Fuzzy LogicFuzzy Logic

Scott GravesScott GravesStephanie CobabeStephanie Cobabe

OutlineOutline

HistoryHistory

ExplanationExplanation

General ApplicationsGeneral Applications

Remote Sensing ApplicationRemote Sensing Application

History of Fuzzy LogicHistory of Fuzzy Logic

1964, Dr. 1964, Dr. LoftiLofti A. A. ZadehZadeh; UC Berkeley; UC Berkeley

History of Fuzzy LogicHistory of Fuzzy LogicGoal:Goal:

Was to develop a model Was to develop a model that could more closely that could more closely describe the natural describe the natural language processlanguage process--

Defined set of rulesDefined set of rulesIF AND THENIF AND THEN

Result known as Result known as ““Fuzzy LogicFuzzy Logic””

History of Fuzzy LogicHistory of Fuzzy LogicDefinition:Definition:

As the complexity of a system increases, it As the complexity of a system increases, it becomes more difficult and eventually becomes more difficult and eventually impossible to make a precise statement about impossible to make a precise statement about its behavior. its behavior.

Provides a very precise approach for dealing Provides a very precise approach for dealing with uncertainty which grows out of the with uncertainty which grows out of the complexity of human behavior.complexity of human behavior.

““Completely true" and "completely false".Completely true" and "completely false".

ExplainedExplainedDesigned to Bridge Modeling Phenomena Designed to Bridge Modeling Phenomena Existing between Quantitative and Existing between Quantitative and QualitativeQualitative

Degree of Membership Degree of Membership

Assigned ValuesAssigned ValuesExtension of Boolean AlgebraExtension of Boolean Algebra

ExplainedExplainedTruth Values:Truth Values:

Range from 0 to 1Range from 0 to 10.0 = Absolute Falseness0.0 = Absolute Falseness1.0 = Absolute Truth1.0 = Absolute TruthAllows Computer to Allows Computer to

Manipulate Information Manipulate Information

Full or Empty?Full or Empty?

Full or Empty?Full or Empty?

Probability vs. Fuzzy LogicProbability vs. Fuzzy Logic

Operates Independently of ProbabilityOperates Independently of Probability

Probability theory measures how likely the Probability theory measures how likely the proposition is to be correct.proposition is to be correct.

Fuzzy logic measures the degree of correctness Fuzzy logic measures the degree of correctness to which the proposition is correct. to which the proposition is correct.

Probability vs. Fuzzy LogicProbability vs. Fuzzy LogicThe important distinction between probabilistic The important distinction between probabilistic information and fuzzy logic is that there is no information and fuzzy logic is that there is no uncertainty about the fullness of the glass but uncertainty about the fullness of the glass but rather about the degree to which it matches the rather about the degree to which it matches the category category ‘‘full' .full' .

Many terms/fuzzy sets, such as 'tall,' 'rich,' 'famous' Many terms/fuzzy sets, such as 'tall,' 'rich,' 'famous' or or ‘‘full,' are valid only to a certain degree when full,' are valid only to a certain degree when applied to a particular individual or situation. applied to a particular individual or situation.

Fuzzy logic tries to measure that degree and to Fuzzy logic tries to measure that degree and to allow computers to manipulate such information. allow computers to manipulate such information.

Probability vs. Fuzzy LogicProbability vs. Fuzzy LogicTerminology:Terminology:

If 0.70 represented a probability value, we If 0.70 represented a probability value, we would read it as there is a 70% chance that would read it as there is a 70% chance that the glass is full, and we have a 70% chance the glass is full, and we have a 70% chance of knowing which group it belongs to. of knowing which group it belongs to.

The glass's degree of membership in the set The glass's degree of membership in the set of full glasses is 0.70, by which we mean that of full glasses is 0.70, by which we mean that if we take the fuzzy set of full glasses and line if we take the fuzzy set of full glasses and line them up, this glass is positioned 70% of the them up, this glass is positioned 70% of the way to the fullest.way to the fullest.

