Consistent Spherical Parameterization

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1 Consstent Sphercal Parameterzaton Arul Asrvatham 1, Eml Praun 1, and Hugues Hoppe 2 1 School of Computng - Unversty of Utah, USA {arul, emlp}@cs.utah.edu 2 Mcrosoft Research, Redmond, USA Abstract. Many applcatons beneft from surface parameterzaton, ncludng texture mappng, morphng, remeshng, compresson, object recognton, and detal transfer, because processng s easer on the doman than on the orgnal rregular mesh. We present a method for smultaneously parameterzng several genus-0 meshes possbly wth boundares onto a common sphercal doman, whle ensurng that correspondng user-hghlghted features on each of the meshes map to the same doman locatons. We obtan vsually smooth parameterzatons wthout any cuts, and the constrants enable us to drectly assocate semantcally mportant features such as anmal lmbs or facal detal. Our method s robust and works well wth ether sparse or dense sets of constrants. 1 Introducton Several applcatons n computer graphcs, such as texture mappng, compresson, surface processng, detal synthess, and object recognton rely on mesh parameterzaton. Parameterzaton refers to computng mappngs between surfaces n 3D and smpler domans such as planar regons, smplcal domans, or spheres. To be useful for these vared applcatons, the mappngs must satsfy several propertes, such as contnuty, bjectvty, smoothness, constrant satsfacton, and acceptable dstrbuton of doman area over the surface. Many parametrzaton technques nvolve parttonng the surface nto smpler peces usng cuts. Such cuts are problematc for applcatons nvolvng a large number of models, snce they can lead to fragmentaton of the map (.e. smaller and smaller peces across whch there s a contnuous map between all models). Moreover, the resultng map generally lacks smoothness along cuts and at ther junctons. Contnuous parameterzatons are only possble between topologcally equvalent models. For the genus-zero models that we consder, the unt sphere s a natural parameterzaton doman snce t s nherently smooth. Allowng the user to specfy constrants s another mportant requrement n many applcatons, snce t provdes a way to ncorporate hgher-level semantc knowledge about the objects. We extend the work of Praun and Hoppe [14] on sphercal parameterzaton to allow smultaneous parameterzaton of multple objects wth pont feature constrants whle guaranteeng contnuty, bjectvty, vsual smoothness, and mnmzng overall dstorton. Usng these consstent sphercal parameterzatons, one can create consstent V.S. Sunderam et al. (Eds.): ICCS 2005, LNCS 3515, pp , Sprnger-Verlag Berln Hedelberg 2005

266 A. Asrvatham, E. Praun, and H. Hoppe Fg. 1.

n a database, openng the door to a large array of applcatons ncludng compresson, converson to

We present the followng contrbutons: Robust constructon of consstent sphercal parameterzatons for several

Methods to avod swrls, and to correct them when they arse.

There have been many methods developed to date; for a survey we refer the reader to Floater and Hormann [3].

topology, such as the unt sphere for genuszero models.

We buld upon the method of Praun and Hoppe [14].

dffcult to guarantee bjectvty of the result.

2 266 A. Asrvatham, E. Praun, and H. Hoppe Fg. 1. Consstent Sphercal Parameterzaton of a collecton of heads geometry mages [6; 14] representng several objects n a database, openng the door to a large array of applcatons ncludng compresson, converson to hardwaresupported subdvson surfaces wth dsplacement maps, natural LODs, etc. We present the followng contrbutons: Robust constructon of consstent sphercal parameterzatons for several surfaces. Constraned sphercal parameterzaton where specfed ponts on a mesh map to gven ponts on the sphere. Methods to avod swrls, and to correct them when they arse. 2 Prevous Work Planar Parameterzaton: The earlest parameterzaton methods establshed mappngs to planar domans. There have been many methods developed to date; for a survey we refer the reader to Floater and Hormann [3]. Unless appled to topologcal dscs, these methods must cut the mesh nto one or several peces, ntroducng dscontnutes n the parameterzaton. Sphercal Parameterzaton: Dscontnutes can be avoded altogether by mappng models to smooth domans of the same topology, such as the unt sphere for genuszero models. Examples of sphercal parameterzaton methods nclude [18; 1; 9;7; 5; 14]. We buld upon the method of Praun and Hoppe [14]. Incorporatng hard constrants nto a sphercal parameterzaton scheme proves to be challengng, because t s dffcult to guarantee bjectvty of the result. We chose to buld upon the approach of Praun and Hoppe [14] because ts herarchcal constructon approach enables a robust soluton. Parameterzaton Constrants. Several methods address the problem of parameterzaton under constrants. Lévy [13] texture maps meshes under soft constrants by ntroducng addtonal terms n the qualty metrc. Ecksten et al. [2] propose a method for satsfyng hard constrants for planar parameterzaton. Kraevoy et al. [11] start wth an unconstraned planar parameterzaton and then move the constraned vertces to ther requred postons by matchng a trangulaton

