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  "body": "\n## Summary\n\nThe dual Radon transform, or back-projection, acts on functions $g$ on the set of hyperplanes.\nIts value at a point $x$ of $\\R^n$ is the integral of $g$ over all hyperplanes through $x$.\nIt is the formal adjoint of the [[Radon transform|VJNZJ352PC]] for the natural measures.\nIt is not the inverse of the Radon transform, but the [[Radon inversion formula|HN93ST5CBT]] writes the inverse as the dual transform composed with a filter.\nIn tomography this is the back-projection step of [[filtered back-projection|SJQDDY9FMX]].\n\n## Explanations\n\n### L1 human\n\nEach line measurement of a scanner says how much material lies along that line, but not where on the line.\nBack-projection smears each measured value back uniformly along its line.\nAdding the smears of all lines through a point gives the value of the dual transform there.\nFor example, if every line carries the value $1$, then every point receives the same total.\nThe resulting picture is a blurred version of the true one, and a filter removes the blur.\n",
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