an explicit fuzzy supervised classification method for multispectral remote sensing images

IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 1, JANUARY 2000 287

An Explicit Fuzzy Supervised Classification Methodfor Multispectral Remote Sensing Images

Farid Melgani, Bakir A. R. Al Hashemy, and Saleem M. R. Taha

Abstract—Fuzzy classification has become of great interestbecause of its capacity to provide more useful information forgeographic information systems. This paper describes an explicitfuzzy supervised classification method which consists of threesteps. The explicit fuzzyfication is the first step where the pixel istransformed into a matrix of membership degrees representingthe fuzzy inputs of the process. Then, in the second step, aMIN fuzzy reasoning rule followed by a rescaling operation areapplied to deduce the fuzzy outputs, or in other words, the fuzzyclassification of the pixel. Finally, a defuzzyfication step is carriedout to produce a hard classification. The classification results onLandsat TM data demonstrate the promising performances of themethod and comparatively short classification time.

Index Terms—Fuzzy classification, fuzzyfication, fuzzy rea-soning rule, remote sensing.

I. INTRODUCTION

A WIDE range of pattern recognition techniques have ap-peared during the last two decades for information extrac-

tion from remotely sensed data. The most popular techniqueis the maximum likelihood classification method known for itsgood performance and robustness. In addition, an approach tothe problem of classification using artificial neural networks hasbeen developed [1]–[4]. However, as Heermann and Khazeniesay about ANN: “This changes the solution process from findingways to understand and represent the problem in a computerlanguage to a task of providing examples of the problem forthe network to learn.” The main advantage of ANN classifica-tion methods is that they are “distribution free.” However, theyhave many problems such that no general criteria for defininga suitable network architecture, dependence on training condi-tions and slow training time [5].

All these methods are used to carry out a hard classifica-tion based on the principle of “one pixel—one class. ” In otherwords, a pixel is either a full member or not of a class. This logicof processing does not represent the best way to deal with datathat are naturally mixed, and so imprecise in nature. The ad-vent of the fuzzy set theory [6] has invaded a lot of fields suchas fuzzy control systems, fuzzy image processing [7] and, in-evitably, fuzzy classification of remote sensed data. Fuzzy clas-sification, or pixel unmixing, estimates the contribution of eachclass in the pixel. It assumes that a pixel is not an indecompos-able unit in the image analysis and, consequently, works on anew principle: “one pixel—several classes” to provide more in-

Manuscript received June 22, 1998; revised November 30, 1998.The authors are with the Electrical Engineering Department, College of En-

gineering, University of Baghdad, Baghdad, Iraq.Publisher Item Identifier S 0196-2892(00)00139-X.

Fig. 1. Raw Landsat TM image (Jordan Valley), band 3 272 × 242 pixel size.

formation about the pixel unlike the hard classification methodswhich are poor in information extraction. A fuzzy c-means clus-tering algorithm has been developed by Bezdeket al. [8] andimplemented by Cannonet al. [9] in unsupervised classifica-tion of remote sensing data. Wang [10] modified the traditionalmaximum likelihood method by implementing a pre-calculatedfuzzy mean and fuzzy covariance matrix. Then, the fuzzy mem-bership degrees are computed by applying the maximum likeli-hood procedure defined on fuzzy class signatures. Another para-metric classification method has been proposed by Foodyet al.[11] to express the class proportions. Adamset al.[12], Foodyetal. [13] introduced models based on a linear mixing. Of course,artificial neural networks have been exploited in fuzzy classifi-cation to relate their outputs to the class contribution in a givenpixel [14], and [15].

In this paper, we propose an explicit fuzzy classificationmethod for multispectral remote sensing images carried outin three steps. An explicit fuzzyfication step for obtaining anestimation of the class contributions in each band assuming aGaussian distribution of the classes. The second step appliesa fuzzy reasoning rule on these fuzzy inputs to obtain, aftera rescaling operation, the fuzzy classification of the scene.Finally, a hard classification can be deduced in the defuzzyfica-tion step in order to illustrate the whole of the cover classes inthe same map. The proposed method will be applied on LandsatTM data, bands 1–3, representing a part of the Jordan Valley(Fig. 1). Results of the fuzzy classification and a comparisonwith the maximum likelihood method of the hard classificationresults will show the efficiency of the method.

