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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $f$ be in the [[Schwartz space|GE020MY43M]] $\\mathcal{S}(\\R^n)$.\nLet $R$ be the [[Radon transform|VJNZJ352PC]] and $R^*$ the [[dual Radon transform|938SXZQS8G]].\nFor a function $g(\\theta, s)$ on $S^{n-1} \\times \\R$, let $\\Lambda g$ be defined by filtering in $s$ with the multiplier $\\abs{\\sigma}^{n-1}$:\n$$(\\Lambda g)(\\theta, s) = \\frac{1}{2\\pi} \\int_\\R \\abs{\\sigma}^{n-1}\\, \\hat g(\\theta, \\sigma)\\, e^{i s \\sigma}\\, d\\sigma, \\qquad \\hat g(\\theta, \\sigma) = \\int_\\R g(\\theta, s)\\, e^{-i s \\sigma}\\, ds.$$\nThen, for every $x \\in \\R^n$,\n$$f(x) = \\frac{1}{2} (2\\pi)^{1-n}\\, R^* (\\Lambda R f)(x).$$\nIf $n$ is odd, $\\Lambda = (-1)^{(n-1)/2} \\partial_s^{n-1}$, and the formula reads\n$$f(x) = \\frac{(-1)^{(n-1)/2}}{2 (2\\pi)^{n-1}} \\int_{S^{n-1}} \\left( \\partial_s^{n-1} R f \\right)(\\theta, x \\cdot \\theta)\\, d\\theta.$$\n\n## Remarks\n\nThe proof starts from the Fourier inversion formula $f(x) = (2\\pi)^{-n} \\int_{\\R^n} \\hat f(\\xi)\\, e^{i x \\cdot \\xi}\\, d\\xi$ in polar coordinates $\\xi = \\sigma\\theta$.\nWriting the integral over $\\sigma \\in \\R$ and $\\theta \\in S^{n-1}$ counts each $\\xi$ twice, which gives the factor $\\frac12$ and the weight $\\abs{\\sigma}^{n-1}$.\nThe [[projection-slice theorem|J7VA040MC3]] replaces $\\hat f(\\sigma\\theta)$ by the Fourier transform of $R_\\theta f$, and the remaining $\\theta$-integral is $R^*$.\n\nExample: for $n = 3$ the formula is $f(x) = -\\frac{1}{8\\pi^2} \\int_{S^2} (\\partial_s^2 R f)(\\theta, x \\cdot \\theta)\\, d\\theta$, so $f(x)$ depends only on the integrals of $f$ over planes close to $x$.\n\nFor even $n$ the multiplier $\\abs{\\sigma}^{n-1}$ is not a polynomial, $\\Lambda$ involves the [[Hilbert transform|NSCEBXM7ZM]] in $s$, and the formula is not local: $f(x)$ depends on integrals over all hyperplanes.\n\n## Explanations\n\n### L1 human\n\nThe Radon transform of a function lists its integrals over all hyperplanes, and the formula gets the function back.\nFirst each profile is sharpened by a filter that boosts high frequencies.\nThen all sharpened profiles are smeared back over space and added up.\nThe result is exactly the original function.\n\n### L2 human\n\nThe formula is the continuous model of [[filtered back-projection|SJQDDY9FMX]], the standard reconstruction algorithm of computed tomography.\nThe filter $\\abs{\\sigma}^{n-1}$ grows at high frequencies, which shows why inversion amplifies noise and why practical filters cut it off.\nIn odd dimensions the inversion is local, a fact used in three-dimensional imaging; in even dimensions, such as the plane, it is not.\nThe same formula, read with the Riesz potential, extends to wider classes of functions and to distributions.\n",
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