FUZZY C-MEANS ALGORITHMS IN REMOTE SENSING

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1 FUZZY C-MEAS ALGORITHMS I REMOTE SESIG Andrej Turčan, Eva Ocelíková, Ladslav Madarász Dept. of Cybernetcs and Artfcal Intellgence Faculty of Electrcal Engneerng and Inforatcs Techncal Unversty of Košce Slovaka Abstract: Fuzzy clusterng s a wdely appled ethod for obtanng fuzzy odels fro data. It has been appled successfully n varous felds ncludng geographcal surveyng, fnance or arketng. A bref overvew on Fuzzy C- Means based algorths and detaled vews on Fuzzy C-Means (FCM) and ts proveent by Gustafson-Kessel (GK) are shown below. Experents on artfcal ade-up data and data fro reote sensng gathered fro probe LADSAT TM7 are ade usng FCM and GK. Keywords: fuzzy clusterng, fuzzy c-eans, reote sensng 1 Introducton 1.1 Clusterng Clusterng s a dvson of data nto groups of slar objects. Each group, called cluster, conssts of objects that are slar between theselves and dsslar to objects of other groups. Representng data by fewer clusters necessarly loses certan fne detals, but acheves splfcaton. It represents any data objects by few clusters, and hence, t odels data by ts clusters. Data odellng puts clusterng n a hstorcal perspectve rooted n atheatcs, statstcs, and nuercal analyss. Fro a achne learnng perspectve clusters correspond to hdden patterns, the search for clusters s unsupervsed learnng, and the resultng syste represents a data concept. Therefore, clusterng s unsupervsed learnng of a hdden data concept. There s a close relatonshp between clusterng technques and any other dscplnes. Clusterng has always been used n statstcs and scence. Typcal

2 applcatons nclude speech and character recognton. Machne learnng clusterng algorths were appled to age segentaton and coputer vson. Clusterng can be vewed as a densty estaton proble. Ths s the subject of tradtonal ultvarate statstcal estaton. Clusterng s also wdely used for data copresson n age processng, whch s also known as vector quantzaton. Clusterng algorths, n general, are dvded nto two categores: Herarchcal Methods (aggloeratve algorths, dvsve algorths) Parttonng Methods (probablstc clusterng, k-edods ethods, k-eans ethods ) Herarchcal clusterng bulds a cluster herarchy. Every cluster node contans chld clusters; sblng clusters partton the ponts covered by ther coon parent. Such an approach allows explorng data on dfferent levels of granularty. Herarchcal clusterng ethods are categorzed nto aggloeratve (botto-up) and dvsve (top-down). An aggloeratve clusterng starts wth one-pont (sngleton) clusters and recursvely erges two or ore ost approprate clusters. A dvsve clusterng starts wth one cluster of all data ponts and recursvely splts the ost approprate cluster. The process contnues untl a stoppng crteron (frequently, the requested nuber k of clusters) s acheved. Data parttonng algorths dvde data nto several subsets. Because checkng all possble subset possbltes ay be coputatonally very consuptve, certan heurstcs are used n the for of teratve optzaton. Unle herarchcal ethods, n whch clusters are not revsted after beng constructed, relocaton algorths gradually prove clusters. 1.2 Reote Earth s survey Satellte reote sensng s an evolvng technology wth the potental for contrbutng to studes of the huan densons of global envronental change by akng globally coprehensve evaluatons of any huan actons possble. Satellte age data enable drect observaton of the land surface at repettve ntervals and therefore allow appng of the extent, and ontorng of the changes n land cover. Evaluaton of the statc attrbutes of land cover and the dynac attrbutes on satellte age data ay allow the types of change to be regonalzed and the proxate sources of change to be dentfed or nferred. Ths nforaton, cobned wth results of case studes or surveys, can provde helpful nput to nfored evaluatons of nteractons aong the varous drvng forces. Fro a general perspectve, reote sensng s the scence of acqurng and analyzng nforaton about objects or phenoena fro a dstance. As huans, we are ntately falar wth reote sensng n that we rely on vsual percepton to provde us wth uch of the nforaton about our surroundngs. As sensors, however, our eyes are greatly lted by senstvty to only the vsble range of

