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Topics in Mathematical Morphology for Multivariate Images Santiago Velasco-Forero

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Topics in Mathematical Morphology for Multivariate Images Santiago Velasco-Forero Topics in mathematical morphology for multivariate images Santiago Velasco-Forero To cite this version: Santiago Velasco-Forero. Topics in mathematical morphology for multivariate images. General Mathematics [math.GM]. Ecole Nationale Supérieure des Mines de Paris, 2012. English. NNT : 2012ENMP0086. pastel-00820581 HAL Id: pastel-00820581 https://pastel.archives-ouvertes.fr/pastel-00820581 Submitted on 6 May 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. !"#$!$%$&'(#&#)!(")(#&($&$()*"+,+-!(# École doctorale nO432 : Sciences des Métiers de l’Ingénieur Doctorat ParisTech THÈSE pour obtenir le grade de docteur délivré par l’École nationale supérieure des mines de Paris Spécialité « Morphologie Mathématique » présentée et soutenue publiquement par Santiago Velasco-Forero le 14 juin 2012 Contributions en morphologie mathématique pour l’analyse d’images multivariées Directeur de thèse : Jesús ANGULO ! Jury M. Dominique JEULIN, Professeur, CMM-MS, Mines ParisTech Président " M. Jos B.T.M. ROERDINK, Professeur, University of Groningen Rapporteur M. Pierre SOILLE, Directeur de recherche, Joint Research Centre of the European Commission Rapporteur # M. Jón Atli BENEDIKTSSON, Professeur, University of Iceland Examinateur M. Fernand MEYER, Directeur de recherche, CMM, Mines ParisTech Examinateur $ M. Jean SERRA, Professeur émérite, ESIEE, Université Paris-Est Examinateur M. Jesús ANGULO, Chargé de recherche, CMM, Mines ParisTech Examinateur ! MINES ParisTech Centre de Morphologie Mathématique, Mathématiques et Systèmes 35, rue Saint-Honoré, 77305 Fontainebleau, France 2 Acknowledgements Iwouldliketoexpressmydeepestgratitudetomyadvisor,Dr. JesúsAngulo,forhisguidance, motivation and the research freedom he gave me during my Ph.D. studies. It was indeed a huge privilege to be his student. I am grateful to Dr. Fernand Meyer, director of the Centre for Mathematical Morphology (CMM) at the Ecole des Mines de Paris, for his continuous support. My research activities were also partially supported by the IHMO Project (ANR-TecSan’07), funded by the French National Research Agency. Iwouldalsoliketothankmythesisreporters(Dr.JosRoerdinkandDr.PierreSoille)fortheir valuable suggestions and contributions that helped me to improve the quality of this document with their objective and pertinent comments. I am highly grateful to Dr. Jón Benediktsson, Dr. Dom- inque Jeulin, Dr. Fernand Meyer and Dr. Jean Serra, who have served in my committee and have contributed many useful comments before and during my dissertation. Much of the inspiration for this work came during a three months visit to the Joint Research Center of the European Commission. I am very grateful to the members of the Institute to the Protection and Security of the Citizen (IPSC), particularly to Dr. Pierre Soille for our several enlightening conversations. At Fontainebleau, I could not have asked for a more fun and interesting group of people to work with: Andres, Bea, Bruno, Catherine, Charles, Christoph, Dominique, El Hadji, Estelle, Etienne, Fernand, François, Guillaume, Hellen, Joana, Jorge, Jean, John, Julie, Louise, Luc, Matthieu, Michel, Petr, Serge, Torben, Vincent, Xiwei, and many others. Thank you all for making CMM a place where I truly enjoyed coming to work. Also, I would like to thank Julie who has been a great source of “bio-energy". She has supported me in hundreds of ways throughout the development and writing of this dissertation. I also take this opportunity to mention my “boricuas" friends: Angel, Ana, Andrea, Catalina, Diana, Karen, Leidy, Marggie, Maider, Oscar, Pedro Alejo, Victor and Yolanda, for encourage me to cross the Atlantic few years ago. At last, but not least, I would like to dedicate this thesis to my family: Esta tesis es para ust- edes.... familia! Carlitos, Foreman, Lola, Papi, Mami, Carloncho, Bubita, Violeta, Paula Victoria y Carolina. Gracias por el constante apoyo, eterno amor y el ejemplo de superación que me han dado, día tras día. 3 4 Résumé Cette thèse contribue au domaine de la morphologie mathématique et illustre comment les statis- tiques multivariées et les techniques d’apprentissage numérique peuvent être exploitées pour con- cevoir un ordre dans l’espace des vecteurs et pour inclure les résultats d’opérateurs morphologiques au processus d’analyse d’images multivariées. En particulier, nous utilisons l’apprentissage super- visé, les projections aléatoires, les représentations