Loop Subdivision Surface Based Progressive Interpolation

Transcription

1 Cheg FH (F), Fa FT, Lai SH et al Loop subdivisio surface based progressive iterpolatio JOURNAL OF COMPUTER SCIENCE AND TECHNOLOGY 24(1): Ja 2009 Loop Subdivisio Surface Based Progressive Iterpolatio Fu-Hua (Frak) Cheg 1, Feg-Tao Fa 1, Shu-Hua Lai 2, Cog-Li Huag 1, Jia-Xi Wag 1 ad Ju-Hai Yog 3 ( ) 1 Departmet of Computer Sciece, Uiversity of Ketucky, Lexigto, KY 40506, USA 2 Departmet of Mathematics ad Computer Sciece, Virgiia State Uiversity, Petersburg, VA 23806, USA 3 School of Software, Tsighua Uiversity, Beijig , Chia cheg@csukyedu; fegtaofa@ukyedu; slai@vsuedu; coglihuag@ukyedu; jiaxiwag@ukyedu; yogjh@tsighuaeduc Received Jue 4, 2008; revised December 11, 2008 Abstract A ew method for costructig iterpolatig Loop subdivisio surfaces is preseted The ew method is a extesio of the progressive iterpolatio techique for B-splies Give a triagular mesh M, the idea is to iteratively upgrade the vertices of M to geerate a ew cotrol mesh M such that limit surface of M would iterpolate M It ca be show that the iterative process is coverget for Loop subdivisio surfaces Hece, the method is well-defied The ew method has the advatages of both a local method ad a global method, ie, it ca hadle meshes of ay size ad ay topology while geeratig smooth iterpolatig subdivisio surfaces that faithfully resemble the shape of the give meshes The meshes cosidered here ca be ope or closed Keywords geometric modelig, Loop subdivisio surface, progressive iterpolatio 1 Itroductio Subdivisio surfaces are becomig popular i may areas such as aimatio, geometric modelig ad games because of their capability i represetig ay shape with oly oe surface A subdivisio surface is geerated by repeatedly refiig a cotrol mesh to get a limit surface Hece, a subdivisio surface is determied by the way the cotrol mesh is refied, ie, the subdivisio scheme A subdivisio scheme is called a iterpolatig scheme if the limit surface iterpolates the give cotrol mesh Otherwise, it is called a approximatig scheme Popular subdivisio schemes such as Catmull-Clark scheme [1], Doo-Sabi scheme [2], ad Loop scheme [3] are approximatig schemes while the Butterfly scheme [4], the improved Butterfly scheme [5] ad the Kobbelt scheme [6] are iterpolatig schemes A iterpolatig subdivisio scheme geerates ew vertices by performig local affie combiatios o earby vertices This approach is simple ad easy to implemet Because of its local property, it ca hadle meshes with a large umber of vertices However, sice o vertex is ever moved oce it is computed, ay distortio i the early stage of the subdivisio will persist This makes iterpolatig subdivisio schemes very sesitive to irregularity i the give mesh I additio, it is difficult for this approach to iterpolate ormals or derivatives O the other had, eve though subdivisio surfaces geerated by approximatig subdivisio schemes do ot iterpolate their cotrol meshes, it is possible to use this approach to geerate a subdivisio surface to iterpolate the vertices of a give mesh Oe method, called global optimizatio, does the work by buildig a global liear system with some fairess costraits to avoid udesired udulatios [7,8] The solutio to the global liear system is a cotrol mesh whose limit surface iterpolates the vertices of the give mesh Because of its global property, this method geerates smooth iterpolatig subdivisio surfaces that resemble the shape of the give meshes well But, for the same reaso, it is difficult for this method to hadle meshes with a large umber of vertices To avoid the computatioal cost of solvig a large system of liear equatios, several other methods have bee proposed A two-phase subdivisio method that works for meshes of ay size was preseted by Zheg ad Cai for Catmull-Clark scheme [9] A method Regular Paper Research work preseted here is supported by NSF of USA uder Grat No DMI The last author is supported by the Natioal Natural Sciece Foudatio of Chia uder Grat Nos ,

