Fuzzy Sets Methods in Image Processing and Understanding: Medical Imaging Applications

311 Pages·2023·6.968 MB·English

by  Isabelle Bloch, Anca Ralescu

Isabelle Bloch
Anca Ralescu
Fuzzy Sets
Methods
in Image
Processing and
Understanding
Medical Imaging Applications
Fuzzy Sets Methods in Image Processing and
Understanding
Isabelle Bloch • Anca Ralescu
Fuzzy Sets Methods in
Image Processing and
Understanding
Medical Imaging Applications
IsabelleBloch AncaRalescu
SorbonneUniversitè,CNRS,LIP6 DepartmentofComputerScience
Paris,France UniversityofCincinnati,ML0030
Cincinnati,OH,USA
LTCI,TélécomParis
InstitutPolytechniquedeParis
Palaiseau,France
ISBN978-3-031-19424-5 ISBN978-3-031-19425-2 (eBook)
https://doi.org/10.1007/978-3-031-19425-2
©SpringerNatureSwitzerlandAG2023
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ToDanandStephan
AncaRalescu
Acknowledgments
Isabelle Bloch’s work on fuzzy sets started in the early 1990s with fuzzy mathe-
maticalmorphology,whichwasacompletelynewtopicbothinimageanalysisand
infuzzysetstheory,andwiththefusionofimageinformation.Shewarmlythanks
Henri Maître for his support and encouragement to investigate research tracks out
ofthemainstreamsincethen.
SheisalsogratefultoDidierDuboisandHenriPradefortheirpositivefeedback
andenlighteningdiscussionsduringmanyworkshopsandconferences.Seeingher
workappreciatedbythesetwoprominentresearcherswasalwaysagreatmotivation
tocontinueexploringnewideas.
ReadingthepapersbyLotfiZadehwasagreatsourceofinspiration,inparticular
toreallyseefuzzysetsassetsinthespatialdomain.Then,withtheaimofexploiting
expert knowledge to help guiding segmentation and recognition in images, in
particularinthefieldofmedicalimaging,sheinvestigatedthefieldofknowledge-
basedimageunderstanding,anditbecamesoonobviousthatspatialrelationshadto
playanimportantrole.However,expertknowledgeisoftenexpressedinalinguistic
way, and for several relations, no satisfactory mathematical definitions existed.
This led her to work on proposing fuzzy models of such relations, mostly based
on mathematical morphology. She benefitted from a sabbatical stay at Berkeley
(October–November 1995 and January 1997) to work in this direction with Mori
Anvari.EnlighteningdiscussionsduringseminarschairedbyLotfiZadehonvarious
subjects related to the theory of fuzzy sets and the applications thereof, and other
nice discussions with several persons of his team, certainly encouraged her to
continue to develop this research line. Although no one was working in the field
ofimageunderstandingandspatialreasoning,shecouldgetinspirationfromthese
seminarsanddiscussions.ShethanksLotfiZadehandMoriAnvarifortheirwarm
hostingduringhersabbaticalperiodatBerkeley,andthe“FondsFrance-Berkeley”
forthefinancialsupportofthissabbatical.
ShewouldliketothankallthePhDcandidates andcolleagues withwhompart
oftheworkpresentedinthisbookwasdone,asacknowledgedbythejointpapers
mentionedinthebibliography.
vii
viii Acknowledgments
Anca Ralescu’s encounter with fuzzy sets and fuzzy logic goes back to 1972
whenshewasastudentinMathematicsattheUniversityofBucharestRomania,and
herclassmateandhusband,DanRalescu,wasinvitedtocollaborateonafuzzysets
project.Theresultofthatprojectwastobetheclassicalbookonfuzzysystemsand
applications coauthored by C.V. Negoita and D.A. Ralescu. Although students in
mathematicsinRomaniahadheardoflogicsotherthantheclassicalBooleanlogic
(thankstothehighlyinspirationalprofessorGrigoreMoisil),itwasveryexcitingto
hearanewtheoryofuncertainty.
Anca’sfirstworkinthefieldoffuzzysetscamemuchlater,afterobtainingher
PhDinMathematics,withthespecialtyinProbabilityTheory,atIndianaUniversity
Bloomington.In1983,shewasappointedasAssistantProfessorintheDepartment
ofMathematicalSciencesattheUniversityofCincinnati.
