Interactive Multiresolution Surface Viewing

Transcription

1 Interactve Multresoluton Surface Vewng Andrew Certan z Jovan Popovć Tony DeRose Tom Duchamp Davd Salesn Werner Stuetzle y Department of Computer Scence and Engneerng Department of Mathematcs y Department of Statstcs Unversty of Washngton Abstract Multresoluton analyss has been proposed as a basc tool supportng compresson, progressve transmsson, and level-of-detal control of complex meshes n a unfed and theoretcally sound way. We extend prevous work on multresoluton analyss of meshes n two ways. Frst, we show how to perform multresoluton analyss of colored meshes by separately analyzng shape and color. Second, we descrbe effcent algorthms and data structures that allow us to ncrementally construct lower resoluton approxmatons to colored meshes from the geometry and color wavelet coeffcents at nteractve rates. We have ntegrated these algorthms n a prototype mesh vewer that supports progressve transmsson, dynamc dsplay at a constant frame rate ndependent of machne characterstcs and load, and nteractve choce of tradeoff between the amount of detal n geometry and color. The vewer operates as a helper applcaton to Netscape, and can therefore be used to rapdly browse and dsplay complex geometrc models stored on the World Wde Web. CR Categores and Subject Descrptors: I.3.5 [Computer Graphcs]: Computatonal Geometry and Object Modelng surfaces and object representatons; J.6 [Computer-Aded Engneerng]: Computer-Aded Desgn (CAD). Addtonal Keywords: Geometrc modelng, wavelets, multresoluton analyss, texture mappng, vewer. 1 Introducton Three-dmensonal meshes of large complexty are rapdly becomng commonplace. Laser scannng systems, for example, routnely produce geometrc models wth hundreds of thousands of vertces, each of whch may contan addtonal nformaton, such as color. Workng wth such complex meshes poses a number of problems. They requre a large amount of storage and consequently are slow to transmt. Addtonally, they contan more faces than can be nteractvely dsplayed on any current hardware. Exstng vewers ether do not deal wth these problems at all, or do so only n crude ways, for example by showng wreframes or by dsplayng only a fracton of the faces durng dynamc vewng, and then swtchng back to surfaces once the moton has stopped. z Department of Computer Scence and Engneerng, Unversty of Washngton, Box , Seattle, WA A more sophstcated way of copng wth both the transmsson and dynamc dsplay problems s to use a precomputed sequence of lower detal approxmatons to the mesh. Such approxmatons can be computed, for example, usng the method of Rossgnac and Borrel [9]. Durng transmsson, a cruder approxmaton s dsplayed whle the next more detaled approxmaton s receved. For dynamc dsplay, one chooses the hghest detal approxmaton compatble wth a desred frame rate. A major dsadvantage of ths approach s that the total amount of data that has to be transmtted and stored s larger than the descrpton of the full resoluton mesh. In fact, there s a tradeoff between granularty (the dfference n resoluton between successve models) on the one hand and transmsson tme and storage requrements on the other hand. Prevous work [1, 5, 6, 10] has demonstrated that, at least n prncple, multresoluton analyss offers a unfed and theoretcally sound way of dealng wth these problems. A multresoluton representaton of a mesh conssts of a smple approxmaton called the base mesh, together wth a sequence of correcton terms called wavelet coeffcents whch supply the mssng detal. The key pont s that truncated sequences of wavelet coeffcents defne approxmatons to the mesh wth fewer faces. Although promsng, prevous work s lackng n at least two ways. Frst, ether color or geometry were represented n multresoluton form, but not both. Second, algorthms for reconstructng and dsplayng multresoluton meshes were much too slow for nteractve use. In ths paper we address both of these defcences. We deal wth complex colored meshes usng separate multresoluton representatons for geometry and color that are combned only at dsplay tme. We also descrbe effcent algorthms and data structures that allow us to ncrementally construct and render lower resoluton approxmatons to the mesh from the color and geometry wavelet coeffcents at nteractve rates. The separaton of color and geometry, together wth our ncremental algorthms, allows the effcent mplementaton of the followng features: Progressve transmsson: We frst transmt and dsplay the base mesh and then transmt the wavelet coeffcents n decreasng order of magntude. As wavelets are receved, they are ncorporated nto the approxmaton, and the approxmaton s perodcally re-rendered. In the examples we have tred the approxmaton rapdly converges to the orgnal mesh (see Color Plates 1(a d)). Only a small penalty s ncurred for progressve transmsson (see Secton 5.1). Performance tunng: By truncatng the color and geometry expansons we can obtan lower detal approxmatons of the mesh wth essentally any desred number of faces. Durng dynamc