3 electroagnetc energy, vewng perspectves dctated by the locaton of our bodes, and the nablty to for a lastng record of what we vew. Because of these ltatons, huans have contnuously sought to develop the technologcal eans to ncrease our ablty to see and record the physcal propertes of our envronent. 2 Fuzzy Clusterng Algorths In classcal cluster analyss each datu ust be assgned to exactly one cluster. Fuzzy cluster analyss relaxes ths requreent by allowng gradual ebershps, thus offerng the opportunty to deal wth data that belong to ore than one cluster at the sae te. Most fuzzy clusterng algorths are objectve functon based. They deterne an optal classfcaton by nzng an objectve functon. In objectve functon based clusterng usually each cluster s represented by a cluster prototype. Ths prototype conssts of a cluster centre and aybe soe addtonal nforaton about the sze and the shape of the cluster. The sze and shape paraeters deterne the extenson of the cluster n dfferent drectons of the underlyng doan. The degrees of ebershp to whch a gven data pont belongs to the dfferent clusters are coputed fro the dstances of the data pont to the cluster centres wth regard to the sze and the shape of the cluster as stated by the addtonal prototype nforaton. The closer a data pont les to the centre of a cluster, the hgher s ts degree of ebershp to ths cluster. Hence the proble to dvde a dataset nto c clusters can be stated as the task to nze the dstances of the data ponts to the cluster centres, snce, of course, we want to axze the degrees of ebershp. Most analytcal fuzzy clusterng algorths are based on optzaton of the basc c-eans objectve functon, or soe odfcaton of t. 2.1 Fuzzy C-Means The Fuzzy C-eans (FCM) algorth proposed by Bezdek [1] as to fnd fuzzy parttonng of a gven tranng set, by nzng of the basc c-eans objectve functonal: J ( Z; U, V) c = = 1k= 1 ( μ ) z k v 2 A where: U = [ μ ] M fc s a fuzzy partton atrx of Z n [ v, v2,, v ] v R V =, 1 K c s a vector of cluster prototypes, to be deterned

4 k v 2 z s dsslarty easure between the saple z and the center v of the specfc cluster of the specfc cluster (Eucldean dstance) ( 1, ) s a paraeter, that deternes the fuzzness ot the resultng clusters k The nzaton of J ( Z; U, V), under the constrant μ = 1, leads to the teraton of the followng steps: and u 1 c = ( D / D jk ) j = 1 μ = 0 2 ( ( μ ) zk ( l) v =, 1 c ( μ ) 1) c = 1 f D > 0 and μ < 0, 1 >, c = 1 μ = 1 The teraton stops when the dfference between the fuzzy partton atrces n two followng teratons s lower than ε. 2.2 Gustafson-Kessel Algorth Gustafson and Kessel extended the standard fuzzy c-eans algorth by eployng an adaptve dstance nor, n order to detect clusters of dfferent geoetrcal shapes n one data set. Each cluster has ts own nor-nducng atrx A. Here we have to eploy the fuzzy covarance atrx F of the -th cluster: F = ( μ ) ( zk v )( zk v ) ( μ ) Algorth s agan based on teraton of the next steps coputng of the cluster covarance atrces: T

5 T ( μ ) ( zk v )( zk v ) F = 1 c ( μ ) coputng of the dstances D 2 A = ( z k v updatng of the partton atrx u ) T 1 n 1 [ det( F ) F ]( z k v ), 1 c, 1 k 1 c 2 ( 1) = ( D / D jk ) j = 1 μ = 0 f D > 0 and μ < 0, 1 >, c = 1 μ = Dfferences between FCM an GK Frst experents were ade on the self-ade data. The reason was to exane dfferences n approach of both algorths and to see the dfferences n the shape of the clusters. Fgure 1: self-ade data set

