Parameterzaton of Quadrlateral Meshes L Lu 1, CaMng Zhang 1,, and Frank Cheng 3 1 School of Computer Scence and Technology, Shandong Unversty, Jnan, Chna Department of Computer Scence and Technology, Shandong Economc Unversty, Jnan, Chna 3 Department of Computer Scence, College of Engneerng, Unversty of Kentucky, Amerca lul_79009@163.com Abstract. Low-dstorton parameterzaton of 3D meshes s a fundamental problem n computer graphcs. Several wdely used approaches have been presented for trangular meshes. But no drect parameterzaton technques are avalable for quadrlateral meshes yet. In ths paper, we present a parameterzaton technque for non-closed quadrlateral meshes based on mesh smplfcaton. The parameterzaton s done through a smplfy-project-embed process, and mnmzes both the local and global dstorton of the quadrlateral meshes. The new algorthm s very sutable for computer graphcs applcatons that requre parameterzaton wth low geometrc dstorton. Keywords: Parameterzaton, mesh smplfcaton, Gaussan curvature, optmzaton. 1 Introducton Parameterzaton s an mportant problem n Computer Graphcs and has applcatons n many areas, ncludng texture mappng [1], scattered data and surface fttng [], mult-resoluton modelng [3], remeshng [4], morphng [5], etc. Due to ts mportance n mesh applcatons, the subject of mesh parameterzaton has been well studed. Parameterzaton of a polygonal mesh n 3D space s the process of constructng a one-to-one mappng between the gven mesh and a sutable D doman. Two major paradgms used n mesh parameterzaton are energy functonal mnmzaton and the convex combnaton approach. Mallot proposed a method to mnmze the norm of the Green-Lagrange deformaton tensor based on elastcty theory [6]. The harmonc embeddng used by Eck mnmzes the metrc dsperson nstead of elastcty [3]. Lévy proposed an energy functonal mnmzaton method based on orthogonalty and homogeneous spacng [7]. Non-deformaton crteron s ntroduced n [8] wth extrapolaton capabltes. Floater [9] proposed shape-preservng parameterzaton, where the coeffcents are determned by usng conformal mappng and barycentrc coordnates. The harmonc embeddng [3,10] s also a specal case of ths approach, except that the coeffcents may be negatve. However, these technques are developed manly for trangular mesh parameterzaton. Parameterzaton of quadrlateral meshes, on the other hand, s Y. Sh et al. (Eds.): ICCS 007, Part II, LNCS 4488, pp. 17 4, 007. Sprnger-Verlag Berln Hedelberg 007
18 L. Lu, C. Zhang, and F. Cheng actually a more crtcal problem because quadrlateral meshes, wth ther good propertes, are preferred n fnte element analyss than trangular meshes. Parameterzaton technques developed for trangle meshes are not sutable for quadrlateral meshes because of dfferent connectvty structures. In ths paper, we present a parameterzaton technque for non-closed quadrlateral meshes through a smplfy-project-embed process. The algorthm has the followng advantages:(1) the method provably produces good parameterzaton results for any non-closed quadrlateral mesh that can be mapped to the D plane; () the method mnmzes the dstorton of both angle and area caused by parameterzaton; (3) the soluton does not place any restrctons on the boundary shape; (4) snce the quadrlateral meshes are smplfed, the method s fast and effcent. The remanng part of ths paper s organzed as follows. The new model and the algorthm are presented n detal n Secton. Test results of the new algorthm are shown n Secton 3. Concludng remarks are gven n Secton 4. Parameterzaton Gven a non-closed quadrlateral mesh, the parameterzaton process conssts of four steps. The frst step s to get a smplfed verson of the mesh by keepng the boundary and nteror vertces wth hgh Gaussan curvature, but deletng nteror vertces wth low Gaussan curvature. The second step s to map the smplfed mesh onto a D doman through a global parameterzaton process. The thrd step s to embed the deleted nteror vertces onto the D doman through a weghted dscrete mappng. Ths mappng preserves angles and areas and, consequently, mnmzes angle and area dstorton. The last step s to perform an optmzaton process of the parameterzaton process to elmnate overlappng. Detals of these