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  "body": "\n## Definition\n\nLet $n \\ge 2$, and let $S^{n-1}$ be the unit [[sphere|P9CM4GQ2T0]] in $\\R^n$.\nFor $\\theta \\in S^{n-1}$ and $s \\in \\R$, the set $\\set{x \\in \\R^n : x \\cdot \\theta = s}$ is a hyperplane, and every hyperplane is of this form.\nLet $\\theta^\\perp = \\set{y \\in \\R^n : y \\cdot \\theta = 0}$, with its $(n-1)$-dimensional Lebesgue measure $dy$.\nLet $f \\colon \\R^n \\to \\C$ be continuous and [[integrable|X7GTF3D98F]] on every hyperplane.\nThe *Radon transform* of $f$ is the function on $S^{n-1} \\times \\R$ given by\n$$(R f)(\\theta, s) = \\int_{\\theta^\\perp} f(s\\theta + y)\\, dy.$$\n\n## Remarks\n\n$R f(\\theta, s)$ is the integral of $f$ over the hyperplane $x \\cdot \\theta = s$.\nSince $(-\\theta, -s)$ gives the same hyperplane, $R f(-\\theta, -s) = R f(\\theta, s)$.\nThe transform is also defined for $f \\in L^1(\\R^n)$, at almost every $(\\theta, s)$, by Fubini's theorem.\nOne often writes $R_\\theta f(s) = R f(\\theta, s)$.\n\nExample: $f(x) = e^{-\\abs{x}^2}$.\nFor $y \\in \\theta^\\perp$, $\\abs{s\\theta + y}^2 = s^2 + \\abs{y}^2$, so $R f(\\theta, s) = e^{-s^2} \\int_{\\R^{n-1}} e^{-\\abs{y}^2}\\, dy = \\pi^{(n-1)/2} e^{-s^2}$.\n\nExample: in $\\R^2$, the indicator of the unit disk has $R f(\\theta, s) = 2\\sqrt{1 - s^2}$ for $\\abs{s} \\le 1$ and $0$ otherwise.\n\nNon-example: $f = 1$ is not integrable on any hyperplane, so it has no Radon transform.\n\nFor $n = 2$ the Radon transform coincides with the [[X-ray transform|V9G9J2C10P]] up to the parametrization of lines.\n",
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