A General Method for Assessing the Uncertainty in Classified Remotely Sensed Data at Pixel Scale

Transcription

1 Proceedngs of the 8th Internatonal Symposum on Spatal Accuracy Assessment n Natural Resources and Envronmental Scences Shangha, P. R. Chna, June 25-27, 2008, pp A General ethod for Assessng the Uncertanty n Classfed Remotely Sensed Data at Pxel Scale Yanchen Bo, 2 + and Jnfeng Wang 3. State Key Laboratory of Remote Sensng Scence, Jontly Sponsored by Bejng Normal Unversty and Insttute of Remote Sensng Applcatons, CAS, 2. Research Center for Remote Sensng and GIS, School of Geography, Bejng Key Laboratory for Remote Sensng of Envronment and Dgtal Ctes, Bejng Normal Unversty, Bejng, 00875, Chna 3. LREIS, Insttute of Geographcal Scence and Natural Resources Research Chnese Academy of Scences, Bejng, 000, Chna Abstract. The uncertanty assessment on the classfcaton of remotely sensed data s a crtcal problem n both academc arena and applcatons. The conventonal soluton to ths problem s based on the error matrx (.e. confuson matrx) and the kappa statstcs derved from the error matrx. However, no spatal dstrbuton nformaton of classfcaton uncertanty can be presented by ths method. A probablty vector-based method has been developed for assessng classfcaton at pxel-scale. However, the use of ths method s severely lmted because the probablty vector can be derved only through the Bayesan classfcaton. In practce, other classfers such as Artfcal Neural Network Classfer, nmum Dstance Classfer, ahalanobs Dstance Classfer and Fuzzy Classfer are wldly used for remote sensng data classfcaton. To assess the uncertanty of thematc maps by these classfers at pxel-scale, a general method s presented n ths paper whch extends the probablty vector-based method to the assessment on the uncertanty classfed by classfers beyond Bayesan classfer. Ths extenson s realzed through a transformaton method, whch transforms the embershp Vector n varous classfers to the transformed probablty vector so that t s comparable to the probablty vector n Bayesan Classfer. The uncertanty measurements could be derved from the probablty vector were evaluated and, the probablty resdual and entropy that derved from the extended probablty vector are used as ndcators to assess the absolute and relatve uncertanty perceptvely. The uncertantes by dfferent classfers are compared at pxel scale. Some examples of the uncertanty of maps from dstance classfers were presented and compared wth that of maps from LC classfer. Keywords: scale, a posteror probablty vector, uncertanty measure.. Introducton Categorcal spatal data set such as land use/land cover and vegetaton type derved from remotely sensed data s the crtcal nputs n ecologcal and envronmental modelng. As the classfed maps from remotely sensed data are uncertan n nature, assessment on the accuracy and uncertanty of remotely sensed data classfcaton s essental to the propagaton and assessment on the uncertanty of models output and, further more, the rsks of decsons based on the models outputs. An essental aspect of the ncreasng sophstcaton of ecologcal and envronmental models s the use of spatally explct nputs and outputs. Thus, the spatal dstrbuton of the classfed maps from remotely sensed data has become a new challenge. The wdely accepted method for assessng the accuracy of thematc maps from remotely sensed data has been the error matrx, or confuson matrx (Congalton and Green, 999). A lmtaton of the error matrx s that t allows only one reference class for each reference ste. Ths s problematc n stes where selecton of + Correspondng author. Tel.: ; fax: E-mal address: boyc@bnu.edu.cn ISBN: , ISBN3:

2 only one class s dffcult or napproprate. A soluton to ths problem s developed by usng fuzzy set theory (Gopal and Woodcock, 994; Woodcock and Gopal, 2000). A common drawback of error matrx based method and fuzzy set theory based method s that t can not provde the spatal dstrbuton of classfcaton uncertanty. Assessng on the uncertanty of classfcaton based on the LC a posteror probablty vector was proposed by Goodchld et al. (992). several uncertanty measures were derved from the LC a posteror probablty vector by Sh (998). However, the practcablty of per-pxel assessment on classfcaton uncertanty based on the a posteror probablty vector was lmtng because the a posteror probablty vector can only be derved through LC classfcaton. For other wdely used classfers n remotely sensed data classfcaton, such as nmum Dstance Classfer ahalabous Dstance Classfer, Artfcal