Improved Gaussian Mixture Model in Video Motion Detection

Transcription

1 JOURNAL OF MULTIMEDIA, VOL. 8, NO. 5, OCTOBER Improved Gaussian Mixure Model in Video Moion Deecion Xie Yong Modern Educaion Technology Cener, Xi'an Inernaional Universiy, 48 Zhangba Norh Road, Xi'an 777, China, Absrac As he classical Gaussian mixure model has some problems of no considering i self s maching degree of Gaussian densiy funcions, model updaing and he bacground in real video moion deecion, made improvemens on he hree aspecs. Opimized Gaussian mixure model s overall archiecure and proposed an improved algorihm according o he analysis of he definiion and disadvanages of classical Gaussian mixure model. Finally, hrough deailed experimen, he resul showed: he improved Gaussian mixure model is faser in model convergence rae in video moion deecion, can quicly adap o he changes of bacground and grealy decreases he fall-ou raio. Index Terms Maching Modal; Video Moion; Framewor Opimizaion; Gaussian Mixure I. INTRODUCTION Efficienly and accuraely absracs he no saionary obecs is a ey o video processing in inelligen video surveillance sysem []. The moion deecion algorihm is mulifarious, he radiional algorihms are opical flow field mehod, frame difference mehod and bacground subracion. The firs wo has some obviously disadvanages. The opical flow field mehod needs repeaedly ieraive operaion, he calculaions are compleing and ime-consuming, i s hard o realize real-ime deecion wihou specific hardware suppor; he algorihm has poor abiliy in ani-amming lie noise so i s rarely used in pracical applicaion [2-5]. The posiion of moion obecs deeced by he frame difference mehod is inaccuracy, is enclosing recangle exrudes in moion direcion and affeced by he obec s velociy and ime inerval beween adacen images, usually canno absrac all he relaively feaure pixel poin, inside of he moion eniy easily generaes caviaions, when he moving obec in he scene has no obviously movemen, he pixel value in he moving obec s area reurns o zero, he moving obec canno be deeced [6-9]. Now he generally used mehod is bacground subracion, differen he inpued video sequence s curren frame and esablished bacground model and absracs he moving obec. Comparing wih oher moion deecion algorihm, i has he advanages lie small daa calculaion, easy o realize and can absracs complee moving obec, as an common mehod of moion deecion, i s wildly used. Now he moion deecion s maor research direcion is o esablish appropriae bacground model o adop o differen change of scene and applicaion environmen [-3]. The Gaussian mixure model is a wildly used bacground model and i s a improvemen of single Gaussian bacground model. The single Gaussian bacground model uses individual Gaussian disribuion o express bacground, can process he simple scene wih small change and secular change [4]. Bu is no applicable o hose scenes wih big change or sudden change of bacground, or he bacground pixel value is mulimodal disribued (lie small and repeaedly moion). Because hen he bacground changes fas, is disribuion is no a ransiion from a relaively seady unimodal disribuion o anoher [5]. Considered he disribuion of bacground pixel value is mulimodal, on he basis of he mehod of unimodal, can uses unimodal se o describe he change of pixel value in complex scene, he Gaussian mixure model exacly uses muliple single Gaussian funcion o describe muli-modal scene bacground. I beer describe he bacground model disribuion in complex scene, has good applicaion effec on processing sudden change of bacground and bacground disurbance. The classical Gaussian mixure proposed by STAUFFER C can solve he he problems lie sudden change of bacground and bacground disurbance, KAEWTRAKULPONG P realized real-ime applicaion on he basis of heory, and had a beer resul in sudden change of bacground and illuminaion variaion. Bu in real applicaion, because of he deficiencies of moion deecion algorihm, here are some