Biomedical Imaging

(which holds if intensity variations are small compared to phase variations) leads to

$I\left({\underset{\_}{r}}_{\perp },z\right)\simeq I\left({\underset{\_}{r}}_{\perp },0\right)\left(1-\frac{z}{k}{\nabla }_{\perp }^2\phi \left({\underset{\_}{r}}_{\perp },0\right)\right),\left(6.134\right)$

holding for small propagation distances z = Δ behind a plane taken to be at the origin z = 0. TIE based phase retrieval – in its simplest form – is the inversion of this equation, and hence computing φ from I(r⊥, z). To this end, the numerical inversion of the Laplacian is required, which can be achieved based on the Fourier representation of the ∇⊥ operator

${\nabla }_{\perp }^{-2}=-{\mathcal{F}}^{-1}\frac{1}{k_x^2+k_y^2}\mathcal{F},\left(6.135\right)$

${\nabla }_{\perp }=-{\mathcal{F}}^{-1}{\underset{\_}{k}}_{\perp }\mathcal{F}.\left(6.136\right)$

When boundary conditions are not important, the phase φ can be solved by implementation of FFT filtering. Alternatively, for special requirements regarding ...