2022-12-25 ☾ *Swapping slices and projections and spaces and frequencies*

The Fourier Slice Theorem is a classical result in signal processing result that establishes an equivalence between projections in the spatial domain and slices in the frequency domain. It's very useful for image reconstruction tasks, particularly in medical imaging (e.g., reconstructing 3D CT and MRI volumes from 2D images).

In 3D volume of data $\mathbf V : \mathbb R^3 \mapsto \mathbb R$, the Fourier Slice Theorem states that (the Fourier transform of a projection of $\mathbf V$) is equal to (an orthogonal slice of the Fourier transform of $\mathbf V$).^{[1]}

[1] | Theorem 1 (Fourier Slice Theorem). $p_\theta(\mathbf V) = \mathcal F^{-1} \circ S_{\theta^\perp} \circ \mathcal F(\mathbf V)$ where $p_\theta(\cdot)$ is a projection in the spatial domain via rays at angle $\theta$, $S_{\theta^\perp}(\cdot)$ is an orthogonal slice in the frequency domain at angle $\theta^\perp$, and $\mathcal F(\cdot)$ is the Fourier transform. |

Simplifications:

The Fourier transform is equivariant to rotation,

^{[2]}so we can assume $\theta = 0$ (i.e., we are projecting onto the $x$-axis) without loss of generality.Projection onto the $x$-axis is invariant to the $y$-intercept, so we can assume $y = 0$ without loss of generality.

[2] | $\mathcal F \circ R = R \circ \mathcal F$ where $R$ is a rotation matrix. |

Let $f : \mathbb R^2 \to \mathbb R$ be a continuous representation of 2D image.

Let $p(x) = \int_{\mathbb R} f(x, y) \mathrm{d}y$ be the projection of $f$ onto the $x$-axis.

$F(k_x, k_y) = \iint_{\mathbb R^2} f(x,y) e^{-2\pi i (xk_x + yk_y)} \mathrm{d}x \mathrm{d}y$ is the 2D Fourier transform of $f$.

Using these definitions, we can derive

$\begin{aligned} S(k_x) &= F(k_x, 0) \\ &= \iint_{\mathbb R^2} f(x,y) e^{-2\pi i (xk_x + y\cancel{k_y})} \mathrm{d}x \mathrm{d}y \\ &= \int_{\mathbb R} \left( \int_{\mathbb R} f(x, y) \mathrm{d}y \right) e^{-2\pi i xk_x} \mathrm{d}x \\ &= \int_{\mathbb R} p(x) e^{-2\pi i xk_x} \mathrm{d}x \\ &= P(k_x) \,, \end{aligned}$

where $P(\cdot)$ is the Fourier transform of the projection $p(\cdot)$. This establishes the 2D version of the Fourier Slice Theorem.

Using the same machinery, we can derive a dual of the Fourier Slice Theorem with projection in the frequency domain and slicing in the spatial domain.^{[3]}

[3] | Theorem 2 (Dual of the Fourier Slice Theorem). $P_\theta(\mathbf V) = \mathcal F^{-1} \circ P_{\theta^\perp} \circ \mathcal F(\mathbf V)$. |

Let $P(k_x) = \int_{\mathbb R} F(k_x, k_y) \mathrm{d}k_y$ be the projectino of $F$ onto the $k_x$-axis.

$f(x, y) = \iint_{\mathbb R^2} F(k_x, k_y) e^{2\pi i (xk_x + yk_y)} \mathrm{d}k_x \mathrm{d}k_y$ be the 2D inverse Fourier transform of $F$.

Similarly, we can derive

$\begin{aligned} s(x) &= f(x, 0) \\ &= \iint_{\mathbb R^2} F(k_x, k_y) e^{2\pi i (xk_x + \cancel{y}k_y)} \mathrm{d}k_x \mathrm{d}k_y \\ &= \int_{\mathbb R} \left( \int_{\mathbb R} F(k_x, k_y) \mathrm{d}k_y \right) e^{2\pi i xk_x} \mathrm{d}k_x \\ &= \int_{\mathbb R} P(k_x) e^{2\pi i xk_x} \mathrm{d}k_x \\ &= p(x) \,, \end{aligned}$

where $s(\cdot)$ is the inverse Fourier transform of $S(\cdot)$. This establishes a duality for slicing and projection in the spatial and frequency domains, respectively.

The most immediate utility of this result is in helping to unify the meaning of *tomography*. By Wikipedia's definition,^{[4]}

The word

tomographyis derived from Ancient Greek τόμος tomos, "slice, section" and γράφω graphō, "to write" or, in this context as well, "to describe."

[4] | https://en.wikipedia.org/wiki/Tomography |

**X-ray:**The Fourier Slice Theorem shows that X-ray projections in the spatial domain are equivalent to orthogonal slices in the frequency domain. Slicing = tomography!**CT:**CTs are reconstructed from multiple X-ray images acquired at different angles, so therefore, they are also tomographic.**MRI:**MRI machines acquire slices of the frequency domain, so they too are tomographic.**Ultrasound:**Acoustic waves are used to acquire slices of the spatial domain in ultrasound. The dual theorem shows that these slices are equivalent to projections in the frequency domain.

In this way, the Fourier Slice Theorem can help to unify the meaning of tomography across many modalities.

© Vivek Gopakrishnan. Last modified: December 25, 2022. Website built with Franklin.jl and the Julia programming language.