2 dsplay, we truncate the expansons at a level of detal that can be rendered wth the desred frame rate. We montor the frame rate and dynamcally modfy the level of detal n response to changng machne load. Automatc texture map generaton: The separaton between color and geometry and the way n whch they are represented allows us to take advantage of texture-mappng hardware, as descrbed n Secton 3.3. Color Plates 1(g) and 1(h) llustrate the gans obtaned by explotng texture mappng. For a gven number of polygons, texture mappng allows dsplay of a far better approxmaton (Color Plate 1(h)), as all the polygons can be dedcated to capturng geometrc detal. Color can always be dsplayed at full resoluton because addng color detal does not ncrease the polygon count. Adaptng to user preferences: Color and geometry expansons can be truncated ndependently. In the absence of texture mappng, the number of faces of the resultng mesh wll depend on the truncaton thresholds. There wll n general be many combnatons of color threshold and geometry threshold that result n approxmately the same number of faces (see Color Plates 1(e g)). Automatcally fndng the combnaton gvng the best lookng approxmaton seems to be a hard problem, as t wll certanly depend on the model tself. Instead, we allow the user to nteractvely choose the tradeoff. To demonstrate our deas, we have bult a prototype vewer runnng as a helper applcaton for Netscape. As demonstrated n the accompanyng vdeotape, our vewer can be used to rapdly browse and dsplay complex geometrc models stored on the World Wde Web. The rest of the paper s organzed as follows. In Secton 2 we present a bref summary of multresoluton analyss of colored meshes. In Secton 3 we descrbe the basc data structures and algorthms for effcently constructng and renderng truncated models. In Secton 4 we sketch the archtecture of our vewer. In Secton 5 we present the results of several numercal experments. Fnally, Secton 6 contans a dscusson and deas for future work. 2 Background In ths secton we frst present a synopss of multresoluton analyss for pecewse lnear functons on trangular meshes. For a more complete exposton, see Stollntz et al. [11]. We then descrbe how to convert an arbtrary colored mesh to a parametrc form amenable to multresoluton analyss. 2.1 Multresoluton analyss The central dea of multresoluton analyss s to decompose a functon nto a low resoluton ( coarse ) part and a sequence of correcton ( detal ) terms at ncreasng resolutons. Multresoluton analyss for functons on R n was formalzed by Meyer [8] and Mallat [7]. Lounsbery [5] and Lounsbery et al. [6] extended multresoluton analyss to a class of functons ncludng functons defned on trangular meshes, whch we call level J pecewse lnear. A functon f defned on a trangular mesh M 0 s called level J pecewse lnear f t s pecewse lnear on the mesh M J obtaned by performng J recursve 4-to-1 subdvsons of the faces of M 0 (see Fgure 1). Let ˆV j denote the vector space of level j pecewse lnear functons on M 0. Let ˆj denote the unque level j pecewse lnear functon assumng value 1 at vertex and value 0 at all other vertces of M j. These level j hat functons form a bass of ˆV j. In the context of multresoluton analyss they are often referred to as scalng functons. The spaces ˆV 0, ˆV 1, : : : form a nested sequence, as requred by multresoluton analyss. Fgure 1 Recursve 4-to-1 splttng of a tetrahedron: (a)m 0, (b) M 1, (c) M 2. Besdes a nested sequence of spaces, the other basc ngredent of multresoluton analyss s an nner product. We use the nner product hf j g = XZ T x2t f (x) g(x) dx, where the sum s taken over all faces of M 0 and dx s the area element, normalzed so that all faces of M 0 have unt area. Gven a nested sequence of functon spaces and an nner product, we can now defne wavelets. The orthogonal complements Ŵ j of ˆV j n ˆV j+1, for 0 j < J, are called orthogonal wavelet spaces. A wavelet bass for ˆV J conssts of the level 0 scalng functons, together wth bases for the wavelet spaces Ŵ 0,..., Ŵ J 1. Gven such a wavelet bass, we can express any level J pecewse lnear functon f on M 0 as a lnear combnaton of scalng functons and wavelets at varous levels. Ideally we would lke the wavelets, together wth the level 0 scalng functons, to form an orthonormal bass for ˆV J. We could then calculate the best k term L 2 