Biomedical Imaging

$C_{\underset{\_}{f}\underset{\_}{f}}(i,j)=C_{\underset{\_}{f}\underset{\_}{f}}(i-j),(7.83)$

in other words, we require statistical invariance under translations, which is actually quite a strong restriction on the object. The imaging process is again modeled as

$\underset{\_}{X}=\left(\underset{\_}{\mathcal{P}}*\underset{\_}{f}\right)+\underset{\_}{\xi },(7.84)$

with the PSF $\underset{\_}{\mathcal{P}}$and independent Gaussian noise ξ with variance σ2. The idea underlying the Wiener filter is to recover the original object f by convolution of the image X with a function h. To this end, h has to be chosen such that the average discrepancy between h ∗ X and f is minimal

$\begin{array}{c}\mathbb{E}\left[{(\underset{\_}{h}*\underset{\_}{X}-f)}^2\right]=\mathbb{E}\left[{\underset{\_}{Z}}^2\right]-2\cdot \mathbb{E}\left[\underset{\_}{Z}\underset{\_}{f}\right]+\mathbb{E}\left[{\underset{\_}{f}}^2\right]\\ ={\displaystyle \sum _i(C_{\underset{\_}{Z}\underset{\_}{Z}}(i,i)}-2C_{\underset{\_}{Z}\underset{\_}{f}}(i,i)+C_{\underset{\_}{f}\underset{\_}{f}}(i,i)),\end{array}(7.85)$

where single components Zi of Z are

$Z_i={\displaystyle \sum _j{h}_j{X}_{i-j}.}(7.86)$