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  "body": "\n## Definition\n\nFor $a \u003e 0$ and $s \\in \\R$ let\n$$A_a = \\begin{pmatrix} a \u0026 0 \\\\ 0 \u0026 \\sqrt{a} \\end{pmatrix}, \\qquad S_s = \\begin{pmatrix} 1 \u0026 s \\\\ 0 \u0026 1 \\end{pmatrix}$$\nbe the parabolic scaling matrix and the shear matrix.\nLet $\\hat\\psi$ denote the [[Fourier transform|DPVM34ADZV]] of $\\psi \\in L^2(\\R^2)$, with variable $\\xi = (\\xi_1, \\xi_2)$.\nA function $\\psi \\in L^2(\\R^2)$ is an *admissible shearlet* if\n$$\\int_{\\R^2} \\frac{\\abs{\\hat\\psi(\\xi)}^2}{\\xi_1^2} \\, d\\xi \u003c \\infty.$$\nThe continuous shearlet system generated by $\\psi$ consists of the functions\n$$\\psi_{a,s,t}(x) = a^{-3/4} \\, \\psi\\left(A_a^{-1} S_s^{-1}(x - t)\\right), \\qquad a \u003e 0, \\ s \\in \\R, \\ t \\in \\R^2.$$\nThe continuous shearlet transform of $f \\in L^2(\\R^2)$ is $\\mathcal{SH}_\\psi f(a, s, t) = \\ip{f}{\\psi_{a,s,t}}$.\n\n## Remarks\n\nThe factor $a^{-3/4} = (\\det A_a)^{-1/2}$ makes $\\norm{\\psi_{a,s,t}}_2 = \\norm{\\psi}_2$.\nDiscrete shearlet systems use scales $a = 2^{-j}$, shears $s = k \\, 2^{-j/2}$ with $k \\in \\Z$, and translations on a lattice.\n\nExample (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 $[-1, 1]$.\nThen $\\hat\\psi$ 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 $\\xi_1 = 0$.\nSo the admissibility integral is finite.\n\nNon-example: the Gaussian $\\psi(x) = e^{-\\abs{x}^2}$ is not admissible.\nIts 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.\n\nFor 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]].\n\n## Explanations\n\n### L1 human\n\nA shearlet is a generating function that is scaled more in one direction than the other, tilted by a shear and moved around the plane.\nThe admissibility condition asks that the function have essentially no energy at low horizontal frequencies, the analogue of a wavelet having zero mean.\n",
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