approxmaton to a functon f 2 ˆV J by keepng the k terms of the expanson wth the largest coeffcents. On the other hand, we want wavelets to have small support so that the contrbuton to the approxmaton from each wavelet term can be rapdly ncorporated nto the model. Unfortunately, orthogonalty of wavelet spaces and small spatal support of wavelets are conflctng goals. As small spatal support s essental for applcatons, we relax the orthogonalty requrement. Lounsbery et al. [6] stpulate a pror the sze k of the support and then construct borthogonal wavelets ˆj that span Ŵj and are as orthogonal as possble to ˆV j. The wavelets obtaned n ths way are called k-dsk wavelets [11]. More precsely, consder a vertex of M j+1 that s located at the mdpont of an edge e of M j. The k-dsk wavelet centered at vertex s a functon of the form X ˆj = ˆj+1 + s j v ˆ j v, (1) v2n k where N k denotes a set of level j vertces n a neghborhood of vertex. The neghborhoods N k are defned recursvely. The neghborhood N 0 for the 0-dsk wavelet conssts of the endponts of e; the neghborhood N k contans the vertces of all trangles ncdent on N k 1 (see Fgure 2). The wavelet consstng of only the levelj + 1 scalng functon s called the lazy wavelet. The coeffcents s j v are chosen to mnmze the norm of the orthogonal projecton of ˆj onto ˆV j. They are determned by solvng the followng system of lnear equatons: X h ˆ j v2n k u j ˆ j v s j v = h ˆ j u j ˆ j+1, for all u 2 N k. Note that the system s local to vertex. The sze of the system for 0-dsk wavelets s only 2 2. For larger values of k the sze of the

3 system depends on the valence of the parent vertces; n regular regons of the mesh where all vertces have valence 6, the system has sze for k = 1, and sze for k = 2. The process of expressng a level J pecewse lnear functon n terms of level 0 scalng functons and wavelets s called flterbank analyss. For a descrpton see Stollntz et al. [11]. Fgure 2 (a) The support of the 1-dsk wavelet ˆj. Dark shaded area: N 0 -neghborhood of center edge; lght shaded area: faces added to form N 1 -neghborhood. (b) The trangles requred to ntroduce ˆj durng reconstructon. (c) The graph of ˆj. 2.2 Converson of colored meshes to multresoluton form Multresoluton analyss of a colored mesh M s based on the premse that M s defned parametrcally by two vector valued level J pecewse lnear functons, a geometry functon f geom and a color functon f color, each mappng a trangular base mesh M 0 nto R 3. Typcally, M wll not be gven n ths form, but nstead n the form of vertces, edges, and faces, vertex postons, and vertex colors. In order to apply multresoluton analyss, M must be converted to parametrc form. We do ths by frst applyng the remeshng algorthm of Eck et al. [1]. The output of the remeshng algorthm s a base mesh M 0 wth a relatvely small number of faces, a parameterzaton : M 0! M, and an approxmaton of by a level J pecewse lnear embeddng f geom : M 0! R 3 of the form f geom = P f ˆJ, where f are vectors n R 3 representng the geometrc postons of the vertces of M J. We next apply the flter bank analyss algorthm of Lounsberyet al. [6] to obtan a wavelet expanson off geom. Note that ths analyss wll generate a vector of three coeffcents for each wavelet, one for each of the three coordnate functons. We sort these coeffcent vectors n order of decreasng length and then store them together wth dentfers for the wavelets (center vertex and level) n a fle called the geometry-wavelet fle. We now turn to multresoluton analyss of color. Color s orgnally gven at the vertces of M, and can be extended to all of M by lnear nterpolaton. The parametrzaton : M 0! M obtaned durng remeshng nduces a color functon on M 0. To construct a level J pecewse lnear approxmaton f color to, we sample at the vertces of M J. As n the case of geometry, we then compute the wavelet expanson of f color by flterbank analyss and store the wavelet coeffcent vectors n order of decreasng length n a color-wavelet fle. The base mesh M 0, ts vertex postons (the coeffcents of the level 0 scalng functons n the expanson of f geom) and ts vertex colors are stored n a base fle. The geometry-wavelet fle, the color-wavelet fle, and the base fle consttute the nput to our multresoluton vewer. 