ShewasgreatlyinfluencedbyProfessorLotfiZadehwhomshehadmetin1976,
especially by his work on linguistic, imprecise quantifiers. Her first works were
ontherepresentationofrulesinvolvingsuchquantifiers.Frequentdiscussionswith
professorZadeh,whomshevisitedoftenattheUniversityofCalifornia,Berkeley,
and who also visited the University of Cincinnati, contributed a great deal to the
directionshetookinherresearch.
FollowingastayoffiveyearsinJapan,firstattheGraduateSchooloftheTokyo
InstituteofTechnologyasguestofProfessorMichioSugeno(famousforhiswork
on fuzzy control), and then at the Laboratory for International Fuzzy Engineering
Research (LIFE), mentored by Professor Toshio Terano, her interests crystalized
around the problem of image understanding. In particular, she was interested in
definingimageunderstandingastheabilitytoverballydescribetheimagecontents.
InchargeoftheImageUnderstandingGroupatLIFE,withhercolleaguesthere,she
contributed to a new definition of spatial relations, to their high-level description
compatible with the way humans describe them. Having read some of the papers
writtenbyDr.IsabelleBloch,shecontactedher,andtheymetin1994.Sincethen,
they have collaborated or exchanged ideas in the domain of image understanding
andapplications,mostrecentlyinconnectionwiththeprojectofwritingthisbook.
The two authors had many discussions and joint works, in particular during
Anca’svisitstoParis,almosteveryMayforseveralyears,whichledtothewriting
of this book. Anca is grateful to Isabelle for this collaboration. They both express
theirwarmthankstoSpringerandtheeditorsfortheirsupport.
Contents
1 Introduction .................................................................. 1
1.1 FuzzySetsandImageUnderstandingUnderImprecision ............ 1
1.1.1 SourcesofImprecision......................................... 1
1.1.2 AdvantagesandUsefulnessofFuzzySets..................... 2
1.1.3 SemanticGap ................................................... 3
1.1.4 AShortReviewofExistingBooks ............................ 3
1.2 Representations ......................................................... 4
1.3 LowLevel—Clustering,Enhancement,Filtering,EdgeDetection... 5
1.4 IntermediateLevel ...................................................... 6
1.5 HigherLevel............................................................. 7
1.5.1 RepresentationsofStructuralInformation..................... 7
1.5.2 Fusion ........................................................... 8
1.5.3 SceneUnderstanding ........................................... 9
1.6 EmergingTopics ........................................................ 9
1.6.1 MiningandRetrieval ........................................... 9
1.6.2 TowardsBipolarity.............................................. 10
1.6.3 TowardsMoreInteractionsBetweenKnowledgeand
ImageInformation.............................................. 10
1.6.4 DeepNeuro-FuzzySystems.................................... 11
References..................................................................... 11
2 Preliminaries ................................................................. 19
2.1 ImprecisioninImagesandRelatedKnowledge....................... 19
2.2 BasicDefinitionsofFuzzySetsTheory................................ 21
2.2.1 FuzzySets....................................................... 21
2.2.2 SetTheoreticalOperations:OriginalDefinitionsof
L.Zadeh......................................................... 23
2.2.3 StructureandTypesofFuzzySets............................. 24
2.2.4 α-Cuts........................................................... 24
2.2.5 Cardinality ...................................................... 25
2.2.6 Convexity ....................................................... 26
ix
x Contents
2.2.7 FuzzyNumber .................................................. 26
2.3 MainOperatorsonFuzzySets ......................................... 29
2.3.1 FuzzyComplementation ....................................... 29
2.3.2 TriangularNormsandConorms ............................... 30
2.3.3 MeanOperators................................................. 35
2.3.4 SymmetricSums................................................ 37
2.3.5 AdaptiveOperators............................................. 38
2.3.6 LogicalConnectives............................................ 39
2.4 LinguisticVariable...................................................... 39
2.4.1 Definition........................................................ 39
2.4.2 ExampleofLinguisticVariable................................ 40
2.4.3 Modifiers........................................................ 40
2.5 TranslatingaCrispOperationintoaFuzzyOperation................ 42
2.5.1 ExtensionPrinciple............................................. 42
2.5.2 CombinationofResultsonα-Cuts............................. 45
2.5.3 TranslatingBinaryTermsintoFunctionalOnes............... 47
2.5.4 Comparison ..................................................... 49
2.6 SummaryoftheMainNotations ....................................... 50