steps are descrbed n the subsequent sectons. For a gven vertex v n a quadrlateral mesh, the one-rng neghbourng vertces of the vertex v are the vertces that share a common face wth v. A one-rng neghborng vertex of the vertex v s called an mmedate neghborng vertex f ths vertex shares a common edge wth v. Otherwse, t s called a dagonally neghborng vertex..1 Smplfcaton Algorthm The computaton process, as well as the dstorton, may be too large f the entre quadrlateral mesh s projected onto the plane. To speed up the parameterzaton and mnmze the dstorton, we smplfy the mesh structure by reducng the number of nteror vertces but try to retan a good approxmaton of the orgnal shape and appearance. The dscrete curvature s one of the good crtera of smplfcaton whle preservng the shape of an orgnal model. In spte of the extensve use of quadrlateral meshes n geometrc modelng and computer graphcs, there s no agreement on the most approprate way to estmate geometrc attrbutes such as curvature on dscrete surfaces. By thnkng of a
Parameterzaton of Quadrlateral Meshes 19 quadrlateral mesh as a pecewse lnear approxmaton of an unknown smooth surface, we can try to estmate the curvature of a vertex usng only the nformaton that s gven by the quadrlateral mesh tself, such as the edge and angles. The estmaton does not have to be precse. To speed up the computaton, we gnore the effect of dagonally neghborng vertces, and use only mmedate neghborng vertces to estmate the Gaussan curvature of a vertex, as shown n Fg.1-(a). We defne the ntegral Gaussan curvature K = K v wth respect to the area S = attrbuted to v by S v n K = K = s = 1 π θ. (1) where θ s the angle between two successve edges. To derve the curvature from the ntegral values, we assume the curvature to be unformly dstrbuted around the vertex and smply normalzed by the area K K =. () S where S s the sum of the areas of adjacent faces around the vertex v. Dfferent ways of defnng the area S result n dfferent curvature values. We use the Vorono area, whch sums up the areas of vertex v s local Vorono cells. To determne the areas of the local Vorono cells restrcted to a trangle, we dstngush obtuse and nonobtuse trangles as shown n Fg. 1. In the latter case they are gven by For obtuse trangles, S 1 A 1 = ( vvk cot( γ ) + vv j cot( δ )). (3) 8 1 SB = vvk tan( γ ), SC = vv j tan( δ ), S A = S S B SC. (4) 8 8 A vertex deleton means the deleton of a vertex wth low Gaussan curvature and the ncdent edges. Durng the smplfcaton process, we can adjust the tolerance value to control the number of vertces reduced. (a) (b) (c) Fg. 1. Vorono area. (a) Vorono cells around a vertex; (b) Non-obtus angle; (c) Obtus angle.
0 L. Lu, C. Zhang, and F. Cheng. Global Parameterzaton Parameterzng a polygonal mesh amounts to computng a correspondence between the 3D mesh and an somorphc planar mesh through a pecewse lnear mappng. For the smplfed mesh M obtaned n the frst step, the goal here s to construct a mappng between M and an somorphc planar mesh U n R that best preserves the ntrnsc characterstcs of the mesh M. We denote by v the 3D poston of the - th vertex n the mesh M, and by u the D poston (parameterzed value) of the correspondng vertex n the D mesh U. The smplfed polygonal mesh M approxmates the orgnal quadrlateral mesh, but the angles and areas of M are dfferent from the orgnal mesh. We take the edges of the mesh M as sprngs and project vertces of the mesh onto the parameterzaton doman by mnmzng the followng edge-based energy functon 1 1 u u j, r 0. (5) r {, j} Edge v v j where Edge s the edge set of the smplfed mesh. The coeffcents can be chosen n dfferent ways by adjustng r. Ths global parameterzaton process s performed on a smplfed mesh (wth less vertces), so t s dfferent from the global parameterzaton and the fxed-boundary parameterzaton of trangular meshes..3 Local Parameterzaton After the boundary and nteror vertces wth hgh Gaussan curvature are mapped onto a D plane, those vertces wth low curvature, are embedded back onto the parametrzaton plane. Ths process has great mpact on the result of the parametrzaton. Hence, t should preserve as many of the ntrnsc qualtes of a mesh as possble. We need to defne