problems: firsly, maching Gaussian modal selecion only considers he modal s weigh wihou considering he modal i self s maching degree of Gaussian densiy funcion. Secondly, i uses oally differen updae mechanism in he bacground model s iniializaion phrase and seady phrase, when he bacground change is very big, he bacground model s re-esablish ime is oo long which affecs he sysem s deecion. Thirdly, he final bacground display only shows he modal wih he bigges weigh wihou considering he effec of oher modals. The Gaussian mixure bacground s modeling process permis he exisence of video moving obec, which is appropriae for he deecion of small and fas moving obec oudoor and wih weaher change. Bu he algorihm has is disadvanages, he deailed analysis is as following: firsly, when maching Gaussian modal 23 ACADEMY PUBLISHER doi:34/mm

2 528 JOURNAL OF MULTIMEDIA, VOL. 8, NO. 5, OCTOBER 23 selecion, his algorihm only considers he weigh of modal wihou considering i self s maching degree of Gaussian densiy funcion, he seleced modal may no be he bes maching. Secondly, he bacground model s iniializaion phrase and he seady phrase uses wo differen updaing mechanism in he Gaussian mixure proposed by STAUFFER C, i has a beer effec when he scene change is no big and he bacground disurbance is small. The paper based on classical Gaussian mixure algorihm and aimed a he disadvanages of Gaussian mixure model algorihm, respecively made a series of improvemen abou maching modal selecion mehod, model s updaing algorihm and bacground display: his paper mainly made exploraive and innovaive wors as following aspecs: Aimed a he problems of maching degree, model updaing and bacground display in moion deecion of classical Gaussian mixure model, when he classical Gaussian mixure model maes modal maching, compares o each modal by urns, once mees X- /, hen regards he sample maches his modal, he selecion paerns has obviously problems, when he modal updaing, he convergence rae of sysem is relaively slow, when he classical Gaussian mixure model algorihm maes bacground display, direcly aes he modal componen value wih he bigges weigh as bacground display, wihou considering oher (B-) componen value is no accurae enough as bacground. So here made improvemens of Gaussian mixure model in hese hree aspecs. The specific mehod of improvemen is: in modal maching, considers he effec of weigh, Gaussian modal iself mean value and variance owards disribuion, maes wice compare of he new samples and he Gaussian modal of he model; aes unified expression in model updaing; reunifies he weigh of he bacground s B modal in bacground display, disribues he bacground s each model componen value by he weigh afer uniformizaion. In order o furher prove he accuracy and efficiency of improved Gaussian mixure model, inegraed considered every modal s disribuion of bacground model, made he displayed bacground closer o he bacground disribuion of real scene. Through deailed experimen, he resul showed, compared wih previous mehod, he model in his ex is faser in convergence rae, can beer adop o he exchange of bacground and grealy decrease he fall-ou raio. II. ANALYSIS ON GAUSSIAN MIXTURE MODEL The bacground modeling s fundamenal of Gaussian mixure model is: according o he differen modal maching s frequency of sample se (he order of pixel poin s gray value formed in ime shaf) and bacground model, consanly updaes he Gaussian disribuion s parameer of every modal in he model, ie, rains he parameer lie weigh, mean value and covariance of every Gaussian disribuion, maes bacground pixel value disribued convergence a one or several Gaussian disribuion, realizes he cluser of bacground pixel value, and realizes he modeling of bacground. A. Model Definiion Based on he bacground modeling mehod of Gaussian mixure model, is modeling obec is every pixel poin s pixel value. In he model, every pixel poin s gray value in he image is regards as a saisics and sochasic process, he