Thesis (PDF) - Signal & Image Processing Lab

the number of gray levels is high, or if the image is very complex, the BTV transform
usually fails. The reason for that is the existence of skeleton (watershed) points
in the BTV data, which have neighbors with potentially very dissimilar alternating
sequences; when eroded, a “trench” may open at these points, because the infimum
of the dissimilar alternating sequences yields a sequence that does not characterize
any of the neighbors. Although some of the “trenches” disappear, when we perform
opening instead of erosion, the results are still not good, see Fig. 4.4.
Let us consider a one dimentional example, given in Table 2.1, in order to under-
stand the origin of the trenches. In this example, x = 5 is a skeleton (or watershed)
point of the BTV transform of function f(x), with two different transform possibil-
ities {−2, 13, −1} and {8, −3, 5}. For x = 4 the alternating sequence is {−2, 13}.
Therefore, the infimum between two neighbors {−2, 13} and {8, −3, 5} is {0}. This
produce the trench at x = 4.
Fig. 4.5 shows the output of the erosion ˆεB on the one dimentional function f(x).
Figure 4.5: Results of BTV-based erosion ˆεB. (a) Function f(x), and (b) Erosion of
f(x).
Solutions for the “trench” problem are proposed in the following sub-sections.
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