---
tag: 140TG7W6A5
kind: definition
name: Shearlet
msc: [42C40, 42C15, 94A08]
defines: F9Q17JRV5K
uses: [DPVM34ADZV, SASP14MA1Z]
sources:
  - url: https://en.wikipedia.org/wiki/Shearlet
---

## Definition

For $a > 0$ and $s \in \R$ let
$$A_a = \begin{pmatrix} a & 0 \\ 0 & \sqrt{a} \end{pmatrix}, \qquad S_s = \begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}$$
be the parabolic scaling matrix and the shear matrix.
Let $\hat\psi$ denote the [[Fourier transform|DPVM34ADZV]] of $\psi \in L^2(\R^2)$, with variable $\xi = (\xi_1, \xi_2)$.
A function $\psi \in L^2(\R^2)$ is an *admissible shearlet* if
$$\int_{\R^2} \frac{\abs{\hat\psi(\xi)}^2}{\xi_1^2} \, d\xi < \infty.$$
The continuous shearlet system generated by $\psi$ consists of the functions
$$\psi_{a,s,t}(x) = a^{-3/4} \, \psi\left(A_a^{-1} S_s^{-1}(x - t)\right), \qquad a > 0, \ s \in \R, \ t \in \R^2.$$
The continuous shearlet transform of $f \in L^2(\R^2)$ is $\mathcal{SH}_\psi f(a, s, t) = \ip{f}{\psi_{a,s,t}}$.

## 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|VEVPDJ8S7B]].

## Explanations

### L1 human

A shearlet is a generating function that is scaled more in one direction than the other, tilted by a shear and moved around the plane.
The admissibility condition asks that the function have essentially no energy at low horizontal frequencies, the analogue of a wavelet having zero mean.