pixel value of pixel poin can be regarded as a vecor sequence, arbirarily pixel poin (x,y) is hisory pixel value can be expressed as: x,..., x l ( x, y) : i r r () There li ( x, y ) refers o he gray value of pixel in ime i. Gaussian mixure model uses K Gaussian disribuion o sand for hose hisory value, so he proporion of pixel x as curren value is: p( x ) q, * ( x, u, i (2) r i r r, i Among hem, w r, is he i weigh of Gaussian disribuion in ime, which reflecs he appearance proporion of Gaussian disribuion; ( xr, ui, r, r, is he Gaussian proporion densiy funcion when he i ime s mean value u( i, ) and he covariance is i, ; px ( ) refers o he proporion of ime pixel as X; K is he number of disribuion. III. PROPOSED SCHEME Figure is a flow diagram of improved Gaussian mixure model, mainly improves he Gaussian mixure algorihm in hree aspecs: he selecion of maching Gaussian modal, he updaing of he model and he display of he bacground. Updae he way Classic MOG es resuls Improved algorihm Inpu image sequence Iniializaion If he curren value in he range of gaussian disribuion Yes To updae a maching gaussian mode: The mean updae Updae he variance Weighs are updaed No maching he gaussian mode o updae he weighs Pic up B from he K gaussian model as bacground Maching mode choice No The minimum weigh of be replaced by he curren disribuion and gauss disribuion Bacground display mode Figure. The flow diagram of improved Gaussian mixure algorihm The process module of blue checs bacground is he flow diagram of classical Gaussian mixure algorihm, he orange process module is he deailed improvemen of Gaussian mixure. The process module of blue checs bacground are respecively maching he selec and udge process of he algorihm s iniializaion, modal updaing, modal maching and bacground modal. The 23 ACADEMY PUBLISHER

3 JOURNAL OF MULTIMEDIA, VOL. 8, NO. 5, OCTOBER orange process module is he maor improve poins of his improved algorihm which includes: he maching modal s selecion, updaing means and bacground display paerns. Aimed a he disadvanages of he Gaussian mixure algorihm, he improved algorihm mainly maes a series of improves on he maching modal selecion, updaing means and bacground display paerns. Firsly, on he maching modal selecion module, combines he modal weigh and i self s maching condiion, maes he udge of Gaussian modal maching; secondly, on updaing means, unifies he updaing means of iniializaion phrase and deecion progress, maes he model a smooh ransiion from iniializaion phrase o seady phrase; finally, on he bacground display paerns, inegraed considers he bacground modal s every modal disribuion and maes he bacground s display can clearly reflec he bacground model s deail disribuion. A. The Selecion of Macching Modal Each modal of he Gaussian mixure model is ordered 2 as w / a from he smalles o he bigges, when he modal maching is processing, comes a new sample X T, compares wih each modal, if only mees X u / 2.6 hen regards he sample is maching o he modal, doesn consider oher no compared modals. This selecion paern has some obviously problems: when he modal maching is processing, he maching modal may no be he bes modal by he curren paern, because he algorihm will ou of he loop when finds one modal, if oher res modal is more suiable han he seleced one for he weigh, his selecion paern is no he bes evidenly. For example: hree Gaussian modal s disribuion and weigh of cerain pixel is showed as figure2. The Gaussian modal s mean value and variance is differen, he weigh decreases successively mode mode mode weigh 5 mode Figure 2. The Gaussian modal and weigh disribuion figure The pixel disribuion a cerain ime is x, if only considers weigh w, and he weigh s bigges modal as maching model, showed as he firs line in figure 3; if only considers wihou considering curren pixel disribuion, hen he bigges modal 2 as maching modal, showed as he second line in figure 2; if boh considers he weigh and iself s