Thesis (PDF) - Signal & Image Processing Lab

4.2. FILTERING USING MULTIPLE MINIMAL PATHS 53
4.2 Filtering using multiple minimal paths
An alternative way to avoid the trenches is to store, for each flat zone, all possible
combinations of the path-variations, and to choose the most suitable one during
filtering. The suitable path-variation is one that has a maximal common-part length.
For the example given in Table 2.1, we can store, for the skeleton point x = 5, both
BTV transform possibilities {−2, 13, −1} and {8, −3, 5}. Graphically, it can be seen
in Fig. 4.6 that the vertex 5, (x = 5), has 2 possible origins - one from the vertex 4
and another from vertex 6. For x = 4, the alternating sequence is {−2, 13}, thus its
father is vertex 3. Therefore, the infimum path-variation of the two neighbors x = 4
and x = 5 is either {0} or {−2, 13}, depending on which option we will choose. We
choose the path-variation that has a maximal common part length - the option where
vertex 4 is the father of vertex 5, that is, {−2, 13}. Of course, this process is more
complex and more memory consuming.
Another possible implementation is to store the information about multiple pos-
sibilities for minimal paths, by enabling connections of the nodes to the multiple
fathers in the TD-Tree. This saves the memory usage, but the price is computational
overhead. In this case, the path-variation will be created a number of times for each
pixel during the filtering, instead of to create it once for each flat zone - before the
filtering. Our implementation uses the first option of storing multiple paths for each
vertex.
A drawback of this approach is that it does not fit all cases, but only those where
the topographic distance is equal in the different paths. Obviously, it ignores a large
number of cases, because the topographic distance does not have to be equal for
multiple paths. Therefore, some trenches remain after the filtering. See the example
given in Fig. 4.12.
This method has another significant drawback - the performance issue. For real-
world images, the number of combinations become very large, and the computation
time and memory requirements are exponentially dependent on the number of flat
zones in the image. Probably, it is possible, by using some heuristic methods, to
reduce significantly the computation requirements, but this issue is out of the scope
of this thesis.