166.

[Cover] [Contents] [Index]

Page 247

on the domain [+∞>(w_r−w_r′)≥0]. The first derivative g′(wr_–wr_′) is generally expressed as:

(6.24)

where h(w_r−w_r′) is called the interaction function. The rationale for including this expression in Equation (6.24) is explained in Section 6.3. The conditions in Equations (6.22) and (6.23) guarantee that if a difference between w_r and w_r, does occur, then the function g(w_r−w_r′) will apply a penalty (in the term of higher energy) to reduce the probability of occurrence of this difference, and the penalty is related to the magnitude of |(w_r −w_r')| as shown below. A simple function for satisfying the above conditions can be a quadratic function such as:

(6.25)

which will penalise every discontinuity. Consequently, Equation (6.25) is likely to generate a surface which is continuous everywhere. Since we are only interested in restoring piecewise continuity, Equation (6.25) can be refined, following Blake and Zisserman (1987), to give:

(6.26)

where λ is a truncation point (described below). Equation (6.26) shows that if the difference w_r−w_r′ is sufficiently large then it is likely to indicate a potential edge. In this case, we would like to pay out lower energy λ instead of (w_r−w_r′)² in order to offer a higher probability for its existence. Figure 6.9 illustrates such a truncation function.

Figure 6.9 A function g(w_r−w_r) truncated by λ.

[Cover] [Contents] [Index]