Thesis (PDF) - Signal & Image Processing Lab

6 CHAPTER 1. INTRODUCTION
the image edges, as can be seen in Figs. 1.1(c) and 1.1(d).
The most popular approach to approximately achieve self-duality in mathematical
morphology is to use alternating dual operators. For instance, opening-closing and
closing-opening filters, or the more general alternating sequential filters (ASF’s) [1, 3].
However, these are not really self-dual, and some of the difficulties related to the lack
of self-duality still appear (as in the above example of Fig. 1.1).
Development of self-dual morphological operators can be found, for instance, in
the works of Serra, [1, chapter 8], Heijmans [6], and Mehnert & Jackway [7]. Al-
though most of these operators are morphological filters (i.e., they are idempotent
and increasing, see [1, 3]), the underlying approaches do not provide for adjunctions,
or designing extensive or anti-extensive operators, see [1, 3]. As a consequence, the
ability to design basic morphological operators (erosions, dilations, openings, and
closings) is lost. Useful morphological tools, such as skeletons, granulometries and
gradients [1, 3], are not available either.
Another important branch of research is that of self-dual connected operators.
Connected operators remove objects from an image without affecting the edges of
the remaining objects. In [8], Heijmans provides a summary of the activity in this
area, and characterizes in depth binary connected operators. A summary and char-
acterization of grayscale connected operators can be found in the works of Serra and
Salembier [9]. A subclass of connected operators are operators based on tree repre-
sentations. Salembier and Garrido proposed a Binary Partition Tree for hierarchical
segmentation in [10, 11]. A tree of shapes was proposed by Monasse and Guichard
[12, 13] (see also [14, 15]). These representations are all self-dual and very powerful.
However, the above methods do not define non-connected morphological operators,
such as an erosion.
1.2 Background and motivation
A new semilattice, based on the so-called Boundary-Topographic-Variation (BTV)
Transform, was defined by Keshet in [16]. The importance of that new semilattice is
that it allows to define self-dual non-connected morphological operators (like open-
ings, erosions, etc.) without need of a reference image.
In contrast, the other morphological approaches have one or more of the following