Combining fuzzy probability and fuzzy clustering for jviultispectral satellite imagery classification

Journal of Science and Technology 54 (3) (2016) 300-313 DOI 10 15625/0866-708»54/3/6463 COMBINING FUZZY PROBABILITY AND FUZZY CLUSTERING FOR JVIULTISPECTRAL SATELLITE IMAGERY CLASSIFICATION Dinh-Sinh iVIai', Le-Hung Trinh, Long Thanh N g o Le Quy Don Technical Universily. No 236 Hoang Quoc Viel Road. Bac Tu Liem . Hanoi 'Email; inaidhthslnliiii.small com Received: 23 June 2015; Accepted for publicaUon; 2 March 2016 ABSTRACT In practice, the classification algorithms and the initialization of the clusters and the initial cenUoid of clusten, have great tnfluence on the stability of the algorithtus, dealing time and classtllcation results. Some algorithms are used commonly in data classification, but their d,sadvantages are low accuracy and unstability such as it-Means algorithm, c-Means algorithm. Iso-data algorithm. Thts paper proposes a method of combining luzzy probability and fiizzy clustering algorithm to overcome these dtsadvantages. The method consists of two steps, first to calculate the number of cluster and the centroid of clusters based fuzzy probability, then to use fuzzy clustering aigoitthm to land-cover classificafion. The results showed that, the accuracy of the land cover classification using muitispecn-al satellite images according to the developed method significantly increases compared with vanous algorithms such as k-Means, Iso-data. Keyword- satelhte ituagery, probability, fuzzy c-means clustenng. I. INTRODUCTION The algorithms applied to image segmentation such as k-Means, c-Means, Iso-data show the same way based on the euchdean distance to detenmne the degree of similarity between the considered objects and cluster centroids. In problems of land cover classification, methods based on statistical parameters have been widely used because they are easy to implement and highly accurate [ 1 - 3 ] . However, these melhods are quite expensive, time consuming and unsuitable. Fuzzy logic has been widely applied in most of scientific and technical fields [4 - 7). lypically in ihe clustenng algonthms, it is fuzzy c-means algorithm (FCM) [8], which is quite common ,n many fields such as image processing, data mtning etc.. With FCM algorithm - a loop IS done ,0 minimize ihc objective funclion by updating the membership function values. Which have lunclion as the weight values that exhibit degree of influence of a data sample oB ciusttrs. However, this algorithm docs nol perf-orm well and is unstable when cenuoids initializing is lardiflercnl from lhe ical centroids. Carnal, .11,1/ iuzzy prooabitliy and fuzzy clustenng far multisaectral satellite Imagery classificatian disadvantTpes'oTprxri™"^ '"P™*"™"" based on FCM algorithm to overcome the th o her I X H / f ,^ '"" . """"""^ •=' "• " i "'"•' " ™ * " " " «"'=•• '=°-«ered pixel, degree of s ^ m t t n L h . " f ""= '"'''™^"'>" * » ' " Position on the mask to calculate the ™^e of th 7 = 1 7 h r ' T ' " = " " " " ' *= -'ghboring pixels, then cafibrate the L et a f n o , ! " " ?:i;'^ ^^^^^ """^ " ^ '"^°""™ f" '"^Se segmentatton problem. Zhao FCM lustenneas s^ en "^ information to improve FCM algonthm. and the authors had to use used to FCM !, L 7 J " r • * / " •" "" ' " ' "" ' '"f""-"'"" 'o eliminate noise and final 2henl;an™nfe°a n 11 , ^ ° ° " ' T . ' A ? ' " : '"""' °" ""= ^"'""^ "f *= membership fimction. combSon of soaia mf T . '' '^°"'™'"°' """' ™'"" classification based on such s o I a D o l v t a ' o n l r , ° ""^ "•""' ™'"e^-These methods have certam limitations r:S5-="=^^-:-^:^si—^^^^ .o.c i^^ :r^=dr-Se^n—;=—t.