{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $f$ be in the Schwartz space $\\mathcal{S}(\\R^2)$, with [[Radon transform|VJNZJ352PC]] $R f$, and write $R_\\theta f(s) = R f(\\theta, s)$.\nLet $\\Phi \\colon \\R \\to \\R$ be an even bounded function with $\\Phi(\\sigma) = 0$ for $\\abs{\\sigma} \u003e 1$, continuous at $0$, with $\\Phi(0) = 1$.\nFor $b \u003e 0$, let $w_b$ be the filter with Fourier transform $\\hat w_b(\\sigma) = \\abs{\\sigma}\\, \\Phi(\\sigma/b)$.\nIts *filtered back-projection* reconstruction with filter $w_b$ is\n$$f_b(x) = \\frac{1}{4\\pi} R^* \\left( w_b * R_\\theta f \\right)(x) = \\frac{1}{4\\pi} \\int_{S^1} (w_b * R_\\theta f)(x \\cdot \\theta)\\, d\\theta,$$\nwhere $*$ is [[convolution|KXC9KHWVGE]] in the variable $s$ and $R^*$ is the [[dual Radon transform|938SXZQS8G]].\n\n## Remarks\n\nThe Fourier convention is $\\hat g(\\sigma) = \\int g(s)\\, e^{-is\\sigma}\\, ds$.\nThe projection-slice theorem gives $\\hat f_b(\\xi) = \\Phi(\\abs{\\xi}/b)\\, \\hat f(\\xi)$, so $f_b \\to f$ as $b \\to \\infty$, uniformly on $\\R^2$.\nWith $\\Phi$ the indicator of $[-1, 1]$ the filter is the Ram–Lak filter, a band-limited ramp.\n\nExample: if $\\Phi$ is the indicator of $[-1, 1]$ and $\\hat f$ vanishes for $\\abs{\\xi} \u003e b$, then $\\hat f_b = \\hat f$ and the reconstruction is exact: $f_b = f$.\n\nNon-example: back-projection without filter, $\\frac{1}{4\\pi} R^* R f$, is not a reconstruction of $f$.\nIt corresponds to $\\hat w(\\sigma) = 1$ instead of $\\abs{\\sigma}$, and its Fourier transform is $\\hat f(\\xi)/\\abs{\\xi}$, a blurred image.\n\nIn practice $R f$ is known only at finitely many angles and offsets, and the integral and the convolution are replaced by sums.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "SJQDDY9FMX",
    "kind": "definition",
    "msc": [
      "44A12",
      "92C55",
      "65R10"
    ],
    "name": "Filtered back-projection",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Tomographic_reconstruction"
      }
    ],
    "tag": "MHC7X73NG9",
    "uses": [
      "VJNZJ352PC",
      "938SXZQS8G",
      "KXC9KHWVGE"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/MHC7X73NG9/",
  "links": [
    {
      "field": "defines",
      "target": "SJQDDY9FMX"
    },
    {
      "field": "uses",
      "target": "VJNZJ352PC"
    },
    {
      "field": "uses",
      "target": "938SXZQS8G"
    },
    {
      "field": "uses",
      "target": "KXC9KHWVGE"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/MHC7X73NG9.md",
  "msc": [
    "44A12",
    "92C55",
    "65R10"
  ],
  "name": "Filtered back-projection",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MHC/MHC7X73NG9.md",
  "status": null,
  "tag": "MHC7X73NG9",
  "uri": "m2://MHC7X73NG9",
  "url": "https://tamnd.github.io/math2/wiki/filtered-back-projection-definition/",
  "writing": {
    "tag": "MHC7X73NG9",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}