maching degree, according o he value of w / x u / a, he bigges modal3 as maching modal, showed as he hird line in figure 3. This shows differen maching modal selecion mechanism can led o differen modal maching resul, considering muliple facors affeced by modal maching, o selec he bes modal maching is he ey o he algorihm improvemen w w/sigma w/sigma/[xo-mu] 5 mode mode mode3 Figure 3. Selecion resul of maching modal The improved algorihm boh considers he weigh and Gaussian modal s mean value and variance s effec on disribuion, maes improvemens for maching modal s selecion mehod as following: compares wice of he new sample wih he model s Gaussian modal: (a) compares X wih each modal successively, when mees X u / 2.6 selecs maching modal, selecs maching modal and maes up new array; if all he modal are no me, hen here are no modal maches he curren disribuion, cass away he modal wih he smalles weigh, esablishes new modal by curren disribuion, he new modal s mean value is he value of curren sample, variance preinsall s a bigger value. (b) compares in he new modal combinaion composed by he mee of w / x u / a w / x u / a, respecively calculaes he value of, selecs he bigges modal as maching modal; oher modal does no selec as maching modal and cass i ou. Maes he value of w / x u / a as maching basis, considers he weigh of (a) modal in he whole model, he bigger weigh w is, he easier of maching; (b) sample and he modal s maching condiion, when x u / a is smaller, i shows he sample and he Gaussian modal is more mached, on he conrary, he bigger value shows he sample and he Gaussian modal is less mached. So when he value is he bigges, i shows he corresponding modal is he mos mached one for he sample, hus o ensure he bes maching modal. B. Updaing Means The updaing of Gaussian mixure model s parameer can be expressed by a uniformed formula (3), ( ) ( ( )). ( ) ( ). ( y( ); ( )) (3) Among hem, () generally refers o parameer. A ime, rae () and conrol facor ( x( ); ( )) deermines he updaing, K() depends on updaing rae. 23 ACADEMY PUBLISHER

4 53 JOURNAL OF MULTIMEDIA, VOL. 8, NO. 5, OCTOBER 23 Mehod : ( ) /, he parameer esimaion is relaed o he number of observed samples, wih he increase of sample s number, he parameer quicly reaches o expecaion value, which is suiable for he updaing in sysem s iniializaion phrase; Mehod 2: when () a, he sysem s parameer is relaed o he sample number in he window which neares lengh is L / a, has no relaions wih previous samples, compares wih iniializaion phrase, he compuing speed is increased, bu he corresponding parameer s convergence rae is also decreased, which is suiable for he parameer updaing when he sysem reaches seady phrase. The lef side in figure 4 is he aen updaing, which shows a speedily convergence of he sysem in iniializaion phrase and reaches seady phrase; he righ side shows he sysem s convergence rae is slower in iniializaion phrase. The classical MOG algorihm respecively aes wo differen updaing mechanism in iniializaion phrase and afer he sysem sabled o reach he expecaion resul X() X() Figure 4. Example of updaing process be effeced, he deecion s error rae would grealy increases even oally canno deec, which is no suiable for applicaion. Showed as figure 6, he sysem s bacground has sudden change in deecion process, i aes a long ime for he sysem o re-convergen o seady phrase, in his period of ime, he sysem deecion resul is unsaisfacory X() Figure 6. Example for updaing process of MOG wih sudden change of he bacground The improved algorihm unifies he wo updaing mehod, which no only considers he speedily convergence of sysem parameer in iniializaion phrase, bu also researches he sysem parameer updaing s calculaion afer sable, maes i can quicly ge expecaion bacground parameer when he bacground has big changes. Figure 7 is he simulaed updaing process uses improved algorihm, shows i can quicly convergen o sable when he sysem has sudden changes X() 3 25 