-^ ^ probabilt t^h^Carr intaUt 'ep" ^ r r " ? T " " f " ' ' '" "^'^^ " " ' - ' " ^ " ' ^ ^ ^ ^ satellite L g c . ^xpenm nt t f he me^h^ °™f ^ ^ ° " * " ' '° "'"^"'^'"« """'^~™' " a i g o n t h m s l i k ^ l s o - d ' a t a ^ ^ l ^ a f s l S s h r ^ L ^ a d ^ n - ' ^ t r o t r - ^ ^ ^ ^ ntethS^s:;wrraSS::ra,;s;;^!:FSzrr"Vr"'''^"'-=^ experimems; Sectton V is eonelus,on and fSmre works elass,ficat,on wuh some 2. BACKGROUND 2.1. Fuzzy Probability Le, us nottce that the probab.lity ofa fuzzy event As F,(R-) could be expressed also in another way as a fczzy set P,(.,)„„[o,,] [1], [3], Its membership mnct.on would be defined for any pe[0,l] by the following fonrtula /',(^)(/i) = H ' ' ^ ( ° - ' ] ' P = ' ' K ) 'f''^(0.1]lp = p(A„}>0 ^ ^ Otherwise ' ' ' It means, the fczzy probab.hty P , (^ , „ „„„„„y , , , ^ , „ ^ , ,^ ,^ ^ ^^^^^^_,. _^ ^ ^ ^ _ cuts of ^ , ^ ( 4 ) , a . (0,1] The following relatton between P , . , holds for any fnzzy event ABF,(.R-)- PiA)=]p(A„)d„. As the luzzy probability F , seems to be too complicated to be used m practice, the crisp probability P will be prefetred in this paper. Now, it will be shown how the fiizzy probability space can be applied to perfom, fltzzy discretization of continuous nsk factors ,n dec,s,on matang under risk. First, let us suppose that consequences of alternatives are affected by only one continuous risk factor Z whose probability distribution is g,ven by a dens,ty fimction/Z). Consider a fiizzy scale . f , , . A , . . . , 4 on the domain of the risk factor. As elements of the ftEzy scale are fiEzy random events, their probabilities P(A,),i=l....,n. are given by: P(A,)= \ A,(z)f(.z)cb ft is easy to check that ^P(A,) = l and P ( / ( , ) > 0 , , = ! , . . . , „ . So, S.pA. a discrete piobabtfity dtsmbution is defined on the given fttzzy scale. If tf,e dens,ty fimction ot the nsk factor Z is not known, a stmilai probability distnbution on the given fiizzy scale can lie derived directly from measured data If measurements z , ,Z j , . . ,z„ of Z are given, m » / l , t h e i , probab,l,ties of the fitzzy scale elements can be set by the formula: m " The fuzzy expected value and the fuzzy standard devtation of such a fiizzy tandom variable Z that takes on values 4 of the given fuzzy scale with probabilities P{A,),i =},...,n [2], are defined by the followtng fomiulas; FEZ = f^P(A,)A 0) FaZ= YiPi-^M-FEZf W 2.2. Fuzzy c-means clustering In general, fuzzy memberships in FCM [8] achieved by computmg the relative distance among the pattems and cluster centroids. Hence, to define the primary membership for a pattern, we defme the membership usmg value of m. The use of ftizzifier gives different objective function as follows, N C (5) in which d^^ = |j.v, - v^ | is Euclidean distance between the pattem x^ and the centroid v,, C is number of clusters and A* is number of patterns. Degree of membership w,^ is determined as in which / - l , , , - , C ; k = ]...., N . Cluster centroids is computed as follows: Z — -ju^y piaoaoiiiiy and fuzzy clustering far multlspeclral satellite Imagery classlficallon in wh,eh / = 1,..., C Next, defitzzification for FCM ,s made as if „,(.v,) > „, (.vJ fory - l,...,C and / / y then x, is assigned to cluster / 3 COMBINmC PROBABILITY THEORY AND FUZZY CLUSTERING SATELLITE IMAGE CLASSIFICATION partttiSnst^stiirbTsid'trt^^r^irrir"'' T-- T *= •>""="• -^ -^ ^ distance in the color space rf„ between the pattem x, and the cenuoid ,• : : : : : ; : : z