A Novel Fuzzy Clustering Method Based on Chaos Small-World Algorithm for Image Edge Detection

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1 A Nove Fuzzy Custerng Method Based on Chaos Sa-Word Agorth for Iage Edge Deteton Mngxn Yuan Sun an Wang Naan Chen Shoo of Mehana Engneerng X an Jaotong Unversty X an Shaanx Provne P.R. Chna xyuan78@ stu.xtu.edu.n Abstrat To sove the fuzzy edge deteton probes n age proessng a nove fuzzy usterng ethod based on haos saword agorth (CSWFCM) s presented. The tradtona fuzzy usterng ethod (FCM) s good at oa searhng apabty but t s senstve to the nta vaue and easy to trap nto oa nu vaue. The sa-word agorth (SWA) nspred by the ehans of sa-word phenoenon s a nove goba searhng agorth whh enabes to enhane the dversty of the popuaton and avod trappng nto oa nu vaue. However the further apabty of sovng opated probes s ted for ts ow effeny of oa short-range searhng operator. In ths paper the haos dsturbane s utzed to prove the searhng effeny of SWA after oa short-range searh and the haos sa-word agorth (CSWA) s used to optze the FCM n age edge deteton. The suaton resuts show that the proposed agorth an orrety detet the fuzzy and exguous edges wth hgher onvergene speed. Keywords fuzzy usterng agorth haos optzaton sa-word agorth edge deteton I. INTRODUCTION Edge deteton s extensvey used to separate the obet fro the bakground n age proessng. A ot of researh has been done n the fed of age segentaton usng edge deteton. Soe of the earest operators detetng edges n an age were proposed by Sobe Prewtt Roberts et. [1]. These operators used oa gradent ethods to detet edges aong a spefed dreton. The oon prese ondtons of these operators are that the edges of an age ust be ear. However the age n reaty s fuzzy and the edges aren t ear. In 1966 Bean and Zadeh frsty put forward fuzzy usterng anayss [2] and the dea of fuzzy usterng quky beaes an effent ethod to sove fuzzy edge deteton n age proessng. Aong nuera fuzzy usterng ethods fuzzy -eans usterng ethod provdes foundaton for other fuzzy usterng ethods fro the theores and appatons and s used wdey. In essene fuzzy -eans usterng ethod s a knd of oa searhng agorth. It s so senstve to the nta vaue that t easy traps nto oa nu vaue. To sove the dsadvantage soe agorths suh as genet agorth (GA) [3] and une evoutonary agorth (IEA) [4] have been used to optze the fuzzy usterng whh resut n good effets to soe extent. However soe dsadvantages of GA and IEA suh as sow onvergene speed and nstabty affet the auray of usterng. In ths paper a nove fuzzy usterng ethod based on the optzaton of haos sa-word agorth s proposed and used n age edge deteton. In the proposed agorth frst the nta popuaton s generated by ogst appng and the haos dsturbane s used to prove the searhng effeny of SWA after oa short-range searh; seondy the FCM s optzed by the haos sa-word; fnay the CSWFCM s used n age edge deteton. The paper s organzed as foows: The fuzzy usterng ethod s presented n Seton II. The sa-word phenoenon and sa-word optzaton agorth are desrbed n Seton III. The haos optzaton agorth s presented n Seton IV. Seton V dsusses the fuzzy usterng ethod based on haos saword agorth. The experents and orrespondng anayses about CSWFCM are desrbed n Seton VI. Fnay Seton VII states soe onusons. II. FUZZY CLUSTERING METHOD Supposng the fnte set X = { x1 x2 xn} s beonged p to the p densona Eudean spae R naey p k = 12 n xk R. FCM parttons X nto fuzzy groups and fnds a usterng enter n eah group suh that the obetve funton based on dstane s na. The obetve funton for FCM s defned as foows: J n 2 ( U = ( µ k ) ( dk ) (1) k = 1 = 1 Where U s the fuzzy atrx of X V s the usterng enter set of X ( 1 + ) s the weght oeffent 1 1 k n dk = xk v s the Eudean dstane between x k and v µ k X s the degree of ebershp of data xk reevant to the th usterng enter v and µ satsfes the foowng restrton: k 1 k n µ k = 1 (2) = 1 The onrete operatng sequenes of FCM are as foows: Step1 Set paraeters: n t = /08 /$ IEEE CIS 2008