X() Figure 5. Example for classical MOG updaing process Figure 5 is he convergence curve when classical MOG updaing, clearly shows he sudden change on updaing mechanism swiches and ransiions. If he sysem has already used he laer updaing mechanism o updae afer iniializaion, if he bacground has big changes, his mehod divided he sysem updaing means arificially would ge some big problems, i would ae longer ime for he parameer o reaches expecaion value when i aes he laer updaing mehod, he deecion resul in his period of ime would Figure 7. Updaing example for improved algorihm in sudden changed bacground The updaing rae () is calculaed as formula:. q c When he sample is mached o he Gaussian modal, q, or q, c is a couning for he number of maching he modal and sample, c c g when he Gaussian modal wih he smalles weigh is replaced by new modal, his modal c reses as. The weigh updaing, w ( ) ( ). ( ) a. q (4) 23 ACADEMY PUBLISHER

5 JOURNAL OF MULTIMEDIA, VOL. 8, NO. 5, OCTOBER The weigh updaing, The variance updaing, ( ) ( ). ( ). x (5) 2 2 ( ) ( ). ( ).( ( )) (6) When sysem iniializes or he bacground has big changes, he number of sample is small, / c, he sysem can quicly convergen in a shor period of ime; when he sysem is sable, he number of sample is big, q.(( a) / c a a, ie,, he sysem s calculaion is small, is performance is improved. IV. BACKGROUND DISPLAY PATTERNS The classical MOG algorihm direcly uses he modal componen value wih he bigges weigh as bacground display, wihou considering oher (B-) componen value as bacground, i s no accuraely in display. A ime, he proporion of pixel gray value as x is: p( x) p( g ) p( x g ). g( x; ) x x i (7) According o Bayes formula, he proporion of pixel x as bacground is: x x x p b gx x x x p x g p g p( b x) p( b g ) p( h x) ( ) p( h x) ( ) ( ) (8) The bacground expecaion value E X B as bacground display is K X X E X B E G P( G B) X. P( B G ) P( G ) K K X B K X X P( B GJ ) P( GI ) PG is he weigh of each modal, re-unifies he weigh of bacground B modal in applicaion, disribues he bacground s each modal componen by uniformed weigh, he bacground display uses he mixure modal re-disribued model as he basis of array display. V. EXPERIMENTAL RESULTS A. Tes Plaform and Daa Sources In order o es he improved algorihm s accuracy, ess i in muliple video and compares wih classical Gaussian mixure bacground modeling mehod (MOG). All he es sequences uses he same parameer seing: K 4,.2, 2.6,.76 The plaform for esing hardware: CPU is Inel Penium Dual E26, basic frequency is.8ghz, RAM is G. The plaform for sofware: Linux Suse.2 Opencv. The daa used in he es conains hree se of esing video: ) uses he sandard sequences offered by hp://cvrr.ucsd.edu/aon/shadow, selecs is indoor scene: K (9) Inelligen-room; 2) he esing video Tree comes wih Opencv and a se of indoor scene Curain; 3) offered simulaed synhesizing daase: hp:// simulaed synhesizes arificially muliple moion deecion s common scene lie breaches shaing, covering, shadowing, mirror reflecing, illuminaion variaing. Maes moion deecion es o each video in same experimen condiion. B. Evaluaion Crierion of Performance The recall and pression is he imporan indicaor of reflecing deecion performance. Recall, he percenage of correc resul o all he correc resul. Pression is he percenage of deeced correc resul o all he deeced resul. The recall: TR Riecall TR FM And he pression is expressed as: TR Pr ecision TR FM The wo s inegrae indicaor: F Score The bacground s error rae: 2. recall. precision recall precision TM FPR TM FM () () (2) (3) TP: foreground of correc deecion, TN: bacground of correc deecion, FP: foreground of false deecion (which should be bacground), FN: bacground of false deecion (which should be foreground). (a) iniial video frame (b) classical MOG deecion resul (c) improved algorihm deecion resul Figure 8. Video moion obec deecion resul of Tree C. Compare and Analysis of he Experimen Resul Figure 8 is he resul for he esing video ree. (a) is he iniial image frame of esing video, (b) is he foreground image deeced by classical MOG, (c) is he deeced foreground image deeced by improved Gaussian mixure algorihm. The experimen resul shows, pars of he foreground deeced by MOG is false deeced shaing pixel poins of he branches, and he foreground deeced by improved algorihm has less pixel poins, which shows he disurbance robusness o bacground of 23 ACADEMY PUBLISHER