0196–2892/00$10.00 © 2000 IEEE

288 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 38, NO. 1, JANUARY 2000

Fig. 2. Block diagram of the explicit fuzzy method.

II. M ETHODOLOGY

Our approach, illustrated in Fig. 2, analyzes the data in orderto extract a maximal amount of information, through an explicitfuzzyfication process, in order to make the decision rule moresimple and reliable. Then, a MIN reasoning rule is carried outto pick out for each class its worst contribution from the set ofmembership degrees of the class among the different bands. Theinferred raw fuzzy outputs are rescaled to provide a fuzzy clas-sification of the pixel. The jump from a fuzzy classification toa hard classification is easily done by a MAX operation repre-senting the defuzzyfication process.

A. Explicit Fuzzyfication Process

The fuzzy domain consists of several fuzzy sets representingthe bands, and each fuzzy set (band) contains fuzzy subsets rep-resenting the cover classes. Each fuzzy subset (cover class c),in a given fuzzy set (band b) is fully defined by its membershipfunction where is the brightness value of the spectralpixel in the band b. The pixel vector in the -dimensionalspace is

We have oriented the choice of the membership function tothe Gaussian distribution because it represents a powerful gen-eral distribution model and involves a minimal extraction com-putational cost for the statistical characteristics of the signatures.Furthermore, the linear models are not able to express correctlythe natural non linear distribution of the classes. The class dis-tributions are totally independent each of the others and, con-sequently, we do not impose any mixing assumption. The in-dependence between the fuzzy subsets (classes) makes us notrespect the usual formal fuzzy requirement

(1)

where is the number of fuzzy subsets.Of course, we can implement a normalization operation to

satisfy (1) but we have not done it for three main reasons:

1) The MIN fuzzy reasoning rule is insensitive to this require-ment.

Proof: The normalization applies the following transfor-mation:

Of course, for ,

However, for ,

We deduce easily that the normalization operation is not nec-essary for the MIN reasoning rule because:

and

if

1) We alleviate the burden of unuseful computations due to thenormalization operation.

2) The class distributions keep their natural aspect.

In the algorithm, we have introduced the notion of modulatedGaussian distribution if prior knowledge of the class extents inthe scene is available. The facultative prior knowledge of theclass extents in the study area is reflected in the size of the classsignatures necessary to carry out a statistics extraction in orderto compute two important parameters to define completely themembership function:

1) The mean () of the class signature which represents theideal pixel of the class, or in other words, the only pointwithout any ambiguity about its membership.

2) The standard deviation () of the class signature, which willdetermine in a way the width of the fuzzy subset.

MELGANI et al.: EXPLICIT FUZZY SUPERVISED CLASSIFICATION METHOD 289

The membership function of class c in band b is

(2)

where is the mean of class c in band b, is the modulatedstandard deviation of class c in band b, and

(3)

where is the standard deviation of class c in band b, andis the modulation factor. If no prior knowledge of the class

extents in the study area is available, the modulation factormust be neutralized to work with a pure Gaussian distribution.It means that

Otherwise, the modulation factor is calculated by the formula

(4)

where is the expected extent of class c in band b, andisthe constant.

In order to reduce the problem that the prior knowledge of theclass extents in the scene is generally vague, we will not use itdirectly in the modulation factor. Instead, we propose to reflectit in the size of the class signatures and to introduce the notionof expected extent of the classes in each band by deducing thenumber of pixels expected to correspond to the mean of eachclass signature. For a given band b, represents the expectednumber of pixels, in the signature histogram, corresponding tothe mean of the class c

(5)

where is the number of classes.The modulation principle is useful to favor the large extent

classes over the lower extent ones and, particularly, in the spec-tral regions where overlaps make the difficult decision. We havededuced by empirical observation that a constant isthe optimal one to satisfy the need to “not strengthen too muchthe strong classes and not weaken too much the weak classes.”Fig. 3 shows the modulation factor versus the class proportionvarying quasi linearly in the interval [0.22, 0.81]. Thus, the mod-ulation function can be approximated by a linear function

(6)

The fuzzyfication process, as illustrated in the Fig. 4, com-putes for a given pixel the membership degrees for each class cand band b from the membership functions fully defined usingthe Gaussian distribution and the results of the statistics extrac-tion. We obtain a matrix of fuzzy inputs of orderwhere is the number of classes and B the number of bands.

