Filtering Operations

Within each z-plane $z=Z_0$ an equidistant sampling is defined by the pixel difference of the screen: $\Delta X_{screen}$, $\Delta Y_{screen}$ according to the fan projection for the static resolution:

$$\Delta X_{obj}\left( z\right) =\frac{\Delta X_{screen}\cdot z}{Z_{screen}}$$ (28)

and

$$\Delta Y_{obj}\left( z\right) =\frac{\Delta Y_{screen}\cdot z}{Z_{screen}}$$ (29)

with $Z_{obs}=0$. According to this distances the local Nyquist frequencies for low pass filtering are given by $f_{Nyx} = \frac{1}{2\Delta X\left( z\right) }$ resp. $f_{Nyy} = \frac{1}{2\Delta Y\left( z\right) }$. Then the low pass filtering for the z-plane $z=Z_0$ can be done at the point $x_0$, $y_0$ by the functions

$$si\left[ \frac{x-x_{0}}{2\cdot \Delta X}\right] =\frac{\sin ... ...Delta X}\right] }{2\pi \cdot \frac{x-x_{0}} {2\cdot \Delta X}}$$ (30)

and

$$si\left[ \frac{y-y_{0}}{2\cdot \Delta Y}\right] =\frac{\sin ... ...Delta Y}\right] }{2\pi \cdot \frac{y-y_{0}} {2\cdot \Delta Y}}$$ (31)

Let be $B\left( x,y,Z_0 \right)$ the picture in z-plane $Z_0$ and $\delta x$ resp. $\delta y$ very small sampling distances. The x-y-filtering then is carried out for the point $x_0$, $y_0$ by the sum

$$B_{p}\left( x_{0},y_{0},Z_{0}\right) = \sum_{i,k}B\left( i\cdot \delta x,ki\cdot \delta y,Z_{0}\right) i\cdot \mbox{ } \ldots$$

$$\ldots si\left[ \frac{i\cdot \delta x-x_{0}} {2\cdot \Delta X_{obj}\left... ... \frac{k\cdot \delta y-y_{0}}{2\cdot \Delta Y_{obj}\left( Z_{0}\right) }\right]$$ (32)

In case a dynamic resolution is wanted $\Delta X_{obj}\left( Z_0 \right)$ has to be substituted by $\Delta X_{objdyn}$.

The not equidistant z-Planes $Z_{obj}\left( i_z \right)$ for sampling are given by equation $\left( \ref{Eq_Zobs7} \right)$. The filtering function for the z-coordinate at the point $z_0 = Z_{obj}\left( i_z=i_0\right)$ is derived as follows:

$$f\left( z,z_{0}\right) =si\left[ \frac{1-\frac{Z_{screen}}{z}}{2\Delta D}-i_{0}\right] \mbox{,}$$ (33)

with

$$\Delta D=\frac{\Delta X_{screen}}{D_{eye}}$$ (34)

Thus the filtering sum for the smaller sampling distances $\delta z$ will be

$$B_{f}\left( x_{0},y_{0},z_{0}\right) =\sum_{n}B_{p}\left( x_{0},y_{0},z\right) \cdot f\left( n\delta z,z_{0}\right)$$ (35)

according $\left( \ref{Eq_Zobs8} \right)$ $Z_{min}<n\delta z<Z_{max}$. This completes the description of the three dimensional filtering operations.

Normally the object image data are given also sampled. Then the analogue function or re-sampling function is achieved by the known sum using si-functions.