TRANSITIONAL OBJECT S SHAPE SIMULATION BY LAGRANGE S EQUATION AND FINITE ELEMENT METHOD

Transcription

1 RANSIIONAL OBJEC S SHAPE SIMULAION BY LAGRANGE S EQUAION AND FINIE ELEMEN MEHOD Raquel R. Pnho João M. R. S. avares LOME Laboratóro de Óptca e Mecânca Expermental FEUP Faculdade de Engenhara da Unversdade do Porto Rua Dr. Roberto Fras s/n Porto Portugal {rpnho tavares}@fe.up.pt Abstract hs paper presents a methodology to obtan transtonal objects shapes by a physcs-based smulaton. Gven two D/3D mages of dfferent objects or of the same object at dfferent nstants usng the Fnte Element Method two models are bult and the Lagrange s equaton s solved to smulate the nvolved deformaton. We used and compared two dfferent fnte elements to buld each objects model: the Sclaroff s soparametrc element and the lnear axal element. eywords Deformable Objects Movement/Deformaton Smulaton Matchng Fnte Element Method Lagrange s Equaton Modal Analyss.. INRODUCION o obtan transtonal objects shapes several technques may be used. However not all the obtaned results are coherent wth the objects physcal propertes. o physcally smulate objects movement/deformaton we use an approach that bulds physcal models for each gven object through the Fnte Element Method (FEM and then solve Lagrange s Equaton (LE (fgure. We have attended to what has already been done by solvng LE among deformable objects mages such as the analytcal determnaton of vbraton modes by Nastar n [] [] [3] the soparametrc fnte element developed by Sclaroff to model objects by the FEM n [4] [5] and the study done by avares wth real objects mages who also used Sclaroff s soparametrc element and groups of lnear axal elements to buld the objects models n [6] [7]. o buld the object s physcal models wth the FEM we use grouped lnear axal elements [6] [8] or Sclaroff s soparametrc element [4] [5] [6]. In ths paper we compare the transtonal objects shapes obtaned by each used model. o match the shapes nodes we use modal analyss [4] Input Data (pxels as nodes of a fnte element model Physcal model s bult wth FEM Grouped lnear axal elements or Sclaroff s Isoparametrc Element can be used Mass and Stfness Evaluaton t t t t MU + CU + U = R Mass and stffness matrces of each model can be evaluated Egenmodes are computed φ = ω Mφ Resoluton of the generalsed egenvalue/vector problem Output ranstonal objects shapes estmaton accordng to physcal propertes Resoluton of the Dynamc Equlbrum Equaton t t t t MU + CU + U = R Dsplacement feld s determned Estmates Estmate dampng matrx C Estmate appled charges on matched or unmatched nodes R Estmate ntal dsplacement U and velocty U Modal Matchng Dsplacements are analyzed n each modal space to match some nodes Fgure : Dagram of the approach used to estmate transtonal objects shapes.