2 Step2 Intaze usterng enter randoy: V (t) = v v v }. { 1 2 Step3 Cauate U (t). 1 1 k n and rando ong-range searh and the transton nforaton s the souton of optzaton ode durng the searh. 2 /( ) d k If d k 0 µ k = 1/ (3) = 1 d k Otherwse f = µ = 1; k f µ = 0. Step4 Cauate V ( t +1). k n 1 v = µ k xk / µ k (4) n k = 1 k = 1 Step 5 Choose an approprate atrx nor to opare V ( t +1) and V (t). If V ( t + 1) V ( t) ε end; otherwse t = t + 1 and go to Step 3 where ε s a postve and sa enough rea nuber. III. SMALL-WORLD OPTIMIZATION ALGORITHM In the 1960s soa psyhoogst Staney Mgra perfored a sa-word experent of trang out short paths through the soa networks of the Unted States [5]. He asked a few hundred peope n Oaha to forward a etter to a target stranger n Boston through persona ontats and ended up wth 60 opeted hans of etters that averaged sx senders the faous sx degrees of separaton [6]. The sa-word phenoenon reveas a ost effetve ode of nforaton transsson of a ot of opex networks n reaty naey a hgh usterng subnets nudng oa ontats nodes and soe rando ong-range shortuts are usefu to nrease effeny of nforaton transsson [7][8]. In 1998 Dunan Watts and Steve Strogatz presented rgorous Watts-Strogatz ode and onstruted sa-word network fro the theory n Nature [9]. Copex networks theory has been wdey pad attentons sne then and suessfuy used n route optzaton [10] dsease propagaton [11] and so on now yet tte dsusson about sa-word phenoenon has been found n the fed of optzaton agorth. Kenberg ponts out that the sa-word phenoenon s an effeny queston about routng agorth where oa knowedge suffes to fnd effetve paths to get to destnaton [12]. Inspred by the ehans of sa-word phenoenon Du frsty produed sa-word agorth (SWA) for funton optzaton [13]. He took the optzaton as a proess that nforaton transts fro anddate node (.e. anddate souton) to opta node (.e. opta souton) n network (.e. searhng spae). The SWA nudes oa short-range searhng operator and rando ong-range searhng operator. The two-densona spae searhng of SWA s sheatay shown n Fg. 1. Fro the fgure we an see that a anddate node oves to opta node through oa short-range searh Fgure 1. Two-densona spae searhng prnpe of SWA Wthout oss of generaty the foowng goba optzaton ode s onsdered: J = n f ( x) = { x1 x2 xq} (5) q x R x Where f (x) s the obetve funton q s the denson of vetor x. x d u ]. [ Aordng to [13] et S be the N - densona transsson node popuaton. node s S s = a 1 a 2 a s the bnary odng of q - densona varabe x. a 1 q a = a 1 a 2 a a {01} 1 dependents on the bnary odng preson σ. a s aed the odng of varabe x and desrbed as a = e( x ) where e ( ) s the odng anner. x s aed the deodng of node a and desrbed as x = e 1 ( ) where e ( ) s the deodng anner. a Defnton 1: defned as foows: a 1 q the deodng anner s 1 x = = + e ( a ) d ( u d) a 2 (6) 2 1 = 1 Defnton 2: s S the neghborhood set of the node s an be defned as foows: ζ ( s ) = { s 0 < s s s S} (7)

3 Where s the Hang dstane. Defnton 3: s S s an be defned as foows: the non- neghborhood set of ζ ( s ) = { s s s > s S} (8) The SWA nspred by the ehans of sa-word phenoenon nudes oa short-range searhng operator Ψ and rando ong-range searhng operator Γ. Supposng R ζ ( s ) s the oa searhng of node s the an aton of Ψ s transttng the nforaton fro node s to the node s ( k +1) whh s the nearest node to the optzaton ode n R. The proess an be desrbed as foows: s ( k + 1) ψ( s ) = { s R f ( e ( s )) = n ( f ( e ( s )))} (9) s ( k ) R ( k ) s( k + 1) ζ ( s ) the an aton of Γ s transttng the nforaton fro s to s ( k +1) at gven preset ong-rang searhng probabty p. The proess an be desrbed as foows: s ( k + 1) Γ( s ) = { s p < p : s ζ ( s ) p = rand()} (10) The onrete operatng proesses of Ψ and Γ are shown n [13]. IV. CHAOS OPTIMIZATION ALGORITHM In SWA the an operaton of Ψ s randoy searhng a node n ζ ( s ). The operatng anner s so easy that the oa searhng effeny of SWA s dereased. To prove the searhng effeny the haos optzaton s used n SWA. Frst akng use of the haratersts of ergodty and randoness of haot varabes the nta popuaton s generated by ogst appng; seondy