\ 0 & a^\alpha\end{pmatrix}\) for some \(\alpha \in (0,1)\). Here \(a>0\), \(s\in \mathbb{R}\) and \(t\in \mathbb{R}^2\) and it is the behaviour of the transform as \(a\to 0+\) that gives the information about the ``edges''. The function \(\psi\) is either chosen so that its Fourier transform is of the form \(\hat \psi(\xi_1,\xi_2)=\hat\psi_1(\xi_1)\hat\psi_2(\frac {\xi_1}{\xi_2})\) or \(\hat \psi(\xi_1,\xi_2)=\hat\psi_1(\xi_2)\hat\psi_2(\frac {\xi_2}{\xi_1})\) where the support of \(\hat \psi_1\) lies in \([-2,-\frac 12]\cup[\frac 12,2]\) and the support of \(\hat\psi_2\) lies in \([-1,1]\). Thus the support of \(\hat\psi_{ast}\) is contained in a pair of trapezoids oriented along a line of slope \(s\). (To get complete reconstruction, an additional coarse-scale transform must be used to take care of functions with frequency support in \([-2,2]\times [-2,2]\).)