Biomedical Engineering Reference
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Figure 11.8: The evolution only by advection leads to attracting a curve (initial
ellipse) to spurios edges, the evolution must be stopped without any reasonable
segmentation result (left). By adding regularization term related to curvature of
evolving curve, the edge is found smoothly (right).
curve evolution is given by Eq. (11.9), which is, of course, only another form of
Eq. (11.8).
Although model (11.8) behaves very well if we are in the vicinity of an edge,
it is sometimes difficult to drive the segmentation curve there. If we start with a
small circular seed, it has large curvature and diffusion dominates advection so
the seed disappears (curve shrinks to a point [22,23]). Then some constant speed
must be added to dominate diffusion at the beginning of the process, but it is not
clear at all when to switch off this driving force to have just the mechanism of
the model (11.8). Moreover, in the case of missing boundaries of image objects,
there is no criterion for such a switch, so the segmentation curve cannot be well
localized to complete the missing boundaries.
An important observation now is that Eq. (11.8) moves not only one partic-
ular level line (segmentation curve) but all level lines by the above mentioned
advection-diffusion mechanism. So, in spite of all previously mentioned seg-
mentation approaches, we may start to think not on evolution of one particular
level set but on evolution of the whole surface composed of those level sets.
This idea to look on the solution u itself, i.e. on the behavior of our segmenta-
tion function, can help significantly.
Let us look on a simple numerical experiment presented in Fig. 11.10
representing extraction of the solid circle depicted in Fig. 11.9. The starting
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