Shearlet

Remarks

The factor $a^{-3/4} = (\det A_a)^{-1/2}$ makes $\norm{\psi_{a,s,t}}_2 = \norm{\psi}_2$. Discrete shearlet systems use scales $a = 2^{-j}$, shears $s = k \, 2^{-j/2}$ with $k \in \Z$, and translations on a lattice.

Example (classical shearlet): let $\hat\psi(\xi) = \hat\psi_1(\xi_1)\,\hat\psi_2(\xi_2/\xi_1)$, where $\hat\psi_1$ is smooth with support in $[-2, -1/2] \cup [1/2, 2]$ and $\hat\psi_2$ is smooth with support in $[-1, 1]$. Then $\hat\psi$ is bounded and supported in the set where $1/2 \le \abs{\xi_1} \le 2$ and $\abs{\xi_2} \le \abs{\xi_1}$, which is bounded and stays away from $\xi_1 = 0$. So the admissibility integral is finite.

Non-example: the Gaussian $\psi(x) = e^{-\abs{x}^2}$ is not admissible. Its Fourier transform is a Gaussian with $\hat\psi(0) \ne 0$, so near $\xi = 0$ the integrand behaves like a positive constant divided by $\xi_1^2$, and the integral diverges.

For images that are smooth apart from smooth edges, suitable discrete shearlet frames give $N$-term approximations with squared error of order $N^{-2}(\log N)^3$, the same rate as curvelets.