6 ext fgures show how the was the data set clustered usng the FCM and GK. It s posble to see that clusters after the FCM clusterng have sphercal shape, whle clusters after the GK clusterng adopted the shape of partcular subset of ponts. Fgure 2: Outcoe of the FCM Fgure 3: Outcoe of the GK algorth

7 3 Experents wth real-world data The data set conssts of ult-spectral Landsat ages (7 densonal data). The selected geographcal area s located n the north part of the cty Kosce, Slovaka. The goal was to dvde age nto 7 partcular types of land: A) urban area, B) rural area, C) barren land, D) agrcultural land, E) nes, F) forest and G) water. Fgure 4: Orgnal age Kosce The data set conssts of saples, where one saple represents area of sze 30 x 30 eters and t represents area of approxately 332 k 2. Experents were ade usng both fuzzy c-eans and Gustafson-Kessel algorths. Fuzznes paraeter was chosen =2. As a coputatonal tool was chosen Matlab nstaled on PC wth 650 MHz processor and 320 MB RAM.

8 Fgure 5: Segentaton obtaned usng FCM Fgure 6: Segentaton obtaned usnggk LEGED: urban area rural area barren land agrcultural area nes forest water

9 4 Results and concluson The results obtaned by classfcaton wth fuzzy c-eans and Gustafson-Kessel algorth are shown n Fg. 5 and Fg 6. As shown, the results generated by the Gustafson-Kessel algorth outperfor those generated wth fuzzy c-eans. The fuzzy clusterng ethods allow classfcaton of the data, where no a pror nforaton s or content s not known. In partcular, the fuzzy ethods allow to dentfy data n ore flexble anner, asgnng to each datu degree of ebershp to all classes. Experents show, that the areas labeled as nes, are probleatc to classfy. Reason for ths s probably n sze of the age area whch s covered by ths class. Or n other words area covered by nes s relatvely sall accordng to the area covered by one pxel. Other possblty for ths, what s also a bg dsadvantage of c-eans based algorths, that they tend to stuck n a local extrees. On the other hand these algorths offer a good tradeoff between accuracy and speed. References [1] Bezdek, J.C.: Pattern Recognton wth Fuzzy Objectve Functon., Plenu Press, ew York, [2] Jan, A.K, Murty, M., Flynn, P.J.: Data Clusterng: A Revew [3] Bonner, R.E.: On Soe Clusterng Technques. IBM, [4] Ball, G.H., Hall, D.J.: ISODATA, A ovel Method of Data Analyss and Pattern Clasfcaton. Standford Res. Insttute, Menlo Park, [5] Fro, F.R., orthouse, R.A.: CLASS, A onparaetrc Clusterng Algorth. Pattern Recognton. [6] Raja, A.,Mester, A., Martverk., P.: Fuzzy Classfcaton Algorths wth Soe Applcatons [7] Barald, A., Blonda. P.: A Survey of Fuzzy Clusterng Algorths for Pattern Recognton. ICSI, TR , 1998 [8] Sddheswar, R., Tur, R.H.: Deternaton of uber Clusters n K-Means Clusterng and Applcaton n Colour Iage Segentaton. [11] Setnes, M., Kayak, U.: Fuzzy Modelng of Clent Preference n Data- Rch Marketng Envroents.

10 [12] Setnes, M., Kayak, U.: Extended Fuzzy Clusterng Algorts. [13] McBratney, A.B., De Grujter, J.J.: A Contnuu approach to sol classfcaton by odfed fuzzy k-eans wth extragrades. [14] Yeung, K.Y., Ruzzo, W.L.: An Eprcal Study on Prncpal Coponent Analyss for Clusterng Gene Expresson Data. [15] Faber, V.: Clusterng and the Contnuous K-Means Algorth. [16] Halkd, M., Batstaks, Y., Vazrganns, M.: On Clusterng Valdaton Technques [16] Babuška, R.: Fuzzy and eural Control.