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{\noindent\normalfont\fontseries{b}\LARGE Preface}
% Preface

\vspace{6mm}\noindent
This edited book on Mathematical Morphology and its Applications to
Signal and Image Processing is divided into eight parts containing the
full papers accepted for oral presentation to the 8$^{\textrm{th}}$
International Symposium on Mathematical Morphology, held in Rio de
Janeiro from October 10$^{\textrm{th}}$ to 13$^{\textrm{th}}$, 2007.

Each part is dedicated to a specific theme of interest. The
distribution of the 38 articles along these themes gives an idea of
the current trends in Mathematical Morphology (MM).

\section*{Lattice theory}

Coincidently, MM and Fuzzy Sets were both born in 1965 and
during the first few decades, both communities
independently made important contributions. Nowadays, we
observe some convergence between them.

Bloch extends the traditional skeleton by influence zones (SKIZ) to
fuzzy sets and shows a possible method of interpolation between fuzzy
sets. Hirata et al.\ present algorithms for the computation of the
basis of binary and gray-scale translation-invariant and
locally-defined operators in the presence of incomplete information
about the kernel. Kiselman introduces upper and lower inverses of
mappings between complete lattices, analyzes their properties and
links with Galois connections, and gives a thorough characterization of
them, as well as an extension to upper and lower quotient of two
mappings. Based on the concept of adjoint conjunctor-implicator pairs,
Popov discusses fuzzy dilation-erosion adjunctions on fuzzy sets,
interval arithmetic with fuzzy numbers, and a fuzzy hit-or-miss
transform. From a partial-order relation defined on tree
representations of images, which provide a complete inf-semilattice
structure, Vichik et al.\ derive self-dual morphological operators
that can be used as filters for noise reduction.

\section*{Geometry and topology}

Despite the fact that discrete geometry is a separate field of
research, it has a large intersection with MM whenever one considers
the applications of MM to digital image processing.

After recalling the definitions and properties of
smallest-neighborhood spaces and the Khalimsky topology on
\ensuremath{\mathbb{Z}^{n}}, Melin introduces a binary operation 
on topological spaces called join, and shows that under local finiteness theses spaces
can be uniquely decomposed as a join of indecomposable spaces. Serra
investigates the conditions for generating granulometries on
\ensuremath{\mathbb{Z}^{n}} using Steiner sets, as well as conditions
on these sets for them to be convex or connected.  In a second
article, Serra proposes a new random closed set model, combining the
generation of random germs and random sets, and applies this model to
the analysis of data on fires that occurred in Malaysia.

\section*{Signal processing}

MM based on lattice theory is a powerful approach to solve nonlinear
problems in signal processing. At the heart of this approach are the
nonlinear operators of dilation and erosion as illustrated in a few of
the contributions below.

Dorini and Leite introduce a scale-space toggle operator,
prove some of its properties, and illustrate its usefulness
for image segmentation. Karantzalos compares the construction
of nonlinear scale space image representations derived from
multiscale levelings constrained by four different markers.
Maragos and Evangelopoulos introduce alternative approaches
based on levelings and texture energy for decomposing an
image into cartoon and texture components.

\section*{Image processing}

The main application of MM is Image Processing. The
success of MM in this field comes from the fact that,
diferently from the linear approach, it is able to manage
the finiteness of the digital image range.
The contributions below include content-based image indexing and retrieval,
texture classification, image simplification, crest line regularization,
and image interpolation for computer-aided animations.

