# Sampling Based on Kalman Filter for Shape from Focus in the Presence of Noise
## 6 Results and Discussion  
### Image Acquisition and Parameter Setting  
For experiments, four objects were used, as shown in Figure 6, consisting of one simulated and three real objects.  
First, a simulated cone image sequence consisting of 97 images, with dimensions of \(360\times 360\) pixels, was acquired. These images were generated using camera simulation software [40].  
The real objects used for experiments were: coin, Liquid Crystal Display-Thin Film Transistor (LCD-TFT) filter, and letter-I. The coin images were magnified images of [PERSON]'s head from the back of the US penny. The coin sequence consisted of 80 images, with dimensions of \(300\times 300\) pixels. The LCD-TFT filter images consisted of microscopic images of an LCD color filter. The image sequence of LCD-TFT filter had 60 images, with the dimensions of \(300\times 300\) pixels each. The third image sequence consisted of letter-I, engraved on the metallic surface. It consisted of 60 images, with dimensions of \(300\times 300\) pixels each. The real objects were acquired through a microscopic control system (MCS) [18]. The system consists of a personal computer integrated with a frame grabber board (Matrox Meteor-II) and a CCD camera (SAMSUNG CAMERA SCC-341) mounted on a microscope (NIKON OPTIPHOT-100S). Computer software obtains images by translating the object plane through a stepper motor driver (MAC 5000), possessing a 2.5 nm minimum step size. The coin and letter-I images were obtained under \(10\times\) magnification, while the LCD-TFT filter images were acquired under \(50\times\) magnification.  
In parameter setting, the standard deviation of the jitter noise for each object was assumed to be ten times the sampling step size of each image sequence, i.e., \(254\) mm, \(6.191\) um, \(1.059\) m, and \(1.529\) um for simulated cone, coin, LCD-TFT filter, and letter-I, respectively. For comparison of 3D shape recovery results, a local window \(7\times 7\) for focus measure operators was used. The total number of iterations \(N\) of the Kalman filter was set as 100.  
For performance comparison, Bayes filter and particle filter were employed [41; 42; 43; 44; 45; 46]. The depth estimation through the Bayes filter is presented in Figure 7.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(\mu_{n}\) & Position of each image frame without jitter noise \\ \(\sigma_{n}\) & Standard deviation of jitter noise \\ \(z_{f}\) & Best focused position through Gaussian approximation in each object point \\ \(\sigma_{f}\) & Standard deviation of Gaussian focus curve \\ \(\zeta\) & The amount of jitter noise before or after filtering \\ \(S\) & Position of each image frame after Kalman filtering \\ \(O\) & Position of each image frame before Kalman filtering \\ \(N\) & Total number of iterations of filters \\ \hline \hline \end{tabular}
\end{table}
Table 1: List of frequently used symbols and notation.  
Figure 6: 10 th frame of experimented objects: (**a**) Simulated cone, (**b**) Coin, (**c**) Liquid Crystal Display-Thin Film Transistor (LCD-TFT) filter, (**d**) Letter-I.