6 532 JOURNAL OF MULTIMEDIA, VOL. 8, NO. 5, OCTOBER 23 improved Gaussian mixure algorihm is beer han iniial MOG algorihm in he aspecs lie overcoming bacground disurbance (branches shaing): Figure 9 is he deecion resul of indoor scene Curain. In his video, he curain is floaing wih wind, which is a big disurbance o he moion deecion. From he es resul, he classical MOG would deecs pars of he floaing curain as foreground, bu he improved algorihm has lile his ind of phenomenon, which only deeced he moion obec. I shows he improved algorihm has beer robusness o bacground disurbance. poins, he bacground model of improved algorihm basically reaches he demand, he esablished bacground model includes due bacground, because he improved algorihm considers oher bacground modal, is srucured model can beer overcome he disurbance made by bacground disurbance lie shaing branches, he deailed color is no disored, which is more fied he real scene. (a) iniial video frame (b) classical MOG deecion resul (c) improved algorihm deecion resul Figure. Video bacground display resul of ree (a) iniial video frame (b) classical MOG deecion resul (c) improved algorihm deecion resul Figure 9. Moion obec deecion resul of Curain Figure is he deecion resul of he scene Inelligen-room. This video is a indoor scene, involved o he bacground change condiion lie scene illuminaion changing. The seleced frame is a he ime when moion obec us came o monior area, from he deecion resul, in MOG algorihm, when he obec us came in, because of he change of bacground, he deecion resul is no good; he improved algorihm, as is bacground model can convergen quicly, even he obec us came in, i can quicly deec he moion obec. Figure 2 is he simulaed daase deecion resul offered by BRUTZER, S. This video is a simulaion scene relaed o he bacground change condiion of muliple moion deecions lie scene illuminaion changing, shadowing, covering and bacground disurbance. The branches is shaing in his scene, because of he cover of buildings, he shadow, cover and illuminaion change is obviously. The seleced frame is he ime when he wo moion obecs shows ogeher, from he deecion resul, in MOG algorihm, when he obecs us came in, because of he bacground s change, he deecion resul is no good, which is affec a lo by he environmen lie shadowing and covering; he improved algorihm, because of he bacground model can convergen quicly, even he obecs us came, i can deec he moion obecs and deec hem prey compleely. (a) iniial video frame (b) classical MOG deecion resul (c) improved algorihm deecion resul Figure. Moion obec deecion resul of Inelligen-room Figure is he bacground frame s display image srucured by he wo algorihms in he Tree video, among hem, (a) is he iniial image frame wih moion obec a is op righ corner, (b) is he single Gaussian bacground display srucured by he Gaussian modal wih he bigges weigh in classical MOG algorihm bacground model, (c) is he bacground frame srucured by Gaussian mixure unified by each modal weigh in bacground model. From he experimen. The bacground model of MOG algorihm has more noisy (a) iniial video frame (b) classical MOG deecion resul (c) improved algorihm deecion resul Figure 2. Moion obec deecing resul of simulaion scene Figure 3 is he index analysis curves of MOG algorihm and improved algorihm, he esing daa is he simulaion daa in common scene offered by BRUTZER,S, selecs he firs 4 frames and ess, draws he analysis curves. The analysis curves is corresponding o Recall, Precision, FPR and F-Score, he blue chain line is he curve of MOG, he blac solid line is for improved algorihm. From he figure, he 23 ACADEMY PUBLISHER