Neural Network Classfer and Fuzzy Classfer, how to assess the uncertanty of classfcaton at pxel-scale s reman problematc. In addton, the varous uncertanty measures were developed based on the LC a posteror probablty vector. Evaluatng and selectng approprate uncertanty measures for uncertanty assessment on remotely sensed data classfcaton at pxel-scale s mportant to the uncertanty assessment practces. The research objectves of ths paper are twofold. Frstly, select the approprate measures for uncertanty assessment on remotely sensed data classfcaton at pxel scale. Secondly, develop a general method for assessng the uncertanty of classfcaton at pxel scale so that the uncertanty measures derved LC a posteror probablty vector can be used to the assessment on the uncertanty of maps classfed by classfers beyond LC at pxel scale. In secton 2, the LC a posteror probablty vector was ntroduced, the varous uncertanty measures derved from a posteror probablty vector were evaluated and approprate measures were selected. In secton 3, the extended probablty vector method were developed to extend the uncertanty measures at pxel scale from the assessment on uncertanty of LC classfcaton to the assessment on the uncertanty of ANN classfcaton, Dstance classfcaton and Fuzzy classfcaton. Secton 4 gves some examples of uncertanty assessment at pxel usng measures from a posteror probablty vector on the maps classfed by classfers beyond LC. Secton 5 summarzes the whole paper and dscussons on further studes were gven. 2. The a Posteror Probablty Vector and the Uncertanty easures 2.. The a Posteror Probablty Vector In the LC classfcaton, the decson of assgnng a pxel to a specfc class s based on the a posteror probablty vector whch presents the condtonal probablty the pxel belongs to every pre-defned class n the classfcaton scheme. ost often, the pxel was assgned to the class wth the maxmum probablty value n the probablty vector. Let the spectral classes for an mage be represented by ω, =... () where s the total number of classes. Accordng to Rchard et al. (998), to determne the class to whch a pxel at a locaton x belongs, the condtonal probabltes are of nterest. p( ω, =,... (2) where the poston vector x s a column vector of brghtness values for the pxel,t descrbes the pxel as a pont n multspectral space wth coordnates defned by the brghtness. The probablty p( ω gves the lkelhood that the correct class sω for a pxel at poston x. Classfcaton s performed accordng to x ω f p( ω > p( ω for all j (3) j.e., the pxel at x belongs to classω f p( ω s the largest. 87

3 Often, the p( ω n () s unknown. Suppose that suffcent tranng data s avalable for each ground cover type, the probablty dstrbuton for a cover type can be estmated. The desred p( ω and the pxω ( ) estmated from tranng data are related by Bayes theorem: p( ω p( x ω ) p( ω )/ p( x) = (4) where p( ω ) s the probablty that classω occurs n the mage. p(x) s the probablty of fndng a pxel from any class at locaton x and can be expressed as: px ( ) = px ( ω) p( ω) = (5) Where the p( ω ) are called a pror probablty and p( ω are a posteror probablty. Usng (2), the classfcaton rule of () s: x ω, f p( x ω ) p( ω ) > p( x ω ) p( ω ) for all j (6) j j where the p( x) has been removed as a common factor. For mathematcal convenence, defne: then (6) s reformed as: g ( x) = ln{ p( x ω ) p( ω )} = ln p( x ω ) + ln p( ω ) (7) x ω, f g ( x) > g ( x) for all j (8) j Ths s the decson rule used n maxmum lkelhood classfcaton. pxω ( ) are referred to as dscrmnate functons. Assume that the probablty dstrbutons for the classes are of multvarate normal dstrbuton, then the dscrmnant functon can be restated as: g x = p x m x m (9) T ( ) ln ( ω) ln ( ) ( ) where m and are the mean vector and covarance matrx of the data n class ω. They were estmated from tranng data. equaton(3)mples that whle the pxel at poston x was assgned to class ω, t was not 00%sure that pxel at x belongs to class ω. Thus, ths s an uncertan decson. The posteror probablty n the dscrmnate functon s an ndcator of the uncertanty n classfcaton decson. In equaton(4), the essence n classfcaton decson s to estmate the condtonal probablty at whch the pxel at x belongs to class ω, =,.... The posteror condtonal probablty pxel at x belongs to every class composed a posteror probablty vector: [ p( ω, p( ω,..., p( ω,..., p( ω ] T (0) 2 Remove the elements of zero value n the posteror probablty vector and rank the elements from large to small, a ranked posteror probablty vector can be formed (Sh, 998): 88 PV ( x) [ p( ω, p( ω,..., p( ω,..., p( ω ] T = () l l2 l lk where k s the number of non-zero elements n equaton (0). In equaton(), p( ω p( ω f < j Uncertanty measures derved from a posteror probablty vector Varous uncertanty measures are derved from posteror probablty vector. Sh (998) defned 4 measures based on the ranked posteror probablty vector. Absolute Uncertanty l lj