References..................................................................... 51
3 FuzzySpatialObjects........................................................ 53
3.1 FuzzySetsintheSpatialDomain...................................... 54
3.2 SetTheoreticalOperations ............................................. 55
3.2.1 DegreeofIntersection.......................................... 55
3.2.2 DegreeofUnionandCovering................................. 59
3.2.3 DegreeofInclusion............................................. 60
3.2.4 DegreeofEquality.............................................. 64
3.3 Topology:Neighborhood,Boundary,andConnectednessof
aFuzzySet .............................................................. 65
3.3.1 FuzzyNeighborhood ........................................... 65
3.3.2 BoundaryofaFuzzySet ....................................... 66
3.3.3 Connectedness .................................................. 67
3.4 FuzzyGeometry......................................................... 67
3.4.1 FuzzyPointsandLines......................................... 67
3.4.2 FuzzyRectanglesandFuzzyConvexPolygons............... 70
3.4.3 FuzzyDisks..................................................... 72
3.4.4 FuzzyGeometricalMeasures .................................. 73
3.5 FuzzyGeometricTransformations..................................... 81
3.5.1 TransformationofaFuzzySetbyaCrispOperation ......... 81
3.5.2 TransformationofaFuzzySetbyaFuzzyOperation ........ 82
References..................................................................... 83
4 FuzzyMathematicalMorphology.......................................... 85
4.1 LatticeStructureofF................................................... 85
4.2 AlgebraicOperators..................................................... 86
4.3 StructuringElementsandBasicMorphologicalOperators ........... 88
Contents xi
4.4 AnExampleinMedicalImaging....................................... 92
4.5 TowardsaFuzzyMathematicalMorphologyToolbox................ 94
4.5.1 NeighborhoodandBoundaryfromFuzzyDilation
andErosion...................................................... 94
4.5.2 FuzzyMorphologicalFilters................................... 95
4.5.3 ConditioningandFuzzyGeodesicOperators ................. 97
4.5.4 FuzzySkeletonandSkeletonbyInfluenceZones............. 101
4.5.5 Fuzzy Median, Application to Interpolation
BetweenFuzzySets ............................................ 112
4.5.6 Extensions....................................................... 115
References..................................................................... 115
5 Fusion ......................................................................... 121
5.1 Definitions............................................................... 121
5.2 FusionSystemsandArchitecturesTypes.............................. 125
5.3 FuzzyModelinginFusion.............................................. 126
5.4 DefiningandEstimatingMembershipFunctions...................... 127
5.5 FuzzyCombination ..................................................... 129
5.6 DecisioninFuzzyFusion............................................... 132
5.7 ExploitingSpatialInformation......................................... 133
5.8 IllustrativeExamples.................................................... 134
References..................................................................... 135
6 SpatialRelations ............................................................. 137
6.1 SetTheoreticalandTopologicalRelations............................. 137
6.1.1 Adjacency....................................................... 138
6.1.2 FuzzyRegionConnectionCalculus............................ 140
6.2 DistancesBetweenImageRegionsorObjects ........................ 142
6.2.1 Representations................................................. 142
6.2.2 ComparisonofMembershipFunctions........................ 143
6.2.3 CombinationofSpatialandMembershipComparisons ...... 147
6.2.4 DiscussionandExamples ...................................... 151
6.3 FuzzyHammingDistance .............................................. 154
6.4 DirectionalRelations.................................................... 160
6.4.1 FuzzyRelationsDescribingRelativePosition ................ 160
6.4.2 CentroidMethod................................................ 161
6.4.3 HistogramofAngles:CompatibilityMethod ................. 161
6.4.4 AggregationMethod............................................ 162
6.4.5 HistogramofForces............................................ 162
6.4.6 ProjectionBasedApproach .................................... 163
6.4.7 MorphologicalApproach....................................... 164
6.4.8 DiscussionandExamples ...................................... 165
6.5 ComplexRelations:Surround,Between,Along,Across,
Parallel,Aligned ........................................................ 168
6.5.1 Surround......................................................... 168
6.5.2 Between......................................................... 169