Projection-Slice Theorem

The projection-slice theorem (also called the Fourier slice theorem or central slice theorem) relates projections of a function to slices of its Fourier transform. It is the theoretical foundation of computed tomography reconstruction.

Statement

The 1D Fourier transform of a projection of a 2D function equals a 1D slice through the 2D Fourier transform of that function, passing through the origin and perpendicular to the projection direction.

For a projection along the $x$-axis:

$$ P(\nu_y) = F(0, \nu_y) $$

where:

• $p(y) = \int_{-\infty}^{\infty} f(x,y) \, dx$ is the projection
• $P(\nu_y) = \mathcal{F}_1\{p(y)\}$ is its 1D Fourier transform
• $F(\nu_x, \nu_y) = \mathcal{F}_2\{f(x,y)\}$ is the 2D Fourier transform

Derivation

Starting with the 1D Fourier transform of the projection:

$$ P(\nu_y) = \int_{-\infty}^{\infty} \left[\int_{-\infty}^{\infty} f(x,y) \, dx \right] e^{-j2\pi \nu_y y} \, dy $$

Since $e^{-j2\pi \nu_y y}$ doesn’t depend on $x$, we can write:

$$ P(\nu_y) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} f(x,y) \, e^{-j2\pi(0 \cdot x + \nu_y \cdot y)} \, dx \, dy = F(0, \nu_y) $$

General Form

For a projection at angle $\theta$:

$$ P_\theta(\nu) = F(\nu \cos\theta, \nu \sin\theta) $$

Each projection gives a radial “spoke” through the origin in the 2D Fourier domain.

Application to CT

1. Acquire projections at many angles $\theta \in [0, \pi)$
2. Each projection’s Fourier transform fills in one radial line
3. Interpolate to a Cartesian grid
4. Apply 2D inverse Fourier transform to recover $f(x,y)$

See Also

• Radon transform
• Computed tomography
• Filtered back-projection
• Fourier transform