3 Algorthms and data structures In ths secton we descrbe the algorthms and data structures that form the bass of our multresoluton vewer. We assume that the colored mesh s represented n multresoluton form,.e., by a base mesh and wavelet expansons of the color and geometry functons. At the full resoluton, the number of faces ofm J s 4 J tmes the number of faces of M 0. The faces of M J can be naturally organzed nto a tree Q. The root of Q has as many chldren as there are faces n the base mesh, whle every other nternal node has four chldren. Each leaf of Q corresponds to a face of M J. Ths tree organzaton was also used by Schroeder and Sweldens [10]. The mesh s rendered by traversng the tree Q, evaluatng f geom and f color at the vertces, and generatng a colored trangle for each leaf. In the absence of texture-mappng hardware, color and geometry are handled dentcally, so we wll couch the dscusson n terms of geometry alone. The use of texture mappng s the topc of Secton 3.3. Frst some termnology: let fgeom r denote the approxmaton to f geom obtaned by summng the scalng functons and the largest r wavelets, and let Q r denote the smallest subtree of Q we for whch fgeom r s lnear on each leaf. For progressve transmsson we frst transmt the base meshm 0 and the coeffcents of the level 0 scalng functons. The assocated tree Q 0 conssts only of the root node and as many leaves as there are faces n the base mesh. As wavelets arrve, we ncrementally grow Q r and update f r geom, and perodcally render the mesh. Use of the wavelet representaton for performance tuned vewng and level-of-detal control s based on the observaton that for small r, the tree Q r wll also be small, and therefore renderng Q r wll result n many fewer trangles than renderng Q. In prncple we could generate approxmatons wth almost any desred number of faces by growng from the base mesh. For effcency reasons we cache trees and vertex postons for a sequence of approxmatons, and then grow the desred tree from the closest approxmaton wth fewer than the desred number of faces. 3.1 Data structures As prevously stated, the prmary data structure used to represent the mesh s a tree Q, whch has as many descendents from the root as there are faces n the base mesh and s a quadtree for all other levels. We represent all nodes of Q, except for the root, wth the followng data structure: type Face = record level: Integer chldren[4]: array of ponter to Face cornervertex[3]: array of ponter to Vertex edgevertex[3]: array of ponter to Vertex end record A face s sad to be of level j f t s a face of M j. The array corner- Vertex has ponters to three vertces of the face, and the arrayedge- Vertex has ponters to three vertces that subdvde the edges of ths face. We represent vertces wth the followng data structure: type Vertex = record parentv[2]: array of ponter to Vertex parentf[2]: array of ponter to Face fgeom: XYZposton fcolor: RGBcolor g: XYZvector hgeom[ ]: array of HatFunctonCoeffcents hcolor[ ]: array of HatFunctonCoeffcents end record The array parentv contans ponters to the two vertces on ether end of the edge that the vertex subdvdes these are calledparent vertces of the vertex. The array parentf contans ponters to the two

4 faces on ether sde of the edge that the vertex subdvdes these are called the parent faces of the vertex. A vertex s sad to be of level j f t was created at the j-th level of subdvson,.e., f ts parent faces are of level j 1. The felds fgeom and fcolor contan the values of fgeom r and fcolor r at the vertex. The role of hgeom and hcolor s explaned n Secton 3.2. Vertces of level j > 0 are ndexed by the base face they le n, together wth ther barycentrc coordnates wthn the face. As t s often necessary to fnd the node representng a vertex from ts ndex, we mantan an auxlary hash table that maps vertex ndces to vertex nodes. Whenever a vertex s created, t s added to the table. 3.2 Algorthms Suppose we have already constructed the face tree Q r and evaluated fgeom r at all ts vertces. Addng a wavelet requres growng Q r nto Q r+1 and evaluatng fgeom. r+1 For effcency reasons we do not reevaluate fgeom r+1 for every new wavelet. Instead we gather a sequence of s wavelets, then evaluate fgeom r+s when the new mesh s rendered. We now descrbe the gather and evaluate stages The gather stage Gatherng a wavelet ˆj wth wavelet coeffcent a j nvolves three steps: 1. Decompose the term ˆj nto a sum of hat functons at level j and j + 1 accordng to Equaton (1). 