Measure of Confidence

A measure of confidence in the detected boundaries can play an important role not only in the estimation of the real boundary but also in the process of detection of individual points on the texture crossings. The texture distance measures described above can be used to evaluate the measure of confidence in the detected boundary.

Figure 2 shows the Bhattacharyya distance between the distributions on both sides of detected contours for a case where the textures are quite (visually) different (a) and another case where the texture is similar and rather complex (c). Bhattacharyya distance varies between $0$ and $1$. We can clearly see that for the case where the textures on both sides are quite distinguishable, the distance is close to $1$while for the complex texture case the value of the distance varies arbitrarily. This observation can be used as a measure of confidence in the detected boundary points.

Figure 2: Bhattacharyya Distance for different and similar texture on both sides of detected boundary points

(a)	(b)
(c)	(d)

Having computed either of these distance metrics for texture we can build up a reliable measure of confidence using a training set. We start by building two distributions for the distances of two textures one for similar textures and one for different textures. Then for a given test sequence and the calculated boundary, using the Bayesian rule, we can optimally determine whether the two texture on both sides are similar or they are effectively two distinct textures. In addition we can calculate the probability of error in such a decision making.

To illustrate this idea, consider the following simple example. The red dots in Figure 3-(a) show the detected boundaries on two Brodatz textures. Figure 3-(b) shows the Bhattacharyya distances between the texture distributions on both sides of the detected boundaries. The distance between two regions selected from the bottom texture is shown in (c) while (d) shows the same distance for two regions selected from both textures. We can observe that in general the distance for the detected boundaries ,(b), is closer to the distance between two regions selected from both sides, (d). Moreover, the distance shown in (c) can be used as a prior knowledge to remove the unreliable points detected. This measure of confidence can be improved using a better learning scheme than the ad-hoc try explained here. Furthermore it applies to any kind of model that we chose for the distribution.

Figure 3: Detected boundaries on two Brodatz textures(a). (b) shows the histogram distances for detected boundary points, (c) is the distance between two regions selected from the bottom texture and (d) gives the distance between two textures selected from the top and bottom areas.

(a)	(b)
(c)	(d)