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  "body": "\n## Problem\n\nLet $f$ be an unknown function on $\\R^n$, $n \\ge 2$, with compact support.\nGiven the [[Radon transform|VJNZJ352PC]] $R f$, or the X-ray transform of $f$, on a set of lines or hyperplanes, possibly with measurement errors, recover $f$.\nIn practice the data are values $R f(\\theta_j, s_k)$ at finitely many directions $\\theta_j$ and offsets $s_k$.\nThe task is then to compute a good approximation of $f$.\n\n## Remarks\n\nWith complete and exact data in $\\R^n$, the problem has a unique solution, given by the [[Radon inversion formula|HN93ST5CBT]]; Radon gave the inversion in the plane in 1917.\nThe problem is an [[inverse problem|PCMW7KN5P4]] that is not [[well-posed|W09B9NZ9B5]] in $L^2$: the inversion contains the filter $\\abs{\\sigma}^{n-1}$, which amplifies high-frequency noise, so some regularization is needed.\nWith finitely many directions, uniqueness fails: for any finite set of directions there is a nonzero smooth function with compact support whose projections in these directions all vanish.\nThe main algorithms are [[filtered back-projection|SJQDDY9FMX]], Fourier reconstruction based on the [[projection-slice theorem|J7VA040MC3]], and iterative algebraic methods such as the Kaczmarz method.\nLimited-angle data and interior data lead to harder versions of the problem, with weaker stability.\n\n## Explanations\n\n### L1 human\n\nA CT scanner sends X-rays through a body from many directions and measures how much each ray is weakened.\nTomographic reconstruction is the task of computing a picture of the inside of the body from these measurements.\nMathematically, one must recover a function from its integrals along lines.\nIt can be done exactly when all lines are known, but real data are finite and noisy, so the methods must balance sharpness against noise.\n",
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