2. For each hat functon n the decomposton, grow the current face tree to accommodate t. A face tree s sad to accommodate a functon f the functon s lnear over each face. Ths process s descrbed more fully below. j0 3. For each hat functon n the decomposton ˆ v, j 0 = j, j + 1, centered at vertex v, update the hgeom feld of v: v.hgeom[j 0 ] += a j sj0 v, where s j0 v s the coeffcent of ˆj 0 v n the decomposton of step 1. The most complcated part of gatherng s growng the current face tree Q r to accommodate a level j hat functon ˆj v centered at a vertex v. We call a level j vertex complete f ts parent faces have been subdvded. (By defnton, all level 0 vertces are complete.) As each vertex has ponters to ts two parent faces (nl f a parent face does not exst), t s easy to test a vertex for completeness. Clearly, Q r can accommodate a hat functon ˆj v f the level j neghbors of vertex v are complete. Thus, there s a smple recursve procedure to make a vertex complete: Make ts two parent vertces complete; Subdvde the two parent faces of the vertex. Whenever a new vertex w s created n the completon process, f r geom s evaluated at the vertex, and the value s recorded n w.fgeom. Snce f r geom s lnear on the edges of Q r, ths evaluaton s accomplshed by averagng the fgeom values of w s parent vertces. Whle ths growng process s smple, t can generate more than the mnmum number of trangles needed to accommodate a hat functon (see Fgure 3) The evaluate stage Recall that wavelets are added n two stages. In the gather stage the face tree s grown so that t contans all the faces necessary to accommodate the new wavelets. At ths stage we also compute the values of the current approxmaton f r geom at the newly ntroduced ver- Fgure 3 Makng a vertex complete: (a) A vertex to be made complete. (The dashed faces are the mnmal number that must be added to make the vertex complete.) (b) The parent vertces are created and made complete by subdvdng ther parent faces. (c) Subdvdng the parent faces of the vertex makes t complete. tex postons. The wavelets are decomposed nto hat functons, and the coeffcent arrays for ther center vertces are updated. The new geometry functon fgeom r+s s not evaluated untl the tree s rendered, at whch tme the contrbutons from all the hat functons are summed n a sngle tree traversal. We wll now descrbe ths evaluaton stage. Let g denote the sum of all the hat functons gathered snce the last evaluaton stage, and let g k denote the partal sum obtaned by addng all the contrbutons from hat functons of levelk or smaller. By constructon g = g L, where L s the maxmum level of any leaf of Q r+s. Note that snce g k s lnear over the faces of level k and above, t s completely determned by ts values at the vertces of Q r+s of level k and less. We now present an nductve procedure to compute the values ofg L at all of the vertces of Q r+s. It s easy to compute the values of g 0 at the level 0 vertces they are the coeffcents of the gathered level 0 hat functons. Next we descrbe how to compute the values ofg k+1 at all vertces of Q r+s of level k + 1 and smaller from the values of g k at all vertces of level k and smaller. Let h k+1 v denote the coeffcent of the level k +1 hat functon centered at v. If v s a vertex of level k or less, then g k+1(v) = g k(v) + h k+1 v. If v s a level k + 1 vertex, then t splts an edge connectng ts two level k parent vertces, Therefore, g k(v) s the average of the values of g k at ts parent vertces, and g k+1(v) = g k(v) + h k+1 v. The calculaton of g L can be performed effcently durng a breadth frst traversal of Q r+s, as summarzed n the pseudocode gven n Fgure Treatment of color As mentoned earler, n the absence of texture-mappng hardware color and geometry are handled dentcally: both color and geometry wavelets are gathered and evaluated as descrbed n the prevous secton. Representng colored meshes n multresoluton form makes t easy to explot texture mappng hardware. The basc dea s to assocate a regon of texture memory wth each face of the base mesh. If the full resoluton model s subdvded to level J, a 2 J 2 J texture map s allocated, but only the lower dagonal s actually used. (To reduce the wasted texture memory, we par adjacent base mesh faces whenever possble. We then allocate a square regon of texture memory to the par.) Snce geometry s represented parametrcally by a pecewse lnear functon over M J, there s a straghtforward soluton for the normally dffcult problem of generatng texture coordnates for arbtrary meshes. The texture coordnates for any vertex are smply the pre-mage of the vertex under the parametrzaton. Therefore, the corner vertces of a base mesh face have texture coordnates (0,0), (1,0), and (0,1), and the texture coordnates for every other vertex are the average of ts parents coordnates. The mage dsplayed n

