---
tag: V9G9J2C10P
kind: concept
name: X-ray transform
msc: [44A12, 92C55, 53C65]
specializes: [YCK709QB06]
sources:
  - url: https://en.wikipedia.org/wiki/X-ray_transform
same_as:
  wikidata: Q8041582
  wikipedia: X-ray transform
---

## Summary

The X-ray transform sends a function $f$ on $\R^n$ to the function that gives the integral of $f$ along each line.
In the plane it is the [[Radon transform|VJNZJ352PC]], up to the parametrization of lines.
In higher dimensions lines are smaller than hyperplanes, and the two transforms differ.
It models what X-ray tomography measures in three dimensions, where the data are line integrals.
It is injective on integrable functions, and in dimension $n \ge 3$ the data are overdetermined, since the set of lines has dimension $2n - 2 > n$.

## Explanations

### L1 human

An X-ray passes through a body along a straight line, and the detector measures the total density along that line.
The X-ray transform collects these totals for all lines.
For example, the integral of a ball of uniform density $1$ and radius $a$ along a line at distance $d < a$ from the center is the chord length $2\sqrt{a^2 - d^2}$.
From all these totals one can recover the density, which is what three-dimensional scanners do.