Consstent Sphercal Parameterzaton 267 of these postons to a trangulaton of the planar mesh formed by the paths between constraned

They demonstrate results nvolvng a farly large number of features.

In addton, we use a multresoluton approach to avod formng an ntal parameterzaton and then optmzng t. Consstent Parameterzaton.

Unlke ther method, we do not mpose a fxed consstent connectvty of the base doman, but only satsfy the gven pont constrants, thereby

[17] and Kraevoy and Sheffer [12], mprove upon the technque of Praun et al. by not requrng the smplcal complex to be specfed a pror.

asymmetrc and would not scale to a large collecton of models.

3 Consstent Sphercal Parameterzaton 267 of these postons to a trangulaton of the planar mesh formed by the paths between constraned vertces. Fnally, they relax ths parameterzaton whle keepng the constrants. They demonstrate results nvolvng a farly large number of features. In contrast, our method works well even wth few provded feature ponts. In addton, we use a multresoluton approach to avod formng an ntal parameterzaton and then optmzng t. Consstent Parameterzaton. Praun et al. [15] consstently parameterze a set of genus-0 models onto a user-specfed smplcal complex. Unlke ther method, we do not mpose a fxed consstent connectvty of the base doman, but only satsfy the gven pont constrants, thereby provdng the map more freedom. Two recent works, by Schrener et al. [17] and Kraevoy and Sheffer [12], mprove upon the technque of Praun et al. by not requrng the smplcal complex to be specfed a pror. However, these new technques do not scale well wth regard to the number of models to be consstently parameterzed. The technque of Schrener et al. s lmted to dealng wth only 2 models, and whle Kraevoy and Sheffer demonstrate consstent parameterzaton among 3 models, ther approach s asymmetrc and would not scale to a large collecton of models. Whereas n all prevous cases the smplcal doman s abstract (wth no nherent geometry), our sphercal doman geometry s explct. Ths mples that paths must be straght arcs on the sphere, rather than arbtrary meanderng mesh paths, whch makes the problem more dffcult. One advantage of the lack of doman connectvty s the opportunty to mprove the map by flppng edges, lke n Delaunay refnement. Step 1 Step 2 Steps 3, 4 Step 5 Step 6 Fg. 2. Steps of the algorthm. For clarty, steps 3-5 show a coarser verson of the remesh grd than the one actually used. The features n step 5 are vertces of the fner grd. Stretch effcences: gargoyle 0.632, bunny 0.697

