Thesis (PDF) - Signal & Image Processing Lab

48CHAPTER 4. THE “TRENCH” PROBLEM AND THE PROPOSED SOLUTIONS
4.1 Filtering using an adaptive structuring element
One of the solutions to the trench problem is, first, to choose smartly only a single
variant of the BTV transform. For instance, if there are two flat zones with the same
topographic distance, which potentially can be the fathers of the current node in
the TD-Tree, choose the one that contains the maximum number of pixels. In our
example shown in Fig. 4.6, vertices 4 and 6 can be fathers of vertex 5. We choose
the vertex 6 to be the father - this option is drawn with a solid line, and discard the
relation to the vertex 4, that is drawn with a dashed line.
Figure 4.6: A tree representing the BTV transform of function f(x).
Second, use an adaptive structuring element, constraining the filtering depth, dur-
ing the filtering operation. That is, while computing the infimum of the neighborhood,
do not take into consideration neighboring pixels that cause the path-variation to be
pruned by more than a given filtering depth. For example, if the filtering depth is
limited to 1, then, while performing erosion of x = 4, we ignore the presence of the
point x = 5 and make a trivial erosion with itself (in the 2D case it usually won’t
happen). For this method the result will be {−2, 13}
The use of an adaptive SE has some drawbacks. For instance, in some applications
it will be necessary to store this adaptive SE for every pixel. But, in case of opening
it can be implemented without the need to store the SE. Another drawback is that
the adaptive SE, although prevents a creation of trenches, limits the performance of