r ' " - '°-^-" ^ ^^-^ - --- -—pa..em%s''bT.h°: and standard deviation FaZ . with/-l,2,...,rf;X = (;c„;r„..,.v) XER' in which r = I if z > 0 othenvise T - 0. Find pattern.,-,, with fl =max„^,, fl, then V, = F, u.v, and X = X>,x . IfX- given a set of candidate points F,, else back lo finding D_ candidate set uK =K then annlv th,-. .,) u , , a,en apply thts algonthm to the set V The centroid malnx V can be initia zed by choosing tite pattems in V, according to the density- of candidales The detailed algonthm consists of the following fout mam steps: Dinh-Sinh Mai, Le-nung innn, uong inann nw Algorithm 1: Find centroids using fuzzy probability Step 1: Initialization l.I Number of cluster C . ( O I ) - 1 2 Compute the FEZ, by the formula (8). 1.3 Compute Ihe FaZ, by the formula (9). Step 2: Finding candidate 2.1. Compute density D, by the formula (10). 2 2. Find pattem -v, with Z), = max,g^g^ ^j 'hen V^. = F,. u x, and X = X\x, Step 3: Check the stop condition: If ^"=11 o r / > C, goto Steps else back to Step 2. Step 4: Given a set of candidate points V^. Overall diagram of finding centroids using fuzzy probability is shown in Figure 1. Algorithm 2: Probability Fuzzy C-means Clustering (PFCM) Step 1: Initialization 1.1 The parameter of fiizzy m, (I <m), error e. 1 2 Initialization centroid V = [v^ ], v, E R" by algorithm 1. Step 2: Compute the flizzy partition matrix U and update centroid V: 2.1 Fuzzy partition matrix U,^ by the formula (6). 2.2. Update the cluster centroid V^ by the formula (7). Step 3: Check the stop condition: If true, go to step 4, otherwise go to step 2. Step 4: Given the clustenng results Figure I Diagram of finding centroids using fiizzy probability. IN^ g g g S i g a g fuzzy prababillty and furry clustenng far miilllspectral satellite Imagery classificatian 4. LAND-COVER CLASSIFICATION USING PFCM .0 . e s M h ' e ^ T o r s T a i l n ^ ' V ' r / f f ? f P™" '™ "f elassification on satellite .magety f r o m m u l t i - s Z r a l s a t d l T e t ? " '^" ' '"e ' 'a lgori thm of PFCM for land cover classiftcation spectral satelhte images consists of the following three main steps: Algorithm 3 : The PFCM algoritim. Step I : Multi-spectial satellite tmagety preprocessing f e X t t i ^ I ™ :i!^;^^,^ " f - ^ - ^^ese n-bauds will be classified mto six classes '• ^ ^ Class 1: Rivers, ponds, lakes. ^- ^ ^ Ciass2- Rocks, bare soil. ^- ' ^ ^ C l a s s S : Fields, grass "^^ ^ ^ Class4: Planted forests, low woods ^' ^ ^ Class5: Perennial tree crops. ^- ^ ^ B C I a s s e : Jungles. Step 3: Compute percentage of the identical region- I ' L s ^ f ' Z l l : : : : ' ™ ' '^ ^ * = - ^ ^ ° ^ " ' - - - '- ^^^on, « be the 2 MuUispectral satellite imagery Figure 2 Overall diagram of classification problem s a t e l i f t e X f a f e r a d l n m X ^ r A l g ' o t i m , : , ; | ' r ' ^ ^ ' = ^ ' " / - - ^' '^e multispcctral corresponding to 6 layers of data these «mto ,ds n ^ y ,'° " " " " aPP'oximate centroids 2. the algorithm 2 will conducj r s s r . e p ' e K „ r " / T " ' ' ^ ' " ' ' " ' ' ' ' ' ' ' ' ' ° ' ' • " I g o n t h m Difference Vegetation Index (NDVl) (241 l o d e M u m c s i ^^"' *'""' ° " Normalized covers n -(l m aetennine six classes representing six types of land Dinh-Sinh Mai, Le-Hung Trinh, Long Thanh Ngo 4.1. Experiments I The studv dataset from Landsat-7™ imagery is region center of Hanoi, Vietnam (2l" 0-15.304-R 105" 29-28 173"E to 20" 52'34.40I"N, 106" 0 9 ' 5 7 . 