Fig. 3. Modulation factor versus class proportion.

TABLE IPIXEL NUMBER OF TRAINING SITES AND

TEST SITES

TABLE IICLASS EXPECTEDEXTENTS IN [%]

For a multispectral pixel , the matrix of fuzzy inputscan be written

(7)

This matrix is then analyzed by the MIN reasoning rule whichis the second step of the method to produce the fuzzy classifi-cation.

B. MIN Fuzzy Reasoning Rule

In dealing with everyday problems, humans often employ asimple but efficient principle which can be summarized in thesefew words “work on the worst assumptions to find the best re-sults.” The fuzzy reasoning rule that satisfies this principle is theMIN rule. On the other hand, the absence of covariance infor-mation makes the fuzzy partitions not necessarily optimal be-cause classes with a high degree of covariance will not be lim-ited to the high fuzzy membership areas. Therefore, high fuzzymembership values more closely represent a maximum possiblevalue rather than a high prior probability of class membership.Consequently, the lowest fuzzy value for all the bands is morelikely to be closest to the “true” fuzzy membership. The MINreasoning rule, applied on the matrix of fuzzy inputs produced


Fig. 4. Architecture of the explicit fuzzy classification method illustrated for two bands and three classes.

by the previous step, will consider, for each class, the member-ship degrees provided by the different fuzzy sets (bands) andpick out the minimal membership degree to represent the classextent in the pixel.

Applying the MIN operation, for which an illustration isgiven in the Fig. 4, to (7), we obtain a primitive fuzzy outputvector

(8)

where

Min with (9)

The fast, simple, and reliable step of the MIN fuzzy reasoningrule is immediately followed by a rescaling operation in orderto normalize the class extents deduced from different fuzzy setsand sharing the same pixel.

The final vector of fuzzy outputs is

(10)

where

(11)

This vector represents the fuzzy classification showing the in-ferred class extents of the pixel, and from which a hard classifi-cation can be deduced by a defuzzyfication step.

We note that the extraction from the data of first order sta-tistical characteristics for deducing the fuzzy membership func-tions and the application of the MIN fuzzy reasoning rule in-volve a partition of the spectral space into fuzzy parallelepipedregions which relate conceptually the method to a fuzzy paral-lelepiped classifier. Furthermore, the explicit fuzzyfication as-sociated to the MIN fuzzy reasoning rule reveal the modular as-pect of the method, as shown in Fig. 4, which allows to add andremove any band from the classification scheme of the studyarea because of the relative independence between the bands.The adding or removal of a cover class will, however, require aminor computational cost due to the adjustment of the modula-tion factors if a prior knowledge of the class extents is available.

C. Defuzzyfication

Next to the fuzzy classification, a hard classification is pro-vided by performing a MAX operation to defuzzyfy the fuzzyoutput into a hard output. We select among the classes mixed inthe pixel the class C with the highest extent such that

and (12)


(a)

(b)

Fig. 5. One-dimensional fuzzy partition, membership degree (MD) versus brightness value (BV), sandy areas (A), agricultural land (B), rangeland (C), barren land(D): (a) using pure Gaussian distribution and (b) using modulated Gaussian distribution.

III. A PPLICATION AND RESULTS

Fig. 1 shows the study area through the band 3 of the LandsatTM with a spatial resolution of 30 × 30 m. The 272 × 242 pixels

of the Jordan Valley include four major cover classes which aresandy areas, agricultural land, range land and barren land re-spectively classes A, B, C, and D. A prior knowledge of the


(a)

(b)

Fig. 6. (a) Two-dimensional fuzzy partition [band 3 brightness value (BV) versus band 2 BV versus membership degree (MD)] of sandy area class (left) andagricultural land class (right). (b) Two-dimensional fuzzy partition [band 3 BV versus band 2 BV versus MD] of range land class (left) and barren land class (right).

region has allowed us to benefit from the modulated Gaussiandistribution. A C language program is used to implement the ex-plicit fuzzy method on a 166 MHz pentium PC.