2 40 J Comput Sci & Techol, Ja 2009, Vol24, No1 proposed by Lai ad Cheg [10] for Catmull-Clark subdivisio scheme avoids the eed of solvig a system of liear equatios by utilizig the cocept of similarity i the costructio process Litke, Levi ad Schröder avoid the eed of solvig a system of liear equatio by quasi-iterpolatig the give mesh [11] However, a method that has the advatages of both a local method ad a global method is ot available yet I this paper a ew method for costructig a smooth Loop subdivisio surface that iterpolates the vertices of a give triagular mesh is preseted The ew method is a extesio of the progressive iterpolatio techique for B-splies [12 16] The idea is to iteratively upgrade the locatios of the give mesh vertices util a cotrol mesh whose limit surface iterpolates the give mesh is obtaied It ca be proved that the iterative iterpolatio process is coverget for Loop subdivisio surfaces Hece, the method is well-defied for Loop subdivisio surfaces The limit of the iterative iterpolatio process has the form of a global method But the cotrol poits of the limit surface ca be computed usig a local approach Therefore, the ew techique ejoys the advatages of both a local method ad a global method, ie, it ca hadle meshes of ay size ad ay topology while geeratig smooth iterpolatig subdivisio surfaces that faithfully resemble the shape of the give meshes The meshes cosidered here ca be ope or closed The remaiig part of the paper is arraged as follows I Sectio 2, we preset the cocept of progressive iterpolatio for Loop subdivisio surfaces o closed meshes I Sectio 3, we prove the covergece of this iterative iterpolatio process Extesio of this techique to ope meshes is cosidered i Sectio 4 Implemetatio issues ad test results are preseted i Sectio 5 Cocludig remarks are give i Sectio 6 2 Progressive Iterpolatio Usig Loop Subdivisio Surfaces for Closed Meshes The cocept of Loop subdivisio surface based progressive iterpolatio for closed meshes ca be described as follows Give a closed 3D triagular mesh M = M 0 To iterpolate the vertices of M 0 with a Loop subdivisio surface, oe eeds to fid a closed cotrol mesh M whose Loop surface passes through all the vertices of M 0 Istead of fidig the relatioship betwee the vertices of M ad the vertices of M 0 directly, we use a iterative process to do the job First, we cosider the Loop surface S 0 of M 0 For each vertex V 0 of M 0, we compute the distace betwee this vertex ad its limit poit V 0 o S 0, D 0 = V 0 V 0, ad add this distace to V 0 to get a ew vertex called V 1 as follows: V 1 = V 0 + D 0 The set of all the ew vertices is called M 1 We the cosider the Loop surface S 1 of M 1 ad repeat the same process I geeral, if V k is the ew locatio of V 0 after k iteratios of the above process ad M k is the set of all the ew V k s, the we cosider the Loop surface S k of M k We first compute the distace betwee V 0 ad the limit poit V k of V k o S k D k = V 0 V k (1) We the add this distace to V k to get V k+1 as follows: V k+1 = V k + D k (2) The set of ew vertices is called M k+1 This process geerates a sequece of cotrol meshes M k ad a sequece of correspodig Loop surfaces S k S k coverges to a iterpolatig surface of M 0 if the distace betwee S k ad M 0 coverges to zero Therefore the key task here is to prove that D k coverges to zero whe k teds to ifiity This will be doe i the ext sectio Note that for each iteratio i the above process, the mai cost is the computatio of the limit poit V k of V k o S k For a Loop surface, the limit poit of a cotrol vertex V with valece ca be calculated as follows: V = β V + (1 β )Q (3) where ad 3 β = ( 3 ( cos 2π ) 2 ) (3) Q = 1 Q i Q i are adjacet vertices of V This computatio ivolves earby vertices oly Hece the progressive iterpolatio process is a local method ad, cosequetly, ca hadle meshes of ay size Aother poit that should be poited out is, eve though this is a iterative process, oe does ot have to repeat each step strictly By fidig out whe the distace betwee M 0 ad S k would be smaller tha the give tolerace, oe ca go directly from M 0 to M k, skippig the testig steps i betwee