5 procedure Evaluate() queue Level 0 faces do whle queue!= empty currentface GetFrstFace(queue) currentlevel currentface.level f IsSubdvded(currentFace) then for each cornervertex v of currentface do v.g += v.hgeom[currentlevel] v.hgeom[currentlevel] 0 end for for each edgevertex e of currentface do f e has two parent faces then f e wll be vsted twce, so add 1/2 per vstg e.g += 0.25 (e.parentv[1].g+e.parentv[2].g) else e.g += 0.5 (e.parentv[1].g+e.parentv[2].g) end f end for for each 2 0, 1, 2, 3 do Append currentfaces.chldren[] to queue else for each cornervertex v of currentface v.fgeom += v.g + v.hgeom[currentlevel] v.g v.hgeom[currentlevel] 0 end for AddToDsplayLst(currentFace) end f end whle end procedure Fgure 4 Fgure 5 llustrates texture mappng. The base mesh has been rendered wth only the scalng functons of f geom, but wth all of the terms of f color. The texture map assocated wth a face of the base mesh s ntalzed by lnearly nterpolatng between the colors at the vertces of the face (.e., the level 0 color scalng functon coeffcents). The texture map s updated as soon as color wavelets are receved, essentally by pantng the wavelet nto the texture map. Snce the addton of color wavelets does not ncrease the trangle count, systems wth texture-mappng hardware color can always dsplay color at ts hghest resoluton. 4 Vewer Archtecture Our vewer, wrtten n OpenGL and Motf for Slcon Graphcs Irs workstatons, s confgured as a helper applcaton for Netscape. When a multresoluton-surface lnk s followed, the vewer applcaton opens an HTTP connecton for the base mesh fle. After recevng the base mesh, the vewer dsplays t n a graphcs wndow (see Fgure 5) and opens two parallel HTTP connectons, one for the color wavelets fle and one for the geometry wavelets fle. As wavelet coeffcents are receved they are ncorporated as descrbed n Secton 3, and the model s perodcally redsplayed. Color Plates 1(a d) llustrate a model at varous stages of transmsson. Assumng a 64Kbs lnk (ISDN speeds), the mages shown represent, from top to bottom, the model after 3 seconds, 17 seconds, 59 seconds, and 180 seconds (the full model). In standard operaton, the qualty of the model dsplayed n the vewer s controlled by the slder labeledframe Tme. When the user s rotatng or translatng the model, the vewer attempts to mantan that frame rate by measurng the polygon performance for the prevous frames and predctng the desred model sze for the upcomng frame. When there s no nteracton, a more refned model s rendered, allowng the user to see more detal. If the refned model takes a sgnfcant tme to render, the renderng s performed n stages, so that the vewer can check for user events durng the renderng. If the user decdes to nteract wth the model whle the vewer s drawng Plate wavelet # geom # color # polys L 2 L 1 type wavelets wavelets error error (a) 0-dsk (b) 0-dsk (c) 0-dsk (d) 0-dsk e e-6 (e) 0-dsk (f) 0-dsk (g) 0-dsk (h) 0-dsk () Lazy e-8 4.7e-7 (j) Lazy (k) 0-dsk (l) 2-dsk Table 1 Statstcs for Color Plate 1 a refned model, renderng s aborted, and the system returns to nteractvty. The qualty of the model can also be controlled n two other ways: the user can explctly set ether the number of geometry and color wavelets to be added to the base mesh, or the number of polygons to be used n creatng the approxmaton. If ether the frame tme or the number of polygons s specfed, the tradeoff between color and geometry s controlled wth the slder labeled Color to Geom. Movng the slder to the left ndcates a preference for geometry detal, whereas movng t to the rght ndcates a preference for color detal. The tradeoff s shown n Color Plates 1(e g), where (e) corresponds to a strong preference for color, (g) corresponds to a strong preference for geometry, and (f) corresponds to a balance between the two. Each of these model conssts of the same number of Gouraud shaded polygons. The color/geometry slder s only actve on machnes wthout texture-mappng hardware. If the machne has texture-mappng hardware, color wavelets do not ncrease the polygon count, so they are always ncluded. Color Plate 1(h) shows the model that can be dsplayed for the same polygon budget used n Plates 1(e g). 5 Results In ths secton we present varous statstcs for the color plates, and we descrbe a number of numercal experments we have performed. Statstcs for the color plates are summarzed n Table 1. Three dfferent types of wavelets were used as ndcated by the second column. All examples were computed usng the same type of wavelet for both color and geometry, although n prncple dfferent types of wavelets could be used. The other columns should be selfexplanatory. The errors reported n the last two columns are normalzed so that the crudest model has error 1. In addton to usng the vewer to create the color plates, we conducted a set of numercal experments to compare the performance of four types of wavelets: lazy, 0-, 1-, and 2-dsk wavelets. The experments focused on the followng factors: Convergence as a functon of number of wavelet coeffcents: For fxed network bandwdth, the rate at whch the transmtted model approaches the orgnal depends on how quckly the error decreases as a functon of the number of wavelet coeffcents. Fgure 6 s a plot of L 2 error n geometry vs. number of coeffcents for the varous types of wavelets for the head model shown n Color Plates 1(e h). The plot of L 1 error s qualtatvely smlar. Smlar results were obtaned for the other two models. Our concluson s that lazy wavelets perform slghtly worse than

