User:NTUL/sandbox/VanderLugtfilter

Image Restoration (8.8)
Image blurring by a point spread function is studied extensively in optical information processing, one way to alleviate the blurring is to adopt Wiener Filter. For example, assume that $$o(x, y)$$ is the intensity distribution from an incoherent object, $$i(x, y)$$ is the intensity distribution of its image which is blurred by a space-invariant point-spread function $$s(x, y)$$ and a noise $$n(x, y)$$ introduced in the detection process:

$$i(x,y)=o(x,y)\otimes s(x,y) + n(x,y)$$

The goal of image restoration is to find a linear restoration filter that minimize the mean-squared error between the true distribution and the estimation $$\hat{o}(x, y)$$. That is, to minimize

$$\epsilon^2=|o-\hat{o}|^2$$

The solution of this optimization problem is Wiener filter:

$$H\left(f_{X}, f_{Y}\right)=\frac{S^{*}\left(f_{X}, f_{Y}\right)}{\left|S\left(f_{X}, f_{Y}\right)\right|^{2}+\frac{\Phi_{n}\left(f_{X}, f_{Y}\right)}{\Phi_{o}\left(f_{X}, f_{Y}\right)}}$$,

where $$S\left(f_{X}, f_{Y}\right), \Phi_o\left(f_{X}, f_{Y}\right), \Phi_n\left(f_{X}, f_{Y}\right)$$ are the power spectral densities of the point-spread function, the object and the noise.

Ragnarsson proposed a method to realize Wiener restoration filters optically by holographic technique like setup shown in the figure. The derivation of the function of the setup is described as follows.

Assume there is a transparency as the recording plane and an impulse emitted from a point source S. The wave of impulse is collimated by lens L1, forming a distribution equal to the impulse response $$h$$. Then the distribution $$h$$ is then split into two parts:


 * 1) The upper portion is first focused (i.e., Fourier transformed) by a lens L2 to a spot in the front focal plan of lens L3, forming a virtual point source generating a spherical wave. The wave is then collimated by lens L3 and produces a tilted plane wave with the form $$r_0e^{-j2\pi \alpha y}$$ at the recording plane.
 * 2) The lower portion is directly collimated by lens L3, yielding an amplitude distribution $$\frac{1}{\lambda f}H(\frac{x_2}{\lambda f},\frac{y_2}{\lambda f})$$.

Therefore, the total intensity distribution is

$$\begin{aligned} \mathcal{I}\left(x_{2}, y_{2}\right)=&\left|r_{o} \exp \left(-j 2 \pi \alpha y_{2}\right)+\frac{1}{\lambda f} H\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)\right|^{2} \\ =& r_{o}^{2}+\frac{1}{\lambda^{2} f^{2}}\left|H\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)\right|^{2}+\frac{r_{o}}{\lambda f} H\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right) \exp \left(j 2 \pi \alpha y_{2}\right)+\frac{r_{o}}{\lambda f} H^{*}\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right) \exp \left(-j 2 \pi \alpha y_{2}\right) \end{aligned}$$

Assume $$H$$ has an amplitude distribution $$A$$ and a phase distribution $$\psi$$ such that

$$H\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)=S\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right) \exp \left[j \psi\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)\right]$$,

then we can rewritten intensity as follows:

$$

\begin{aligned} I\left(x_{2}, y_{2}\right)=& r_{o}^{2}+\frac{1}{\lambda^{2} f^{2}} S^{2}\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)+\frac{2 r_{o}}{\lambda f} S\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right) \cos \left[2 \pi \alpha y_{2}+\psi\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)\right]. \end{aligned} $$

Note that for the point at the origin of the film plane ($$(x_2,y_2)=\mathbf{0}$$), the recorded wave from the lower portion should be much stronger than that from the upper portion because the wave passing through the lower path is focused, which leads to the relationship $$r_0<<\frac{1}{\lambda f}$$.

In Ragnarsson' s work, this method is based on the following postulates:

I=r_{o}^{2}+\frac{1}{\lambda^{2} f^{2}} S^{2} \end{aligned}$$. \Delta I \left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)=\frac{2 r_{o}}{\lambda f} S\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right) \cos \left[2 \pi \alpha y_{2}+\psi\left(\frac{x_{2}}{\lambda f}, \frac{y_{2}}{\lambda f}\right)\right]. \end{aligned}$$.
 * 1) Assume there is a transparency, with its amplitude transmittance $$t$$ proportional to $$s(x, y)$$, that has recorded the known impulse response of the blurred system.
 * 2) The maximum phase $$\phi$$ shift introduced by the filter is much smaller than $$2\pi$$ radians so that $$e^{j\phi}\approx 1+j\phi$$.
 * 3) The phase shift of the transparency after bleaching is linearly proportional to the silver density $$D$$ present before bleaching.
 * 4) The density is linearly proportional to the logarithm of exposure $$\begin{aligned}
 * 1) The average exposure $$\bar{I}$$ is much stronger than varying exposure $$\begin{aligned}

By these postulates, we have the following relationship:

$$\Delta t\propto\Delta \phi \propto\Delta D \propto\Delta(logI)\approx \frac{\Delta I}{I}$$.

Finally, we get a amplitude transmittance with the form of a Wiener filter:

$$\Delta t\propto\frac{S^\ast}{r_0^2\lambda^2 f^2 + \left\vert S \right\vert^2}$$.