Table Of Content

Mathematical Foundations of Image Processing and Analysis 2 To Blandine, Flora and Pierre-Charles Series Editor Jean-Pierre Goure Mathematical Foundations of Image Processing and Analysis 2 Jean-Charles Pinoli Firstpublished2014inGreatBritainandtheUnitedStatesbyISTELtdandJohnWiley&Sons,Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permittedundertheCopyright,DesignsandPatentsAct1988,thispublicationmayonlybereproduced, storedortransmitted,inanyformorbyanymeans,withthepriorpermissioninwritingofthepublishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentionedaddress: ISTELtd JohnWiley&Sons,Inc. 27-37StGeorge’sRoad 111RiverStreet LondonSW194EU Hoboken,NJ07030 UK USA www.iste.co.uk www.wiley.com ©ISTELtd2014 TherightsofJean-CharlesPinolitobeidentifiedastheauthorofthisworkhavebeenassertedbyhimin accordancewiththeCopyright,DesignsandPatentsAct1988. LibraryofCongressControlNumber:2014939771 BritishLibraryCataloguing-in-PublicationData ACIPrecordforthisbookisavailablefromtheBritishLibrary ISBN978-1-84821-748-5 PrintedandboundinGreatBritainbyCPIGroup(UK)Ltd.,Croydon,SurreyCR04YY Contents PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxv PART 5. TWELVE MAIN GEOMETRICAL FRAMEWORKS FOR BINARY IMAGES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 CHAPTER21.THE SET-THEORETICFRAMEWORK . . . . . . . . . . . . . 3 21.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 21.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 3 21.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 3 21.3.MainnotionsandapproachesforIPA. . . . . . . . . . . . . . . . . . . 4 21.3.1.Pixelsandobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 21.3.2.Pixelandobjectseparation . . . . . . . . . . . . . . . . . . . . . . 4 21.3.3.Localfiniteness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 21.3.4.Settransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 21.4.MainapplicationsforIPA . . . . . . . . . . . . . . . . . . . . . . . . . 6 21.4.1.Objectpartitionandobjectcomponents . . . . . . . . . . . . . . . 6 21.4.2.Set-theoreticseparationofobjectsandobjectremoval . . . . . . . 6 21.4.3.Countingofseparateobjects. . . . . . . . . . . . . . . . . . . . . . 7 21.4.4.Spatialsupportsbordereffects . . . . . . . . . . . . . . . . . . . . 7 21.5.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 CHAPTER22.THE TOPOLOGICALFRAMEWORK . . . . . . . . . . . . . . 9 22.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 22.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 9 22.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 9 22.2.2.SpecialclassesofsubsetsofRn . . . . . . . . . . . . . . . . . . . . 9 22.2.3.Felltopologyforclosedsubsets . . . . . . . . . . . . . . . . . . . . 10 vi MathematicalFoundationsofIPA2 22.2.4.Hausdorfftopologyforcompactsubsets . . . . . . . . . . . . . . . 10 22.2.5.Continuityandsemi-continuityofsettransformations . . . . . . . 12 22.2.6.Continuityofbasicset-theoreticandtopologicaloperations . . . . 12 22.3.MainnotionsandapproachesforIPA. . . . . . . . . . . . . . . . . . . 12 22.3.1.TopologiesinthespatialdomainRn . . . . . . . . . . . . . . . . . 12 22.3.2.TheLebesgue–(Cˇech)dimension . . . . . . . . . . . . . . . . . . . 13 22.3.3.Interiorandexteriorboundaries . . . . . . . . . . . . . . . . . . . . 13 22.3.4.Path-connectedness . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 22.3.5.Homeomorphicobjects. . . . . . . . . . . . . . . . . . . . . . . . . 14 22.4.MainapplicationstoIPA . . . . . . . . . . . . . . . . . . . . . . . . . . 14 22.4.1.Topologicalseparationofobjectsandobjectremoval. . . . . . . . 14 22.4.2.Countingofseparateobjects. . . . . . . . . . . . . . . . . . . . . . 15 22.4.3.Contoursofobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 22.4.4.Metricdiameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 22.4.5.Skeletonsofproperobjects . . . . . . . . . . . . . . . . . . . . . . 16 22.4.6.Dirichlet–Voronoidiagrams . . . . . . . . . . . . . . . . . . . . . . 18 22.4.7.Distancemaps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 22.4.8.Distancebetweenobjects . . . . . . . . . . . . . . . . . . . . . . . 19 22.4.9.Spatialsupport’sbordereffects . . . . . . . . . . . . . . . . . . . . 20 22.5.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 CHAPTER23.THE EUCLIDEANGEOMETRICFRAMEWORK . . . . . . . 23 23.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 23 23.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 23 23.2.2.Euclideandimension . . . . . . . . . . . . . . . . . . . . . . . . . . 24 23.2.3.Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 23.2.4.Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 23.2.5.Eigenvalues,eigenvectorsandtraceofamatrix . . . . . . . . . . . 