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Mathematical Morphology
Mathematical morphology Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception http://cmp.felk.cvut.cz/˜hlavac, [email protected] Outline of the talk: Point sets. Morphological transformation. Skeleton. Erosion, dilation, properties. Thinning, sequential thinning. Opening, closing, hit or miss. Distance transformation. Morphology is a general concept 2/55 In biology: the analysis of size, shape, inner structure (and relationship among them) of animals, plants and microorganisms. In linguistics: analysis of the inner structure of word forms. In materials science: the study of shape, size, texture and thermodynamically distinct phases of physical objects. In signal/image processing: mathematical morphology – a theoretical model based on lattice theory used for signal/image preprocessing, segmentation, etc. Mathematical morphology, introduction 3/55 Mathematical morphology (MM): is a theory for analysis of planar and spatial structures; is suitable for analyzing the shape of objects; is based on a set theory, integral algebra and lattice algebra; is successful due to a a simple mathematical formalism, which opens a path to powerful image analysis tools. The morphological way . The key idea of morphological analysis is extracting knowledge from the relation of an image and a simple, small probe (called the structuring element), which is a predefined shape. It is checked in each pixel, how does this shape matches or misses local shapes in the image. MM founding fathers 4/55 Georges Matheron (∗ 1930, † 2000) Jean Serra (∗ 1940) Matheron, G. Elements pour une Theorie del Milieux Poreux Masson, Paris, 1967. Serra, J. Image Analysis and Mathematical Morphology, Academic Press, London 1982. -
Lecture Notes in Statistics 183 Edited by P
Lecture Notes in Statistics 183 Edited by P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, S. Zeger Michel Bilodeau Fernand Meyer Michel Schmitt (Editors) Space, Structure and Randomness Contributions in Honor of Georges Matheron in the Field of Geostatistics, Random Sets and Mathematical Morphology With 121 Figures Michel Bilodeau Fernand Meyer Michel Schmitt Ecole des Mines de Paris Ecole des Mines de Paris Ecole des Mines de Paris Centre de Morphologie Centre de Morphologie Direction des recherches Mathématique Mathématique 60 Boulevard Saint Michel 35 rue Saint Honoré 35 rue Saint Honoré 75272 Paris Cedex 06 77305 Fontainebleau 77305 Fontainebleau France France France [email protected] [email protected] [email protected] Library of Congress Control Number: 2005927580 ISBN 10: 0-387-20331-1 Printed on acid-free paper ISBN 13: 978-0387-20331-7 © 2005 Springer Science+Business Media, Inc. All rights reserved. This work may not be translated or copied in whole or in part without the written per- mission of the publisher (Springer Science+Business Media, Inc., 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. -
Revised Mathematical Morphological Concepts
Advances in Pure Mathematics, 2015, 5, 155-161 Published Online March 2015 in SciRes. http://www.scirp.org/journal/apm http://dx.doi.org/10.4236/apm.2015.54019 Revised Mathematical Morphological Concepts Joseph Ackora-Prah, Yao Elikem Ayekple, Robert Kofi Acquah, Perpetual Saah Andam, Eric Adu Sakyi, Daniel Gyamfi Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana Email: [email protected], [email protected], [email protected], [email protected], [email protected], [email protected] Received 17 February 2015; accepted 12 March 2015; published 23 March 2015 Copyright © 2015 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract We revise some mathematical morphological operators such as Dilation, Erosion, Opening and Closing. We show proofs of our theorems for the above operators when the structural elements are partitioned. Our results show that structural elements can be partitioned before carrying out morphological operations. Keywords Mathematical Morphology, Dilation, Erosion, Opening, Closing 1. Introduction Mathematical morphology is the theory and technique for the analysis and processing of geometrical structures, based on set theory, lattice theory, topology, and random functions. We consider classical mathematical mor- phology as a field of nonlinear geometric image analysis, developed initially by Matheron [1], Serra [2] and their collaborators, which is applied successfully to geological and biomedical problems of image analysis. The basic morphological operators were developed first for binary images based on set theory [1] [2] inspired by the work of Minkowski [3] and Hadwiger [4]. -
Fundamenta Morphologicae Mathematicae * 1. Introduction