Membership/Fuzzy SetsMembership/Fuzzy Sets

General applicationsGeneral applications

Automotive (Cruise control, automatic Automotive (Cruise control, automatic transmissions, antitransmissions, anti--lock brakes)lock brakes)

Home (Rice cooker, washing machines)Home (Rice cooker, washing machines)

Building (Elevators, air conditioners)Building (Elevators, air conditioners)

Fuzzy logic in remote sensing Fuzzy logic in remote sensing analysisanalysis

ClassificationClassificationConventionalConventionalFuzzyFuzzy

Process of fuzzy classificationProcess of fuzzy classification

Conventional ClassificationConventional Classification

BVBV’’s 0s 0--30 classified as water30 classified as waterBVBV’’s 31s 31--60 classified as forested wetland60 classified as forested wetlandBVBV’’s 61s 61--90 classified as upland forest90 classified as upland forest

Fuzzy ClassificationFuzzy Classification

BVBV’’s < 24 = membership value of 1 for waters < 24 = membership value of 1 for waterBVBV’’s of 30 = 0.5 water and 0.5 forested wetlands of 30 = 0.5 water and 0.5 forested wetlandBVBV’’s of 60 = 0.7 forested wetland and 0.3 upland forests of 60 = 0.7 forested wetland and 0.3 upland forestBVBV’’s > 70 have a membership value of 1 for upland s > 70 have a membership value of 1 for upland forestforest

Process of fuzzy classificationProcess of fuzzy classification

Class definitionClass definitionEstablishing training sitesEstablishing training sitesFuzzy mean parameterFuzzy mean parameterFuzzy covariance parameterFuzzy covariance parameterMembership functionsMembership functionsClass partitioningClass partitioning

Training sitesTraining sites

Select training sites can Select training sites can be a combination of pure be a combination of pure and mixed of materialand mixed of material

Training classes (e.g. Training classes (e.g. grass, forest, water, grass, forest, water, ectect.) .) are defined by mean and are defined by mean and covariance parameterscovariance parameters

Fuzzy mean is defined byFuzzy mean is defined by

Finds the average value of a pixelµc* is the fuzzy mean of training class cfc(xi) is the membership of pixel xi to training class cxi is the value of pixel i

Fuzzy covariance is defined byFuzzy covariance is defined by

Finds the average value of the product of two pixelsVc* is the fuzzy covariancefc(xi) is the membership of pixel xi to training class c

Membership functionMembership function

In order to perform fuzzy class partition, a In order to perform fuzzy class partition, a membership function must be defined for membership function must be defined for each classeach class

This function allows each pixel to be This function allows each pixel to be assigned a membership valueassigned a membership value

Definition of membership functionDefinition of membership function

Where fc(x) is the membership of pixel x, to class cPc*(x) is the maximum likelihood probability of pixel x, to class cm is the number of classes

Class partitioningClass partitioning

Each unknown pixel is assigned a membership value describing Each unknown pixel is assigned a membership value describing how close the pixel is to the training class mean how close the pixel is to the training class mean

Fuzzy membership matrixFuzzy membership matrix

Pixels A,B,D, and E are relatively homogeneousPixels A,B,D, and E are relatively homogeneousPixels C,F, G, and H are mixedPixels C,F, G, and H are mixed

Final productFinal product

From this matrix a series of maps can be From this matrix a series of maps can be created created

Fuzzy classified maps (Typically supervised Fuzzy classified maps (Typically supervised classification)classification)

Map showing only pixels that have a forested value Map showing only pixels that have a forested value > 0.70 and grass value of >0.20> 0.70 and grass value of >0.20

A hard classification can still be shown by changing A hard classification can still be shown by changing the maximum value in each column to 1 and all other the maximum value in each column to 1 and all other values to 0values to 0

Supervised Supervised Classification Using Classification Using

Fuzzy LogicFuzzy Logic

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