6 100 Lazy 0-dsk 1-dsk 2-dsk 10 Error (log) # of coeffs (log) Fgure 6 L 2 error vs. number of wavelet coeffcents. 100 Lazy 0-dsk 1-dsk 2-dsk 10 Error (log) Fgure 5 The multresoluton vewer. k-dsk wavelets, but there seems to be no sgnfcant dfference between varous values of k. Convergence as a functon of number of polygons: For fxed polygon dsplay rate and update frequency, the vsual appearance of the model depends on how quckly the error decreases as a functon of the number of polygons n the model. Fgure 7 s a plot of the L 2 error vs. number of polygons for the same head model. Agan, the correspondng plot for thel 1 error s qualtatvely smlar. Color Plates 1( l) llustrate the vsual fdelty for the earth model when dfferent types of wavelets are used to produce a model wth a fxed polygon count. Table 5 ndcates that the error for ths number of polygons s actually less for lazy wavelets than for 2-dsk wavelets, due to the large number of polygons that a 2- dsk wavelet may ntroduce. Color plate 1() s the full-resoluton earth model, subdvded to level-6. The next three color plates, 1(j l), depct the earth reconstructed to approxmately 5500 polygons usng lazy wavelets, 1(j), 0-dsk wavelets, 1(k), and 2-dsk wavelets, 1(l). Although there are vsual dfferences between the mages, t s not clear whch s preferable. Our concluson s agan that lazy wavelets perform slghtly worse than k-dsk wavelets numercally, but there apparently s no sgnfcant dfference between varous values ofk. Vsually, there s no clear preference. Numercal stablty: In the conversons to and from multresoluton form some numercal error s nevtable. Whle the numercal stablty propertes of orthogonal wavelet constructons are # of polys (log) Fgure 7 L 2 error vs. number of polygons. relatvely well understood, stablty of borthogonal schemes lke ours s less clear. Lackng theory to gude us, we ran the followng experment on the earth model. For each of the four types of wavelets we performed wavelet analyss followed by wavelet synthess on a level J = 6 verson of the model. For lazy wavelets, the relatve error n the vertex postons was on the order of the machne precson. For 0-, 1-, and 2-dsk wavelets, the relatve errors were on the order of , 0.001, and When we reran the experment usng a level 3 verson of the earth, the relatve error for 2-dsk wavelets was reduced to 1 n Our concluson s that wavelets wth smaller supports are lkely to be more stable numercally than those wth larger ones, and that stablty becomes ncreasngly mportant as the number of levels ncreases. Speed: Wavelets wth larger support clearly take longer to add. There are potentally more new faces to ntroduce, and there are always a greater number of vertces whose hat functon coeffcents need to be updated. We ran a seres of tmng experments and found that on average each type of wavelet could be added (the gather stage) at the followng rate: lazy, 2700 coeffcents per second; 0-dsk, 2300; 1-dsk, 1200; 2-dsk, 600. The tme for the evaluate stage was unchanged relatve to wavelet sze, whch was expected. Overall, we conclude that 0-dsk wavelets combne good vsual fdelty for a gven number of coeffcents and for a gven number

7 of polygons, wth good numercal stablty and computaton tme. These fndngs, however, are prelmnary, and requre further confrmaton. 5.1 Data encodng As mentoned n the ntroducton, there s a small penalty for representng a mesh n multresoluton form. Snce the wavelet coeffcents are sorted n magntude order for progressve transmsson, we need to transmt wth each coeffcent the vertex dentfer for the center of the wavelet. Ths nformaton could be made mplct f the complete model was transmtted before any processng or dsplay took place. We use a smple encodng whch represents the coeffcent for a color or geometry wavelet wth three floatng pont numbers, together wth a word of nformaton for the vertex dentfer. Ths represents a 33% penalty for the beneft of progressve transmsson. A suggeston for reducng ths penalty s descrbed below. 