Thesis (PDF) - Signal & Image Processing Lab

Chapter 4
The “trench” problem and the
proposed solutions
The BTV-domain morphological operators (erosion, opening etc.) proposed by Keshet
in [16] are self-dual. This means that those operators treat bright and dark “objects”
similarly. However, when applying those operators, a “trench” problem may arise.
This chapter studies this problem, and proposes solutions for it.
The BTV-domain morphological filtering approach involves three steps. First, the
BTV transform is performed on the input image, using the TD-tree representation
of the image. Then the transformed image is simplified using morphological filtering,
based on the partial ordering of images of alternating sequences, defined in Section
2.5.4. Finally output image is constructed from the processed image by an inverse
BTV transform, using Equation (2.21).
Figs. 4.1 and 4.3 show the output of the erosion ˆεB on two test images. Figs. 4.2
and 4.4 show the corresponding opening ˆγB of the same two test images.
Notice that the results of self-dual erosion and opening (Figs. 4.1-4.2) on the
synthetic image are as expected. The corresponding erosion has the effect of shrinking
all the elements of the image, regardless to their contrast. The opening rounds the
corners that SE can not get into in all the elements of the image, bright and dark.
However, the results on the natural image are very strange; the output shows
many “trenches” that do not appear to have any justification to exist.
These are typical results. When the number of gray levels is small, and the image
is not very complex, the BTV-based erosion returns useful results. However, when
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