4 268 A. Asrvatham, E. Praun, and H. Hoppe 3 Approach Gven a set of meshes and correspondng feature ponts, we form a consstent parameterzaton of all the meshes. Our approach has two major parts. Frst, we fnd good sphercal feature locatons, such that the fnal maps have low dstorton and dstrbute the sphere area adequately to the varous parts of the nput meshes. Second, we create a constraned sphercal parameterzaton for each surface, forcng the feature ponts to map to the computed locatons. Here s the algorthm n more detal (see also Fgure 2). Let M (=1..n) be the ntal meshes, P be a parameterzaton, and F a set of sphercal feature locatons. 1. ( P 1, F ) := UnconstranedSphercalParam( M 1 ) //Intal feature locatons F on sphere usng one model. 2. For =2..n, P := ConstranedSphercalParam( M, F ) //Parameterze all models usng those ntal locatons. R 3. For =1..n, M := Remesh( M, P ) //Remesh to n geometry mages wth dentcal connectvtes. R 4. * R R R M := { M 1, M 2,, M n } //Concatenate to sngle mesh wth vertex coordnates n R 3n. 5. (P R R,F) := UnconstranedSphercalParam( M * ) //Fnd good feature locatons consderng all models. 6. For =1..n, P := ConstranedSphercalParam( M, F) //Compute fnal parameterzatons usng these locatons. The man new procedure s the constraned sphercal parameterzaton used n steps 2 and 6. It s descrbed n detal n Secton 4. The remanng steps are adapted from earler work [14]. In step 3 we need to exactly represent the feature ponts, so we snap the closest grd samples to the sphercal locatons F. R Step 5 nvolves the constructon of a progressve mesh for the specal mesh M * wth geometry n R 3n. We modfy the quadrc error metrc [4] to sum each of the n errors n R 3. We also modfy the sphercal parameterzaton [14] to sum the n stretch energes from the sphere to the n mesh geometres n R 3. 4 Constraned Sphercal Parameterzaton We adapt the method of [14] to work wth constrants as follows. Durng coarse-tofne refnement, we can smply fx the sphercal locaton of feature vertces. The dffculty s to bootstrap the algorthm by creatng a vald startng state that satsfes all constrants. Specfcally, we must create a progressve mesh representaton of the surface where the base doman contans only feature vertces and s trangulated the same way as the sphercal features (see Fgure 3).

Consstent Sphercal Parameterzaton 269 To satsfy ths, we need only fnd a sphercal trangulaton and a correspondng embeddng of ts arcs onto the mesh, gven by a set of non-ntersectng paths.

"Trangulatons" of features on the sphere (left) and mesh (mddle); base mesh after smplfcaton (rght) to produce a progressve mesh, but we keep the feature vertces, and only allow vertces on the

We lnk together pars of feature ponts, wth great crcle arcs on the sphere, and paths on the mesh, untl we complete a full trangulaton.

To guarantee both termnaton and topologcal equvalence of the two trangulatons, we mantan consstent orderng of neghbors around each vertex n the partally completed graph, and we avod addng any

5 Consstent Sphercal Parameterzaton 269 To satsfy ths, we need only fnd a sphercal trangulaton and a correspondng embeddng of ts arcs onto the mesh, gven by a set of non-ntersectng paths. Once we have such a path network, we smplfy the mesh Fg. 3. "Trangulatons" of features on the sphere (left) and mesh (mddle); base mesh after smplfcaton (rght) to produce a progressve mesh, but we keep the feature vertces, and only allow vertces on the feature paths to be collapsed nto other path or feature vertces [16]. To produce the path network on the 3D surface, we use a method smlar to those of Praun et al. [15] and Kraevoy et al. [11]. We lnk together pars of feature ponts, wth great crcle arcs on the sphere, and paths on the mesh, untl we complete a full trangulaton. The paths (and arcs) cannot ntersect each other except at feature vertces (and ther sphercal locatons). The algorthm proceeds n greedy fashon, selectng the best par from a pool of canddates. To guarantee both termnaton and topologcal equvalence of the two trangulatons, we mantan consstent orderng of neghbors around each vertex n the partally completed graph, and we avod addng any arcs/paths closng cycles before we have lnked all the vertces n a spannng tree [15]. The canddate pool s populated ntally by shortest paths between all the feature pars, computed usng a Djkstra search on mesh vertces. When we select the best canddate path we check to see f t ntersects any paths already nserted n the network and f t does we re-compute t usng a restrcted search. These restrcted searches can also use edge mdponts n addton to mesh vertces (though wth a cost penalty), correspondng to nsertng Stener vertces n the orgnal mesh [11]. To mprove the geometrc qualty of the trangulaton we employ a set of heurstcs to avod and fx swrls (Secton 4.1) and we flp edges by replacng them wth the other dagonal of the quad formed by the two adjacent patches. Ths s done only when the new path s shorter and the new confguraton s vald on the sphere. 4.1 Dealng wth Swrls Sometmes the paths appear bad to a human observer because they take unnecessarly long routes around other feature vertces. We call a swrl the local confguraton of these long paths (Fgure 4). Praun et al. [15] note that swrls cannot be repared usng 1-rng relaxatons of patches around a vertex, and propose several heurstcs to prevent them. Unfortunately we do not beneft from a userprovded base doman, so some of ther heurstcs do Fg. 4. Swrl on the horse leg: the whte patch (rght) has to connect to B, but t does so around A. It cannot be straghtened sne t would have to move over A and AB

