Biomedical Imaging

In a more concise form, the function f(x) can be reconstructed by calculation of

$f\left(\underset{\_}{x}\right)={\left(2\pi \right)}^{-n+\frac{1}{2}}{\displaystyle \int \text{d}t{t}^{n-1}}{\displaystyle \int \text{d}{\underset{\_}{n}}_{\theta }{e}^{i\underset{\_}{x}\cdot {\underset{\_}{n}}_{\theta }t}}\left({\mathcal{F}}_s\mathcal{R}f\right)\left({\underset{\_}{n}}_{\theta },t\right).\left(4.24\right)$

Hence, the function f(x) can be obtained by Fourier transforming the one-dimensional, measured intensity profiles, arranging them in the n-dimensional Fourier space followed by an inverse Fourier transform, according to the FST.

However, FST based reconstruction is not very suitable for incomplete data, when information is reduced to only a few lines, see Figure 4.8. When the number of projection angles is insufficient, or the angles are not evenly distributed (the sample holder, for example, can easily obstruct some angles creating a missing wedge, see Section 4.5.1), FST based reconstruction ...