4 Absolute Uncertanty was defned as: U ( x) = p( ω /[ p( ω / x)] (2) A l l n equaton(2), U A ranges n a nterval (0, + ). The larger the U A,the more certan the class ω l was assgned to the pxel at x. Relatve Uncertanty Relatve uncertanty descrbes the msclassfcaton of pxel at x between class ω l and classω lj. The relatve uncertanty was defned as: U ( x,, j) = p( ω / x) p( ω / x) (3) R l lj U (,, ) R x j n equaton (3) ranges n a nterval of [0,].The larger the value of UR ( x,, j ),the easer the pxel at x to be dscrmnated between class ω and ω. xture degree of pxel xture degree of pxel measures the degree that a pxel s a mxed pxel. It s defned as: l lj n j= px ( ω ) px ( ω ) = for all j px ( ω ) j (4) Where px ( ω ) s the probablty that pxel at x belongs to classω. The smlar the elements n the posteror probablty vector, the smaller the value, and the more possblty that the pxel x s a mxed pxel. Evdence Incompleteness In the process of remotely sensed data classfcaton, some pxels are hard to determne ther classes and were labeled as unclassfed because the evdence for classfcaton s ncomplete. The evdence ncompleteness ( Θ ) was defned as: k p( ωl + Θ= (5) = Where the value of Θ range n the nterval of [0,]. Large value of evdence ncompleteness ndcates hgh possblty that the pxel was labeled as unclassfed. Relatve axmum Probablty Devaton(Clark Lab, 200) There s another uncertanty measure developed by Clark Lab (200) that we call the Relatve axmum Probablty Devaton. The Relatve axmum Probablty Devaton was defned as: U sum max = (6) Where max stand for the maxmum value n probablty vector. Sum stands for the sum of all elements n probablty. s the number of elements n probablty vector. Probablty Entropy In addton to the uncertanty measures defned by Sh (998), probablty entropy derved from posteror probablty entropy was wldly used as a uncertanty measures of remotely sensed data classfcaton ( asell, 994; Foody, 996 ). Probablty entropy s a measure of uncertanty and nformaton formulated n terms of probablty theory, whch expresses the relatve support assocated wth mutually exclusve alternatve classes. When two or more alternatve classes have non-zero probabltes 89

5 assocated wth them then each probablty s n conflct wth the others. When there s a fnte set of alternatve classes the expected value of conflct s gven by the probablty entropy (Foody, 996). Ths may be used to descrbe the varatons n class membershp probabltes assocated wth each pxel. The probablty entropy ndcates further requred nformaton content to assgn a pxel to a class wth 00% confdence. The probablty entropy was expressed as: = (7) H ( p ) p ( ω x )*log p ( ω x ) =,... 2 For the convenence of vsual presentaton, the relatve probablty entropy was developed by asell et al.(994)and was used n practce. The relatve probablty entropy was defned as: H ( p) H( p)/ H ( p)*00 R = (8) max 3. Selectng Uncertanty easures and Extendng Posteror Probablty Vector 3.. Selectng measures for uncertanty assessment at pxel scale To make the assessment on the uncertanty of remotely sensed data classfcaton at pxel scale be comprehensve and comparable, approprate uncertanty measures have to be selected. The absolute uncertanty defned by Sh (998) and the Relatve axmum Probablty Devaton present smlar uncertanty nformaton. In the defntons of these two measures, the element wth maxmum value n the probablty vector s emphaszed. These two measures are smple and straghtforward n descrbng the absolute uncertanty n assgnng a pxel to a specfc class. The frst drawback of these two measures s that the relatonshps between elements n probablty vector were not consdered. ost often, relatonshp between elements n probablty vector, especally the relatonshp between the maxmum one and the second maxmum one, provde much more uncertanty nformaton. Take a classfcaton wth 3 classes as an example, two posteror probablty vectors [0.5, 0.3, 0.2] and [0.5, 0.4, 0.] present qute dfferent uncertanty nformaton. Though the absolute uncertanty and the relatve maxmum probablty devaton of these two probablty vector are the same, the former mples less uncertanty then the later one. Another