Ugail H (2007): "3D Data Modelling and Processing using Partial Differential Equations", Advances and Applications of Dezert-Smarandache Theory for

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1 Ugail H (7): "3D Daa Modelling and Proessing using Parial Differenial Equaions", Advanes and Appliaions of Dezer-Smarandahe Theory for Plausible and Paradoxial Reasoning for Informaion Fusion, Bulgarian Aademy of Sienes (ed.), pp

2 3D Daa Modelling and Proessing using Parial Differenial Equaions H. Ugail Shool of Informais Universiy of Bradford Bradford BD7 DP, UK Absra In his paper we disuss ehniques for 3D daa modelling and proessing where he daa are usually provided as poin louds whih arise from 3D sanning devies. The pariular approahes we adop in modelling 3D daa involves he use of Parial Differenial Equaions (PDEs). In pariular we show how he oninuous and disree versions of ellipi PDEs an be used for daa modelling. We show ha using PDEs i is inuiively possible o model daa orresponding o omplex senes. Furhermore, we show ha daa an be sored in ompa forma in he form of PDE boundary ondiions. In order o demonsrae he mehodology we ulise several examples of praial naure. Keywords: 3D Daa Modelling, Parial Differenial Equaions, Daa Sorage. Inroduion Today here exis a wide variey of 3D sanning ehnologies whih enable us o generae he 3D geomery of omplex real world senes. The abiliy o generae suh powerful digial represenaions of obes hrough poin loud daa has enormous number of poenial uses. These inlude generaion of virual environmens for errain visualisaion, resoraions of arhived obes wih senimenal values and visualisaion of obes wihin virual environmens whih would oherwise be diffiul or impossible (for example maor organs in he human body). An example of suh 3D digial model represenaion is ha of Mihelangelo s David produed by Sanford s Digial Mihelangelo Proe[8]. This pariular geomery model onsiss of 56 million riangles and is reonsrued from 4 range maps using a disane field wih mm auray.. I is worh poining ou ha he proessing of daa afer aquisiion is an imporan sep in he proess of generaing he omplee digial represenaion of a given obe or sene. Thus, here are imporan onsideraions o be given in he proessing of daa. These inlude he abiliy o be able o represen he daa hrough an effiien mehanism by means of an underlying mahemaial sruure and he abiliy o be able o sore he daa in a ompa form. Exising mehods for proessing 3D apured daa involve he use of splines for daa fiing[7,], polygonal mesh modelling[3] and subdivision[9]. The ehniques based on hese mehods an be eiher applied o he 3D apured daa alone or heir ombinaions are uilised for reonsruing he geomery of he shape onaining he original poin loud daa. For example, mesh based mehods of surfae reonsruion[] enable generaion of smooh surfaes using an ordered D array of deph samples, whih are someimes referred as range image. Anoher mehod involves daa inerpolaion ehniques based on he original samples using alpha shapes[6], alhough suh inerpolaion ehniques may no serve useful for reonsruing obes from noisy daa. Oher ehniques involve he use of polygonal mesh modelling ehniques[4] based on evolving level se using 3D signed disane funions[3]. Variaions of hese ehniques also uses 3D radial basis funions based on level se formulaions[]. In his paper we disuss ehniques for 3D daa proessing using Parial Differenial Equaions (PDEs). We show how boh oninuous and disree PDEs an be uilised o reonsru smooh surfaes onaining he poin loud daa whih are apured from a 3D apure devie. Furhermore, we show how suh daa an be sored effiienly using he underlying mahemaial represenaion based on he PDE formulaions. For he sake demonsraion of our work we uilise ehniques based on PDEs whereby we ake an ellipi PDE based on he sandard Biharmoni equaion whih is widely known as he PDE mehod[,]. Thus, we show how we an represen an exising geomery of an obe as auraely as possible wih minimal shape daa informaion. A series of examples demonsraing he ehniques are presened. The paper is organized as follows. Firs in Seion we presen he mahemaial desripion of PDE ehniques for 3D modelling for boh he oninuous and disree ases. Then in Seion 3 we disuss some praial examples. Finally we provide some onluding remarks. Ellipi PDEs for 3D daa modelling In geomeri design, i is ommon praie o define urves and surfaes using a parameri represenaion. Thus, surfaes are defined in erms of wo parameers

