Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image pr
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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image pr

E erre Defuzzi57346cation cr iter ia and classi57346cation Fuzzy sets and Systems ol 108 pp 159178 1999 A Runk el and M Glesner A set of axioms or defuzzi57346cation str ategies to ards theor of ational defuzzi57346cation oper ators Proc of 2nd IEEE

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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image pr




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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Fuzzy Sets and Fuzzy echniques Lecture 11 Defuzzication Nataa Sladoje Centre or Image Analysis Uppsala Univ ersity ebr uar 28, 2007 Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the

fuzzication An application in image processing Outline Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Liter ature an Leekwijc k, E.E. erre “Defuzzication: cr iter ia and

classication”, Fuzzy sets and Systems ol. 108, pp 159–178, 1999. .A. Runk el and M. Glesner “A set of axioms or defuzzication str ategies to ards theor of ational defuzzication oper ators”, Proc. of 2nd IEEE Inter n. Conf on Fuzzy Systems pp 1161–1166, 1993. Ralescu, “A er age le el of fuzzy set”, Proc. of IPMU Madr id pp 190–194, 2000 E. Ro enta, Spircu, “A er aging procedures in defuzzication processes”, Fuzzy sets and Systems ol. 136, pp 375–385, 2003. L. Rondeau, R. Ruelas L. Le vr at, M. Lamotte “A defuzzication method respecting the

fuzzication”, Fuzzy Sets and Systems 86 ol. 86, 311–320, 1997. Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Denition Denition Defuzzication is process that maps fuzzy set to cr isp set. Defuzzication has attr acted ar less attention than other processes in olv ed in fuzzy systems and technologies Defuzzication is sometimes not seen as par of the

core of fuzzy system; fuzzy system “ends where uncer tainty and imprecision end. Defuzzication is “just the last step”; it is an interf ace with cr isp models of the or ld. Defuzzication is model of synthesis theref ore it is completely opposite to the main concept of fuzzy set theor
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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Approaches Defuzzication

to point. Defuzzication to set. Gener ating good representativ of fuzzy set. Reco er ing cr isp or iginal set. Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Cr iter ia Defuzzication to point Classication is made with respect to the mathematical str ucture of the univ ersal set Arbitrar univer ses (1) Core selection cr iter ion (2) Scale in ar iance (ordinal, inter al,

atio relativ absolute) Or dered univer ses (3) Monoton (4) -conor cr iter ion The set of real umber (5) -T anslation (6) -Scaling (7) Contin uity Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Cr iter ia Defuzzication to set Defuzzication should contain the er nel of fuzzy set. Defuzzication should be consistent with afne tr ansf or mations Defuzzication

should preser natur al order in the set of fuzzy sets If then or fuzzy sets and their er agings and Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to point list of methods Maxima methods and der iv ativ es Selection of an element from the core of fuzzy set as defuzzication alue Main adv antage is simplicity Distr ib ution methods and der iv ativ es Con ersion of the

membership function into prob- ability distr ib ution, and computation of the xpected alue Main adv antage is contin uity proper ty Area methods The defuzzication alue divides the area under the membership function in tw (more or less) equal par ts
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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to point list of methods Maxima methods Random choice

of maxima Prob core core otherwise First of maxima, Last of maxima, Middle of maxima FOM min core LOM max core or MOM it holds that core MOM core MOM if core is odd core MOM core MOM if core is en Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to point list of methods Distr ib ution methods (1) Centre of vity (Set of real umbers) COG max min max min Mean of maxima (Set of

real umbers) MeOM core core Basic defuzzication distr ib ution BADD max min max min where reects the condence in the system. Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to point list of methods Distr ib ution methods (2) Gener aliz ed le el set defuzzication GLSD where is the condence to the system, is the umber of -cuts and is the

er age alue of the th -cut. Inde ed centre of vity ICOG where is selected threshold belo which all membership alues are set to ero Fuzzy mean (combines agg regation and defuzzication) FM where is the umber of fuzzy output sets is the deg ree of consistency obtained inf erence ules and is some umer ical alue associated with the output fuzzy sets (often MOM ). Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in

image processing Defuzzication to point list of methods Area methods Centre of area (CO A) CO is the alue that minimiz es the xpression CO inf sup CO
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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to point Cr iter ia fullment (1) (2) (3) (4) (5) (6) (7) FOM/LOM es es max es MOM es es es COG es es es es CO es es es es Fuzzy Sets and Fuzzy

echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to set er aging procedures -cuts chosen at ar ious le els er age -cuts based on an integ ation of set-v alued function, called udo-A umann integ ation. Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the

fuzzication An application in image processing Defuzzication to set er aging procedures -cuts or and are er often used; attempts to chose some -cut or are made; attempts to dene er aging which is not necessar ily an -cut are made Example: The separating po wer of fuzzy set is set supp with compact suppor and non-z ero Lebesgue measure supp )) 0), such that the function dv dv attains its suprem um or it. denotes the complement of with respect to the suppor of SP is an er aging procedure Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches

Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to set er aging procedures Example: Ar ithmetic mean of tw sets: or tw er aging procedures their ar ithmetic mean )( )@ is an er aging procedure It is usually denoted or an is an er aging procedure Ho er it is not alw ys (intuitiv ely) good.
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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set

Defuzzication respecting the fuzzication An application in image processing Defuzzication to set er aging procedures ossib ly better choice is sho wn belo w: Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to set er age -cuts or set alued function udo-A umann integ al is dened as dm dm where :; is integ ab le is the space of integ ab le

selector of Note that the integ al of set-v alued function is set. Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to set er age -cuts Let fuzzy set be giv en membership function Sets are cuts of the fuzzy set or Selectors are inf and sup Then, the er age -cut of is inf sup Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia

Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication to set er age -cuts Often, Not alw ys Example: otherwise In this case is not alw ys an -cut of or an Example: otherwise In this case Since or er
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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing

Defuzzication respecting the fuzzication What is the idea? Assume: univ ersal set being the set of real umbers; all par ameters of the fuzzication are kno wn; alue is fuzzied and ector char acter izing fuzzy membership alues corresponding to each of the linguistic ter ms used, is kno wn. Dene defuzzication so that is reco ered from Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An

application in image processing Defuzzication respecting the fuzzication Example: Let linguistic ter ms be represented fuzzy sets as sho wn: Let be giv en. is the ector of fuzzy membership alues corresponding to ar tial solutions of the prob lem are the sets containing elements such that or each 4. In this case The solution is dened as the intersection of all par tial solutions: and in this case is obtained. Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set

Defuzzication respecting the fuzzication An application in image processing Defuzzication respecting the fuzzication Ho does it look in reality? Constr aints: All the par ameters of the fuzzication are kno wn. The alue is such that or some In reality giv en is arely such, i.e ., there is no fullling Instead, the fuzzy representation of “candidate or is gener ated, using and fuzzy sets representing linguistic ter ms (i.e ., using kno wledge about fuzzication). obtain cr isp solution defuzzication (to point) of its fuzzy

representation is perf or med. Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication respecting the fuzzication bit more gener al approach In case when or giv en or all “candidate or is ound an optimization (minimization) of the function: ))
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Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia

Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication respecting the fuzzication bit more gener al approach Assume: Functions (used or fuzzication) are tr iangular The core of each is singleton, denoted or giv en and kno wn look or such that is equal to with eight of In other ords minimiz es Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set

Defuzzication respecting the fuzzication An application in image processing Defuzzication respecting the fuzzication bit more gener al approach Analytical solution leads to defuzzication the height method which computes the bar ycentre of membership function er nels eighted the corresponding membership deg rees: Recall that the fuzzication assumed is based on triangular membership functions Fuzzy Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set

Defuzzication respecting the fuzzication An application in image processing Defuzzication respecting the fuzzication bit more gener al approach one more step Constr aints on fuzzication: or all there are at most tw (linguistic ter ms) such that 0. The er nel of each is singleton. are monotonically increasing or and monotonically decreasing or The sum of membership alues of each element to all the fuzzy sets is equal to 1. All ha the same shape (are dened the same monotonically increasing function .t. and 1). Fuzzy Sets and Fuzzy echniques Nataa

Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication respecting the fuzzication bit more gener al approach one more step The aim is to nd such that )( where the function appropr iately adjusts the eights in the case when non-linear fuzzication is applied, and is under the assumptions abo sho wn to be equal to The defuzzication method that is in this de eloped is
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Fuzzy

Sets and Fuzzy echniques Nataa Sladoje Liter ature Denition, Approaches Cr iter ia Defuzzication to point Defuzzication to set Defuzzication respecting the fuzzication An application in image processing Defuzzication in image processing Defuzzication eature distance minimization Liter ature: N. Sladoje Lindb lad, and I. Nystrm, “Defuzzication of spatial fuzzy sets eature in ar iance”, submitted or pub lication, un 2005. Lindb lad and N. Sladoje “F eature Based Defuzzication at Increased Spatial Resolution”, Proc.

of 11th Inter n. or kshop on Combinator ial Image Analysis (IWCIA2006), Ber lin, Ger man Lecture Notes in Computer Science ol. 4040, pp 131–143, Spr inger-V er lag, 2006. Lindb lad, N. Sladoje and Luki c, “F eature Based Defuzzication in and Using Scale Space Approach”. Proc. of 13th Inter n. Conf erence on Discrete Geometr or Computer Imager (DGCI2006), Sz eged, Hungar Lecture Notes in Computer Science ol. 4245, pp 379–390, Spr inger-V er lag, 2006.