drawback of these two uncertanty measures s that values range n a unlmted nterval between (0, + ), whch make t hard to be presented by a gray scale mage. The relatve uncertanty compares the possble msclassfcaton of pxel between every two classes. In relatve uncertanty, relatonshp between elements n probablty vector s consdered. However, because the dfferences of posteror probablty between every two classes have to be calculated, the relatve uncertanty was presented by many posteror probablty dfferences together. Take a classfcaton wth 5 classes as an example, 0 relatve uncertanty values need to be calculated, and for a classfcaton wth 0 classes, 45 relatve probablty values have to be calculated! The mxture degree of pxel () descrbes the degree to whch the pxel beng classfed s a mxed classfcaton. Whle the dfferences between elements n probablty vector are large enough, the value s senstve to the mxture degree of pxel. However, f the maxmum of elements n two probablty vector are the same, s not senstve. Take the two probablty vector [0.6, 0.4, 0.0] and [0.6, 0.2, 0.2] agan as example, both values of these two probablty vector are.33. The evdence ncompleteness s subject to the posteror probablty threshold set n the LC classfcaton. In the LC classfcaton, a probablty threshold was set. If the maxmum value of the probablty vector of a pxel s less than the threshold, then there s no suffcent evdence to assgn ths pxel to any classes. Thus, dfferent posteror probablty threshold comes up wth dfferent mxture degree for same pxel. The probablty entropy was wldly used as an uncertanty measure. As all the elements n the probablty vector were accounted n the calculaton of probablty entropy, all the nformaton n the probablty vector was fully exploted. Because probablty entropy mples the varaton and dsperson of the elements n 90

6 probablty vector (asell, 994), both the nformaton expressed by relatve uncertanty and the mxture degree of pxel were contaned n the probablty entropy. An mportant mert of probablty entropy s that t expresses the uncertanty contaned n probablty vector usng a sngle value. Though the probablty entropy s not senstve to the class that pxel was fnally assgned(zhu, 997), because the pxels to be classfed are always assgned to the class wth maxmum probablty value n the probablty vector, thus the absolute uncertanty and the relatve uncertanty nformaton were both mpled n the probablty entropy. The probablty entropy has been accepted as a good measure of classfcaton uncertanty(zhu,997; Foody, 992; asell, 994). In summary, the absolute uncertanty and maxmum relatve probablty devaton both ndcate the absolute uncertanty nformaton contaned n probablty vector, but no relatve uncertanty nformaton mpled because the relatonshps between elements n probablty vector were not accounted n. Take the uncertanty measures developed by Sh (998) as a whole, they present both absolute and relatve uncertanty nformaton contaned n probablty vector, however need many measures smultaneously. Probablty entropy expresses the uncertanty nformaton contaned n the probablty vector usng a sngle value. Because the relatonshps between elements n probablty vector were accounted n probablty entropy, the relatve uncertanty nformaton contaned n probablty vector was mpled n probablty entropy. Thus, we use the relatve maxmum probablty devaton and the probablty entropy to present the absolute and relatve uncertanty contaned n probablty vector respectvely Extendng the a posteror probablty vector As was dscussed n secton 2, the posteror probablty vector could only be derved through LC classfcaton. In practce, many others classfers ncludng ANN classfer, Fuzzy classfer and varous dstance classfers were wdely used n remotely sensed data classfcaton. In these classfers, there are also measures that present the membershps pxel belongs to every class (Foody, 996; 2000), and can be taken as an approxmaton of posteror probablty that pxel belongs to gven class. Another type of classfer wldly used n remotely sensed data classfcaton s spectral dstance based classfers, ncludng nmum Dstance Classfer and ahalanobs Dstance Classfer (Rchards and Ja, 998). Lke n the LC classfcaton where the membershps of a pxel to every class were presented by the posteror probabltes of the pxel belong to every class, the membershps of a pxel to every class n dstance classfers were presented by the Eucldean dstances or ahalanobs dstances from the feature of pxel to the features of classes estmated from tranng data. Thus, the membershps