27 23.2.6.Matrixnorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 23.3.MainnotionsandapproachesforIPA. . . . . . . . . . . . . . . . . . . 29 23.3.1.Affinetransformations . . . . . . . . . . . . . . . . . . . . . . . . . 29 23.3.2.Specialgroupsofaffinetransformations . . . . . . . . . . . . . . . 30 23.3.3.Linearandaffinesub-spacesandGrassmannians . . . . . . . . . . 31 23.3.4.Linearandaffinespans . . . . . . . . . . . . . . . . . . . . . . . . . 31 23.4.MainapplicationstoIPA . . . . . . . . . . . . . . . . . . . . . . . . . . 32 23.4.1.Basicspatialtransformations . . . . . . . . . . . . . . . . . . . . . 32 23.4.2.Hyperplanes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 23.4.3.Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 23.4.4.Minkowskiadditionandsubtraction . . . . . . . . . . . . . . . . . 34 23.4.5.Continuityandsemi-continuitiesofEuclideantransformations . . 34 23.5.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Contents vii CHAPTER24.THE CONVEXGEOMETRICFRAMEWORK . . . . . . . . . . 37 24.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 24.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 37 24.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 37 24.3.MainnotionsandapproachesforIPA. . . . . . . . . . . . . . . . . . . 37 24.3.1.Convexobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 24.3.2.Hausdorfftopologyforcompactconvexobjects. . . . . . . . . . . 40 24.3.3.Compactpoly-convexobjects . . . . . . . . . . . . . . . . . . . . . 41 24.3.4.Star-shapedobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 24.3.5.Simplices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 24.4.MainapplicationstoIPA . . . . . . . . . . . . . . . . . . . . . . . . . . 42 24.4.1.Convexdeficiencysetandconcavities . . . . . . . . . . . . . . . . 42 24.4.2.Functionsrelatedtoconvexandstar-shapedobjects . . . . . . . . 43 24.4.3.Delaunaytriangulation . . . . . . . . . . . . . . . . . . . . . . . . . 44 24.5.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 CHAPTER25.THE MORPHOLOGICAL GEOMETRICFRAMEWORK . . . 47 25.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 25.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 47 25.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 47 25.3.MathematicalnotionsandapproachesforIPA . . . . . . . . . . . . . . 47 25.3.1.Morphologicaldilationanderosion . . . . . . . . . . . . . . . . . . 48 25.3.2.Morphologicalclosingandopening . . . . . . . . . . . . . . . . . 48 25.3.3.Setpropertiesofmorphologicaldilation,erosion, closingandopening . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 25.3.4.Morphologicalregularobjects. . . . . . . . . . . . . . . . . . . . . 49 25.3.5.Continuityofthemorphologicaloperations . . . . . . . . . . . . . 50 25.4.MainnotionsandapproachesforIPA. . . . . . . . . . . . . . . . . . . 51 25.4.1.Morphologicaltransformations . . . . . . . . . . . . . . . . . . . . 51 25.4.2.Parallelobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 25.4.3.Federersets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 25.5.MainapplicationstoIPA . . . . . . . . . . . . . . . . . . . . . . . . . . 52 25.5.1.Objectcontoursandmorphologicalboundaries . . . . . . . . . . . 52 25.5.2.Objectfilteringandmorphologicalsmoothing. . . . . . . . . . . . 53 25.5.3.Morphologicalskeleton . . . . . . . . . . . . . . . . . . . . . . . . 54 25.5.4.Ultimateerosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 25.5.5.Morphing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 25.6.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 CHAPTER26.THE GEOMETRICAND TOPOLOGICALFRAMEWORK . . 57 26.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 26.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 57 viii MathematicalFoundationsofIPA2 26.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 57 26.2.2.ManifoldsorlocallyEuclideanspaces . . . . . . . . . . . . . . . . 58 26.2.3.Manifoldswithborder . . . . . . . . . . . . . . . . . . . . . . . . . 58 26.2.4.Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 26.2.5.Compactandclosedmanifolds . . . . . . . . . . . . . . . . . . . . 59 26.2.6.LipschitzmanifoldsandLipschitzsets . . . . . . . . . . . . . . . . 60 26.3.MathematicalapproachesforIPA . . . . . . . . . . . . . . . . . . . . . 60 26.3.1.Unitballandunitcube,toriiandannulii . . . . . . . . . . . . . . . 60 26.3.2.Points,curvesandsurfaces . . . . . . . . . . . . . . . . . . . . . . 61 26.3.3.Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 26.3.4.Homeomorphicandhomotopicobjects. . . . . . . . . . . . . . . . 62 26.4.MainapplicationstoIPA . . . . . . . . . . . . . . . . . . . . . . . . . . 63 26.4.1.Contour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 26.4.2.Topologicalcontent. . . . . . . . . . . . . . . . . . . . . . . . . . . 64 26.4.3.TheLebesgue(-Cˇech)dimensionofhomeomorphic orhomotopicobjects. . . . . . . . . . . . . . . . . . . . . . . . . . . 