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  "body": "\n## Example\n\nLet $\\mu \\colon \\R^2 \\to [0, \\infty)$ be the X-ray attenuation coefficient of a slice of a body, a function with compact support.\nAn X-ray beam that crosses the slice along a [[line|0MP9BCF6DP]] $L$ enters with intensity $I_0$ and leaves with intensity $I$.\nBy the Beer–Lambert law of absorption, $I = I_0 \\exp\\left(-\\int_L \\mu \\, ds\\right)$, where $ds$ is arc length on $L$.\nSo each measurement gives the number\n$$-\\log \\frac{I}{I_0} = \\int_L \\mu \\, ds = (R\\mu)(L),$$\nthe value of the [[Radon transform|VJNZJ352PC]] $R\\mu$ at the line $L$.\nA scanner measures $R\\mu(L)$ for many lines $L$, and the task is to recover $\\mu$ from these values.\n\nThis is an [[inverse problem|PCMW7KN5P4]]: the forward map $\\mu \\mapsto R\\mu$ is known and linear, and one must invert it from data.\nWhen $R\\mu(L)$ is known for all lines, $\\mu$ is determined, and the [[Radon inversion formula|HN93ST5CBT]] reconstructs it.\nThe inversion is unstable, because it involves a derivative, a \"ramp filter\" $\\abs{\\omega}$ in the Fourier variable.\nSo small errors at high frequencies in the data become large errors in $\\mu$.\nWith data from only a limited range of directions the instability is much worse.\n\n## Explanations\n\n### L1 human\n\nA CT scanner cannot look inside the body directly.\nIt only measures how much each X-ray beam is weakened, and that number is the total density along the beam.\nFrom thousands of such totals along lines in many directions, a computer rebuilds the density at each point.\nThis \"rebuild the cause from its effects\" is a typical inverse problem.\nNoise in the measurements is the main enemy, so practical reconstruction methods filter the data before they invert it.\n",
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