can also compose a Dstance Vector [ d( ω, d( ω,..., d( ω,..., d( ω ] T (9) 2 In the classfcaton decson of dstance classfers, pxels were assgned to the class wth the mnmum dstance n the dstance vector. Lke the probablty vector, the dstance vector contans the uncertanty nformaton of the dstance-based classfcaton. However, because the value of elements n dstance vector ranges n a unlmted nterval, rather than n the nterval of [0,] n probablty vector, dstance vectors from dfferent classfcatons are not comparable, and the measures of uncertanty dscussed n secton 2 can not derved drectly from the dstance vector. Take three classes classfcaton as an example. Gven the means vector estmated from the tranng data s[30,60, 90], the spectral DN value of pxel A s 70, and pxel B 00. Based on the mnmum dstance classfcaton, the dstance vector of pxel A and pxel B are [40, 0, 20] and [70,50, 0] respectvely. Clearly t s unreasonable to draw a concluson that ths two dstance vectors present same absolute uncertanty just because they have same mnmum element. Thus, t s necessary to convert the elements n the dstance vector to the nterval of [0,] whle the relatve relatonshp between elements are preserved. The conversaton was made usng the followng equaton: 9

7 / d( ω pc( ω = / d( ω (20) = Applyng the equaton (20) to the example above, we can see that the conversed probablty vector for pxel A s [0.57, 0.43, 0.286], and pxel B the [0.534, 0.333, 0.333]. From these two probablty vector, t s reasonable to draw a concluson that the absolute uncertanty of pxel A s less than that of pxel B. By the converson, uncertanty measures from posteror probablty vector n LC can be extended to the assessment on uncertanty of classfcaton from neural network classfer and dstance classfers at pxel scale. Thus, we call the actvaton level vector and the conversed dstance vector together the Extended probablty vector. 4. Example The land cover mappng by multspectral remotely sensed data classfcaton was used here to llustrate the uncertanty assessment at pxel scale presented n ths paper. The standard false color composed magery of Landsat Thematc apper n Laner Lake, U.S. n Fgure. Land cover types n the magery nclude agrcultural feld, bare ground, grass land, confers, decduous, water, urban and the developed. Because there are some cloud and the shade of cloud n ths mage, the classes n classfcaton nclude the cloud and the shade the magery was classfed by LC classfer and numum Dstance classfer respectvely usng same tranng samples. The classfed land cover maps were shown n fgure. The posteror probablty vector was saved n the LC classfcaton. The dstance vector was saved n the mnmum dstance classfcaton and was conversed to the probablty vector usng equaton (28). The axmum Relatve Probablty Devaton and the Probablty Entropy for both LC classfcaton and nmum Dstance classfcaton were calculated and presented n Fgure 2. Fg.. The Landsat T magery and the land cover maps classfed usng axmum Lkelhood Classfer and nmum Dstance Classfer: (a) The Landsat T magery n Laner Lake, U.S.A (b) Land cover map classfed usng LC (c) Land cover map classfed usng nmum Dstance Classfer (d) Land cover legend 92

8 The absolute uncertanty of landcover map classfed usng LC was presented usng the maxmum relatve probablty devaton n Fgure 2 (a). The pxels wth uncertanty larger than 0.5 manly locate n the urban and the developed, bare ground, grass and agrculture feld and the boundary of water. These spatal dstrbuton of absolute uncertanty was caused by the spectral ambguty between urban and the bare ground, and between grass land and agrculture feld. Pxels wth uncertanty between [0.3, 0.5] locates n the confers and the decduous. Ths spatal dstrbuton was caused by the mxed pxels of the confers and the decduous. The absolute uncertanty decreases from the objects center to ther boundares. The relatve uncertanty of land cover map classfed usng LC was presented usng probablty entropy n Fgure 2 (b), n whch the pxels wth hgh probablty entropy locate n the boundary of the water, the clouds, the shades, the confers and the decduous. The uncertantes of these pxels were manly caused by pxel mxture. Fg. 2. Uncertanty of maps classfed usng LC and nmum Dstance Classfer: (a) The axmum Relatve Probablty Devaton of land cover map classfed usng LC; (b) Probablty entropy of land cover map classfed usng LC; (c) axmum relatve probablty devaton of land cover map classfed usng nmum Dstance classfer; (d) Probablty entropy of land cover map classfed usng