Applcaton of Learnng Machne Methods to 3 D Object Modelng Crstna Garca, and José Al Moreno Laboratoro de Computacón Emergente, Facultades de Cencas e Ingenería, Unversdad Central de Venezuela. Caracas, Venezuela. {cgarca, jose}@neurona.cens.ucv.ve http://neurona.cens.ucv.ve/laboratoro Abstract. Three dfferent machne learnng algorthms appled to 3D object modelng are compared. The methods consdered, (Support Vector Machne, Growng Grd and Kohonen feature Map) were compared n ther capacty of modelng the surface of several synthetc and expermental 3D objects. The prelmnary expermental results show that wth slght modfcatons these learnng algorthms can be very well adapted to the task of object modelng. In partcular the Support Vector Machne Kernel method seems to be a very promsng tool. 1 Introducton Object modelng s a very mportant technque of computer graphcs and has been object of study for more than two decades. The technque has found a broad range of applcatons, from computer-aded desgn and computer drawng to mage analyss and computer anmaton. In the lterature several approaches to object modelng can be found. One approach s to look for a rgd model that best fts the data set, an alternate one s to deform a model to ft the data. The later ones are known as dynamcally deformable models. They were frst ntroduced by Kass, Wtkn and Terzopoulos [8] and have created much nterest snce then because of ther clay lke behavor. A complete survey can be found n [7]. To acheve 3D object modelng, t s desrable that the appled method show a great flexblty n terms of shape and topology representaton capacty. A deformable model wth ths characterstc s the smplex mesh developed by Delngette [4][5]. It s a non parametrc deformable model, that can take vrtually any shape of any topology wth a very low computatonal cost as compared to other deformable models wth the same flexblty. It s well known that several machne learnng algorthms posses the ablty to nduce effcently, n a more or less automatc manner, arbtrary surfaces and topology preservng mappngs on arbtrary dmensonal nosy data. The proposed approach to 3D object modelng takes advantage of ths low cost surface representaton capacty nherent to these learnng algorthms.
In ths paper we descrbe the applcaton of the Support Vector Kernel Method (SVM) [3], the Kohonen Self-Organzng Feature Maps [9] and the Growng Grd [6] machne learnng algorthms to model objects from data sets that contan nformaton from one or more 3D objects. In general the applcaton of the algorthms start wth a cloud of 3D data ponts and no a pror nformaton regardng the shape or topology of the object or objects n the scene. These data ponts are appled to the learnng algorthms, that wth smple modfcatons on the learnng rules, generate adaptvely a surface adjusted to the surface ponts. In case of the Kohonen Feature Map and the Growng Grd a sphercal network s randomly ntalzed n the nteror of the cloud of ponts. Then the learnng rule s appled and the network deforms and grows (Growng Grd) untl t reaches stablty at the surface of the cloud of ponts. In case of the Support Vector Kernel Method the data ponts are mapped to a hgh dmensonal feature space, nduced by a Gaussan kernel, where support vectors are used to defne a sphere enclosng them. The boundary of the sphere forms n data space (3D) a set of closed surfaces contanng the cloud of ponts. As the wdth parameter of the Gaussan kernel s ncreased, these surfaces ft the data more tghtly and splttng of surfaces can occur allowng the modelng of several objects n the scene. At the end of each of these processes we wll have a model for each object. The organzaton of the paper s as follows: n the second secton, an overvew of the appled learnng machne methods and ther modfcatons for 3D object modelng are dscussed. In the thrd secton expermental results are presented and fnally n the fourth secton the conclusons and further work are descrbed. 2 Learnng Machne Methods 3D Object modelng s an ll defned problem for whch there exst numerous methods [2],[7],[10],[12]. In the present approach, learnng machne methods that produce data clusterng are appled to surface modelng. These clusterng methods can be based on parametrc models or can be non parametrc. Parametrc algorthms are usually lmted n ther expressve power,.e. a certan cluster structure s assumed. In ths work experments of surface modelng wth two parametrc (Kohonen Feature Map and Growng Grd) and a non parametrc (SVM Kernel Method) clusterng algorthms are presented and compared. In what follows a bref dscusson of each learnng method s made. 2.1 Support Vector Kernel Method The dea behnd the applcaton of the support vector formalsm [3] to object modelng follows the SVM clusterng method n [1]. Let {x} be a data set of N 3D ponts representng the object n the scene. Usng a nonlnear transformaton φ from the nput 3D space to some hgh dmensonal feature space, we look for the smallest enclosng sphere of radus R, descrbed by the constrants:
φ 2 2 ( ) a R x (1) Where use s made of the Eucldean norm and a s the center of the sphere. Soft constrants are ncorporated by addng slack varables ξ ι 2 2 ( x ) a R + ξ ξ 0 φ ; (2) Ths problem s solved n the formalsm of Lagrange by ntroducng and mnmzng the Lagrangan L 2 ( x ) a β ξ µ + C 2 2 = R R + ξ φ ξ (3) Where µ, β are postve Lagrange multplers, C s a constant and the last term s a penalty term. The statonarty of the Lagrangean wth respect to R and ξ ι leads to the followng relatons: a β =1 (4) ( x ) = βφ (5) β C µ = (6) The Karush, Kuhn and Tucker complementary condtons result n: R ξ µ = 0 (7) 2 2 ξ φ ( x ) a (8) + β From these relatons t s easy to verfy that a pont x wth ξ > 0 s outsde the sphere n feature space, such ponts have µ = 0 and β = C. A pont wth ξ = 0 s nsde or on the surface of the sphere n feature space. To be on the surface t must
have β not equal to zero. Ponts wth 0 < β < C wll be referred to as Support Vectors. The above relatons allow the dervaton of the Wolf dual of the Lagrangan: ( x ) φ( x ) β β φ( x ) φ( x ) W = β φ.. (9) j j j and the problem s solved by maxmzng the dual. The dot products φ(x ). φ(x j ) can be convenently replaced by a sutable Mercer kernel K(x,x j ) n ths way the Wolf dual can be rewrtten as ( x, x ) β K ( x x ) W = K β, β (10) j j j The Lagrange multplers β are obtaned by maxmzng ths expresson. Ths s computatonally done by the applcaton of the SMO algorthm [11]. In the approach to object modelng wth SVM the Gaussan kernel s employed K ( ) 2 x, x = exp q x x j j (11) s In feature space the square of the dstance of each pont to the center of the sphere R 2 2 ( x ) = φ ( x ) a (12) The radus of the sphere s { R( x ) x s a support vector} R = (13) In practce the average over all support vectors s taken. The surface of the clouds of ponts n 3D data space s gven by the set: { R( x) R} x = (14)
2.2 Kohonen Feature Map The Kohonen Feature Map [9] s an unsupervsed learnng machne that typcally conssts of one layer of neurons n a network of constraned topology. In ts learnng phase the weght vectors are randomly ntalzed. Durng learnng, for every nput vector the Best Matchng Unt (BMU) s determned. The BMU and a number of unts n a neghborhood of the BMU, n the constraned topologcal network, are adjusted n such a way that the weght vectors of the unts resemble the nput vector more closely. The unts surroundng the BMU are adjusted less strongly, accordng to the dstance they have to the BMU. The weght vectors w j (t) are adjusted by applyng the Kohonen Learnng Rule: w j ( ) ( t + ) = w ( t) + ε( t).φ ( t). x( t) w ( t) 1 (15) j j j where the learnng rate ε(t) s a lnear tme decreasng functon, x(t) s the nput vector at tme t, and Φ j (t) s the neghborhood functon wth the form: ( ) 2 2 t = exp j σ ( t) Φ j / (16) Here j s the poston of the BMU n the topologcal network and the poston of a unt n ts neghborhood. The wdth parameter σ(t) s also a lnear tme decreasng functon. It can be noted that snce the learnng rate and the wdth parameter both decrease n tme the adjustments made on the weght vectors become smaller as the tranng progresses. On a more abstract level, ths means that the map wll become more stable n the later stages of the tranng. It can be seen that the result of learnng s that the weght vectors of the unts resemble the tranng vectors. In ths way the Kohonen Feature Map produces a clusterng of the n-dmensonal nput vectors onto the topologcal network. In order to model the surface of the nput data ponts, two modfcatons to the Kohonen Feature Map are ntroduced: (a) The constraned topologcal network chosen s a sphercal grd. (b) the weght vectors of the BMU and ts neghbors are actualzed only f the nput vector s external to the actual sphercal grd. The mplementaton used three dfferent rates to decrease parameters ε and σ: a frst rapd decreasng stage a mddle slower but longer stage and a fnal short fne tunng stage.