3 Fuhua (Frak) Cheg et al: Loop Subdivisio Surface Based Progressive Iterpolatio 41 3 Covergece of the Iterative Iterpolatio Process for Closed Meshes The proof eeds a fact about the eigevalues of the product of positive defiite matrices This fact is preseted i the followig lemma Lemma 1 Eigevalues of the product of positive defiite matrices are positive Proof The proof of Lemma 1 follows immediately from the fact that if P ad Q are square matrices of the same dimesio, the P Q ad QP have the same eigevalues (see, eg, [17], p14) To prove the covergece of the iterative iterpolatio process for Loop subdivisio surfaces, ote that at the (k + 1)-st step, the differece D k+1 ca be writte as: D k+1 = V 0 V k+1 = V 0 (β V k+1 + (1 β )Q k+1 ) where Q k+1 is the average of the adjacet vertices of V k+1 Q k+1 = 1 Q k+1 i By applyig (2) to V k+1 ad each Q k+1 i, we get Q k+1 i = Q k i + D k, Q k i D k+1 = V 0 (β V k + (1 β )Q k ) ( β D k + 1 β ) D k Q i ( = D k β D k + 1 β ) D k Q i [D k+1 1, D k+1 2,, D k+1 m ] T = (I B) where Q k is the average of the adjacet vertices of V k I matrix form, we have D k 1 D k 2 D k m D 0 1 = (I B) k+1 D 0 2 D 0 m where m is the umber of vertices i the give matrix, I is a idetity matrix ad B is a matrix of the followig form: β 1 1 β β i i β i β m The matrix B has the followig properties: 1) b ij 0, ad j=1 b ij = 1 (hece, B = 1); 2) there are i + 1 positive elemets i the i-th row, ad the positive elemets i each row are equal except the elemet o the diagoal lie; 3) if b ij = 0, the b ji = 0 Properties 1) ad 2) follow immediately from the formula of D k+1 i (3) Property 3) is true because if a vertex V i is a adjacet vertex to V j the V j is obviously a adjacet vertex to V i Due to these properties, we ca write the matrix B as B = DS where D is a diagoal matrix 1 β β D = 2 1 β m 0 m ad S is a symmetric matrix of the followig form: 1 β β 1 S = i β i 1 1 β i m β m 1 β m D is obviously positive defiite We will show that the matrix S is also positive defiite, a key poit i the covergece proof Theorem 1 The matrix S is positive defiite Proof To prove S is positive defiite, we have to show the quadric form f(x 1, x 2,, x m ) = X T SX is positive for ay o-zero X = (x 1, x 2,, x m ) T Note that if vertices V i ad V j are the edpoits of a edge e ij i the mesh, the s ij = s ji = 1 i the

4 42 J Comput Sci & Techol, Ja 2009, Vol24, No1 matrix S Hece, it is easy to see that f(x 1, x 2,, x ) = e ij 2x i x j + m i β i 1 β i x 2 i where e ij i the first term rages through all edges of the give mesh O the other had, if we use f ijr to represet a face with vertices V i, V j ad V r i the mesh, the sice a edge i a closed triagular mesh is shared by exactly two faces, the followig relatioship holds: m (x i + x j + x r ) 2 = 4x i x j + i x 2 i f ijr e ij where f ijr o the left had side rages through all faces of the give mesh The last term i the above equatio follows from the fact that a vertex with valece is shared by faces of the mesh Hece, f(x 1, x 2,, x ) ca be expressed as f(x 1, x 2,, x ) = f ijr m (x i + x j + x r ) 2 ( i β i ) i x 2 i 1 β i 2 From (3), it is easy to see that β 1 β 3 5 for 3 Hece, f(x 1, x 2,, x m ) is positive for ay oe zero X ad, cosequetly, S is positive defiite Based o the above lemma ad theorem, it is easy to coclude that the iterative iterpolatio process for Loop subdivisio is coverget Theorem 2 The iterative iterpolatio process for Loop subdivisio surface is coverget Proof The iterative process is coverget if ad oly if absolute value of the eigevalues of the matrix P = I B are all less tha 1, or all eigevalues λ i, 1 i m, of B are 0 < λ i 1 Sice B = 1, we have λ i 1 O the other had, sice B is the product of two positive defiite matrices D ad S, followig Lemma 1, all its eigevalues must be positive Hece, the iterative process is coverget 4 Extesio to Ope Meshes Loop subdivisio surface based progressive iterpolatio techique ca be used for ope meshes as well Actually the same advatages hold for ope meshes too Before we preset Loop subdivisio surface based progressive iterpolatio techique for ope meshes, we eed to review subdivisio rules for the boudaries of a ope mesh first Two