what t means by ntrnsc qualtes for a dscrete mesh. In the followng, the mnmal dstorton means best preservaton of these qualtes..3.1 Dscrete Conformal Mappng Conformal parameterzaton preserves angular structure, and s ntrnsc to the geometry and stable wth respect to small deformatons. To flatten a mesh onto a twodmensonal plane so that t mnmzes the relatve dstorton of the planar angles wth respect to ther counterparts n the 3D space, we ntroduce an angle-based energy functon as follows E A = j N ( ) α j (cot 4 βj + cot ) u 4 u j. (6) where N () s the set of mmedate one-rng neghbourng vertces, and α j, βj are the left and opposte angles of v, as shown n Fg. -(a). The coeffcents n the
Parameterzaton of Quadrlateral Meshes 1 formula (6) are always postve, whch reduces the overlappng n the D mesh. To mnmze the dscrete conformal energy, we get a dscrete quadratc energy n the parameterzaton and t depends only on the angles of the orgnal surface..3. Dscrete Authalc Mappng Authalc mappng preserves the area as much as possble. A quadrlateral mesh n 3D space usually s not flat, so we cannot get an exact area of each quadrlateral patch. To mnmze the area dstorton, we dvde each quadrlateral patch nto four trangular parts and preserve the areas of these trangles respectvely. For nstance, n Fg. -(b) the quadrlateral mesh v v jvk v j+ 1 s dvded nto trangular meshes Δv v j v j+ 1, Δvv jvk, Δvv kv j + 1 and Δvvv j k j + 1 respectvely. Ths changes the problem of quadrlateral area preservng nto that of trangular area preservng. The mappng resulted from the energy mnmzaton process has the property of preservng the area of each vertex's one-rng neghbourhood n the mesh, and can be wrtten as follows E x = j N ( ) γ j δ j (cot + cot ) u v v j u j. (7) where γ j, δ j are correspondng angles of the edge ( v, v j ) as shown n Fg. -(c). The parameterzaton dervng from E x s easly obtaned, and the way to solve ths system s smlar to that of the dscrete conformal mappng, but the lnear coeffcents now are functons of local areas of the 3D mesh. (a) (b) (c) Fg.. Edge and angles. (a) Edge and opposte left angles n the conformal mappng; (b) Quadrlateral mesh dvded nto four trangles; (c) Edge and angles n the authalc mappng..3.3 Weghted Dscrete Parameterzaton Dscrete conformal mappng can be seen as an angle preservng mappng whch mnmzes the angle dstorton for the nteror vertces. The resultng mappng wll preserve the shape but not the area of the orgnal mesh. Dscrete authalc mappng s area preservng whch mnmzes the area dstorton. Although the area of the orgnal
L. Lu, C. Zhang, and F. Cheng mesh would locally be preserved, the shape tends to be dstorted snce the mappng from 3D to D wll n general generate twsted dstorton. To mnmze the dstorton and get better parameterzaton results, we defne lnear combnatons of the area and the angle dstortons as the dstorton measures. It turns out that the famly of admssble, smple dstorton measures s reduced to lnear combnatons of the two dscrete dstorton measures defned above. A general dstorton measure can thus always be wrtten as E = qe + (1 q) E. (8) A where q s a real number between 0 and 1. By adjustng the scalng factor q, parameterzatons approprate for specal applcatons can be obtaned..4 Mesh Optmzaton The above parameterzaton process does not mpose restrcton, such as convexty, on the gven quadrlateral mesh. Consequently, overlappng mght occur n the projecton process. To elmnate overlappng, we optmze the parameterzaton mesh by adjustng vertex locaton wthout changng the topology. Mesh optmzaton s a local teratve process. Each vertex s optmzed for a new locaton n a number of teratons. q Let u be the q tmes teraton locaton of the parameterzaton value u. The optmsaton process to fnd the new locaton n teratons s the followng formula u u u u u = ( ) + ( ),0< + < 1 n q 1 q 1 n q 1 q 1 q q 1 j k u λ 1 1 n λ 1 n λ λ = =. (9) where uj, uk are the parameterzaton values of the mmedate and dagonal neghbourng vertces respectvely. It s found that vertex optmzaton n the order of "worst frst" s very helpful. We defne the prorty of the vertex follows u u u u σ = λ ( ) + λ ( ) n q 1 q 1 n q 1 q 1 j k = 1 n = 1 n X. (10) The prorty s smply computed based on shape metrcs of each parameterzaton vertex. For a vertex wth the worst qualty, the hghest prorty s assgned. Through experments, we fnd that more teratons are needed f vertces are not overlapped n an order of "frst come frst serve". Besdes, we must pont out that the optmzaton process s local and we only optmze overlappng vertces and ts one-rng vertces, whch wll mnmze the dstorton and preserve the parameterzaton results better. 3 Examples To evaluate the vsual qualty of a parameterzaton we use the checkerboard texture shown n Fg. 3, where the effect of the scalng factor q n Eq. (8) can be found. In