the oa searh of ndvdua s perfored by haos dsturbane after oa shortrange searh. Consderng the Logst Equaton s ore onvenent and the auated aount s ess than other haos teraton equatons we use the foowng (11) to generate nta popuaton S (0). z( k + 1) = η z( k)(1 z( k)) (11) Where η s the haos attrator the haos spae s [0 1] when η = 4 z s the haos varabe at kth teraton z (01) and z { }. Let υ be the tota teraton tes of appng. The haos dsturbane s defned as foows: * ( 1 α ) β αβ β ( k) = + (12) Where Θ( β = x ) s the opta haos varabe x s the urrent opta souton of optzaton ode Θ ( ) s the appng fro souton spae to haos spae et Θ ( ) be the nverse appng naey x( k) = Θ ( β ) β s the haos varabe at kth teraton β s the haos varabe after haos dsturbane α s the adustent oeffent and defned as foows: p k α = 1 (13) k Where p s an nteger and p = 2 n ths paper k s the teraton tes. Let γ be the tota teraton tes of haos dsturbane. V. FCM BASED ON CHAOS SMALL-WORLD Aordng to the obetve funton J ( U we an know that the fna target of fuzzy usterng s to obtan the usterng enter V and fuzzy atrx U of fnte set X. Sne V and U s nterreated we an take ether of the as the optzaton varabe. Consderng the oputaton ost we hoose to ode on V by bnary n ths paper. The obetve funton J ( U s taken as the ftness funton of optzaton probe. Based on the above fuzzy uster ethod sa-word optzaton agorth and haos optzaton agorth we an dept a nove FCM based on haos sa-word as foows: Step1 Intazaton: bnary odng preson σ usterng enter nuber weght oeffent popuaton sze N neghborhood sze ong-range searhng probabty p tota teraton tes of appng υ tota teraton tes of haos dsturbane λ axa generaton k k 0. Step2 Extrat datu. Step3 Generate nta popuaton of usterng enters S (0) aordng to (11). ax

4 Step4 Exeute the optzaton of popuaton S = { s1 s2 sn } based on sa-word optzaton agorth. Step 4.1 S S( k) ; Step 4.2 Fnsh rando ong-range searh aordng to gven p : s ( k + 1) Γ( s ) s S 1 N ; Step 4.3 Fnsh oa short-range searh : s ( 1) ( k + ψ s ( k + 1)) 1 N ; Step 4.4 If J ( e ( s ( k + 1))) < J ( e ( s )) then s ( ) k s ( k + 1) 1 N. Step 5 Exeute the optzaton of S based on haos dsturbane. Step 5.1 Choose fx( µ N) ndvduas whose ftness vaues are saer n S and for the popuaton S = { s1( k) s2 s } = fx( µ N) where fx ( ) s the round funton; Step 5.2 Transforng fro odng spae to haos spae: s S 1 fx( µ N) x e( s ) ; β Θ( x( k)) ; Step 5.3 Exeute haos dsturbane to β aordng to (12) (13). Step 6 Judge whether the ternatng ondton s satsfed. If not go on the foowng proess otherwse output the opta uster enters and end. Step 7 k k + 1 and go to Step 4. VI. SIMULATION RESULTS AND ANALYSIS To verfy the vadty of our proposed agorth n age edge deteton two ages of pxes 2 naey Lena and Caeraan are tested wth MATLAB7.0 on an Inte Pentu IV 2.99GHz oputer wth 512MB RAM. We opare the deteton resuts aong FCM GAFCM and CSWFCM. In CSWFCM s 2 N s 30 s 1 p s 0.8 σ s 10-5 υ s 300 λ s 30 and k ax s 50. In GAFCM the popuaton sze s 30 rossover probabty s 0.8 utaton probabty s 0.1 and axa generaton s 50. The paraeters of FCM are the sae as the orrespondng paraeters n CSWFCM. Durng the age edge deteton four vaues are hosen for the uster nuber naey and 5. Consderng the randoness eah agorth s tested for twenty tes. Tabe I s the perforane oparson of edge deteton aong three agorths. Fro the tabe we an see that wth the nrease of usterng enter nuber the average and best ftness J ( U both dereases whh ndates that the fuzzy usterng effets are better and better. Ang at any ertan usterng enter nuber the ftness of CSWFCM s better than FCM and GAFCM. The dfferenes of average and best ftness are not obvous when s 2 3 and 5 however when s 4 FCM aways traps nto oa nu vaue GAFCM and CSWFCM an avod the deeptve probe and get better resuts. A n a through tabe I we an see that CSWFCM an onverge to a goba opta vaue and get better edge deteton effet than other two ethods to soe extent. TABLE I. PPERFORMANCE COMPARISON OF EDGE DETECTION AMONG THREE ALGORITHMS Iage C J(UV) FCM GAFCM CSWFCM ena aeraan Average Best Average Best Average Best Average Best Average Best Average Best Average Best Average Best In order to see the edge deteton resuts n reaty we take the ena age as an exape and see the dfferenes of edge deteton aong three agorths when =3 and =4. Fg. 2 s the orgna ena age. Fg.3 and Fg. 4 are the oparsons of edge deteton about ena age aong three agorths when =3 and =4 respetvey. Fgure 2. Orgna ena age