Fundamenta Informaticae 41 (2000) 1-31 1 !OS Press Fundamenta Morphologicae Mathematicae * John Goutsias t Center for Imaging Science and Department of Electrical and Computer· Engineering The Johns Hopkins University Baltimore, MD 21218, U. S. A. [email protected] Henk J .A.M. Heijmans Centre for Mathematics and Computer Science P.O. Box 94079 1090 GB Amsterdam, The Netherlands [email protected] Abstract. Mathematical morphology is a geometric approach in image processing and anal ysis with a strong mathematical f!avor. Originally, it was developed as a powerful tool for shape analysis in binary and, later, grey-scale images. But it was soon recognized that the underlying ideas could be extended naturally to a much wider class of mathematical ob jects, namely complete lattices. This paper presents, in a bird's eye view, the foundations of mathematical morphology, or more precisely, the theory of morphological operators on complete lattices. Keywords: adjunction, alternating sequential filter, Boolean function, closing, complete lattice, connectivity, connected operator, dilation, erosion, flat operator, fuzzy set, grain operator, granulometry, grey-scale image, idempotence, mathematical morphology, morpho logical filter, morphological operator, multivalued image, opening, thresholding. 1. Introduction In 1975, a seminal book by Georges Matheron, entitled Random Sets and Integral Geometry, appeared in the literature [20]. This book laid down the foundations of a novel technique for • This work was supported in part by NATO Collaborative Research Grant CRG.971503. The work of the first author was also supported by the Office of Naval Research, Mathematical, Computer, and Information Sciences Division, under ONR Grant N00014-90-1345, and by the National Science Foundation, under NSF Award #9729576. -
The Birth of Mathematical Morphology
ISMM 2000 , Xeros Center Palo-Alto, June 2000 The Birth of Mathematical Morphology Georges Matheron and Jean Serra G. Matheron, J. Serra Ecole des Mines de Paris ( 2000 ) Birth of Math. Morph. 1 ISMM 2000 , Xeros Center Palo-Alto, June 2000 Context • Before considering how Mathematical Morphology originated in 1964, we will describe briefly the backgrounds of its two founders at the beginning of that year, • even though they were hardly aware of the theoretical turn they would take a few months later. G. Matheron, J. Serra Ecole des Mines de Paris ( 2000 ) Birth of Math. Morph. 2 ISMM 2000 , Xeros Center Palo-Alto, June 2000 First Protagonist • Georges Matheron was a mining engineer from the Corps des Mines, 34 years old, working with the B.R.G.M. ( i.e. geological survey) , in Paris. • At the end of 1963, he had just achieved the doctrinal body of his theory concerning the estimation of deposits, called Geostatistics, and started in Algeria ten years before. G. Matheron, J. Serra Ecole des Mines de Paris ( 2000 ) Birth of Math. Morph. 3 ISMM 2000 , Xeros Center Palo-Alto, June 2000 First Protagonist (II) • It was time for a break and for investigating new horizons beyond the mining world. • His publication of reports paused at that time, decreasing from fifty in ten years to none in ten months. He wouldn't return to geostatistics until January 1969, five years later, via universal kriging. G. Matheron, J. Serra Ecole des Mines de Paris ( 2000 ) Birth of Math. Morph. 4 ISMM 2000 , Xeros Center Palo-Alto, June 2000 Second Protagonist • Jean Serra was a civil engineer from the Ecole des Mines de Nancy. -
Newsletternewsletter
No. 88 June 2014 IAMGIAMG NewsletterNewsletter Official Newsletter of the International Association for Mathematical Geosciences Contents his year - 2014 - marks the 150th anniversary of the publication of Jules Vernes Voyage au centre de la Terre, the first (?) President’s Forum ............................................................................3 science fiction novel with a geologic subject. It seems almost Association Business .......................................................................4 Tlike yesterday but time passes and some of us are getting older just 2014 Krumbein and Griffiths Awards ..............................................4 like the literature. IAMG is 46 years old, and your Newsletter editor New Commission: IAMG Future Publication Strategy ...................4 has been a member for 40 years (which is hard for me to believe). Honorary Life Member: John Harbaugh .........................................4 Many of the founding members of Indexing of IAMG Annual Conference Proceedings .......................4 From the Editor IAMG and many of those who have Request for Proposals to Host IAMG 2017 .....................................4 From the Editor been active for many years are still Distinguished Lecturer News ...........................................................5 From the Editor continuing with a full schedule. 2014 George Matheron Lecture .......................................................5 Secretary General Frits Agterberg, Studies in Mathematical Geosciences Series ..................................6 -
Propriétés Effectives De Matériaux Architecturés Justin Dirrenberger
Propriétés effectives de matériaux architecturés Justin Dirrenberger To cite this version: Justin Dirrenberger. Propriétés effectives de matériaux architecturés. Autre. Ecole Nationale Supérieure des Mines de Paris, 2012. Français. NNT : 2012ENMP0045. pastel-00797363 HAL Id: pastel-00797363 https://pastel.archives-ouvertes.fr/pastel-00797363 Submitted on 6 Mar 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ABCBD École doctorale n◦432: Sciences des Métiers de l’Ingénieur (SMI) Doctorat ParisTech THÈSE pour obtenir le grade de docteur délivré par l’École Nationale Supérieure des Mines de Paris Spécialité “Science et Génie des Matériaux” présentée et soutenue publiquement par Justin DIRRENBERGER le 10 décembre 2012 Propriétés effectives de matériaux architecturés Effective properties of architectured materials Directeurs de thèse: Samuel FOREST Dominique JEULIN Jury M. Yves BRÉCHET, Professeur, SIMAP, Grenoble-INP Président M. Albrecht BERTRAM, Professeur, Otto-von-Guericke-Universität Magdeburg Rapporteur M. Rémy DENDIEVEL, Professeur, SIMAP, Grenoble-INP Rapporteur M. Michel BORNERT, Ingénieur en Chef des Ponts, Eaux et Forêts, Navier, École des Ponts-ParisTech Examinateur M. Pierre GILORMINI, Directeur de recherche, PIMM, Arts et Métiers-ParisTech / CNRS Examinateur M. Marc THOMAS, Ingénieur de recherche, DMSM, ONERA Examinateur M. -