3 u and v so ha any poin on he surfae is given by an expression of he form, ( u, () Equaion () an be viewed as a mapping from a domain Ω in he ( u, parameer spae o Eulidean 3-spae. In he ase of he PDE surfaes his mapping is defined by means of a parial differenial operaor, Luv m ( ) F( u,, () where he ellipi parial differenial operaor L is of degree m. Thus, effeively, surfae design is reaed as an appropriaely posed boundary-value problem wih appropriae boundary ondiions imposed on Ω, he boundary of Ω. The hosen PDE is solved sube o a se of boundary ondiions whih are usually defined a he edges of he surfae pah. I is imporan o highligh ha he ellipi PDE operaor disussed in Equaion () is a smoohing operaor in whih given a se of daa poins his operaor produes new inerpolaion poins whih are in some sense an average of he surrounding daa poins. Hene, his proess ensures ha he surfaes generaed hrough he ellipi PDEs are smooh and fair. Below we disuss how daa modelling an be underaken hrough he PDE formulaion. In pariular, we disuss boh oninuous and disree versions of he PDE operaor and is appliabiliy for 3D daa modelling.. Daa Modelling using Coninuous PDEs In he ase of using oninuous PDEs, in priniple, one an ake any PDE based on he ellipi operaor. For he sake of demonsraion here we disuss he PDE based on he Biharmoni operaor. Thus we ake, ( u, ( u) os( n ( u)sin( n, (4) where n is an ineger. The form of (u) sube o he general boundary ondiions is given as, nu nu nu e + ue + 3e + 4 nu ( u) ue, (5) where,, 3 and 4 are given by, 4 P P P 3 [ P ( e + e + e ) n n ( ne + e + ne e ) n ne P ( e e )] / d, s [ P ( e + ne n) n n ( ne + e ne ) Ps ( e e + ) n n ( e e e )] / d, P 4n [ P ( e e + e e ) n n ( ne + e + ne e ) s ne n ( e e )] / d, 4n [ P ( ne + e ne ) n 4n ( e + ne ne ) s ( e e e ) n ( e e + e )] / d, 4n where d e 4n e e +. Given he above explii soluion sheme he unknowns,, 3 and 4 an be deermined by he imposed ondiions a u i where u i. (6) (7) (8) (9) u + v ( u,. (3) The above equaion an be solved sube o boundary ondiions on he funion ( u, and is firs derivaives whih an be imposed a he edges of he surfae pah orresponding o he daa se. Alernaively, he boundary ondiions an be defined suh ha funion ( u, values an be speified a he edges and some inermediae posiions wihin he surfae pah orresponding o he daa poins. There are various ehniques for solving he equaions of he form desribed in (3). These ehniques range from analyi soluions hrough o sophisiaed numerial ehniques. For example, for periodi boundary ondiions (e.g. u, v π ) a losed form soluion of Equaion (3) allows ( u, o be wrien as, Figure Boundary ondiions and he surfae generaed for he Biharmoni equaion.

4 Figure shows he shape of he surfae generaed whereby he Biharmoni Equaion () is solved sube o he ondiions shown in Figure. Noe here we have aken four boundary ondiions where he ondiions p and p define he edges of he surfae pah while he ondiions d and d define some inermediae posiions wihin he surfae pah. One should noe ha he resuling surfae pah onains all he boundary urves shown in Figure, hus produing a smooh inerpolaion of all he boundary urves. Furhermore, i is also imporan o highligh ha he resuling surfae pah is solely onrolled by he four boundary urves. shown in Figure () and (d). This proess is hen repeaed o generae he surfaes shown in Figure (e) and (f). One an see ha his proess of generaing a smooh mesh is very effiien and inuiive. I is also noeworhy ha he resuling meshes, hrough he appliaion of he disree PDE operaor, is generally smooh.. Daa Modelling using Disree PDEs In his seion we show how he disree ellipi PDE operaor an be uilised o reonsru smooh shapes orresponding o a given disree daa se arising from a 3D sanning devie. Taking he form of he ellipi operaor disussed in Equaion (), one an simply wrie down is disree finie differene approximaion for a given neighborhood of daa poins. For example, for he Laplae operaor, by denoing a node in he square finie differene mesh by he inegers i,, wih mesh spaing h so ha he oordinaes u ih and v h, we have, i+, i, + + h + h, whih we an wrie down as, [ i+, i, + ]. () 4 Equaion () an be onvenienly represened in he form of a marix known as he Harmoni mask and is of he form, () (d) 4. In a similar fashion for he Biharmoni equaion he mask akes he following form, Figure shows how his ehnique an be uilised for generaion of smooh surfaes. Saring from a give daa se as shown in Figure and he disree Biharmoni operaor is applied o generae he surfaes (e) (f) Figure Surfaes generaed using he disree Biharmoni equaion. 3 Examples of 3D Daa Modelling In his seion we show examples involving real life appliaions of 3D daa modelling using he PDE approahes disussed earlier. In pariular, we show wo examples, he firs one demonsraing he ase of he oninuous PDE and he seond example demonsraing he disree ase.