Andaló et al.\ introduce a new shape descriptor based
on tensor scale, show how it can improve content-based
image retrieval and provide a faster computation of tensor scale
by exploiting the Euclidean Image-Foresting Transform.
Aptoula and Lefèvre combine the complementary information extracted by morphological covariance and granulometry
in order to improve texture classification, and propose an extension
to color data introducing an ordering based on luminance and saturation.
Meyer and Angulo introduce the so-called bilevelings of an image defined on
an hexagonal grid, prove that they can be characterized in term of
so-called micro-viscous operators that appear to form adjunctions between set
of functions defined on vertices and edges, and verify that
they lead to higher levels of simplification than with ordinary levelings.
Retornaz and Marcotegui present a robust technique using
morphological numerical residues for automatic localization of textual information
in general image database through connected component extraction.
Vachier and Meyer enlarge the concept of viscous transform
to increasing operator, introduce three viscous dilations and
show how dotted thin crest lines in gray-tone images may
be reconnected and smoothed using these
transformations. Vidal et al.\ extend to mosaic images their previous work on an
interpolation technique for binary images based on morphological
median sets, by processing all the levels of
the hierarchical region-based tree that represents
the region structure of the mosaic image.

\section*{Connectivity}

Among the major developments in MM in the last
two decades are the interrelated subjects of connectivity classes and
connected operators. Connectivity classes, in their general
lattice-theoretical formulation, not only unify in a single axiomatic
framework useful but disparate notions of connectivity, but also
include new interesting definitions of connectivity not previously
possible. Connected operators, on the other hand, have become very
popular in image analysis applications due to the fact that these
operators can preserve edge information by working at the level of the
image flat zones, which are defined using connectivity criteria.

Crespo examines in detail the equivalence between two important
classes of connected operators, namely, set levelings and
adjacency-stable operators. Dimiccoli and Salembier propose the use of
image inpainting to improve the appearance of images processed by
connected operators. Naegel et al.\ propose a general definition of
vector-attribute filters for gray-level images and describe its
application in detection and segmentation tasks.

\section*{Watershed segmentation}

The watershed method represents a powerful paradigm in image
segmentation. Ever since its introduction three decades ago, it has
attracted the interest of many investigators and practitioners, and
research activity in this area continues unabated today.

All\'{e}ne et al.\ study the links that exist between the watershed
and minimum cuts, minimum spanning forests, and shortest-path forests.
Angulo and Jeulin propose a watershed-based stochastic segmentation
methodology. Audigier and Lotufo use the Image Foresting Transform and
the Tie-Zone concept to establish the relationship between several
discrete watershed transforms. Ten Caat et al.\ address the
visualization of electroencephalography (EEG) data by means of a
watershed-based clustering method. Cousty et al.\ study several
properties of watersheds in edge-waited graphs. Papa et al.\ present a
classification method based on optimum-path forests. Stawiaski et al.\ present a technique to compute approximate geodesics and minimal
surfaces.

\section*{Texture and geometrical segmentation}

The articles in this part further examine the segmentation problem,
by using geometric and texture criteria, within or without a watershed
framework.

Angulo presents a method to calculate texture gradients and use them
for joint color and texture segmentation. Consularo et al.\ present
an efficient method for interactive image segmentation based on
inexact graph matching. Cord et al.\ propose morphological and linear
approaches to the characterization and segmentation of random
textures. Noyel et al.\ introduce new connectivity classes that are
able to improve flat-zone segmentation of hyperspectral images. Sofou
and Maragos investigate problem of image segmentation via PDEs,
focusing on a generalized flooding procedure using geometric and
textural information.

\section*{Algorithms and architectures}

The last, but not by any means least, part of the book is devoted
to an aspect of MM that allowed it to become widespread and popular in
both research and practice, namely, efficient algorithms and
architectures for implementation of MM operators.

Bergo and Falc\~{a}o introduce an algorithm that expresses any Imaging
Foresting Transform operator as a series of independent computations,
facilitating thus parallel implementations of such operators. Menotti
et al.\ present a linear time and space algorithm to compute the
component tree of one-dimensional signals, from which they derive an
efficient gray-level image multithresholding method. Ouzounis and
Wilkinson present a concurrent implementation of a previously
developed Max-Tree algorithm, which implements anti-extensive
attribute filters based on second-generation connectivity. Vaz et al.\ describe an efficient method to implement MM operators using convex
and symmetric structuring elements, including Euclidean discs and
spheres.

\vspace{0.7cm}
\begin{flushright}
	Gerald Banon, Junior Barrera, and Ulisses Braga-Neto
\end{flushright}
\end{document}