7 JOURNAL OF MULTIMEDIA, VOL. 8, NO. 5, OCTOBER performance index is grealy increased han MOG afer he algorihm was improved. Recall Recall.2.2 Recall - Plo Frame FPR - Plo Frame Figure 3. Precision Precision.2.2 Precision - Plo Frame MOG F- Score Plo Proposed Frame The performance analysis curves of MOG and improved algorihm Figure 4 is he real-ime screensho deecion resul when he algorihm is used in highway real-ime monior sysem. The experimen chooses he real-ime video of Huning highway, PEG-2, resoluion is D(74 576); seen as he figure, deeced he unusual video even is oo slow (<4m/h). According o applicaion feedbacs, he algorihm operaes well in he sysem. Figure 4. Applicaion resul screensho of he highway VI. CONCLUSION This ex improves classical Gaussian (MOG) on he basis of deeply research on he bacground modeling echniques, o overcome he conradicion of is convergence rae and deecion accuracy. The algorihm made full use of sample informaion, designed he fuse rule of algorihm, he experimen showed, his algorihm can beer fied he complex scene changes han previous algorihms and has high deecion accuracy and recall. In he nex sep of wor, he research will focus on harmony exure informaion, eliminaing he adverse effecs of shadowing, and researches on he ransplanaion mehod from algorihm o DSP and CUDA plaform, realizes he promoing of sysem s sabiliy and processing capaciy while reducing energy inensiy and volume. REFERENCES [] Marcel Waldvogel, Rober Rinaldi. Efficien opology-aware overlay newor. ACM SIGCOMM Compuer Communicaion Review, Vol. 33(), pp. -6, 23. [2] Ion Soica, Rober Morris, David Liben-Nowell, e al., Chord: A Scalable Peer-o-Peer Looup Proocol for Inerne Applicaions, IEEE/ACM Transaions on Neworing, Vol. (), pp.7-32, 23. [3] S. Ranasamy, P. Francis, M. Handley, e.al., A Scalable Conen Addressable Newor, In Annual Conference of he ACM Special Ineres Group on Daa Communicaions (ACM SIGCOMM 2), San Diego, CA, USA, Augus 2. [4] A. Kumar, S. Merugu, J. Xu, and X. Yu, Ulysses: A robus, low-diameer, low-laency peer-o-peer newor, European ransacion on elecommunicaions, vol. 5(6), pp , 24. [5] I. Clare, O. Sandberg, B. Wiley, e al. Freene: a disribued anonymous informaion sorage and rerieval sysem. Proceedings of he ICSI Worshop on Design Issues in Anonymiy and Unobservabiliy 23. pp , 23. [6] Klingberg, R. Manfredi. Gnuella proocol developmen: Gnuella [EB/OL]. hp:// rfc-gnuella.sourceforge.ne /developer/esing/ index.hml [7] D. Malhi, M. Naor, and D. Raacza, Viceroy: A scalable and dynamic emulaion of he buerfly, Proc. of ACM PODC, 22. [8] J. Xu, On he fundamenal radeoffs beween rouing able size and newor diameer in peer-o-peer newors, Proc. of IEEE Infocom 23, vol.-3, pp , 23. [9] C. Qu, W. Nedl, and M. Kriesell, Cayley DHTs - a group-heoreic framewor for analyzing dhs based on cayley graphs, Parallel and Disribued Processing and Applicaions. Second Inernaional Symposium, ISPA 24, 24. [] D. J. Was and S. H. Srogaz, Collecive Dynamics of Small-world Newors, Naure, 998. [] K. Aberer, L. O. Alima, A. Ghodsi, S. Girdziausas, S. Haridi, and M. Hauswirh, The essence of p2p: a reference archiecure for overlay newors, Fifh IEEE Inernaional Conference on Peer-o-Peer Compuing, 25. [2] WATTS D J. Small Worlds - The Dynamics of Newors beween Order and Randomness, ACM SIGMOD Record, vol. 97(3), pp , 22. [3] Hung-Chang Hsiao, Yung-Chih Lin, and Hao Liao. Building Small-World Peer-o-Peer Newors Based on Hierarchical Srucures, IEEE ransacions on parallel and disribued sysems, vol. 2(7), pp , 29. [4] F. T. Leighon, Inroducion o Parallel Algorihms and Archiecures: Arrays, Trees, Hypercubes, Morgan Kaufmann, 992. [5] W. Xiao and B. Parhami, Cayley graph as models of deerminisic small-world newors, Informaion Processing Leers, vol. 97(3), pp.5-7, ACADEMY PUBLISHER