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Introduction to Mathematical Morphology
COMPUTER VISION, GRAPHICS, AND IMAGE PROCESSING 35, 283-305 (1986) Introduction to Mathematical Morphology JEAN SERRA E. N. S. M. de Paris, Paris, France Received October 6,1983; revised March 20,1986 1. BACKGROUND As we saw in the foreword, there are several ways of approaching the description of phenomena which spread in space, and which exhibit a certain spatial structure. One such approach is to consider them as objects, i.e., as subsets of their space of definition. The method which derives from this point of view is. called mathematical morphology [l, 21. In order to define mathematical morphology, we first require some background definitions. Consider an arbitrary space (or set) E. The “objects” of this :space are the subsets X c E; therefore, the family that we have to handle theoretically is the set 0 (E) of all the subsets X of E. The set p(E) is incomparably less arbitrary than E itself; indeed it is constructured to be a Boolean algebra [3], that is: (i) p(E) is a complete lattice, i.e., is provided with a partial-ordering relation, called inclusion, and denoted by “ c .” Moreover every (finite or not) family of members Xi E p(E) has a least upper bound (their union C/Xi) and a greatest lower bound (their intersection f7 X,) which both belong to p(E); (ii) The lattice p(E) is distributiue, i.e., xu(Ynz)=(XuY)n(xuZ) VX,Y,ZE b(E) and is complemented, i.e., there exist a greatest set (E itself) and a smallest set 0 (the empty set) such that every X E p(E) possesses a complement Xc defined by the relationships: XUXC=E and xn xc= 0. -
Newsletternewsletter Official Newsletter of the International Association for Mathematical Geology
No. 71 December 2005 IAMGIAMG NewsletterNewsletter Official Newsletter of the International Association for Mathematical Geology Contents oronto - welcoming city on the shores of Lake Ontario and site of IAMG 2005 this last August. The organizers of the confer- ence chose well: Hart House of the University of Toronto, an President’s Forum .................................................................................3 Tivy-draped, Oxbridge style college building, recalling the British her- IAMG Journal Report ...........................................................................4 itage, was a very suitable venue for plenary talks in the “Great Hall”, topical sessions in various smaller rooms, spaces for work sessions, Association Business ............................................................................4 posters and the “music room” for reading, computer connections, and The Matrix Man ....................................................................................5 listening to the occasional piano player. Member News .......................................................................................5 Downtown Toronto turned out to be a nicely accessible city with Conference Reports ...............................................................................6 many places within walking distance from the university, and a good public transportation system of Toronto Photo Album ............................................................................7 subways and bus lines to reach Student Affairs ....................................................................................10 -
Learning Deep Morphological Networks with Neural Architecture Search APREPRINT
LEARNING DEEP MORPHOLOGICAL NETWORKS WITH NEURAL ARCHITECTURE SEARCH APREPRINT Yufei Hu Nacim Belkhir U2IS, ENSTA Paris, Institut Polytechnique de Paris Safrantech, Safran Group [email protected] [email protected] Jesus Angulo Angela Yao CMM, Mines ParisTech, PSL Research University School of Computing National University of Singapore [email protected] [email protected] Gianni Franchi U2IS, ENSTA Paris, Institut Polytechnique de Paris [email protected] June 16, 2021 ABSTRACT Deep Neural Networks (DNNs) are generated by sequentially performing linear and non-linear processes. Using a combination of linear and non-linear procedures is critical for generating a sufficiently deep feature space. The majority of non-linear operators are derivations of activation functions or pooling functions. Mathematical morphology is a branch of mathematics that provides non-linear operators for a variety of image processing problems. We investigate the utility of integrating these operations in an end-to-end deep learning framework in this paper. DNNs are designed to acquire a realistic representation for a particular job. Morphological operators give topological descriptors that convey salient information about the shapes of objects depicted in images. We propose a method based on meta-learning to incorporate morphological operators into DNNs. The learned architecture demonstrates how our novel morphological operations significantly increase DNN performance on various tasks, including picture classification and edge detection. Keywords Mathematical Morphology; deep learning; architecture search. arXiv:2106.07714v1 [cs.CV] 14 Jun 2021 1 Introduction Over the last decade, deep learning has made several breakthroughs and demonstrated successful applications in various fields (e.g. -