kids of boudary rules have bee preseted for Loop subdivisio i the literature [18 21] I this paper, we follow the rules preseted i [18, 19] These rules, together with the Loop subdivisio schemes, geerate a smooth surface that is C 1 cotiuous at the boudaries [18,19] For these rules to work, the vertices o the boudary are divided ito two categories: regular vertices ad extraordiary vertices A boudary vertex is called a regular vertex if its valece is 4, as the oe show i Fig1(a) Otherwise, a boudary vertex is called a extraordiary vertex For each existig boudary vertex, a ew vertex is computed as a liear combiatio of the existig vertex ad its two eighbors with weights 3/4, 1/8 ad 1/8, respectively This vertex formula applies to both regular vertices ad extraordiary vertices For each boudary edge, a ew edge vertex is geerated i two ways If the edpoits of the edge are both regular or both extraordiary, the the ew vertex is just the average of the edpoits If oe of them is regular ad the other oe is extraordiary, the the ew vertex is a liear combiatio of the regular vertex ad the extraordiary vertex with weights 5/8 ad 3/8, respectively, as i Fig1(c) Fig1 (a) Regular vertex (b) (d) Boudary subdivisio rules (e) (j) Limit poit forumlas The solid circular poits are regular vertices ad rectagle poits are extraordiary vertices

5 Fuhua (Frak) Cheg et al: Loop Subdivisio Surface Based Progressive Iterpolatio 43 Limit poits are computed usig two formulas, oe for regular vertices ad oe for extraordiary vertices, as i Figs 1(e) ad 1(f) These formulas require both eighbors to be regular vertices O boudaries of the iitial mesh, oe vertex could have either regular or extraordiary vertices There are totally 6 differet cofiguratios For each cofiguratio, we ca get a ew limit poit formula by combiig the boudary subdivisio rules with the stadard limit poits These 6 limit poit formulas are show i Figs 1(e) 1(j) Note that boudary subdivisio ivolves oly the boudary vertices Therefore iterpolatio ca be performed for boudary vertices first Oce we have all the boudary vertices, iterpolatio of iterior vertices is the performed without makig ay chages to the boudary vertices The fial mesh will iterpolate both the boudary vertices ad the iterior vertices exactly Iterpolatio of the boudary vertices is doe usig our progressive iterpolatio techique The covergece of the iterpolatio is guarateed A mesh could have several discoected closed boudaries, such as 6 i the pipe model i Fig3(a) For each closed boudary, a liear system of equatios ca be built based o the limit poit formulas for boudary vertices V B = EV B Sice every row is from oe of the 6 limit poit formulas, the E must be a strictly diagoally domiat matrix which meas j=1,j i e ij < e ii The eigevalues λ i of E satisfy λ i 1 for j=1 e ij = 1 Thus the eigevalues of E are i (0, 1] The advatage of our techique is very desirable No matter how may discoected boudaries there are, iterpolatio is doe for all boudaries at the same time just through the local geometric operatios It avoids explicitly solvig several liear equatios separately Iterpolatio of iterior vertices also uses the progressive iterpolatio techique Its covergece is also provable Let V = V B V I be the vertex set of the iitial mesh M, where V B ad V I are the set of boudary vertices ad iterior vertices, respectively If we add oe extra vertex q to M ad coects every boudary vertices with q, we get a closed mesh M with vertex set V C = V q Applyig the vertex limit poit formula (3) to iterior vertices, we get a liear equatio: V I = W V = W I V I + W B V B W I is a submatrix of W cosistig of V I colums of W correspodig to the iterior vertices W B is a similar submatrix correspodig to the boudary vertices W I is similar to B for closed meshes It ca be decomposed ito oe diagoal matrix D I ad a symmetric matrix S I For this ew closed mesh M, it is okay to apply the progressive iterpolatio techique developed i the previous sectios Therefore, the followig equatios hold = F V C That is, V I V B q V C W I 0 = W B 0 V B w q α q q It is clear that W is just a