5 well as have an equal number of urve poins for eah profile urve. One he profile urves are available in he appropriae forma hey are hen pu ino groups of four wih ommon urves in beween so ha appropriae number of funion alls o Equaion (3) an be deermined. Figure 3 shows he resuling surfae pah generaed using he urves shown in Figure 3 arising from he for he san daa. As one an easily visualise, he reonsrued surfae learly represens he fae orresponding o he faial boundary urves. Figure 3 Generaion of smooh surfae of a human fae from apured 3D daa using he oninuous Biharmoni equaion. In he firs example we show he modelling of a human fae from 3D apured daa. Figure 3 shows a series of profile urves whih have been auomaially exraed from he 3D san daa. One he urves are exraed, for eah urve, we hen fi a spline of he form Pi Ci Bi where B i is a ubi polynomial and C i are i he orresponding onrol poins. This proess of urve fiing o he exraed disree urve daa enables us o have a smooh urve passing hrough he disree daa as Figure 4 Generaion of a errain surfae from apured 3D daa using he disree Biharmoni equaion. In he seond example we show he modelling of a errain surfae of a rural field whih is apured using a 3D range sanning devie. Figure 4 shows he poin loud daa orresponding o he errain. The proess of fiing a surfae for his daa involves an iniial riangulaion of he surfae. This iniial riangulaion is performed using he Delaunay riangulaion whih provides an iniial mesh. Then for a given poin, in he daa se, based on is neighbouring poins, he disree Biharmoni operaor is applied. This produes a smooh inerpolaion of he original mesh poins.

6 Figure 4 shows he resuling surfae pah generaed using he using he disree PDE operaor for he san daa shown in Figure 4. As one an easily visualise, he reonsrued surfae learly represens he original san daa. 4 Conlusions In his paper we have disussed ehniques for 3D daa modelling and proessing where he daa are usually provided as poin louds whih arise from 3D sanning devies. For his purpose we have adoped he mehod of Parial Differenial Equaions (PDEs). Thus, we have shown ha boh he oninuous and disree versions ellipi PDEs an be used for daa modelling. To demonsrae our ehniques we have shown how daa orresponding o real world examples an be modelled using PDEs. As par of he fuure work we are ineresed in developing PDE based ehniques whih uilses differen versions of PDEs where geomery onsruion and smoohing an be seleively performed on a given san daa. We also wish o underake furher work in order o develop PDE based ehniques for fas proessing and effiien sorage of very large daa ses. [8] M. Levoy, K. Pull B. Curless, S. Rusinkiewiz, D. Koller, L. Pereira, M. Ginzon, S. Anderson, J. Davis, J. Ginsberg, J. Shade, and D. Fulk. The Digial Mihelangelo Proe: 3D sanning of large saues, in SIGGRAPH, Compuer Graphis Proeedings, pp. 3 44, ACM,. [9] C. T. Loop. Smooh Subdivision Surfaes based on Triangles, Maser's Thesis, Deparmen of Mahemais, Universiy of Uah, 987. [] J. Monerde and H. Ugail, A General 4h-Order PDE Mehod o Generae Bézier Surfaes from he Boundary, Compuer Aided Geomeri Design, 3() :8-5, 6. [] L. Piegl, Reursive Algorihms for he Represenaion of Parameri Curves and Surfaes, Compuer Aided Design, 7(5):5-9, 985. [] G. Turk, M. Levoy, Zippered Polygon Meshes from Range Images, Proeedings of SIGGRAPH 94, ACM, 994. [3] R. Whiaker, A Level-se Approah o 3D Reonsruion from Range Daa, Inernaional Journal of Compuer Vision, 9(3) :3-3, 998. Aknowledgemens The auhor wishes o aknowledge he finanial suppor reeived from he UK Engineering and Physial Sienes Researh Counil grans EP/C58/ and EP/D7/ hrough whih his work was ompleed. Referenes [] M. I. G. Bloor and M.J. Wilson, Generaing Blend Surfaes using Parial Differenial Equaions, Compuer Aided Design, (3):65-7, 989. [] J. C. Carr, R. K. Beason, J. B. Cherrie, T. J. Mihell, W. R. Frigh, B. C. MCallum and T. R. Evans, Reonsruion and Represenaion of 3D Obes wih Radial Basis Funions, Proeedings of SIGGRAPH, ACM,. [3] E. Camull and J. Clark, Reursively Generaed B- Spline Surfaes on Arbirary Topologial Meshes, Compuer Aided Design, (6):35-355, 978. [4] Y. Chen and G. Medion Desripion of Complex Obes from Muliple Range Images Using an Inflaing Balloon Model, Compuer Vision and Image Undersanding, 6(3):35-334, 995. [5] T. K. Dey, J. Giesen, J. Hudson, Delaunay Based Shape Reonsruion from Large Daa, in Proeedings of Symposium on Parallel and Large Daa Visualizaion and Graphis Surfaes and Parallel Rendering, ACM,. [6] H. Edelsbrunner, E. P. Muke, Three-Dimensional Alpha Shapes, ACM Transaions on Graphis, 3() :43-7, 994. [7] G. Farin, From onis o NURBS, A Tuorial and Survey, IEEE Compuer Graphis and Appliaions, (5):78-86, 99.