Puupintojen Laadunvalvonta Ja Luokittelu
Image thresholding During the thresholding process, individual pixels in an image are marked as “object” pixels if their value is greater than some threshold value (assuming an object to be brighter than the background) and as “background” pixels otherwise. This convention is known as threshold above. Variants include threshold below, which is opposite of threshold above; threshold inside, where a pixel is labeled "object" if its value is between two thresholds; and threshold outside, which is the opposite of threshold inside (Shapiro, et al. 2001:83). Typically, an object pixel is given a value of “1” while a background pixel is given a value of “0.” Finally, a binary image is created by coloring each pixel white or black, depending on a pixel's label. Image thresholding Sezgin and Sankur (2004) categorize thresholding methods into the following six groups based on the information the algorithm manipulates (Sezgin et al., 2004): • histogram shape-based methods, where, for example, the peaks, valleys and curvatures of the smoothed histogram are analyzed • clustering-based methods, where the gray-level samples are clustered in two parts as background and foreground (object), or alternately are modeled as a mixture of two Gaussians • entropy-based methods result in algorithms that use the entropy of the foreground and background regions, the cross-entropy between the original and binarized image, etc. • object attribute-based methods search a measure of similarity between the gray-level and the binarized images, such as fuzzy shape similarity, edge coincidence, etc. • spatial methods [that] use higher-order probability distribution and/or correlation between pixels • local methods adapt the threshold value on each pixel to the local image characteristics." Mathematical morphology • Mathematical Morphology was born in 1964 from the collaborative work of Georges Matheron and Jean Serra, at the École des Mines de Paris, France. -
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]. -
Prelucrarea Imaginilorimaginilor Curscurs 88 Morphologicalmorphological Imageimage Processingprocessing
PrelucrareaPrelucrarea ImaginilorImaginilor CursCurs 88 MorphologicalMorphological ImageImage ProcessingProcessing Utilizate in extragerea componentelor unei imagini la reprezentare si descriere: Extragere contur Schelet Invelitoare convexa Filtrare Subtiere Curatare ... 2/21 GrayGray--ScaleScale imagesimages -- EroziuneEroziune,, DilatareDilatare Functiile pentru valorile nuantelor de gri: XX(x,y) Æ obiect BB(x,y) Æ element structural DilatareaDilatarea luilui XX prin BB, notată cu XX ℗ BB este: (XX ℗ B)B (s,t) = MaxMax{XX(s-x,y-t)++BB(x,y) / ((s-x),(y-t))∈XX, (x,y)∈BB} EroziuneaEroziunea lui X de către BB, notată cu XXΘΘBB este: (XX Θ B)B (s,t) = MinMin{XX(s+x,y+t)--BB(x,y) / ((s+x),(y+t))∈XX, (x,y)∈BB} 3/21 …… GrayGray--ScaleScale imagesimages - Eroziune, Dilatare ... proprietăţi: ExempluExemplu:: X,X, XX ℗ B,B, XX Θ BB :: Dualitate (eroziunea şi dilatarea sunt duale faţă de complementare notată cu XC): (X(XC ℗℗ ⌐B)B) (s,t) == (X(X ΘΘ B)B)C (s,t) ,, unde: XXC == --XX (x,y) si ⌐BB == BB (-x,-y).. 4/21 … Morphological Image Processing - Transformări uzuale derivate Deschiderea lui X faţă de B, notată cu XB este : XB = (X Θ B) ℗ B; B Închiderea lui X faţă de B, notată cu X este: XB = (X ℗ B) Θ B; Dualitate (Deschiderea şi Închiderea ) : B C C ((X ) = (X(X )) (⌐B) 5/21 … Morphological Image Processing - Transformări uzuale derivate Netezire morfologica : B gg = (XB) ; Efect: atenuare a nuantelor deschise/inchise si a zgomotului: 6/21 … Morphological Image Processing - Transformări uzuale derivate Gradientul morfologic : gg = (X ℗ B) – (X Θ B); Top-Hat : hh = X - XB; 7/21 … Morphological Image Processing - Transformări uzuale derivate Segmentare texturala : Se aplica operatorul de inchidere utilizand succesiv elemente structurale mai mari decat elementele de textura mici; Se aplica operatorul de deschidere utilizand un element structural mai mare decat distanta dintre elemntele de textura mari; Avand o regiune deschisa in stanga si una inchisa in dreapta, vom folosi un prag simplu pentru a rezulta granita dintre cele doua texturi. -
Medium Resolution
