Methodology in Deciding Coupling Distance

Following the previous observation, we would like to quantify how the two fully immersed

spheres interact with each other without physical contact. We propose to use the coupling distance,

tic

s , to signify at what interstitial distance that the target sphere is set into motion. .

Ideally, we can directly search the vt – sti diagram for the point where vt is first elevated from vt

= 0 to determine the coupling distance. However, the severe fluctuations observed in Fig. 4.17 (a) and (b) invalidate this simple approach. Instead, we employed changes in the location of the target sphere, st, from its original status to determine the coupling distance, which bypassed the problematic time differentiation of this slow motion. The st data obtained from the circular Hough

slight fluctuation at the beginning of the process persists as boxed, which closely correlates with the results of the error analysis presented in Section 3.2.3.

Fig. 4.18: Example of the position-separation profile of the target sphere for detecting the onset of its motion.

As discussed in Section 3.2.3 and Fig. 3.15, the most probable fluctuation of the error, Δ, was evaluated to be 0.2 pixels around some center of fluctuation. Its corresponding physical scale can be easily determined by multiplying it with the physical-to-image conversion ratio, D/d. Next, we want to determine the center of fluctuations. An evaluation scheme that adopts varying number of data points are now proposed. In practice, we consider the first N data points of the position (st) time sequence to compute its mean and standard deviation (STD). By increasing N, we are averaging over a greater extent of position data. For a ‘true’ static target motion, increase of N would help to average out the spurious fluctuation from image processing. Thus, a local minimum of STD would

displacement in st from the center of fluctuation never exceeds Δ in its physical scale, this

displacement is considered illusory due to fluctuation. The very target location that exceeds this threshold is employed to determine the desired coupling distance by the corresponding sti.

Needless to say, a reasonable mean of fluctuation plays a crucial role in the success of this methodology, which requires a more detailed discussion in the following. Firstly, the STD results obtained for the data shown in Fig. 4.18 using increasing N, or equivalently decreasing sti, are shown in Fig 4.19. The trend for STD is reasonable. Initially, the STD starts from 0 as N = 1 and it jumps to a peak and gradually steps down. This overshooting suggests that the mean is highly sensitive to data fluctuation when the sample size is low. With more samples under consideration, the STD becomes more stable and reaches a local minimum indicated by the arrow. It is this sample size that is thought to give the most reasonable mean. Apart from this, the following trend keeps quite stable for a long range. If we ever use a point in this range to determine the data extend of average, the result should be acceptable as well. Note that a local minimum does not always occur so early and so discernably but if it does occur, it is observed to always reside in the ‘stable’ range stated above.

Fig. 4.19: Example of the trend of STD used to determine the center of fluctuation. The very head section of this trend is enlarged in the rectangle. The data point indicated by the arrow is a local minimum, and its index is thought to be the data extent that gives the most reasonable mean.

After a mean of fluctuation at the beginning of the -s s_{t ti} profile is determined as the blue

horizontal line added to Fig. 4.20, the estimated error,  , is superimposed after unit conversion as the red line. The intersection of this elevated line with the profile should represent the onset of target motion with the uncertainty in locating the sphere taken into account. The corresponding interstitial distance, sti, of the intersection thus determines the coupling distance, s_ti^c.

Fig. 4.20: Center of fluctuation and the threshold imposed onto the trend of STD. The st-intercept of the horizontal blue line is the center of fluctuation, and the threshold (red line) is elevated from the blue line by an amount of Δ in its physical scale.

However, when the region about the intersection is enlarged and examined in Fig. 4.21, eight

cover a range of sti for 5mm and it is difficult to exactly determine the point of intersection when interfered by this fluctuation. Thus, we now set out to evaluate the data trend in a statistical manner.

In Fig. 4.21, we firstly select point A from which all the preceding points are below the elevated threshold, Similarly, point B is selected from which all the following points are above the threshold. All the points with their sti falling between those for A and B are chosen as a basic set (as boxed by the rectangle) for fitting a linear -s s_{t ti} line through them. In doing so, N-paired

downstream and upstream data points are included stepwisely for alternative linear fits. For example, with N = 1, the point A1 and B1 are included and then we continue for N = 2 to further include the points A2 and B2. Thus a sequence of slopes for the line fitted recursively to the expanded data points is generated. The corresponding sequence of STD is also computed. The same procedure introduced above in determining the mean of fluctuation is again applied here to determine the most feasible set of data points which gives the best linear fit. In some situations, a proper N can never be found and 5-paired downstream and upstream data points will be automatically appended to the basic set for a linear fit. The so-called best-fitted line then gives a specific intersection with the elevated threshold line, and the desired coupling distance is given by the sti - coordinate of the intersection.

Fig. 4.21: Determining the intersection of the position-separation profile (data points) and the threshold (horizontal line) in a statistical manner. The boxed data points are taken as a basic set for fitting a linear line through them.