Transport of Intensity Equation

General form of TIE:

$$-\frac{2 \pi}{\lambda} \frac{\partial I(x,y)}{\partial z}= \nabla\bot \cdot I(x,y) \nabla\bot \phi(x,y)$$

For of pure phase object:

$$ -\frac{2 \pi}{\lambda I(x,y)} \frac{\partial I(x,y)}{\partial z}= \nabla_\bot^2 \phi(x,y) $$

Solution for solving TIE

FFT method

标准泊松方程的FFT解法：

$$\begin{array}{l}
Y = \nabla_\bot^2 X \ \Rightarrow \Im \left { Y \right } =\frac{\Im \left {X \right }} { 4{\pi ^2} \left({f_x}^2 + {f_y}^2 \right)} \ \Rightarrow X=\Im^{-1} \left {\Im \left { Y \right }{ 4{\pi ^2} \left({f_x}^2 + {f_y}^2 \right)} \right} \end{array}$$

TIE的FFT解法需要解2次泊松方程：

1. 化TIE为泊松方程

$$\begin{array}{l} Y=-\frac{2 \pi}{\lambda} \frac{\partial I(x,y)}{\partial z}\ \nabla\bot^2 X = \nabla\bot \cdot I(x,y) \nabla_\bot \phi(x,y) \end{array}$$

可以求解出$X$.

1. 再次化标准泊松方程，$X$为常量，$\phi$为变量。

$$\begin{array}{l} \nabla\bot \cdot I(x,y) \nabla\bot \phi(x,y)=\nabla\bot^2 X\ \Rightarrow I(x,y) \nabla\bot \phi(x,y) = \nabla\bot X\ \Rightarrow \nabla\bot^2 \phi(x,y)= \nabla\bot \frac{\nabla\bot X}{I(x,y) }\ \end{array}$$

再次解泊松方程便可求出$\phi$。