---
tag: J7VA040MC3
kind: statement
role: theorem
name: Projection-slice theorem
msc: [44A12, 42B10, 92C55]
uses: [VJNZJ352PC, DPVM34ADZV, P9CM4GQ2T0]
about: [VJNZJ352PC, DPVM34ADZV]
sources:
  - url: https://en.wikipedia.org/wiki/Projection-slice_theorem
same_as:
  wikidata: Q7249438
  wikipedia: Projection-slice theorem
---

## Statement

Let $n \ge 2$ and let $f \in L^1(\R^n)$.
Let $\hat f(\xi) = \int_{\R^n} f(x)\, e^{-i x \cdot \xi}\, dx$ be its [[Fourier transform|DPVM34ADZV]].
Let $\theta$ be in the unit [[sphere|P9CM4GQ2T0]] $S^{n-1}$, and let $R_\theta f(s) = R f(\theta, s)$ be the [[Radon transform|VJNZJ352PC]] of $f$ in the direction $\theta$.
Then $R_\theta f \in L^1(\R)$, and for every $\sigma \in \R$,
$$\int_\R R_\theta f(s)\, e^{-i s \sigma}\, ds = \hat f(\sigma \theta).$$

## Remarks

The proof is one application of Fubini's theorem: write $x = s\theta + y$ with $s \in \R$ and $y \in \theta^\perp$, so that $x \cdot \sigma\theta = s\sigma$ and $dx = ds\, dy$.

Example: for $f(x) = e^{-\abs{x}^2}$ one has $R_\theta f(s) = \pi^{(n-1)/2} e^{-s^2}$, whose one-dimensional Fourier transform is $\pi^{n/2} e^{-\sigma^2/4}$.
This is $\hat f(\sigma\theta) = \pi^{n/2} e^{-\abs{\sigma\theta}^2/4}$, as the theorem says.

In words: the one-dimensional Fourier transform of a projection is the restriction of the $n$-dimensional Fourier transform to a line through the origin.
Since these lines cover $\R^n$, the theorem shows at once that $R f$ determines $f$ for $f \in L^1(\R^n)$, by injectivity of the Fourier transform on $L^1$.

## Explanations

### L1 human

Project a two-dimensional picture onto a line by adding up the values along each perpendicular line.
The theorem says that the frequencies of this one-dimensional profile are exactly the frequencies of the whole picture along that direction.
Projections in all directions therefore give all the frequencies of the picture, and hence the picture itself.

### L2 human

The theorem is the bridge between the [[Radon transform|VJNZJ352PC]] and Fourier analysis.
Combined with Fourier inversion in polar coordinates, it gives the [[Radon inversion formula|HN93ST5CBT]] and the ramp filter $\abs{\sigma}^{n-1}$ of [[filtered back-projection|SJQDDY9FMX]].
It also shows that the Radon transform is injective on $L^1(\R^n)$ and explains the moment conditions in the [[range conditions for the Radon transform|WASXFTTKK6]].
In practice it suggests Fourier reconstruction: compute the Fourier transforms of the projections, interpolate them onto a grid, and invert one $n$-dimensional Fourier transform.