Parameterzaton of Quadrlateral Meshes 3 fact, whle q s equal to 0 or 1, the weghted dscrete mappng s dscrete conformal mappng and authalc mappng separately. We can fnd few parameterzaton methods of quadrlateral meshes, so the weghted dscrete mappng s compared wth dscrete conformal mappng and authalc mappng of quadrlateral meshes wth q = 0 and q = 1 n Eq. (8) separately. Fg. 3-(a) and (e) show the sampled quadrlateral meshes. Fg. 3-(b) and (f) show the models wth a checkerboard texture map usng dscrete conformal mappng wth q = 0. Fg.3-(c) and (g) show the models wth a checkerboard texture map usng weghted dscrete mappng wth q = 0. 5. Fg. 3-(d) and (h) show the models wth a checkerboard texture map usng dscrete authalc mappng wth q = 1. It s seen that the results usng weghted dscrete mappng s much better than the ones usng dscrete conformal mappng and dscrete authalc mappng. (a) (b) (c) (d) (e) (f) (g) (h) Fg. 3. Texture mappng. (a) and (e) Models; (b) and (f) Dscrete conformal mappng, q=0; (c) and (g) Weghted dscrete mappng, q=0.5; (d) and (h) Dscrete Authalc mappng, q=1. The results demonstrate that the medum value (about 0.5) can get smoother parameterzaton and mnmal dstorton energy of the parameterzaton. And the closer q to value 0 or 1, the larger the angle and area dstortons are. 4 Conclusons A parameterzaton technque for quadrlateral meshes s based on mesh smplfcaton and weghted dscrete mappng s presented. Mesh smplfcaton
4 L. Lu, C. Zhang, and F. Cheng reduces computaton, and the weghted dscrete mappng mnmzes angle and area dstorton. The scalng factor q of the weghted dscrete mappng provdes users wth the flexblty of gettng approprate parametersatons accordng to specal applcatons, and establshes dfferent smoothness and dstorton. The major drawback n our current mplementaton s that the proposed approach may contan concave quadrangles n the planar embeddng. It s dffcult to make all of the planar quadrlateral meshes convex, even though we change the trangular meshes nto quadrlateral meshes by deletng edges. In the future work, we wll focus on usng a better objectve functon to obtan better solutons and developng a good solver that can keep the convexty of the planar meshes. References 1. Levy, B.: Constraned texture mappng for polygonal meshes. In: Fume E, (ed.): Proceedngs of Computer Graphcs. ACM SIGGRAPH, New York (001) 417-44. Alexa, M.: Mergng polyhedron shapes wth scattered features. The Vsual Computer. 16 (000): 6-37 3. Eck, M., DeRose, T., Duchamp, T., Hoppe, H., Lounsbery, M., Stuetzle, W.: Multresoluton analyss of arbtrary meshes. In: Mar, S.G., Cook, R.(eds.): Proceedngs of Computer Graphcs. ACM SIGGRAPH, Los Angeles (1995) 173-18 4. Allez, P., Meyer, M., Desbrun, M.: Interactve geometry remeshng. In: Proceedngs of Computer Graphcs.ACM SIGGRAPH, San Antono (00) 347-354 5. Alexa, M.: Recent advances n mesh morphng. Computer Graphcs Forum. 1(00) 173-196 6. Mallot, J., Yaha, H., Verroust, A.: Interactve texture mappng. In: Proceedngs of Computer Graphcs, ACM SIGGRAPH, Anahem (1993) 7-34 7. Levy, B., Mallet, J.: Non-dstorted texture mappng for sheared trangulated meshes. In: Proceedngs of Computer Graphcs, ACM SIGGRAPH, Orlando (1998) 343-35 8. Jn, M., Wang, Y., Yau, S.T., Gu. X.: Optmal global conformal surface parameterzaton. In: Proceedngs of Vsualzaton, Austn (004) 67-74 9. Floater, M.S.: Parameterzaton and smooth approxmaton of surface trangulatons. Computer Aded Geometrc Desgn.14(1997) 31-50 10. Lee, Y., Km, H.S., Lee, S.: Mesh parameterzaton wth a vrtual boundary. Computer & Graphcs. 6 (006) 677-686