The Beginnings of CMM Jean Serra
The beginnings of CMM Jean Serra To cite this version: Jean Serra. The beginnings of CMM. 2017. hal-01532876 HAL Id: hal-01532876 https://hal.archives-ouvertes.fr/hal-01532876 Submitted on 3 Jun 2017 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The beginnings of CMM Jean Serra May 16, 2017 1963-1967 : Before CMM It is difficult to understand the way CMM has evolved into the institution it is today, if we do not look back a few years before its creation. At BRGM (Geological Survey) Georges Matheron was an engineer at the French Geological Survey, in Paris, and he was establishing a new theory for ore bodies es- timation, called geostatistics. In 1963 I started my PhD thesis under his direction, while working at Iron and Steel Institute, near Metz. A thesis which started by mining estimations and finished by the birth of mathematical morphology. I was Matheron's cadet for 10 years, meeting quite often at Nancy where I was his teaching assistant for courses on probability. But some times I would go to Paris to meet with him. When one would enter his office, one could see very big room, with a small wooden table at the center, which engulfed in a huge cloud of smoke coming from a pipe, and in the middle of the cloud, Matheron. -
Mathematical Morphology: a Modern Approach in Image Processing Based on Algebra and Geometry* Henk J
SIAM REVIEW Vol. 37, No. I, pp. 1-36. March 1995 © 1995 Society for Industrial and Applied Mathematics 001 MATHEMATICAL MORPHOLOGY: A MODERN APPROACH IN IMAGE PROCESSING BASED ON ALGEBRA AND GEOMETRY* HENK J. A. M. HEUMANSt Abstract. Mathematical morphology is a theory of image transformations and image functionals which is based on set-theoretical, geometrical, and topological concepts. The methodology is particularly useful for the analysis of the geometrical struct~re in an i'.11age. The main goal of this paper is to give an impression of the underlying philosophy and the mathematical theones which are relevant to this field. The following topics are discussed: introduction to mathematical morphology; generalization to complete lattices; morphological filters and their construction by iteration; geometrical aspects of morphology (e.g., convexity, distance, geodesic operators, granulometries, metric dilations, distance transform, cost functions); and extension of binary operators to grey-scale images. Key words. image analysis, mathematical morphology, hit-or-miss operator. Minkowski sum and difference, dilation, erosion, opening, closing, Boolean function, complete lattice, adjunction, morphological filter, iteration of morphological operators, ,I.- (respectively, t-) continuous operator, metric, gauge function. convex set, geodesic op erator, granulometry, metric dilation, distance transform, cost function, grey-scale morphology, fiat function operator AMS subject classifications. 69K52. 06835, 51K99, 52A22 1. Introduction. In the early sixties two researchers at the Paris School of Mines in Fontainebleau, Georges Matheron and Jean Serra, worked on a number of problems in min eralogy and petrography. Their main goal was to characterize physical properties of certain materials, e.g., the permeability of a porous medium, by examining their geometrical structure. -
Mathematical Morphology
Mathematical morphology Václav Hlaváč Czech Technical University in Prague Czech Institute of Informatics, Robotics and Cybernetics 160 00 Prague 6, Jugoslávských partyzánů 1580/3, Czech Republic http://people.ciirc.cvut.cz/hlavac, [email protected] also Center for Machine Perception, http://cmp.felk.cvut.cz Outline of the talk: Point sets. Morphological transformation. Skeleton. Erosion, dilation, properties. Thinning, sequential thinning. Opening, closing, hit or miss. Distance transformation. Morphology is a general concept 2/56 In biology: the analysis of size, shape, inner structure (and relationship among them) of animals, plants and microorganisms. In linguistics: analysis of the inner structure of word forms. In materials science: the study of shape, size, texture and thermodynamically distinct phases of physical objects. In signal/image processing: mathematical morphology – a theoretical model based on lattice theory used for signal/image preprocessing, segmentation, etc. Mathematical morphology, introduction 3/56 Mathematical morphology (MM): is a theory for analysis of planar and spatial structures; is suitable for analyzing the shape of objects; is based on a set theory, integral algebra and lattice algebra; is successful due to a a simple mathematical formalism, which opens a path to powerful image analysis tools. The morphological way . The key idea of morphological analysis is extracting knowledge from the relation of an image and a simple, small probe (called the structuring element), which is a predefined shape. It is checked in each pixel, how does this shape matches or misses local shapes in the image. Mathematical morphology pioneers 4/56 Georges Matheron Jean Serra Ivan Saxl Josef Mikeš ∗1930, †2000 ∗1940 ∗1936, †2009 ∗1946 Matheron, G.