5 (a) FCM (a) FCM (b) GAFCM (b) GAFCM () CSWFCM Fgure 3. Coparsons of edge deteton about ena age aong three agorths when =3 () CSWFCM Fgure 4. Coparsons of edge deteton about ena age aong three agorths when =4

6 As the best ftness J ( U about ena age of FCM GAFCM and CSWFCM s ose when =3 we an see that edge detetons based on above three agorths a get good effets and the dfferenes of edge detetons aren t obvous fro Fg. 3. When =4 t s obvous that the edge detetons of GAFCM and CSWFCM are better than FCM for thers goba optzaton apaty and the deteton of CSWFCM has ore deta and earer ontour suh as the ontnutes of fae and hat br than FCM and GAFCM. GAFCM and CSWFCM avod oa nu and get good resuts to soe extent and the onvergene speed of CSWFCM s quker than GAFCM. VII. CONCLUSIONS In ths paper a nove fuzzy usterng ethods based on haos sa-word agorth s presented to sove the age edge deteton and the edge deteton resuts of two ages are opared wth FCM and GAFCM. Aordng to the suaton experents we an draw the foowng onusons: (1) Inspred fro the ehans of sa-word phenoenon the sa-word agorth s an effent optzaton agorth and haos optzaton further proves the searhng effeny of SWA; (2) FCM based on the optzaton of CSWA avods the senstvty to the nta vaue of usterng enter and trappng nto oa nu vaue; (3) The age proessng resuts of CSEFCM are better at deta and earness of ontour than FCM and GAFCM; (4) CSWFCM has the hgher onvergene speed and stabty than FCM and GAFCM. ACKNOWLEDGMENT The support of the proet of ehana anufaturng equpent key aboratory of Shan x provne (Grant No. 03JF06) s gratefuy aknowedged. REFERENCES Fgure 5. Convergene urves of best ftness about ena age of three agorths when =3 Fgure 6. Convergene urves of best ftness about ena age of three agorths when =4 Fg. 5 and Fg. 6 are the onvergene urves of best ftness about ena age of FCM GAFCM and CSWFCM when =3 and =4 respetvey. When =3 athough the best ftness of three agorths s ose the onvergene speed of CSWFCM s quker than FCM and GAFCM and the onvergene of FCM has futuaton as shown n Fg. 5. When =4 fro Fg. 6 we an see that FCM traps nto oa nu vaue [1] Rafae C. Gonzaez Rhard E. Woods. Dgta age proessng Pubshng House of Eetrons Industry Beng pp Jan [2] R. Bean R. Kaaba and L.A. Zadeh. Abstraton and pattern assfaton Journa of Matheata Anayss and Appatons vo. 13 no. 1pp [3] C. Caro S. Antono. A naver-stokes based strategy for the aerodyna optzaton of a turbne asade usng a genet agorth n proeedngs of ASME Turbo Expo New Oreans USA: ASME pp. 1~8. [4] Z. L. Guo S. A. Wang. A nove une evoutonary agorth norporatng haos optzaton Pattern Reognton Letters vo. 27 no. 1 pp [5] J. C. Jaes C. C. Carson. It s a sa word Nature vo. 393 no. 4 pp [6] D. J. Watts. Sx Degrees:The sene of a onneted age New York: W.W. Norton & Copany pp [7] J. Kenberg. Navgaton n a sa word Nature vo. 406 no. 8 pp [8] J. Kenberg. The sa-word phenoenon: an agorth perspetve In Pro. The 32nd ACM Syposu on Theory of Coputng Portand Oregon May 2000 pp [9] D. J. Watts S. H. Strogatz. Coetve dynas of sa-word networks Nature vo. 393 no. 4 pp [10] J. Y. Zeng W. J. Hsu. Opta routng n a sa-word network n Pro Sxth Internatona Conferene on Parae and Dstrbuted Coputng Appatons and Tehnooges Daan Chna De pp [11] N. Zerkr J. P. Cere. Statsta and dynaa study of dsease propagaton n a sa word network Physa Revew E vo. 65 no. 2 pp ~ [12] J. Kenberg. The sa-word phenoenon and deentrazed searh SIAM News vo. 37 no. 3 pp [13] H. F. Du X. D. Wu and J. Zhuan. Sa-word optzaton agorth for funton optzaton n Pro. 2nd Internatona Conferene on Natura Coputaton X an Chna. Sep pp