The output of all these experiments are object boundaries. We have set up experiments using synthetic images to qualitatively evaluate the method.

For our purposes we created a synthetic image that has one target object in the center surrounded by a background. To be noticed is that the target object has both convex and concave parts to it. Further, it also has some high curvature points to make the boundary finding process non-trivial. To make it even more complicated we have smoothed the image using a Gaussian kernel so that the edges become fuzzy. On top of that white noise was added which would again affect the boundaries the most as they were smoothed out. Thus, in a way it represents most of the troubles associated with structure segmentation in biomedical images. Fig 8(a) shows an example of such a synthetic image. The advantage of synthetic images is that we know exactly the actual boundary location.

For evaluation we used the following procedure. First, the true boundary was evenly sampled into 256 points. The boundary finding process was initialized with a boundary that is spatially far away from the true boundary. Depending upon the situation it resulted in an output boundary. This was then evenly sampled into as many points. To find out how closely the output boundary approximated the true boundary, we need to calculate the distance between them. To solve the problem of pointwise correspondence, we keep one of the boundaries fixed, and vary the starting point of the other boundary point by point, calculating at each step the total distance as a sum of the distances between each corresponding points. Thus for example, in the first instance, point 1 of the first boundary is matched with point 1 of the second, point 2 with point 2 of the second and so on. Next the second boundary was shifted by one point so that point 1 of the first goes with point 2 of the second, point 2 matches with point 3 of the second and point 256 compares with point 1 of the second boundary. Thus we end up with 256 values of probable distances between the two boundaries. The minimum of these is considered to be the actual value of the distance.

The comparisions were done using three versions of the objective function. When only the gradient based term in the objective function was used, we have the traditional gradient based boundary finding. The second method only uses the region based term, where information only from the region classified image is used. Finally, the proposed method, uses both of the above terms in a combined way. For all the cases optimization was carried out using the conjugate gradient method. Another important thing that needs to be mentioned is that for any reading on any one of the plots the experiment was repeated ten times under exactly the same settings, and then the average was taken to filter out minor variations that may arise. One needs to do it because even though the noise distribution remains the same, the pixel values of the noise could be different.

Mon Mar 21 19:45:28 EST 1994