2 [5] [6] [7] [9] [] whch generally obtans satsfactory matchng results although not all nodes may be successfully matched. However to solve LE all nodes should be matched so we propose a soluton based on the neghbourhood crteron that overcomes ths matchng problem. o solve LE we dd not consder any addtonal nformaton about the represented objects or of the movement/deformaton nvolved and so we had to estmate the ntal dsplacement and velocty as well as the mplct appled charges. In ths paper we also menton how these requred parameters were estmated. hs approach can be used to do physcs-based morphng to reconstruct 3D objects from D slces to smulate objects collson to nterpolate objects data etc.. FINIE ELEMENS USED In the presented work we consdered two dfferent object s type: D contour and 3D surface. o model each gven object we appled the FEM by usng Sclaroff s soparametrc element or a group of lnear axal elements [6]. Usng Sclaroff s soparametrc fnte element the bult models for the gven D contours delmt a vrtual object wth elastc propertes and the obtaned model s lke an elastc membrane [4] [6]. When a 3D surface object s modelled by the same element t s as f each feature pont s covered by a blob of rubbery materal [4] [6]. Instead f a group of lnear axal elements s used to buld the D/3D models then the contours/surfaces are shallow models whose edges are D fnte elements. So all the cells nterors are gnored [4] [6]. When a shape s modelled wth Sclaroff s soparametrc element t s not necessary to determne the pxels order; however when lnear axal elements are used t s necessary to determne the pxels correct order so that the groupng s correctly made [6]. In ths case to obtan the correct order a D Delaunay algorthm can be used [6] []. o buld the Sclaroff s element nterpolaton matrx H (whch relates the dstances between objects nodes Gaussan functons are used: X X /( σ g ( X = e where X s the functon s n -dmensonal center and σ ts standard devaton (whch controls data nteracton. he nterpolaton functons h are then gven by: m h ( X = a g ( X k k k = where a k are the nterpolaton coeffcents wth value (one at node and (zero at all other nodes and m s the number of nodes. he nterpolaton coeffcents a k can be determned by nvertng matrx G defned as: g( x g( xm G =. gm( x gm( xm hs way the nterpolaton matrx of Sclaroff s soparametrc element for a D shape wll be: h hm H( X = h h. m he mass and stffness matrces M and respectvely are then bult as usually [4] [5] [6] [9]. hus for D shapes we obtan: M aa aa ab M = M and = aa ba bb where M = ρπσ aa g aaj k jl kl kl ( + ξ πβ α = = a a xˆ yˆ g abj baj k jl kl kl kl kl = ( y ( y y x x ˆkl k x l = ˆkl k l and ρ s the materal s densty α β and ξ are obtaned from materal modulus of elastcty E and Posson s rato υ : υ E ( υ υ α = β = and ξ =. υ ( + υ ( υ ( υ For a 3D shape H wll be [4] [5] [6]: h hm H( x = h hm h h m and the 3D Sclaroff s soparametrc element mass matrx wll be defned as: Μ M = Μ Μ where M = ρπ σ A GA= ρπ σ G GG the elements of G are the square roots of G s elements and the stffness matrx s gven by: 3 = where are symmetrc submatrces gven by: j 3 + ξ xˆ + ξ ( yˆ + zˆ kl kl kl = π σβ a a g j k jl kl kl 3 + ξ yˆ + ξ ( xˆ + zˆ kl kl kl = π σβ a a g j k jl kl kl

3 3 + ξ zˆ + ξ ( xˆ + yˆ kl kl kl = π σβ a a g 33j k jl kl kl 3 π β( α + ξ a a xˆ yˆ g j k jl kl kl kl kl = 3 π β( α + ξ a a xˆ zˆ g 3j k jl kl kl kl kl and = 3 π β( α + ξ a a yˆ zˆ g 3j k jl kl kl kl kl =. he determnaton of the mass and stffness matrces for the group of axal lnear elements s a FEM standard and usual process (see for example [6] [8] []. he used dampng matrx C s a lnear combnaton of the mass and stffness matrces: C = ˆ α M + ˆ β where ˆα and ˆβ are respectvely the mass and stffness constrants determned by the desred crtcal dampng [3]. 3. MODAL MACHING o match the ntal and target shapes nodes each generalzed egenvalue/vector problem s solved: Φ= MΦΩ where for a D model wth m nodes: u ω u m Φ= [ φ φm ] = and Ω= v. ω m v m he mode shape vector φ descrbes the dsplacement ( uv of each node due to the vbraton mode and n the dagonal matrx Ω the frequency of vbratons squares are ncreasngly ordered. After buldng each modal matrx by comparng the dsplacement of each node n the respectve modal egenspace some nodes can be matched. o do so an affnty matrx Z s bult wth entres: Z j = u u j + v v j and best matches are ndcated by the mnmum values of assocated lnes and columns. he affnty between nodes and j wll be (zero f the match s perfect and wll ncrease as the match worsens. he process of matchng 3D shapes s entrely analogous [4] [6]. 4. RESOLUION OF HE DYNAMIC EQUILIBRIUM EQUAION In ths work to obtan the transtonal objects shapes attendng to physcal propertes we solve the Lagrange s Equaton: t t t t MU + CU + U = R ( wth the nodal dsplacements U descrbed as: U = X X where X s the poston of the th node n the ntal shape and X n the target one and U s the assocated dsplacement. o solve the dynamc equlbrum equaton several ntegraton methods can be used. In ths paper we present results obtaned by the Mode Superposton Method [9] [] [] [3]. 4.. Mode Superposton Method hs method allows the resoluton of LE ( usng only part of the model s modes n the computaton process. hs measure reduces the computatonal cost because t despses the movement s local components essentally assocated to nose. hs way the computaton nvolved s accelerated wthout great loss of nformaton as generally the hgh order frequency modes have very lttle effect on the movement [4] [6]. he Mode Superposton Method proposes the transformaton between the generalzed coordnates Φ used to transform the modal dsplacements X nto nodal dsplacements U and vce-versa: Ut ( = Φ Xt ( and therefore obtan the correspondng uncoupled equlbrum equatons: X ( t+φ CΦ Xt ( +Ω Xt ( =Φ Rt ( ( where X and X are respectvely the frst and second order dervatves of the modal dsplacement vector and: Φ Φ=Ω where I represents the dentty matrx. 5. ESIMAES In ths secton we menton the soluton adopted for some problems related to the lack of nformaton about the objects and about the deformaton nvolved. We wll present the solutons found to estmate the ntal dsplacement and velocty as well as the nodes mplct appled charges (successfully matched or not. 5. Intal Dsplacement and Velocty he used ntegraton method requres the ntal dsplacement and velocty vectors. he soluton found to estmate the frst one s to consder t as a part of the expected modal dsplacement:

4 X = cx( X X (3 where X represents the th component of the modal ntal dsplacement and c X s a constrant defned by the user accordng to each applcaton case. Smlarly the ntal modal velocty was estmated as a part of the ntal modal dsplacement: X = cvx (4 where c V s also an user defned constrant. As bgger values of c X e c V are used larger are the obtaned dsplacements and the target shape can be reached wth fewer steps [9] []. Note that n the results to be presented we have employed equal values of c X and c V for all nodes. 5. Implct Appled Charges he mplct appled charges on each matched node are supposed to be proportonal to the expected nodal dsplacement: R = k( X X j (5 where R s the appled charges vector th component X s the th node s poston n the target shape and X j n the j th shape and k s a global stffness constrant (once agan consdered equal for all components Unmatched Nodes When all nodes aren t successfully matched we can not use (5 and we have to estmate the mplct charges on the unmatched nodes by other means. So we used the neghbourhood crteron accordng to whch nodes must preserve ther order durng the deformaton process: we apply on the unmatched nodes Horn s estmated global rgd transformaton [6] [4]. hen we lnk each unmatched node s rgd transformed to ts neghbour nodes that are wthn a predefned dstance (fgure. B B' A' C' cx X = R f k k (6 X = f k = and unless k s null the ntal modal dsplacement of the matched nodes equals (3. For each unmatched node of the frst shape the ntal modal dsplacement s nfluenced such as the mplct appled charges by the nodes of the target shape that are near to the rgd transformed nodes. It should be notced that ths soluton s also based on the neghbourhood crteron. 6. EXPERIMENAL RESULS he used approach was ncluded n a software platform bult to develop and test mage processng and computer graphcs algorthms (for a detaled presentaton see [5]. hat system ntegrates some publc doman lbrares as for computer graphcs algorthms the V he Vsualzaton oolkt [5] [6]. For the D shapes we consdered contour objects to reduce the computatonal cost (although ths methodology also works wth nner objects ponts the results would be smlar [6]. For the frst example consder contours (composed by 3 nodes and (8 nodes obtaned from two real heart mages [6]. We used polyethylene as vrtual materal the global stffness constrant k = the tme step t = crtcal dampng levels between.5% and 3% and 75% of the modes n the LE resoluton. Under these crcumstances f Sclaroff s soparametrc element s used then 6 nodes are matched (fgure 3 and the fnal contour can be approached n 7 steps (fgure 4; otherwse f grouped lnear axal elements are used 4 nodes are matched (fgure 5 and the fnal contour s approached n 6 steps (fgure 6. Please note that the results presented wth Sclaroff s element or wth grouped axal elements probably are not the better ones. If we obtan a larger number of successful matches the estmated shapes can be more realstc (fgures 7 and 8. A B C Fgure : Used crteron to estmate the appled charges f A s matched wth A C wth C and B s an unmatched node. he ntal dsplacement soluton presented n (3 s also not defned for unmatched nodes. Once the mplct appled charges have been estmated we recursvely specfy the ntal dsplacement vector as part of the appled charges vector. So: Fgure 3: Matchng between contours and when Sclaroff s element s used. Fgure 4: All ntermedate shapes obtaned wth Sclaroff s element.

