{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $f \\in L^1(\\R^n)$.\nLet $\\hat f(\\xi) = \\int_{\\R^n} f(x)\\, e^{-i x \\cdot \\xi}\\, dx$ be its [[Fourier transform|DPVM34ADZV]].\nLet $\\theta$ be in the unit [[sphere|P9CM4GQ2T0]] $S^{n-1}$, and let $R_\\theta f(s) = R f(\\theta, s)$ be the [[Radon transform|VJNZJ352PC]] of $f$ in the direction $\\theta$.\nThen $R_\\theta f \\in L^1(\\R)$, and for every $\\sigma \\in \\R$,\n$$\\int_\\R R_\\theta f(s)\\, e^{-i s \\sigma}\\, ds = \\hat f(\\sigma \\theta).$$\n\n## Remarks\n\nThe proof is one application of Fubini's theorem: write $x = s\\theta + y$ with $s \\in \\R$ and $y \\in \\theta^\\perp$, so that $x \\cdot \\sigma\\theta = s\\sigma$ and $dx = ds\\, dy$.\n\nExample: for $f(x) = e^{-\\abs{x}^2}$ one has $R_\\theta f(s) = \\pi^{(n-1)/2} e^{-s^2}$, whose one-dimensional Fourier transform is $\\pi^{n/2} e^{-\\sigma^2/4}$.\nThis is $\\hat f(\\sigma\\theta) = \\pi^{n/2} e^{-\\abs{\\sigma\\theta}^2/4}$, as the theorem says.\n\nIn words: the one-dimensional Fourier transform of a projection is the restriction of the $n$-dimensional Fourier transform to a line through the origin.\nSince these lines cover $\\R^n$, the theorem shows at once that $R f$ determines $f$ for $f \\in L^1(\\R^n)$, by injectivity of the Fourier transform on $L^1$.\n\n## Explanations\n\n### L1 human\n\nProject a two-dimensional picture onto a line by adding up the values along each perpendicular line.\nThe theorem says that the frequencies of this one-dimensional profile are exactly the frequencies of the whole picture along that direction.\nProjections in all directions therefore give all the frequencies of the picture, and hence the picture itself.\n\n### L2 human\n\nThe theorem is the bridge between the [[Radon transform|VJNZJ352PC]] and Fourier analysis.\nCombined with Fourier inversion in polar coordinates, it gives the [[Radon inversion formula|HN93ST5CBT]] and the ramp filter $\\abs{\\sigma}^{n-1}$ of [[filtered back-projection|SJQDDY9FMX]].\nIt also shows that the Radon transform is injective on $L^1(\\R^n)$ and explains the moment conditions in the [[range conditions for the Radon transform|WASXFTTKK6]].\nIn practice it suggests Fourier reconstruction: compute the Fourier transforms of the projections, interpolate them onto a grid, and invert one $n$-dimensional Fourier transform.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "VJNZJ352PC",
      "DPVM34ADZV"
    ],
    "kind": "statement",
    "msc": [
      "44A12",
      "42B10",
      "92C55"
    ],
    "name": "Projection-slice theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7249438",
      "wikipedia": "Projection-slice theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Projection-slice_theorem"
      }
    ],
    "tag": "J7VA040MC3",
    "uses": [
      "VJNZJ352PC",
      "DPVM34ADZV",
      "P9CM4GQ2T0"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/J7VA040MC3/",
  "links": [
    {
      "field": "about",
      "target": "VJNZJ352PC"
    },
    {
      "field": "about",
      "target": "DPVM34ADZV"
    },
    {
      "field": "uses",
      "target": "VJNZJ352PC"
    },
    {
      "field": "uses",
      "target": "DPVM34ADZV"
    },
    {
      "field": "uses",
      "target": "P9CM4GQ2T0"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/J7VA040MC3.md",
  "msc": [
    "44A12",
    "42B10",
    "92C55"
  ],
  "name": "Projection-slice theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/J7V/J7VA040MC3.md",
  "status": null,
  "tag": "J7VA040MC3",
  "uri": "m2://J7VA040MC3",
  "url": "https://tamnd.github.io/math2/wiki/projection-slice-theorem/",
  "writing": {
    "tag": "J7VA040MC3",
    "kind": "statement",
    "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
      }
    ]
  }
}