Table Of ContentSPRINGER BRIEFS IN APPLIED SCIENCES AND
TECHNOLOGY COMPUTATIONAL INTELLIGENCE
Patricia Melin
Gabriela E. Martinez
Extension of the
Fuzzy Sugeno
Integral Based on
Generalized
Type-2 Fuzzy Logic
123
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Patricia Melin Gabriela E. Martinez
(cid:129)
Extension of the Fuzzy
Sugeno Integral Based
on Generalized Type-2
Fuzzy Logic
123
Patricia Melin Gabriela E.Martinez
Division of Graduate Studies Division of Graduate Studies
andResearch andResearch
TijuanaInstitute of Technology TijuanaInstitute of Technology
Tijuana, BajaCalifornia, Mexico Tijuana, BajaCalifornia, Mexico
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefs inApplied SciencesandTechnology
ISSN 2625-3704 ISSN 2625-3712 (electronic)
SpringerBriefs inComputational Intelligence
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Preface
In this book, an extension of the aggregation operator of the generalized interval
type-2Sugenointegralbymeansofgeneralizedtype-2fuzzylogicispresented.The
main goal of extending the Sugeno integral aggregation operator, is to give it the
abilitytohandlehigherlevelsofuncertaintybyaddinganynumberofsourcesand
types of information in a wide variety of applications. In addition, for demon-
strating that the extended aggregation operator with generalized type-2 fuzzy logic
presentsabetterperformancethanthetraditionaloperatororextendedoperatorwith
interval type-2 fuzzy logic. As part of the objectives, the proposed method was
implemented in a modular neural network applied to face recognition and in an
edgedetector.Comparisonsoftheresultsobtainedwithrespecttootheraggregation
operatorswerealsomade.Duringthedevelopmentoftheextensionoftheoperators
ofageneralizedtype-2fuzzysystemwereappliedtothefuzzydensities,thelambda
calculation, fuzzy measurements and the fuzzy Sugeno integral.
This method offers the best performance in applications of pattern recognition
and edge detection, but the extension of the Sugeno integral can be used in any
application where it is necessary to add numerical information.
This book can be considered as a reference for future research in which are
involved operators that use some type of measure and is organized as follows: In
Chap. 1, we begin by offering a brief introduction to the potential use of the
aggregation operators in real applications.
WedescribeinChap.2,thebasicconceptsoftype-1fuzzylogic,intervaltype-2
and generalized type-2 fuzzy logic are presented. Also we explained the Sugeno
integral aggregation operator and the Interval type-2 Sugeno integral, which serve
as a reference for developing the proposed method. Additionally, we explain in
detail the concepts of Modular neural networks and edge detectors. The method-
ology used for developing the generalized type-2 Sugeno integral, is presented in
Chap. 3. In Chap. 4 offers presents simulation results with benchmark faces
v
vi Preface
databases and benchmark images to illustrate the advantages of the proposed
generalized type-2 Sugeno integral method and finally, Chap. 5 presents the con-
clusions and future work from the obtained results.
Tijuana, Mexico Patricia Melin
Gabriela E. Martinez
Contents
1 Introduction to the Type-2 Fuzzy Sugeno Integral . . . . . . . . . . . . . . 1
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Basic Theory for the Type-2 Fuzzy Sugeno Integral. . . . . . . . . . . . . 5
2.1 Fuzzy Logic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Type-1 Fuzzy Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Type-2 Fuzzy Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Generalized Type-2 Fuzzy Logic . . . . . . . . . . . . . . . . . . . 7
2.2 Fuzzy Measures and the Sugeno Integral . . . . . . . . . . . . . . . . . . . 12
2.2.1 Sugeno Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2.2 Sugeno Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Fuzzy Measures and Sugeno Integral
for Interval Type-2 Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 14
2.3 Edge Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Edge Detector Based on Gradient. . . . . . . . . . . . . . . . . . . 17
2.4 Modular Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Methodology for Face Recognition. . . . . . . . . . . . . . . . . . 19
2.4.2 Face Databases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.3 Pre-processing with the Morphological Gradient (MG)
Edge Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.4 Cross Validation Method . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.5 Training of the MNN. . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.6 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 Proposed Method for the Type-2 Fuzzy Sugeno Integral . . . . . . . . . 29
3.1 Generalized Interval Type-2 Sugeno Integral . . . . . . . . . . . . . . . . 29
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
vii
viii Contents
4 Simulation Results of the Type-2 Fuzzy Sugeno Integral . . . . . . . . . 37
4.1 Morphological Gradient Edge Detector Using
GT2SI (MG + GT2SI) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 GT2SI in Modular Neural Networks (GT2SI + MNN) . . . . . . . . . 39
4.2.1 Simulation Results Using ORL Database . . . . . . . . . . . . . 44
4.2.2 Simulation Results Using Cropped Yale Database . . . . . . . 45
4.2.3 Comparison of Results with Other
Aggregation Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . 45
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5 Conclusions and Future Work on the Generalized Type-2 Fuzzy
Sugeno Integral. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Appendix A... .... .... .... ..... .... .... .... .... .... ..... .... 57
Appendix B... .... .... .... ..... .... .... .... .... .... ..... .... 59
Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 65
List of Figures
Fig. 1.1 Representation of the aggregation operator.. .... .... ..... .. 2
Fig. 2.1 Generalized type-2 membership function ... .... .... ..... .. 8
Fig. 2.2 FOU of a general type-2 membership function... .... ..... .. 8
Fig. 2.3 a plane in generalized type-2 membership functions... ..... .. 9
Fig. 2.4 Cuts at certain points of the GMF. .... .... .... .... ..... .. 9
Fig. 2.5 Diagram of Generalized type-2 fuzzy logic system based
on a planes. .... .... ..... .... .... .... .... .... ..... .. 10
Fig. 2.6 Interval type-2 and generalized type-2 membership
functions... .... .... ..... .... .... .... .... .... ..... .. 11
Fig. 2.7 Intervaltype-2fuzzydensitiesforspecificvaluesl =0.02,
cðx Þ
1
lcðx Þ = 0.4, lcðxÞ = 0.95 and FOU l = 0.2 . .... .... ..... .. 15
2 3
Fig. 2.8 Matrix of 3 (cid:2) 3 of the index Z, that is indicating the
i
calculation of the gradient in the four directions.. .... ..... .. 18
Fig. 2.9 Proposed architecture for face recognition... .... .... ..... .. 20
Fig. 2.10 a Original faces of the ORL database, b image after applying
MGT1FLS, c image after applying MGIT2FLS .. .... ..... .. 22
Fig. 2.11 a Original faces of the Cropped Yale database, b images after
applying the MGT1FLS method, c images after applying the
MGIT2FLS method... ..... .... .... .... .... .... ..... .. 22
Fig. 2.12 ArchitectureofeachmonolithicneuralnetworkusingtheORL
database ... .... .... ..... .... .... .... .... .... ..... .. 24
Fig. 2.13 Procedure for recognition ... .... .... .... .... .... ..... .. 25
Fig. 3.1 Generalized membership function trigausstype-2.. .... ..... .. 30
Fig. 3.2 Alpha cuts of the information sources.. .... .... .... ..... .. 31
Fig. 4.1 Sphere synthetic image..... .... .... .... .... .... ..... .. 38
Fig. 4.2 Diagram that represent the integration of gradients.... ..... .. 39
Fig. 4.3 Sphere reference image..... .... .... .... .... .... ..... .. 40
Fig. 4.4 MG edge detected of the sphere image. .... .... .... ..... .. 40
ix
