We still consider the restoration algorithm, in which the following conductivity is used for the perturbed problem:
Then we have the following asymptotic result [18]:
(2.42) |
This result proves that the segmented image can be approximated by if is small. We now assume that the edge set is of codimension in . From the point of view of applications, it is completely natural to assume that the edges are flat in the image. In order to have coherent notations, we will further denote by the edge set. We assume that is known, e.g. provided by the crack detection algorithm previously seen.
We can rewrite the approximated segmentation problem as follows:
(2.45) |
(2.46) |
For numerical reasons, it can be very difficult to solve directly problem , and even problem for too small values of . Indeed the conditioning of the system to be solved goes to infinity when . In order to overcome this issue, we will expand the solution of problem into a power series of .