IAMGIAMG No. 75 December 2007 NewsletterNewsletter Official Newsletter of the International Association for Mathematical Geology Contents t’s about time! We finally will have a female president: Vera President’s Forum ............................................................................3 Pawlowsky-Glahn from the University of Girona in Spain Association Business .......................................................................4 Iwill take over as IAMG president in 2008. As Frits points out IAMG Distinguished Lecturers ........................................................4 in the President’s Forum she is well qualified for the job, and we Forthcoming IAMG Name Change .................................................5 look forward to her “reign”. This is a first for the IAMG, but other Membership Campaign ...................................................................5 organizations have preceded us in electing female presidents. The largest one, AAPG, has had only one, Robbie Gries, in Slate of candidates for the 2008-2012 IAMG Council ...............6-11 2001. At SEPM Mary Kraus will take over the presidency in Responsibilities of Officers and Councilors .................................. 11 the coming year. A cursory survey of a few other associations Member News ...........................................................................11-12 with easily accessible data shows three female presidents in SPE Where Are They Now .................................................................... 11 (DeAnn Creig in 1998, -
A Tribute to Prof. Dominique Jeulin Jean Serra, François Willot
Special topic on multiscale modeling of granular media: a tribute to Prof. Dominique Jeulin Jean Serra, François Willot To cite this version: Jean Serra, François Willot. Special topic on multiscale modeling of granular media: a tribute to Prof. Dominique Jeulin. Image Analysis and Stereology, International Society for Stereology, 2019, 38 (1), pp.1. 10.5566/ias.2146. hal-02102998 HAL Id: hal-02102998 https://hal.archives-ouvertes.fr/hal-02102998 Submitted on 18 Apr 2019 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. Image Anal Stereol 2019;38:1–2 doi:10.5566/ias.2146 Editorial SPECIAL TOPIC ON MULTISCALE MODELING OF GRANULAR MEDIA: A TRIBUTE TO PROF. DOMINIQUE JEULIN JEAN SERRA1 AND FRANC¸ OIS WILLOT2 1Laboratoire d’Informatique Gaspard-Monge, ESIEE, Paris-Est University, France, 2Centre for Materials, PSL Research University, Mines ParisTech, France e-mail: [email protected],[email protected] ABSTRACT A few words on the present special topic, devoted to the multiscale modeling of granular media, and published in honor of Prof. Dominique Jeulin’s enduring contribution to the wide field of image analysis, random structures and material science. -
Mathematical Morphology Operators
IMAGE FILTERING WITH MATHEMATICAL MORPHOLOGY OPERATORS DR. – ING. GABRIELE CAVALLARO, JUELICH SUPERCOMPUTING CENTRE (JSC) DEPUTY HEAD - HIGH PRODUCTIVITY DATA PROCESSING RESEARCH GROUP 18.05.2020, IMAGE PROCESSING FOR REMOTE SENSING (INVITED LECTURE), TECHNICAL UNIVERSITY OF BERLIN LECTURER Gabriele Cavallaro . Since 2013 working in the field of remote sensing and image processing . Holds PhD in Electrical and Computer Engineering (University of Iceland - 2016) − BS and MS Degrees in Telecommunications Engineering (University of Trento – 2011,2013) . Postdoctoral researcher at Forschungszentrum Juelich - Juelich Supercomputing Centre (since 2016) − Deputy Head of the High Productivity Data Processing group − The group focuses on application-driven parallel and scalable machine learning methods that exploit innovative high performance and distributed computing technologies . Research interests − Processing and analysis of high dimensional remote sensing data − Parallel and scalable machine (deep) learning with HPC . Contact − Email: [email protected] − Website: https://www.gabriele-cavallaro.com/ Page 2 FORSCHUNGSZENTRUM JUELICH (FZJ) Multi-Disciplinary Research Centre of the Helmholtz Association in Germany (Juelich Supercomputing Centre - JSC) [1] Holmholtz Association Facility on 2.2 . Selected Facts 2 − One of EU’s largest inter-disciplinary research centres (~6000 employees) − Special expertise in physics, materials science, nanotechnology, neuroscience and medicine & information technology (HPC & Data) Page 3 JUELICH SUPERCOMPUTING CENTRE (JSC) Evolution of Supercomputers and Research activities . JSC is part of the Institute for Advance Simulation (IAS) − Operate supercomputers (->TOP500) and cloud systems − Unique support and research environment . R&D work − Algorithms, computational science, performance tools − Scientific big data analytics and data management − Exascale labs (e.g., NVIDIA) and Co-design projects (DEEP) . With General purpose processors . -
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.