submatrix of F F is decomposed ito D C ad S C S C depeds oly o the topology of the mesh M is part of M Therefore, S I is just a mior of a positive defiite matrix S C Now it is clear that W I satisfies the coverget coditio for progressive iterpolatio The examples i Fig3 show iterpolatio results of ope meshes 5 Results The progressive iterpolatio process is implemeted for Loop subdivisio surfaces o a Widows platform usig OpeGL as the supportig graphics system Quite a few cases have bee tested Some of the closed cases (a hog, a rabbit, a tiger, a statue, a boy, a turtle ad a bird) are preseted i Fig2 All the data sets are ormalized, so that the boudig box of each data set is a uit cube For each closed case, the give mesh ad the costructed iterpolatig Loop surface are show The sizes of the data meshes, umbers of iteratios performed, maximum ad average errors of these cases are collected i Table 1 V I Table 1 Loop Surface Based Progressive Iterpolatio: Test Results Model No of No of Max Error Ave Error Vertices Iteratios Hog Rabbit Tiger Statue Boy Turtle Bird Ope mesh examples are show i Fig3 The performace is about the same as the closed mesh examples For istace, for the face model (299 vertices) show i Fig3(c), it takes 10 iteratios to reach a error of for boudary vertex iterpolatio ad also 10 iteratios to reach a error of

6 44 J Comput Sci & Techol, Ja 2009, Vol24, No1 Fig2 Examples of progressive iterpolatio usig Loop subdivisio surfaces (G-mesh give mesh; I-surface iterpolatig Loop surface) (a) (c) (e) (g) (i) (k) (m) G-mesh (b) (d) (f) (h) (j) (l) () I-surface

7 Fuhua (Frak) Cheg et al: Loop Subdivisio Surface Based Progressive Iterpolatio 45 Refereces Fig3 Ope mesh examples (I-surface iterpolatig Loop surface) Red solid dots are boudary vertices of the give mesh (a) (c) Give mesh (b) (d) I-surface (e) Differet view for iterior vertex iterpolatio by the ew pogressive iterpolatio techique From these results, it is easy to see that the progressive iterpolatio process is very efficiet ad ca hadle large meshes with ease This is so because of the expotetial covergece rate of the iterative process Aother poit that ca be made here is, although o fairess cotrol factor is added i the progressive iterative iterpolatio, the results show that it ca produce visually pleasig surface easily 6 Cocludig Remarks A progressive iterpolatio techique for Loop subdivisio surfaces is preseted ad its covergece is proved The limit of the iterative iterpolatio process has the form of a global method Therefore, the ew method ejoys the stregth of a global method O the other had, sice cotrol poits of the iterpolatig surface ca be computed usig a local approach, the ew method also ejoys the stregth of a local method Cosequetly, we have a subdivisio surface based iterpolatio techique that has the advatages of both a local method ad a global method The ew techique works for both ope ad closed meshes Our ext job is to ivestigate progressive iterpolatio for CatmullClark ad Doo-Sabi subdivisio surfaces Ackowledgemet Triagular meshes used i this paper are dowloaded from the Priceto Shape Bechmark[22] [1] Catmull E, Clark J Recursively geerated B-splie surfaces o arbitrary topological meshes Computer-Aided Desig, 1978, 10(6): [2] Doo D, Sabi M Behaviour of recursive divisio surfaces ear extraordiary poits Computer-Aided Desig, 1978, 10(6): [3] Loop C Smooth subdivisio surfaces based o triagles [Master Thesis] Dept Math, Uiv Utah, 1987 [4] Dy N, Levi D, Gregory J A A butterfly subdivisio scheme for surface iterpolatio with tesio cotrol ACM Tras Graphics, 1990, 9(2): [5] Zori D, Schro der P, Sweldes W Iterpolatig subdivisio for meshes with arbitrary topology Computer Graphics, A Cof Series, 1996, 30: [6] Kobbelt L Iterpolatory subdivisio o ope quadrilateral ets with arbitrary topology Comput Graph Forum, 1996, 5(3): [7] Halstead M, Kass M, DeRose T Efficiet, fair iterpolatio usig Catmull-Clark surfaces I Proc SIGGRAPH 1993, Aaheim, USA, August 1 6, 1993, pp47 61 [8] Nasri A H Surface iterpolatio o irregular etworks with ormal coditios Computer Aided Geometric Desig, 