tensorielles et les transformations conditionnelles pour concevoir de nouveaux types d’ordres multivariés et de nouveaux filtres morphologiques pour les images multi/hyperspectrales. Nos contributions clés incluent les points suivants : • Exploration et analyse d’ordres supervisés, basés sur les méthodes à noyaux. • Proposition d’un ordre non supervisé, basé sur la fonction de profondeur statistique calculée par projections aléatoires. Nous commençons par explorer les propriétés nécessaires à une image pour assurer que l’ordre ainsi que les opérateurs morphologiques associés, puissent être interprétés de manière similaire au cas d’images en niveaux de gris. Cela nous amènera à la notion de décomposition en arrière plan / premier plan. De plus, les propriétés d’invariance sont analysées et la convergence théorique est démontrée. • Analyse de l’ordre supervisé dans les problèmes d’appariement par forme de référence, qui correspond à l’extension de l’opérateur tout-ou-rien aux images multivariées grâce à l’utilisation de l’ordre supervisé. • Discussion sur différentes stratégies pour la décomposition morphologique d’images. Notam- ment, la décomposition morphologique additive est introduite comme alternative pour l’analyse d’images de télédétection, en particulier pour les tâches de réduction de dimension et de clas- sification supervisée d’images hyperspectrales. • Proposition d’un cadre unifié basé sur des opérateurs morphologiques, pour l’amélioration de contraste et pour le filtrage du bruit poivre-et-sel. • Introduction d’un nouveau cadre de modèles Booléens multivariés par l’utilisation d’une for- mulation en treillis complets. Cette contribution théorique est utile pour la caractérisation et la simulation de textures multivariées. 5 6 Abstract This thesis contributes to the field of mathematical morphology and illustrates how multivariate statistics and machine learning techniques can be exploited to design vector ordering and to include results of morphological operators in the pipeline of multivariate image analysis. In particular, we make use of supervised learning, random projections, tensor representations and conditional transformations to design new kinds of multivariate ordering, and morphological filters for color and multi/hyperspectral images. Our key contributions include the following points: • Exploration and analysis of supervised ordering based on kernel methods. • Proposition of an unsupervised ordering based on statistical depth function computed by ran- dom projections. We begin by exploring the properties that an image requires to ensure that the ordering and the associated morphological operators can be interpreted in a similar way than in the case of grey scale images. This will lead us to the notion of background/foreground decomposition. Additionally, invariance properties are analyzed and theoretical convergence is showed. • Analysis of supervised ordering in morphological template matching problems, which corre- sponds to the extension of hit-or-miss operator to multivariate image by using supervised ordering. • Discussion of various strategies for morphological image decomposition, specifically, the ad- ditive morphological decomposition is introduced as an alternative for the analysis of remote sensing multivariate images, in particular for the task of dimensionality reduction and super- vised classification of hyperspectral remote sensing images. • Proposition of an unified framework based on morphological operators for contrast enhance- ment and salt-and-pepper denoising. • Introduces a new framework of multivariate Boolean models using a complete lattice formula- tion. This theoretical contribution is useful for characterizing and simulation of multivariate textures. 7 8 Contents Contents 9 List of Figures 13 1 Introduction 17 1.1 Motivation ......................................... 17 1.2 Order and Mathematical morphology .......................... 20 1.3 Why do we need order? .................................. 22 1.4 Multivariate Ordering ................................... 23 1.5 Thesis overview and main contributions ......................... 25 ILearningOrderingforMultivariateMathematicalMorphology 29 2 Short review on morphological operators 31 2.1 Introduction ......................................... 31 2.2 Scalar images ........................................ 31 2.3 Morphological transformations .............................. 32 2.3.1 Dilation and erosion ................................ 32 2.3.2 Opening and closing ................................ 33 2.3.3 Contrast mappings ................................. 36 2.3.4 Morphological center
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