Mathematical Morphology

From Theory to Applications

Laurent Najman, Hugues Talbot, Laurent Najman, Hugues Talbot

• Publisher: Wiley-ISTE
• Year: 2013
• Print ISBN: 9781848212152
• eBook ISBN: 9781118600856
• Edition: 1
• Topic: Technology & Engineering
• Subtopic: Signals & Signal Processing
• Index: Technology & Engineering

Mathematical Morphology allows for the analysis and processing of geometrical structures using techniques based on the fields of set theory, lattice theory, topology, and random functions. It is the basis of morphological image processing, and finds applications in fields including digital image processing (DSP), as well as areas for graphs, surface meshes, solids, and other spatial structures. This book presents an up-to-date treatment of mathematical morphology, based on the three pillars that made it an important field of theoretical work and practical application: a solid theoretical foundation, a large body of applications and an efficient implementation.

The book is divided into five parts and includes 20 chapters. The five parts are structured as follows:

• Part I sets out the fundamental aspects of the discipline, starting with a general introduction, followed by two more theory-focused chapters, one addressing its mathematical structure and including an updated formalism, which is the result of several decades of work.
• Part II extends this formalism to some non-deterministic aspects of the theory, in particular detailing links with other disciplines such as stereology, geostatistics and fuzzy logic.
• Part III addresses the theory of morphological filtering and segmentation, featuring modern connected approaches, from both theoretical and practical aspects.
• Part IV features practical aspects of mathematical morphology, in particular how to deal with color and multivariate data, links to discrete geometry and topology, and some algorithmic aspects; without which applications would be impossible.
• Part V showcases all the previously noted fields of work through a sample of interesting, representative and varied applications.

PART I

Foundations

Chapter 1

Introduction to Mathematical Morphology ¹

In this chapter we endeavor to introduce in a concise way the main aspects of Mathematical Morphology, as well as what constitutes its field. This question is difficult, not so much as a technical matter but as a question of starting point. Historically, mathematical morphology began as a technique to study random sets with applications to the mining industry. It was rapidly extended to work with two-dimensional (2D) images in a deterministic framework first with binary images, then gray-level and later to color and multispectral data and in dimensions > 2. The framework of mathematical morphology encompasses many various mathematical disciplines from set theory including lattice theory, random sets, probabilities, measure theory, topology, discrete and continuous geometry, as well as algorithmic considerations and finally applications.

The main principle of morphological analysis is to extract knowledge from the response of various transformations which are generally nonlinear.

One difficulty in the way mathematical morphology has been developed and expanded [MAT 75, SER 82, SER 88c] (see also [HEI 94a, SCH 94, SOI 03a]) is that its general properties do not fall within the general topics taught at school and universities (with the exception of relatively advanced graduate-level courses). Classical mathematics define a function as an operator associating a single point in a domain with a single value. A contrario, in morphology we associate whole sets with other whole sets. The consequences of this are important. For instance, if a point generally has zero measure, this is not generally the case for sets. Consequently, while a probability of the presence of a point may be zero, this is not the case for a set.

In addition, we can compare morphology to other image processing disciplines. For instance, linear operator theory assumes that images are merely a multidimensional signal. We also assume that signals combine themselves additively. The main mathematical structure is the vector space and basic operators are those that preserve this structure and commute with basic rules (in this case, addition and multiplication by a constant). From this point deriving convolution operators is natural; hence it is also natural to study Fourier or wavelet transforms. It is also natural to study decomposition by projections on basis vectors. This way is of course extremely productive and fruitful, but it is not the complete story.

Indeed, very often a 2D image is not only a signal but corresponds to a projection of a larger 3D ‘reality’ onto a sensor via an optical system of some kind. Two objects that overlap each other due to the projectio...

Table of contents

• Cover
• Dedication
• Title Page
• Copyright
• Preface
• Part I. Foundations
• Part II. Evaluating and Deciding
• Part III. Filtering and Connectivity
• Part IV. Links and Extensions
• Part V. Applications
• Bibliography
• List of Authors
• Index