mnmum dstance classfer The absolute uncertanty and relatve uncertanty of land cover map classfed usng the mnmum dstance classfer were presented n Fgure 2 (c) and fgure 2 (d) respectvely. The maxmum relatve probablty devaton and the probablty entropy were derved from the probablty vector converted from the dstance vector produced by the mnmum dstance classfer. Due to the characterstcs of the mnmum dstance classfer, the classes wth smlar spectral characterstcs, such as urban areas, clouds and bare ground, showed hgh uncertanty. The water showed less uncertanty as ts spectral characterstcs has dstnctve dfference wth other classes. Fgure 2 showed that the uncertanty of maps classfed usng dfferent classfers are qute dfferent even usng same tranng samples. 5. Conclusons The uncertanty assessment on the classfcaton of remotely sensed data s a crtcal problem n both academc arena and applcatons. The conventonal soluton to ths problem s based on the error matrx (.e. confuson matrx) and the kappa statstcs derved from the error matrx. However, no spatal dstrbuton nformaton of classfcaton uncertanty can be presented by ths method. A probablty vector-based method 93

9 has been developed for assessng classfcaton at pxel-scale. However, the use of ths method s severely lmted because the probablty vector can be derved only through the maxmum lkelhood classfcaton and because lack of the comprehensve and comparable uncertanty measures derved from the posteror probablty. The researches n ths paper focused on the solutons to overcome these two drawbacks. Frstly, the uncertanty measures from the probablty vector were evaluated. The maxmum relatve probablty devaton and the probablty entropy were selected to present the absolute uncertanty and relatve uncertanty respectvely. Secondly, the actvaton level vector produced n the feedforward neural network wth backpropagaton algorthm classfcaton and the dstance vector produced n the dstance classfers were converted to probablty vector so that all elements n t are n an nterval of [0,] and are comparable to the probablty vector produced n the L classfer. By ths converson, uncertanty measures for uncertanty assessment on remotely sensed data classfcaton usng LC at pxel scale can be used n the uncertanty assessment on classfcaton usng neural network classfer and dstance classfers at pxel scale. The examples of land cover mappng by classfyng Landsat T multspectral remotely sensed data showed that the maxmum relatve probablty devaton and the probablty entropy can present the spatal dstrbuton of absolute uncertanty and relatve uncertanty well. The spatal dstrbutons of uncertanty of maps classfed usng dfferent classfers are qute dfferent. 6. Acknowledgements Ths research were supported by the grants from Natonal Basc Research Program of Chna (Project No.2007CB ), State Key Lab. of Remote Sensng Scence, NSFC ( ),and the Program for Changjang Scholars and Innovatve Research Team n Unversty(PCSIRT, No.IRT0409) 7. References [] Clark Lab, Idrs Gude to GIS and mage processng, 200, 2. [2] R.G. Congalton and K. Green, Assessng the accuracy of remotely sensed data: Prncples and practces. Lews Publshers, 999: 37. [3] G.. Foody, N.A. Campbell, et al, Dervaton and applcatons of probablstc measures of class membershp from the maxmum lkelhood classfcaton. Photogrammetrc Engneerng and Remote Sensng, 992: [4] G.. Foody, Approaches for the producton and evaluaton of fuzzy land cover classfcatons from remotely sensed data. Internatonal Journal of Remote Sensng, 996: [5] G.. Foody, appng land cover from remotely sensed data wth a softened feed forward neural network classfcaton. Journal of Intellgent and Robotc Systems, 2000: [6] S. Gopal, and C. E. Woodcock, Accuracy Assessment of Thematc aps Usng Fuzzy Sets I: Theory and ethods. Photogrammetrc Engneerng and Remote Sensng, 994: [7] F. asell, C. Conese and L. Petkov, Use of probablty entropy for the estmaton and graphcal representaton of the accuracy of maxmum lkelhood classfcatons. ISPRS Journal of Photogrammetry and Remote Sensng, 994: [8] Rchards and Ja, Remote Sensng dgtal mage analyss: an ntroducton. Sprnger, 998. [9] Wenzhong Sh, Theory and ethods for Error Processng n Spatal Data. Scence Press, 998. [0] C.E. Woodcock, and S. Gopal, Fuzzy set theory and thematc maps: accuracy assessment and area estmaton. Internatonal Journal of Geographcal Informaton Scence, 2000: [] A.X. Zhu, easurng uncertanty n class assgnment for natural resource maps under fuzzy logc. Photogrammetrc engneerng and Remote Sensng997: