The Central Slice Theorem

The Central (/Fourier) Slice Theorem

The Central or Fourier Slice Theorem ( or projection-slice theorem, abbreviated CST) is the basis for Fourier-based inversion techniques. We keep the discussion restricted to 2 dimension ( $$\mathbf{x}\in \mathbb{R}^{2}, t\in \mathbb{R} , \boldsymbol{\omega}\in \mathbb{S}^{1}$ ). Let us take the function f(x,y) and take the Fourier transform of its Radon transform, but only regarding its t affine parameter, the rounded brackets indicate variables after the transform:

$$ \mathfrak{F}_{t}\left [\mathfrak{R}f\left ( t,\vartheta \right ) \right ]\left ( r,\vartheta \right )=\int_{-\infty }^{\infty }\mathfrak{R}f\left ( t,\vartheta \right )e^{-itr}dt=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f\left ( t\boldsymbol{\omega} +s\boldsymbol{\omega}_{\perp } \right )e^{-itr}dtds$

Let us choose new integration variables x and y:
$$ t=\boldsymbol{\omega}\textbf{x}=x\cos \left ( \vartheta \right )+y\sin \left ( \vartheta \right )$
$$ s=\boldsymbol{\omega_{\perp} }\textbf{x}=-x\sin \left ( \vartheta \right )+y\cos \left ( \vartheta \right )$
The Jacobian equals of course to 1:
$$ \begin{vmatrix} \partial_{x}t & \partial_{y}t\\ \partial_{x}s &\partial_{y}s \end{vmatrix}=\cos ^{2} \left ( \vartheta \right )+\sin ^{2} \left ( \vartheta \right )=1$
Thus:
$$ \int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f\left ( x,y \right )e^{-i\left ( x\cos \vartheta +y\sin \vartheta \right )r}dxdy=\int_{-\infty }^{\infty }\int_{-\infty }^{\infty }f\left ( x,y \right )e^{-ix r\cos \vartheta }e^{-iry\sin \vartheta \right )}dxdy=\mathfrak{F}_{x,y}\left [ f \right ]\left ( r\cos \vartheta ,r\cos \vartheta \right )$

On the RHS of the equation is a 2D Fourier transform and does not include Radon transformation, and its frequency space variables are also 2D and given in polar coordinates. The 1D Fourier transform in the t affine parameter of the Radon transform gives the 2D Fourier transform of our original function with variables $$ \left (\xi_{1} ,\xi_{2} \right )= \left (r\cos \vartheta ,r\sin \vartheta \right )$ . If our tomograph operates with equally spaced sampling in variables $$ $ \left(t,\boldsymbol{\omega} \right )$ the resulting 2D Fourier transform is sampled in the following points:

It would be straightforward to start using this result as a basis for a numerical inversion scheme. We could 1D Fourier transform our sinogram for every fixed angle and take the 2D inverse Fourier transform of the obtained dataset. Fourier transform itself is a standard numerical technique.

The popularity of the Fourier transform in signal and image processing can largely be attributed to the existence of the Fast Fourier Transform (FFT) algorithm. FFT in the simple case needs a Cartesian grid, in this case for the 2D inverse Fourier transform the input data is given in polar coordinates. For the so called direct Fourier reconstruction we need to resample the polar grid to Cartesian.

The relatively low abundance of the direct Fourier reconstruction lies in the inaccuracy of resampling and of the uneven distribution of information. The Central Slice Thereom can further be manipulated for better numerical stability, the derivation is given in the next section.

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