65 26.4.4.TheDescartes–Euler–Poincaré’snumberandthe Bettinumbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 26.4.5.Someparticularbasicmanifolds . . . . . . . . . . . . . . . . . . . 66 26.5.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 CHAPTER27.THE MEASURE-THEORETICGEOMETRICFRAMEWORK 71 27.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 27.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 71 27.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 71 27.2.2.TheGaussmeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 27.2.3.ThePeano–Jordanmeasures . . . . . . . . . . . . . . . . . . . . . . 72 27.2.4.Measuresandcontents . . . . . . . . . . . . . . . . . . . . . . . . . 73 27.2.5.OutermeasuresandBorelsets . . . . . . . . . . . . . . . . . . . . 74 27.2.6.Finiteandσ-finitemeasures . . . . . . . . . . . . . . . . . . . . . . 75 27.2.7.Nullsets,negligiblesetsandcompletemeasures . . . . . . . . . . 75 27.2.8.Atomsandatomicmeasures . . . . . . . . . . . . . . . . . . . . . . 75 27.2.9.Then-dimensionalLebesguemeasure . . . . . . . . . . . . . . . . 75 27.2.10.Them-dimensionalHausdorffmeasure . . . . . . . . . . . . . . . 78 27.2.11.Jordansets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 27.3.MainapproachesforIPA . . . . . . . . . . . . . . . . . . . . . . . . . . 79 27.3.1.Rectifiableobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 27.3.2.Paralleldilatedobjects . . . . . . . . . . . . . . . . . . . . . . . . . 81 27.3.3.TheMinkowskicontents . . . . . . . . . . . . . . . . . . . . . . . . 82 27.3.4.TheFréchet–Nikodym–Aronszajndistance . . . . . . . . . . . . . 83 27.3.5.Caccioppolisets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 27.4.ApplicationstoIPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 27.4.1.Perimetermeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Contents ix 27.4.2.Invariantmeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 27.4.3.Them-dimensionalFavardmeasure . . . . . . . . . . . . . . . . . 86 27.4.4.Comparisonofobjects . . . . . . . . . . . . . . . . . . . . . . . . . 87 27.5.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 CHAPTER28.THE INTEGRAL GEOMETRICFRAMEWORK . . . . . . . . 89 28.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 28.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 89 28.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 89 28.2.2.Geometricfunctionals . . . . . . . . . . . . . . . . . . . . . . . . . 90 28.2.3.IntrinsicvolumesandMinkowskifunctionalsoncompact convexobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 28.2.4.Contentfunctionalsonfiniteunionsofcompactconvexobjects . . 92 28.2.5.Hadwiger’scharacterizationtheorem . . . . . . . . . . . . . . . . . 93 28.2.6.Particularm-dimensionalcontentfunctionals . . . . . . . . . . . . 93 28.2.7.Continuityofgeometricfunctionals . . . . . . . . . . . . . . . . . 94 28.3.MainapproachesforIPA . . . . . . . . . . . . . . . . . . . . . . . . . . 95 28.3.1.TheFavardmeasureandCauchy–Crofton’sformulas. . . . . . . . 95 28.3.2.Cauchy–Crofton’sformulasforcompact, poly-convexobjects . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 28.3.3.Cauchy–Crofton’sformulasfora k-dimensionalcountablyrectifiablemanifold . . . . . . . . . . . . . 97 28.3.4.Intersectionswithlowerdimensionalaffinesubspaces . . . . . . . 98 28.3.5.Thecovariogramofameasurableobject . . . . . . . . . . . . . . . 99 28.4.ApplicationstoIPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 28.4.1.p-dimensionalaffinesections . . . . . . . . . . . . . . . . . . . . . 100 28.4.2.m-dimensionalcontentfunctionalsforn“1,2and3 . . . . . . . 100 28.4.3.Steiner’sformulasforn“1,2and3 . . . . . . . . . . . . . . . . . 101 28.4.4.Cauchy-Crofton’sformulasindimension2and3 . . . . . . . . . . 102 28.4.5.Feretdiametersandareas . . . . . . . . . . . . . . . . . . . . . . . 104 28.4.6.Otherdiameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 28.4.7.Cauchy’sprojectionformulas . . . . . . . . . . . . . . . . . . . . . 105 28.4.8.Cabo–Baddeley’slinealtransformation . . . . . . . . . . . . . . . 107 28.4.9.Crofton–Hadwiger’schordpowerformula . . . . . . . . . . . . . . 107 28.4.10.Miles–Lantuéjoul’scorrectionmethod . . . . . . . . . . . . . . . 108 28.4.11.Hadwiger’srecursiveformulaforthe Descartes–Euler–Poincaré(DEP)number . . . . . . . . . . . . . . . 109 28.5.Additionalcomments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 CHAPTER29.THE DIFFERENTIAL GEOMETRICFRAMEWORK . . . . . 111 29.1.Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 29.2.Mathematicalconceptsandstructures . . . . . . . . . . . . . . . . . . 111 29.2.1.Mathematicaldisciplines. . . . . . . . . . . . . . . . . . . . . . . . 111

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