6 270 A. Asrvatham, E. Praun, and H. Hoppe not work n our case. Furthermore, our settng s harder snce the sphercal locatons of the feature ponts obtaned n steps 1 and 5 of the algorthm (Secton 3) may be qute dfferent from preferred sphercal locatons consderng the geometry of the current mesh. We develop a more robust set of swrl-avodng heurstcs, as well as a method to remove swrls after they have appeared. Swrl-Avodng Heurstcs. The man tool we use n selectng new paths to nsert n the network s ther rankng n the prorty queue. Paths are ranked accordng to ther surface length (shorter s better), but they are occasonally penalzed (placed at the end of the queue) when certan condtons occur. Such condtons nclude: The current path lnks non-extreme vertces, and there are stll some extremtes left unconnected. Extremtes are features wth large average dstance to ther nearest neghbors (such as legs, arms, etc.). We start the spannng tree constructon by lnkng such features. If left unconnected they mght cause swrls snce paths lnkng other vertces go around the base of the extremty, equally lkely on the correct as on the wrong sde. The sphercal mage of the path would create sphercal trangles wth very small angles (<10 degrees n our examples). In these cases the wndng order of the 3 feature locatons on the sphere s not relable. Furthermore, sknny trangles make the sphercal optmzaton less robust. Faled sdedness tests for neghborng features. We check whether the projecton of a feature vertex onto the path s on the same sde as the projecton of the correspondng feature pont onto the arc on the sphere. If some of the vertces are on dfferent sdes on the mesh and the sphere, we try to force the path to le on the correct sde of nearby feature vertces. To do ths we add temporary constrant paths from the path endponts to the neghborng feature. We now trace the shortest path on the mesh. The path so traced wll be on the correct sde of all the features n the connected component of the neghbor. However, ths mght not always be possble as the addton of the temporary constrant paths mght form a cycle enclosng ether the source or the destnaton vertex on dfferent sdes on the mesh and the sphere. In such cases, we add the path to the end of the queue. Unswrl Operator. In addton to usng heurstcs that avod swrls (as Praun et al. dd [15]), we have also developed a method to dentfy and remove them n the rare cases that they do occur. Paths between two features ncdent to many other long paths (wth hgh ratos between actual length and geodesc dstance between endponts) are lkely to be the center of swrls. To fx them, we remove all paths ncdent to the two vertces, and then replace them n a new order, ntroducng frst paths that were prevously bad. 5 Results and Applcatons Fgure 1 llustrates the basc approach for applcatons makng use of large model databases. To create the database, a few representatve models are selected and consstently parameterzed usng our algorthm, n order to obtan good sphercal locatons for the 22 feature ponts. In our example we used the 5 heads shown out of a set of 8. The remanng models can be subsequently added to the database by runnng con-

Consstent Sphercal Parameterzaton 271 straned sphercal parameterzaton.

On the rght sde of Fgure 1 we show the average of our set of heads, and the frst three prncpal components (vsualzed added to the average head). Fg. 5.

The two left mages show the sphercal parameterzaton and regular remesh usng the locatons obtaned usng the cow model alone.

left are combned wth the geometry of the head n Consstent parameterzatons can the mddle to produce the one on the rght also be used to transfer mesh propertes,

Tmng results (n mnutes) for the heads hgh-frequency components of a example (Fgure 1), bunny and gargoyle example multresoluton mesh representaton), normals,

Fgure 6 demonstrates examples of color and normals transfer Steps between two heads. Models 1 2 5 6 Total tme Table 1 shows tmng results for Fg.