2.3 Growng Grd Method Ths model [6] s an enhancement of the feature map. The man dfference s that the ntally constraned network topology grows durng the learnng process. The ntal archtecture of the unts s a constraned topologcal network wth a small number of unts. A seres of adaptaton steps, smlar to the Kohonen learnng rule, are executed n order to update the weght vectors of the unts and to gather local error nformaton at each unt. Ths error nformaton s used to decde where to nsert new unts. A new unt s always nserted by splttng the longest edge connecton emanatng from the unt wth maxmum accumulated error. In dong ths, addtonal unts and edges are nserted such that the resultng topologcal structure of the network s conserved. The mplemented process nvolves three dfferent phases: an ntal phase where the number of unts s held constant allowng the grd to stretch, a phase where new unts are nserted and the grd grows, and fnally a fne tunng phase. 3. Expermental Results The learnng algorthms for modelng are appled on several synthetc objects represented n the form of clouds of 3000 ponts each obtaned from the applcaton of a 3D lemnscate wth 4, 5 and 6 foc. The ntal and fnal parameters used for the Kohonen Feature Map were: ε 0 = 0.5, ε f = 0.0001, σ 0 = 4, σ f = 0.01, wth a total 10 6 teratons. For the Growng Grd: ε 0 = 0.08, ε f = 0.05, a constant σ = 0.7, the number of teratons s dstrbuted as: 500 * grd sze n the stretchng phase; 500 * grd sze n the growng phase; and 200 * grd sze teratons for fne tunng. The parameters n the SVM algorthm are C = 1.0 and q = 2 (Fg. 1 and Fg. 3), q = 4 (Fg. 2) and q = 0.00083 (Fg. 4). In fgure 1 the orgnal surface (5 foc lemnscate) and the surface models resultng from the applcaton of the three learnng methods (a) Kohonen Feature Map (b) Growng Grd and (c) SVM algorthms are shown. The Kohonen Map conssted on a sphercal topologcal network wth 182 unts randomly ntalzed n the nteror of the cloud of ponts. The Growng Grd was also a sphercal network wth 6 ntal unts randomly ntalzed n the nteror of the cloud, the network was grown up 162 unts. The SVM model was constructed wth 49 support vectors. It can be apprecated that the three algorthms acheve a reasonable modelng of the orgnal object. The best results are produced by the Growng Grd and SVM methods.
Fg. 1. Results of the three machne learnng methods n the modelng of a surface from a sold generated by a 5 foc lemnscate. (a) Kohonen Feature Map (b) Growng Grd and (c) SVM Kernel method. In fgure 2 the orgnal surface (6 foc lemnscate) and the surface models resultng from de applcaton of the three learnng methods (a) Kohonen Feature Map (b) Growng Grd and (c) SVM algorthms are shown. The Kohonen Map conssted on a sphercal topologcal network wth 266 unts randomly ntalzed n the nteror of the cloud of ponts. The Growng Grd was also a sphercal network wth 6 ntal unts randomly ntalzed n the nteror of the cloud the network and grown up to 338 unts. The SVM model was constructed wth 77 support vectors. Fg. 2. Results of the three machne learnng methods n the modelng of a surface from a sold generated by a 6 foc lemnscate. (a) Kohonen Feature Map (b) Growng Grd and (c) SVM Kernel method.
It can be apprecated that agan n ths experment the three algorthms acheve a reasonable modelng of the orgnal object. The best results are produced by the Growng Grd and SVM methods. In fgure 3 the orgnal surface (4 foc lemnscate) and the surface models resultng from the applcaton of the two learnng methods (a) Growng Grd and (b) SVM algorthms are shown. For ths case the Growng Grd was a sphercal network wth 6 ntal unts randomly ntalzed n the nteror of the cloud of ponts. The network was grown up 134 unts. The SVM model was constructed wth 38 support vectors. In ths experment the object s of partcular nterest snce t conssts of two parts joned by a pont, an approxmaton to a scene of two separate objects. In ths case both the Kohonen feature Map and the Growng Grd do not produce a good model of the surface. However, t can be apprecated that the SVM method acheves a very good model and s clearly superor. In fgure 4 the orgnal surface (expermental data from the left ventrcle of a human heart echocardogram) and the surface models resultng from the applcaton of the two learnng methods (a) Growng Grd and (b) SVM algorthms are shown. For ths case the Growng Grd was a sphercal network wth 6 ntal unts randomly ntalzed n the nteror of the cloud of ponts. The network was grown up 282 unts. The SVM model was constructed wth 46 support vectors. Fg. 3. Results of two machne learnng methods n the modelng of a surface from a sold generated by a 4 foc lemnscate. (a) Growng Grd and (b) SVM Kernel method.