3 i r E ) m Figure 3, its area: (d) (e) 0 Figure 3. Study data of Hanoi: a) Band 1; b) Band 2; c) Band 3, d) Band 4; e) Band 5; f) E Table I Resultsof land cover classification inHanoi. Class 1 2 3 4 5 6 PFCM (%) 4.5263 13.3306 22.8785 26.1571 214329 11 6747 FCM (%) 4.8804 14.3340 22.0145 25.6635 20.3387 12.7688 Iso-data (%) 5.5866 15.7601 19.3886 24.3211 20.2183 14.7253 K-means (%) 9.9213 16.9050 16.7701 21.2313 19.0090 16.1633 M 1 ib) fc) 1 1 praaabllity and fuzzy clusterina for multlspectral satellite Imagery classification Percentage Chart o«>i a»,2 a,s,3 a.ss4 a,„5 a«,6 mnan .FCM i s o j , , , , « „ „ Figure 4. The result of algonthms- PFCM, FCM, Iso-dala and K-Mea„s Figiiiv.l Result of land cover classiti, icanon a) K-Mca„s. b| Iso-da,.,. c) FC M. dl PFCM Dinh-Sinh Mai, ^^. ,„ .„ , resi,l,?„VpFr»7cA5«T '"I'^'" ^ ' " ' '^'''""'' '" "'^'='' '"'• *'• ('^''"'' W) ^e classificadon results ot PFCM, FCM, Iso-dati, and k-Means algorithms, respectively. Figure 4 and Table I compare classification results behveen PFCM, FCM, Iso-data and k-Means. There is a significant difference between Uie algorithms of PFCM, FCM, k-Means and Iso-data in classify,ng based on est,matmg the area of regions. This result showed that the area of the layere ,s d,fferent, the btggest difference between k-Means and PFCM. To assessmg the performance of the algoriflmis on the expenmental images we analyzed the results on the basis of several validity indexes. We considered flie different validity indexes such as the Bezdeks pattition coefficient (PC-I) [22], Classtfication Enttopy index (CE-I) [23,24] and Kappa mdex. The values of these validity indexes are shown in the Table 2. Table 2 The vanous vahdity indexes on the LANDSAT-7 images of Hanoi area. Validity Index CE-1 PC-I Kappa K-means 0.9869 0.6982 0.4182 Iso-data 0.5872 0.7282 0.4882 FCM 0.1972 0.8628 0.7628 PFCM 0.1317 0.8893 0.9156 Note that the validity indexes are proposed to evaluate the quality of clustering. The better algorithms have smaller values of CE-I and larger value of PC-I, Kappa. The results in Table 2 show that the PFCM have better quality clustering than the other typical algorithm such as FCM, K.-means and Iso-data. 4.2. Experiments 2 The authors using Landsat-7 satellite image data, which taken Lamdong area on 12/02/2010, 12" 13'0I.88"N, 107° 33'27.5n"E to 11° 37'40.927"N, 108" 49'49.252"E and square of area: 3393.7 hectares, see in Figure 6. The results are shown in Figure 7 in which (a), (b), (c) and (d) are the classification results i of PFCM, FCM, Iso-data and K-means algorithms, respectively. Figiu-e 8 and Table 3 compare d classification results between PFCM, FCM, Iso-data and k-Means. There is a significan^fl difference between the algorithms of PFCM, FCM, k-Means and Iso-data in classifying based a estimating the area of regions. In Figure 7, the results show that PFCM algorithm noi reduction quite good, while K-means algorithm is much the most noise. Table 4 show that tl PFCM have better quality clustering than the other typical algorithm such as FCM, K-owans a Iso-data. In summary, from two test areas, these deviations can be explamed tbat ibe boundary of l water and soil classes are usually quite clear, while the vegetation classra are often 6onfiised | between grasses and trees. With satelhte imagery resoluhott 30na30m, die differences ofi classification results can be acceptable m assessment of land co|^fflija Bfige area, reducing | costs compared to other methods. This result not only co^g^^^i^^^lfciHrt tte l a ^ covff f fluctuations but also supports urban planning, natural-lesofflm^MBwBKiltt^tgd so on. Combining