Thesis (PDF) - Signal & Image Processing Lab

8 CHAPTER 1. INTRODUCTION
disadvantages:
1. Are not really self-dual (see [1, 3]).
2. Allow to design filters (see [9, 8, 10, 11]), but not openings (or erosions, etc.).
3. Need a reference image (like in the reference semilattices, developed by Keshet
[4], and further studied by Heijmans and Keshet in [5]).
The erosion and opening operators that are based on the BTV Transform suffer
from a “trench” problem. The “trench” problem arises, especially in complex gray-
scale images, when the operators are applied (see Fig. 4.3(b)). Our initial goal in
this research was to solve this problem. Afterwards, the next goal was to design
improved self-dual morphological operators that are not necessary connected, and to
check application possibilities for the new operators.
1.3 Original contributions
Initially, an efficient implementation of self-dual morphological operators in Boundary-
Topographic-Variation (BTV) domain, is proposed. In addition, the “trench” problem
introduced by the BTV morphological operators is studied and a few solutions are
proposed.
We also noticed that an efficient implementation of these morphological operators
is achieved by first transforming the input image into an appropriate tree represen-
tation.
While working on our research, another, more robust semilattice (without trench
problems) was introduced by Keshet [17]. This new semilattice is based on yet another
tree representation, the tree of shapes, proposed by Monasse and Guichard [12, 13]
(see also [14, 15]). The tree of shape is slightly different from that of the BTV.
Notice that both the BTV and the shape tree semilattices, and their corresponding
morphological operators, can be directly associated to self-dual tree representations.
There are many other self-dual tree representations in the literature. For instance,
Salembier and Garrido proposed a Binary Partition Tree for hierarchical segmenta-
tion in [10, 11]. These tree representations are usually used for obtaining connected
filtering operations on an image; however, they do not yield non-connected operators,