Fg. 4. Results of two machne learnng methods n the modelng of a surface from expermental data (left ventrcle of a human heart echocardogram) (a) Growng Grd and (b) SVM Kernel method. 4. Conclusons and Future Work Ths work compared the applcaton of three machne learnng algorthms n the task of modelng 3D objects from a cloud of ponts that represents ether one or two objects. The experments show that the Kohonen Feature Map and the Growng Grd methods generate reasonable models for sngle objects wth smooth spherodal surfaces. If the object posses pronounced curvature changes n ts surface the modelng produced by these methods s not very good. An alternatve to ths result s to allow the number of unts n the network to ncrease together wth a systematc prune mechansm of the edges n order to take account of the abrupt changes on the surface. Ths modfcatons are theme of further work n case of the Growng Grd algorthm. On the other hand, the expermental results wth the Support Vector Kernel Method are very good. In the case of sngle smooth objects the algorthm produces a sparse (small number of support vectors) model for the objects. A very convenent result for computer graphcs manpulatons of the object. Ths extends to the case wth two objects n whch the method s able to produce models wth splt surfaces. A convenent modfcaton of the SVM algorthm would be to nclude a better control on the number of support vectors needed n the model. Ths possblty could hnder the roundng tendency observed n the SVM models and allow the modelng of abrupt changes of the surface as seen on the data of the echocardogram. To model multple objects t s necessary, n the Kohonen and Growng Grd methods, the applcaton of a splttng algorthm to the topologcal network. Ths
splttng algorthm s related to the systematc prune of the edges and s also theme of further work. The data sets for multple objects are not a representaton of a real scene. In a real scene the clusters of data ponts wll be connected by background nformaton (walls and floor). The future work wll also nclude the extenson of the actual algorthms to make them applcable to real scenes. Fnally t must be noted that n all cases the computatonal costs of the algorthms are not very hgh a fact that can lead to real tme mplementatons. References 1. A. Ben-Hur, D. Horn, H.T. Segelmann, V. Vapnk, A Support Vector Method for Clusterng. Internatonal Conference on Pattern Recognton 2000. 2. M. Bro Nelsen: Actve Nets and Cubes. Techncal Report, Insttute of Mathematcal Modelng Techncal Unversty of Denmark, November 1994. 3. N. Crstann, J. Shawe-Taylor, An Introducton to Support Vector Machnes. Cambrdge Unversty Press, 2000. 4. H. Delngette, General Object Reconstructon based on Smplex Meshes. Techncal Report 3111, INRIA, France. (1997) 5. H. Delngette, Smplex Meshes: A General Representaton for 3D Shape Reconstructon. Techncal Report 2214, INRIA, France. (1994) 6. B. Frtzke, Some Compettve Learnng Methods. Insttute for Neural Computaton, Ruhr- Unverstat Bochum, Draft Report, 1997. 7. S.F.F. Gbson, B. Mrtch, A Survey of Deformable Modelng n Computer Graphcs. Mtsubsh Electrc Research Laboratory Techncal Report, November (1997) 8. M. Kass, A. Wtkn, D. Terzopoulos, Snakes: Actve contour models. Int. J. Of Computer Vson. 2 (1998) 321 331 9. T. Kohonen, Self-Organzaton Assocatve Memory. (3 rd Edton). Sprnger-Verlag, Berln 1989. 10. D. Metaxas, Physcs based Deformable Models: Applcatons to Computer Vson, Graphcs, and Medcal Imagng. Kluwer Academc, Boston, 1997. 11. J. Platt, Fast Tranng of Support Vector Machnes Usng Sequental Mnmal Optmzaton. http://www.research.mcrosoft.com/~jplatt 12. K. Yoshno, T. Kawashma and Y. Aok: Dynamc Reconfguraton of Actve Net Structure, Proceedngs of Asan Conference n Computer Vson, pp. 159 162, 1993.