fuzzy probabi/ity and fuzzy clustenng far multlspectral satellite imagery classificatian figure 7 Rcsull orcliistering a| K-means b) Iso-diila. cl FCM. dl PFCM. Dinh-Sinh Mai, Le-Hung Trinh, Long Thanh Nao^ Table 3. Results of land cover classification in Lamdong (%). Class 1 2 3 4 5 6 PFCM (%) 8.3890 20.0359 19.8786 15.7184 19.0240 16.9540 FCM (%) 9.2619 19.4947 18.5279 15.0599 20.1407 17.5149 Iso-data (%) 12.3099 17.8593 15.9935 13.9328 21.2678 18.6368 K-meam (%) 17.1510 15.5828 13.4288 11.4685 21.3346 21.0342 Percentase Chart crass 1 Cbss 2 Class 3 Class 4 Oass 5 Class 6 • PFCM • FCM • Iso-data • k-Meani Figure 8. The result of algorithms: PFCM, FCM, Iso-data and K-Means. Table 4. The various validity indexes on the LANDSAT-7 images of Lamdong area Validity Index CE-I PC-I Kappa K-meam 0 9629 J 0S8id| Iso-data ;_ a6581 O.^^I|^ffiM^ reu .&pn ifJS»^'' PFCM 0.1429 0JS91 0.8S99 This paper presents a for ftizzy clustermf ilfo] the proposed algordiin classification Basi^ d classification were doni satellite images, detection. itnh^ &eoiy as the initial st^ im9ge T^he results showed that class of land cover Lts of land cover to other types of ;> Iffiid cover change Combining fuzzy prababltitv and fuzzy clustering for multlspectral satellite Imagery classification The next goal ,s to implement further research on die Landsat-8 satellite images, hyper- spectral satelhte tmagery for enwronmental classtfication. assessment of land surface temperature changes, speed-up the proposed methods based on CPUs platfotms 9. REFERENCES Lotfi Zadeh - Fuzzy Set Theoiy and Probability Tlieoty: Wltat ,s the Relationship? Intemattonal Encyclopedia of Stattstical Science (2011) 563-566. Jana Talasova and Ondrej Pavlaeka - Fuzzy Probability Spaces and Their Applicattons ,n lJec,s,on Mal<,ng, Austnan Joumal of Stattstics 35 (2006) 347-356 Denis de Brucq, Olivier Colot, Amand Sombo - Identical Foundatton of Probability heoty and Fuzzy Set Theoty, Proceedings of the Fifth Intemational Conference on Information Fusion II, USA, 2002, pp 1442-1449. Naoko lino, Kisei Kinoshita and Chikara Kanagaki - Satellite Images of an pollutants and ofThe P h T '^""™°™"»' '•'"'""on »"'i disaster prevention. IntetBational Archives ot he Photogrammetry, Remote Senstng and Spatial Informatton Scence XXXVI Patt 6 Tokyo Japan, 2006. ' Cihlar J. - Land cover n,app,ng of large areas ftom satellites: status and research priorities Inl J. remote sensing 21 (6 & 7) (2000) 1093-1114. pnonties, Stavrakoudis. D. O., Galidaki, G N., Gitas, I. Z., and Theocharis, J. B. - Enhancing the In^tp^tabil,^ of Genenc Fuzzy Classifiers in Land Cover Classificat on l l 2OTO,7P |277-r284' " ' " ^ """^ '*' '"" ^ ° " ^ " ' °" C°-Pu.at.onal Intelligence, Alt Asghar Torahi, Suresh Chand Ra, - Land Cover Class,fication and Forest Change Analysts. Us,ng Satelhte Imagety - A Case Smdy in Dehdez Area of Zagros Mountata fn Iran. Joumal ofGeographic Information System 3 (2011) l-ll B""!-"ountam in James C. Bezdek, Robett Ehrlich, Wilham Full - FCM- Tire Fuzzy C-Meams clustering algorithm, Computers and Geosciences 10 (2) (1984) 191-203. clustenng , ° n T , T V " °°°'""" " • ^''™«"1"^» E., Philips W. - An improved flizzy clustenng ProTs:^g,'2'oioT2r9T52"'"°"- "'' '^'' ' "»™' -™' ' » " ' - - ™ ' ™ ^ ' 10. Zhao Lt, X,aom,ng Zhou - Class,ficafions Mod,fication Based FCM wnh Spatial TroCiSn.'pp^sir"'""'"' '"'=™'--' °^"f«e"«™„ru„rd?: ' '• on'cTor ta'ce^'l 'd^Sn",' T ^ f ^"""^ " ' ™ """''' ^=6-=- . . ° " Algonthm Based c lmtcaurEn^„ 'eenS '2"(20"B)T8^ ' " ™ ' ° " " ' ""^^' "' ^ ^ P " - - " 12. Hamed Shamsi and Had, Seyedarab, - A Mod,fied Fuzzy C-Mea„s Clus.enng w„h Soattal ^S^ZM^::SZ^TT-- '"'-'-''"-' orcompu.er%;:^ -rj Dinh-Sinh Mai, Le-Hung Trinh. Long Thanh Ngo 13. Long Thanh Ngo and Dinh Dung Nguyen - Land cover classification using interval type-2 fuzzy clusienne for multi-spectral satellite imagery, IEEE Conference on Systems, Man. and Cybernetics. 