5 get closer to the target shape slowly (compare fgures 3 4. Once agan the obtaned results can get better f a larger number of successful matches s obtaned. 8. CONCLUSIONS AND FUURE WOR Fgure 5: Matchng between contours and when axal elements are used. Fgure 7: Matchng between contours and when Sclaroff s element s used and 7 nodes are successfully macthed. Fgure 6: Intermedate shapes obtaned n steps of wth axal elements. Fgure 8: All ntermedate shapes obtaned wth Sclaroff s element. Now consder surfaces 3 ( nodes and 4 (7 nodes obtaned from two real heart mages [6]. We used polyethylene as vrtual materal the global stffness constrant k = and the same tme step crtcal dampng and percentage of consdered modes. When Sclaroff s soparametrc element s used to buld the surfaces models 33 nodes are successfully matched as n fgure 9. When grouped lnear axal elements are used only 8 nodes are matched (fgure and the ntermedate shapes hs paper proposes a physcs-based approach to obtan the transtonal shapes of mage represented objects. o buld the gven objects models we used Sclaroff s soparametrc element or grouped lnear axal elements. We verfed that generally the models bult by lnear axal elements are more flexble and less dense than the ones bult wth the Sclaroff s element [6]. hs can make t more dffcult to do objects deformaton smulaton as the model s more elastc wtch can cause nstablty and so the obtaned shapes may not converge to the target shape. If all nodes of the gven objects were successfully matched then the mplct appled charges were consdered as proportonal to the dstances between each node and ts par. But when not all the shapes nodes were matched we ntroduced a soluton based on the neghbourhood crteron. So for each unmatched node we appled the estmated rgd transformaton between the gven shapes and appled t to the unmatched nodes then the obtaned pont s nfluenced by the matched nodes that are n a predefned dstance. In ths paper the dynamc equlbrum equaton has been solved usng the Mode Superposton Method. hs Fgure 9: Modal Matchng between surfaces 3 (left and 4 (rght when Sclaroff s soparametrc element s used. Fgure : 5 th Intermedate shape obtaned wth Sclaroff s element. Fgure : th Intermedate shape obtaned wth Sclaroff s element. Fgure : Modal Matchng between surfaces 3 (left and 4 (rght when grouped axal lnear elements are used. Fgure 3: 5 th Intermedate shape obtaned wth grouped axal lnear elements. Fgure 4: th Intermedate shape obtaned wth grouped axal lnear elements.

6 method allows that only a part of the model s modes are used whch reduces the computatonal effort (wth accuracy decrease but dependng on the applcaton s needs the obtaned results may be satsfactory. o apply the methodology proposed t was necessary to estmate the mplct appled charges. he adopted soluton can be mproved through the search of alternatve models to represent these charges (always attendng to the possblty of unmatched nodes. Another future task s the necessary valdaton of ths methodology wth real applcaton examples. For example n the doman of computer graphcs ths methodology may be used to smulate vrtual realty namely n the case of collson between deformable objects or to nterpolate objects data. 9. ACNOWLEDGMENS he frst author would lke to thank the support of the PhD grant SFRH / BD / 834 / 3 of the FC - Fundação de Cênca e ecnologa n Portugal.. REFERENCES [] C. Nastar and N. Ayache Fast Segmentaton rackng and Analyss of Deformable Objects. 4 th. Internatonal Conference on Computer Vson Berln Germany [] C. Nastar Modèles Phsques Déformables et Modes Vbratores pour l'analyse du Mouvement non-rgde dans les Images Multdmensonnelles L'École Natonale des Ponts et Chaussées 994. [3] C. Nastar N. Ayache Frequency-Based Nonrgd Moton Analyss: Applcaton to Four Dmensonal Medcal Images IEEE ransactons on Pattern Analyss and Machne Intellgence [4] S. Sclaroff Modal Matchng: A Method for Descrbng Comparng and Manpulatng Dgtal Sgnals MI 995. [5] S. Sclaroff A. Pentland Modal Matchng for Correspondence and Recognton IEEE ransactons on Pattern Analyss and Machne Intellgence [6] J. avares Análse de Movmento de Corpos Deformáves usando Vsão Computaconal FEUP. [7] J. avares J. Barbosa and A. J. Padlha Matchng Image Objects n Dynamc Pedobarography. RecPad' - th Portuguese Conference on Pattern Recognton Porto Portugal [8] L. Merovtch Elements of vbraton analyss (Mcgraw-Hll 986. [9] R. Pnho Determnação do Campo de Deslocamentos a partr de Imagens de Objectos Deformáves Unversdade do Porto. [] R. Pnho J. avares Resolução da Equação Dnâmca de Equlíbro entre Imagens de Objectos Deformáves. VII Congresso de Mecânca Aplcada e Computaconal Évora Portugal [] V. Foley H. Fener Computer graphcs (Addson- Wesley 99. []. Bathe Fnte element procedures (Prentce-Hall 996. [3] R. Cook D. Malkus M. Plesha Concepts and applcatons of fnte element analyss (Wley 989. [4] B. Horn Closed-Form Soluton of Absolute Orentaton usng Unt Quaternons Journal of the Optcal Socety Amerca A [5] W. Schroeder. Martn B. Lorensen he vsualzaton toolkt (3 rd. Edton tware. [6] W. Schroeder. Martn he V user s gude (tware Inc. 3.