{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nThis is the second-generation curvelet system on $\\R^2$.\nLet $W \\colon (0,\\infty) \\to [0,\\infty)$ and $V \\colon \\R \\to [0,\\infty)$ be smooth windows with $W$ supported in $(1/2, 2)$ and $V$ supported in $[-1, 1]$, such that\n$$\\sum_{j \\in \\Z} W(2^j r)^2 = 1 \\text{ for } r \u003e 0, \\qquad \\sum_{l \\in \\Z} V(t - l)^2 = 1 \\text{ for } t \\in \\R.$$\nFor an integer $j \\ge 0$, let $\\varphi_j$ be the function whose [[Fourier transform|DPVM34ADZV]] is, in polar coordinates $(r, \\theta)$ of the frequency plane,\n$$\\hat\\varphi_j(r, \\theta) = 2^{-3j/4} \\, W(2^{-j} r) \\, V\\left(\\frac{2^{\\floor{j/2}}\\theta}{2\\pi}\\right).$$\nLet $R_\\theta$ be the rotation of $\\R^2$ by the angle $\\theta$.\nFor $l$ with $0 \\le l \u003c 2^{\\floor{j/2}}$, put $\\theta_l = 2\\pi \\cdot 2^{-\\floor{j/2}} \\, l$.\nFor $k = (k_1, k_2) \\in \\Z^2$, put $x_k^{(j,l)} = R_{\\theta_l}^{-1}(k_1 2^{-j}, k_2 2^{-j/2})$.\nThe *curvelet* at scale $2^{-j}$, orientation $\\theta_l$ and position $x_k^{(j,l)}$ is\n$$\\varphi_{j,l,k}(x) = \\varphi_j\\left(R_{\\theta_l}\\left(x - x_k^{(j,l)}\\right)\\right), \\qquad x \\in \\R^2.$$\nThe curvelet coefficients of $f \\in L^2(\\R^2)$ are $c(j,l,k) = \\ip{f}{\\varphi_{j,l,k}}$.\n\n## Remarks\n\nEach $\\hat\\varphi_j$ lives on a polar wedge with radial size about $2^j$ and angular width about $2^{-j/2}$.\nIn space, $\\varphi_j$ is then a needle of length about $2^{-j/2}$ and width about $2^{-j}$.\nThis is parabolic scaling: width $\\approx$ length$^2$.\nThe number of orientations doubles at every second scale.\nTogether with a coarse-scale family at low frequencies, the curvelets form a tight frame of $L^2(\\R^2)$ in the sense that $\\sum \\abs{c(j,l,k)}^2 = \\norm{f}_2^2$.\n\nExample: any pair of smooth windows with these supports and partition conditions, such as the Meyer-type windows used for the Meyer wavelet, gives a curvelet system.\n\nNon-example: if the angular window is not narrowed with scale, that is, $V(2^{\\floor{j/2}}\\theta/(2\\pi))$ is replaced by a fixed angular window, the elements keep the same aspect ratio at all scales.\nThey then behave like directional wavelets and lose the parabolic scaling, which is the defining feature of curvelets.\n\n## Explanations\n\n### L1 human\n\nCurvelets are built in frequency space from wedges of a polar grid.\nFiner scales use wedges farther out and narrower in angle, and the inverse Fourier transform turns each wedge into a thin oriented needle in the plane.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "VEVPDJ8S7B",
    "kind": "definition",
    "msc": [
      "42C40",
      "42C15",
      "94A08"
    ],
    "name": "Curvelet",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Curvelet"
      }
    ],
    "tag": "XJMGD9V64J",
    "uses": [
      "DPVM34ADZV",
      "SASP14MA1Z"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/XJMGD9V64J/",
  "links": [
    {
      "field": "defines",
      "target": "VEVPDJ8S7B"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    },
    {
      "field": "uses",
      "target": "SASP14MA1Z"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/XJMGD9V64J.md",
  "msc": [
    "42C40",
    "42C15",
    "94A08"
  ],
  "name": "Curvelet",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/XJM/XJMGD9V64J.md",
  "status": null,
  "tag": "XJMGD9V64J",
  "uri": "m2://XJMGD9V64J",
  "url": "https://tamnd.github.io/math2/wiki/curvelet-definition/",
  "writing": {
    "tag": "XJMGD9V64J",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}