A. Statistics Extraction

For each class, we have selected a set of pixels according toour knowledge of the region to build the class signatures. Twothirds of each set were used as training site and the last third astest site. A report of the pixel number of the training and testsites is given by Table I.

We can observe, from Table II, that the expected extent of theclasses vary from a band to another because the deduced numberof pixels associated to each mean changes from a band to an-other. Furthermore, the bands 1 and 2 are correlated because ofthe typical biophysical nature of the region.

B. Fuzzy Partition

Fig. 5(a) shows a one-dimensional (1-D) fuzzy partition ofthe spectral space (bands 1, 2, and 3) using a pure Gaussian dis-

tribution. We observe that the fuzzy subsets are strongly over-lapped involving an overloading on the spectral space, and par-ticularly in the band 3 between classes B, C, and D. The het-erogeneity of the class B (agricultural land) is well expressedby its broad distribution over the bands. Using the modulatedGaussian distribution [Fig. 5(b)], the fuzzyness (ambiguity) islessened, because of the overlap reduction, in the way to favorin the critical spectral regions the large extent classes and putat a disadvantage the small extent classes. Band 3 shows thatclasses C and D are almost covered by the class B. The effect ofthe MIN fuzzy reasoning rule on the 1-D fuzzy partition is in-vestigated by a two-dimensional (2-D) fuzzy partition to provethe capability of this rule to provide reliable results despite ofthe overlap degree between the classes. Fig. 6 shows the resultsof the rule applied on band 3 versus band 2, illustrating clearlythe partition of the spectral space into fuzzy parallelepiped re-gions which represent the multispectral references for the fuzzyand hard classifications of the scene. We observe that class A isnearly separated from the other bands, and two zones of conflict


(a) (b)

(c) (d)

Fig. 7. Fuzzy classification of the study area; (a) sandy area class, (b) argricultural land class, (c) range land class, and (d) barren land class.

(a) (b)

Fig. 8. Hard classification of the study area, black (sandy areas), dark gray (agricultural land), gray (range land), and light gray (barren land); (a) by explicite fuzzymethod and (b) by maximum likelihood method.

are declared between classes B and C in one hand, and D and Cin the other hand, revealing the reliable fuzzy spectral zones.

C. Fuzzy Classification

A fuzzy classification of the whole scene has been carriedout to show the extent of each class throughout the study area.

Fig. 7(b)–(d) reveals a nonnegligible mixture between classes B,C, and D and particularly at the West and East parts of the pho-tograph. As discussed previously, it was expected because ofthe biophysical nature of the classes (agricultural land, range-land, and barren land) which involves a possibility of mixturebetween them. The Jordan Valley is a good example of this par-ticular mixture. However, Fig. 7(a) shows nearly a hard extent


TABLE IIICONFUSIONMATRIX OF THE TRAINING SITE RESULTS FOR THEEXPLICIT FUZZY METHOD

(a)

(b)

TABLE IVCONFUSIONMATRIX OF THE TEST SITE RESULTS FOR THEEXPLICIT FUZZY METHOD

(a)

(b)

of the class A excepting some sporadic pixels in the agriculturalregion.

D. Hard Classification

In order to strengthen the evaluation of the method, a com-parison between the hard classification provided by the explicitfuzzy method and that provided by the maximum likelihoodmethod has been made. Fig. 8 provides a visual result of thehard classification by both methods. We notice that the imageproduced by the explicit fuzzy method is more smooth and thatthis last has favored the class C (range land) in the west part ofthe study area showing a better similarity to the ground truth,whereas the maximum likelihood method has emphasized on

the class D (barren land). But in a general point of view, the twographic results are very near.