1991, 8: [9] Zheg J, Cai Y Y Iterpolatio over arbitrary topology meshes usig a two-phase subdivisio scheme IEEE Tras Visualizatio ad Computer Graphics, 2006, 12(3): [10] Lai S, Cheg F Similarity based iterpolatio usig CatmullClark subdivisio surfaces The Visual Computer, 2006, 22(9): [11] Litke N, Levi A, Schro der P Fittig subdivisio surfaces I Proc Visualizatio 2001, Sa Diego, USA, Oct 21 26, 2001, pp [12] de Boor C How does Agee s method work? I Proc 1979 Army Numerical Aalysis ad Computers Coferece, ARO Report 79-3, Army Research Office, pp [13] Li H, Bao H, Wag G Totally positive bases ad progressive iteratio approximatio Computer & Mathematics with Applicatios, 2005, 50: [14] Li H, Wag G, Dog C Costructig iterative o-uiform B-splie curve ad surface to fit data poits Sciece i Chia (Series F), 2003, 47(3): (i Chiese) [15] Qi D, Tia Z, Zhag Y, Zheg J B The method of umeric polish i curve fittig Acta Mathematica Siica, 1975, 18(3): (i Chiese) [16] Delgado J, Pe a J M Progressive iterative approximatio ad bases with the fastest covergece rates Computer Aided Geometric Desig, 2007, 24(1): [17] Magus I R, Neudecker H Matrix Differetial Calculus with Applicatios i Statistics ad Ecoometrics New York: Joh Wiley & Sos, 1988 [18] Hoppe H, DeRose T, Duchamp T, Halstead M, Ji H, McDoald J, Schweitzer J, Stuetzle W Piecewise smooth surface recostructio I Proc the 21st Aual Coferece o Computer Graphics ad Iteractive Techiques (SIGGRAPH 94), Orlado, USA, July 24 29, 1994, pp [19] Schweitzer J E Aalysis ad applicatio of subdivisio surfaces [PhD Dissertatio] Uiversity of Washigto, Seattle, 1996 [20] Zori D, Schro der P, DeRose T, Kobbelt L, Levi A, Sweldes W Subdivisio for modelig ad aimatio I SIGGRAPH 2000 Course Notes, ACM SIGGRAPH, Bosto, USA, 2006, pp30 50

8 46 J Comput Sci & Techol, Ja 2009, Vol24, No1 [21] Bierma H, Levi A, Zori D Piecewise smooth subdivisio surfaces with ormal cotrol I Proc SIGGRAPH 2000, New Orleas, USA, July 23 28, 2000, pp [22] Shilae P, Mi P, Kazhda M, Fukhouser T The Priceto shape bechmark I Proc Shape Modelig It l, Jue 7 9, 2004, pp Fuhua (Frak) Cheg is professor of computer sciece ad director of the Graphics & Geometric Modelig Lab at the Uiversity of Ketucky He holds a PhD degree from the Ohio State Uiversity, 1982 His research iterests iclude computer aided geometric modelig, computer graphics, parallel computig i geometric modelig ad computer graphics, approximatio theory, ad collaborative CAD He is o the editorial board of Computer Aided Desig & Applicatios ad Joural of Computer Aided Desig & Computer Graphics Feg-Tao Fa is curretly a PhD cadidate i the Departmet of Computer Sciece at the Uiversity of Ketucky His research iterests iclude computer graphics, geometric modelig, GPU techiques ad computer visio Shu-Hua Lai is curretly a assistat professor of computer sciece at the Virgiia State Uiversity He received his PhD degree i computer sciece from the Uiversity of Ketucky i 2006 His research iterests iclude computer graphics ad computer aided geometric modelig Cog-Li Huag is curretly a Master cadidate i the Departmet of Computer Sciece, the Uiversity of Ketucky He received a BS degree i computer sciece from Sichua Uiversity, Chia His research iterests iclude computer graphics, solid modelig ad computer visio Jia-Xi Wag is curretly workig for Avet, Ic She received her BE degree of computer sciece from North Chia Electric Power Uiversity i 2005, ad her MS degree of computer sciece from Uiversity of Ketucky i 2008 Her research iterests iclude computer graphics ad computer etworks Ju-Hai Yog is curretly a professor i School of Software at Tsighua Uiversity, Chia He received his BS ad PhD degrees i computer sciece from Tsighua Uiversity, Chia, i 1996 ad 2001, respectively He held a visitig researcher positio i the Departmet of Computer Sciece at Hog Kog Uiversity of Sciece & Techology i 2000 He was a post doctoral fellow i the Departmet of Computer Sciece at the Uiversity of Ketucky from 2000 to 2002 His research iterests iclude computer-aided desig, computer graphics, computer aimatio, ad software egieerig