2012. pp 2371-2376. 14. Sinh Dmh Mai. Long Thanh Ngo - Interval Type-2 Fuzzy C-Means Clustering wilh Spatial Information for Land-Cover Classification, The 7th Asian Conference on Intelligent Information and Database Systems, part 1, Spnnger LNAI 9011, 2015, pp.387- 397- 15. Trinh Le Hung. Mai Dmh Sinh - Detection and classification of oil spills in envisat asai imagery using adaptive filter and ftizzy logic. Journal of Petroviemam 5 (2014) 49-55. 16- Rauf K. S.. Valentin V. G , Leonid P. P. - Fuzzy clustering methods in Multi-spectral Satellite Image Segmentation. International Joumal of Computing 8 (2009) 87-94. 17 Han J. C . Chi K. n. . and Yeon Y- K. - Land Cover Classification of IKONOS Multlspectral Satellite Data' Neuro-ftizzy, Neural Network and Maximum Likelihood Methods. Lecture Notes m Computer Science, (3642), 2005, pp 251-262 18. Gordo O.. Martinez E , Gonzalo C , Arquero A - Classification of Satellite Images by means of Fuzzy Rules generated by a Genetic Algorithm. Latin America Transactions Revista IEEE America Latina 9 (!) (2013) 743-748 19. Genitha C. H and Vam K. - Classification of satellite images using new Fuzzy cluster centroid for unsupen'ised classification algorithm, IEEE Conference on Information and Communication Technologies. 2013, pp. 203-207, 20- Shackelford A K. and Davis C. H - A ftizzy classification approach for high-resolution multlspectral data over urban areas, IEEE International Geoscience and Remote Sensing Symposium 3 (2002) 1621-1623. 21 Enc K Forkuo. Adubofour Frimpong - Analysis of Forest Cover Change Detection, International Journal of Remote Sensing Applications 2 (4) (2012) 82-92. 22 Wang W. and Zhang Y. - On fuzzy cluster validity indices, Fuzzy Sets and Systems 158 (2007)2095-2117- 23 Bezdek J . Pal N- - Some new indexes of cluster validity IEEE Transactions on Systems, Man and Cybernetics 3 (1998) 301-315 24. Ray D- Jackson and Alfredo R. Huete - Interpreting vegetation indeces Preventive veterinary Medicine, Elsevier 11(1991) 185-200. TOM TAT KET HOP XAC SUAT MCi VA PHAN CUM MC) PHAN LOAI ANH VE TINH DA PHO Mai Dinh Smh', Tnnh Le Hiingva Ngo Thanh Long Hoc vien Ky ihmii Quan su, 236 Hoang Qi,6c Viet, Bdc Tu Uem. Ha Noi. Viet Nam Email iJMidinhMnhfw.-mail com Canibii-iing luzzy prababililv and fuzzy clustering for multlspectral satellite imagery classification cum bl^iul -'^ - "tv^ ""'; * " - '°*° P " " '°""' "^ "^ ">« " ° >« ' " W cum va trong tam eac oaT M6t sT,h^^, r '*" f" * ' °" "'•'* " " * " ' ' '•""• **> g ' - ^* » - ^ t qua phan hing I?d° h i r ,h™' " ' . i " "^ •"" "'=" ""•"' P'^'" '"^i "«> "«". " i - e nhuoc diem cua sTnhuac d e^ ^ n ^ T h f ™ ' h" ""^ f ' " " " * ™ *"'< ' " ^ . P " " cum ma ii khic phuc mot cac cum d ™ ? - ' ^ - '^'"',""8 P^P "ay bao g6m 2 buoc. thii nbS, tiri, toan s6 cum va ttong tam u T c t Z ' l T T V : " t ' - " . ^ ""•« ' ' " '" P"*" '^""' •"" « PMn loai lop phtl' Cac p h u r „ h 4 p d l x u K ' ^ ^ 1 1 ' " " '" ' ' ° '"'• "^P-P"" =* " W l l * ve tinh da ph6 theo pnuong phap de xuat tang dang ke khi so sanh vd, mot so thuat loan ph6 nhu k-Means, Iso-data. 7i> Ichoa: imh ve ttnh, xac suiit, phan cum mo c-Mcans.