7 Consstent Sphercal Parameterzaton 271 straned sphercal parameterzaton. Once the database s created, the consstent parameterzaton can be used for tasks such as classfcaton and retreval based on prncpal component analyss. On the rght sde of Fgure 1 we show the average of our set of heads, and the frst three prncpal components (vsualzed added to the average head). Fg. 5. Parameterzaton qualty mproves after optmzng the map takng nto account all models Fgure 5 demonstrates the role of steps 3-6 of the algorthm. The two left mages show the sphercal parameterzaton and regular remesh usng the locatons obtaned usng the cow model alone. The qualty mproves (rght) when the feature locatons are computed usng all the Fg. 6. The texture and normals of the head on the models. left are combned wth the geometry of the head n Consstent parameterzatons can the mddle to produce the one on the rght also be used to transfer mesh propertes, such as geometrc detal (the Table 1. Tmng results (n mnutes) for the heads hgh-frequency components of a example (Fgure 1), bunny and gargoyle example multresoluton mesh representaton), normals, colors, or texture colatve for the dfferent models n the set (Fgure 2). The tmngs for steps 2 and 6 are cumuordnates. Fgure 6 demonstrates examples of color and normals transfer Steps between two heads. Models Total tme Table 1 shows tmng results for Fg our method. For small meshes, step Fg s the most expensve snce t nvolves several geometres smultaneously. How- ever, when the orgnal meshes are dense, step 6 becomes the most expensve one. 6 Summary We have presented a robust algorthm for parameterzng genus-zero models onto a sphere n the presence of feature constrants. The central part of the algorthm, constraned sphercal parameterzaton s guaranteed to produce topologcally equvalent sphercal trangulatons and mesh patch parttons, and avods awkward swrl confgu-

8 272 A. Asrvatham, E. Praun, and H. Hoppe ratons through a collecton of novel heurstcs. The regularly sampled consstent geometry mages that can be obtaned usng our parameterzatons allow dgtal geometry processng applcable to many real-world applcatons. References 1. Alexa, M Mergng polyhedral shapes wth scattered features. The Vsual Computer, 16(1), pp Ecksten, I., Surazhsky, V. and Gotsman, C Texture mappng wth hard constrants. Eurographcs 2001, pp Floater, M. and Hormann K Recent advances n surface parameterzaton. Multresoluton n geometrc modelng Garland, M. and Heckbert, P Surface smplfcaton usng quadrc error metrcs. ACM SIGGRAPH 97, pp Gotsman, C., Gu, X. and Sheffer, A Fundamentals of sphercal parameterzaton for 3D meshes. SIGGRAPH 2003, pp Gu, X., Gortler, S., And Hoppe, H Geometry mages. ACM SIGGRAPH 2002, pp Gu, X., Wang, Y., Chan, T., Thompson, P, and Yau, S.-T Genus zero surface conformal mappng and ts applcaton to bran surface mappng., Informaton Processng Medcal Imagng Guskov, I., Vdmče, K., Sweldens, W., and Schröder, P Normal meshes. ACM SIGGRAPH 2000, pp HAKER, S., ANGENENT, S., TANNENBAUM, S., KIKINIS, R., SAPIRO, G., AND HALLE, M Conformal surface parameterzaton for texture mappng. IEEE TVCG, 6(2), pp Hoppe, H Progressve meshes. ACM SIGGRAPH 96, pp Kraevoy, V., Sheffer, A. and Gotsman, C Matchmaker: constructng constraned texture maps. SIGGRAPH 2003, pp Kraevoy, V., and Sheffer, A Cross-parameterzaton and compatble remeshng of 3D models. SIGGRAPH 2004, to appear. 13. Lévy, B Constraned texture mappng for polygonal meshes. ACM SIGGRAPH 2001, pp Praun, E. and Hoppe, H Sphercal parameterzaton and remeshng. ACM SIGGRAPH 2003, pp Praun, E., Sweldens, W. and Schröder, P Consstent mesh parameterzatons. ACM SIGGRAPH 2001, pp Sander, P., Snyder, J., Gortler, S., and Hoppe, H Texture mappng progressve meshes. ACM SIGGRAPH 2001, pp Schrener, J., Asrvatham, A., Praun, E., and Hoppe, H Inter-surface mappng. ACM SIGGRAPH SHAPIRO, A. AND TAL, A Polygon realzaton for shape transformaton. The Vsual Computer, 14 (8-9), pp

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