A comparison based on numbers is given in Tables III–V. Wenote that the average accuracy is the mean of the classificationaccuracies obtained for each class, whereas the overall accuracyrepresents the ratio of the true pixel number to the total pixelnumber for the whole of the classes. The classification of thetraining samples [Table III(a) and (b)] reveals that the maximumlikelihood method is a bit better than the explicit fuzzy methodwith an average accuracy of 79.52 against 75.15% and an overallaccuracy of 78.21 against 77.38%. These results prove that themaximum likelihood method has the advantage to produce anefficient adaptation to the training sites due to the use of second


TABLE VTOTAL CLASSIFICATION TIME

order statistics (covariance matrices). However, the results ofthe classification of the test samples show a slight decrease ofnearly 2% in the accuracies for the explicit fuzzy method (OA

75.54% and AA 74.71%) and a nonnegligible decreaseof approximately 5% in the accuracies for the maximum likeli-hood method (OA 72.89% and AA 75.56%) which provesa better generalization ability of the explicit fuzzy method be-cause of the use of first order statistics making the method lessdependent on the training samples.

Another important parameter in the comparison is the totalclassification time of the scene which shows that the explicitfuzzy method is a very fast classification method. As listed inTable V, the explicit fuzzy method needed a total classificationtime of 12.36 s, nearly seven times shorter than the time neededby the maximum likelihood method. As we can see, the explicitfuzzy method provides an encouraging classification accuracyand a far shorter classification time, which is one of the majorhandicaps of the Maximum Likelihood classifier because of itsnecessary complex matrix computations.

IV. CONCLUSIONS

We have proposed in this work a new method for fuzzy super-vised classification which reveals promising performances, andparticularly, a fast classification time, which becomes a factor ofgreat importance with the advent of new generations of sensorsproviding broad collections of data (for example, HIRIS with192 spectral bands). The strength of the method is certainly itssimplicity and its optimal extraction of the data through the ex-plicit fuzzyfication, exploited efficiently by the MIN fuzzy rea-soning rule. Another advantage of the explicit fuzzyfication isits modular architecture involving a high flexibility for easilyinserting new bands or removing bands without disturbing theremaining parts of the classifier. Furthermore, the implementa-tion of the explicit fuzzy method on artificial neural networkswould not pose any difficulty.

ACKNOWLEDGMENT

The authors would like to thank the Royal Jordanian Geo-graphic Center for providing the raw images and the reviewersfor their helpful and valuable comments and suggestions.

REFERENCES

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Farid Melgani was born in Batna, Algeria, on Au-gust 25, 1971. He received the State Engineer de-gree in electronics from the University of Batna, Al-geria, in 1994, and the M.Sc. degree in electrical en-gineering from the University of Baghdad, Iraq, in1999.

He was with the International Computer Services,Algeria, in 1994 and 1995. He is currently with theDepartment of Electrical Engineering, Universityof Baghdad, and his current research interests arepattern recognition applications to signal image

processing.

Bakir A. R. Al Hashemy received the B.Sc. (Hons.) degree in electrical andelectronic engineering from Leeds University, U.K., in 1967, the PostgraduateDiploma in computer management studies from London University, London,U.K., in 1972, the M.Sc. degree in computer science from the University ofWales, U.K., in 1975, and the Ph.D. degree in computer studies from Lough-borough University of Technology, U.K., in 1978.

He is currently a Professor of Computer Studies in the Electrical EngineeringDepartment, Baghdad University, Iraq. His research interests lies in te area ofcomputer graphics, parallel processing, parallel algorithms, numerical analysis,and image processing.

Saleem M. R. Tahawas born in Baghdad, Iraq, onMay 21, 1956. He received the B.Sc. degree in elec-trical engineering with elective courses in computerscience, the Higher Diploma degree in electronicsand communications and the M.Sc. degree in digitalelectronic systems from the University of Baghdadin 1978, 1980, and 1982, respectively.

He is currently an Assistant Professor at the Uni-versity of Baghdad. He has contributed and authoredmany papers in the areas for hybrid circuit designs,digital signal processing, digital measurement intru-

mentation, microcomputer applications, computer-aided design, computer sim-ulation, and biomedical engineering.

an explicit fuzzy supervised classification method for multispectral remote sensing images

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