6 Dscusson and future work We have extended prevous work on multresoluton analyss of meshes n two ways. Frst, we have shown how to perform multresoluton analyss of colored meshes by separately analyzng shape and color. Second, we have developed effcent algorthms and data structures that allow us to ncrementally construct lower resoluton approxmatons to colored meshes at nteractve rates. We have ntegrated these algorthms n a prototype mesh vewer that supports progressve transmsson, dynamc dsplay at a constant frame rate ndependent of machne performance and load, and the ablty to nteractvely trade off the amount of detal n geometry and color. The separaton of geometry and color also allows us to make effcent use of texture-mappng hardware. In future work we ntend to nvestgate: Multresoluton edtng: In analogy to Fnkelsten and Salesn s work on multresoluton curves [2] we plan to extend our multresoluton vewer to allow edtng of meshes at dfferent levels of detal. Other wavelets: We currently use pecewse lnear wavelets to represent geometry and color. When modelng smooth objects or objects wthout sharp color transtons, use of smooth wavelets may result n better compresson. Automatc tradeoff between color and geometry: If there s no texture-mappng hardware, addng wavelets for ether color or geometry wll ncrease the number of polygons that have to be rendered. When there s an upper bound on the number of polygons, for example durng dynamc vewng, one has to choose between color detal and geometry detal. Currently the tradeoff s left to the user. Heurstcs for automatcally choosng a tradeoff that results n a vsually close approxmaton would be useful. Comparson to progressve meshes: In smultaneous work Hoppe [3] has ntroduced the noton ofprogressve meshes to address the dffcultes of storage, transmsson, and dsplay of complex meshes. The basc dea s to record the changes a mesh optmzer [4] makes as t smplfes a mesh. Snce the orgnal mesh can be recovered by runnng the record of changes n reverse, the progressve mesh representaton s the smplest mesh together wth the record of changes n reverse order. The relatve advantages and dsadvantages of such an approach need further study. Better encodng: The wavelet coeffcents for a partcular model typcally span a large dynamc range, makng floatng pont an obvous choce for encodng ther values. Better use of bandwdth and storage could be made, however, by takng advantage of the wavelets beng sorted n magntude order. Fxed pont numbers could be transmtted for each coeffcent, wth the scale nformaton beng transmtted only as t changes. Ths mprovement could potentally elmnate the overhead ncurred for progressve transmsson. Acknowledgments Ths work was supported n part by the Natonal Scence Foundaton under grants CCR , DMS , and DMS , by an Alfred P. Sloan Research Fellowshp (BR-3495), an NSF Presdental Faculty Fellow award (CCR ), an ONR Young Investgator award (N ), and ndustral gfts from Interval, Mcrosoft, and Xerox. Head models courtesy of Cyberware. References [1] Matthas Eck, Tony DeRose, Tom Duchamp, Hugues Hoppe, Mchael Lounsbery, and Werner Stuetzle. Multresoluton analyss of arbtrary meshes. In Robert Cook, edtor, SIGGRAPH 95 Conference Proceedngs, Annual Conference Seres, pages ACM SIGGRAPH, Addson Wesley, August held n Los Angeles, Calforna, August [2] Adam Fnkelsten and Davd Salesn. Multresoluton curves. Computer Graphcs (SIGGRAPH 94 Proceedngs), 28(3): , July [3] H. Hoppe. Progressve meshes. In SIGGRAPH 96 Conference Proceedngs, Annual Conference Seres. ACM SIGGRAPH, Addson Wesley, August held n New Orleans, Lousana, August [4] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle. Mesh optmzaton. In J.T. Kajya, edtor, SIGGRAPH 93 Conference Proceedngs, Annual Conference Seres, pages ACM SIG- GRAPH, Addson Wesley, August held n Anahem, Calforna, August [5] J. Mchael Lounsbery. Multresoluton Analyss for Surfaces of Arbtrary Topologcal Type. PhD thess, Department of Computer Scence and Engneerng, Unversty of Washngton, September Avalable as ftp://cs.washngton.edu/pub/graphcs/lounsphd.ps.z. [6] Mchael Lounsbery, Tony DeRose, and Joe Warren. Multresoluton analyss for surfaces of arbtrary topologcal type. Submtted for publcaton, Prelmnary verson avalable as Techncal Report b, Department of Computer Scence and Engneerng, Unversty of Washngton, January, Also avalable as ftp://cs.washngton.edu/pub/graphcs/tr931005b.ps.z. [7] Stephane Mallat. A theory for multresoluton sgnal decomposton: The wavelet representaton. IEEE Transactons on Pattern Analyss and Machne Intellgence, 11(7): , July [8] Yves Meyer. Ondelettes et fonctons splnes. Techncal report, Sémnare EDP, École Polytechnque, Pars, [9] J. Rossgnac and P. Borrel. Mult-resoluton 3D approxmatons for renderng. In B. Falcdeno and T.L. Kun, edtors,modelng n Computer Graphcs, pages Sprnger-Verlag, June-July [10] Peter Schröder and Wm Sweldens. Sphercal wavelets: Effcently representng functons on the sphere. In Robert Cook, edtor, SIG- GRAPH 95 Conference Proceedngs, Annual Conference Seres, pages ACM SIGGRAPH, Addson Wesley, August held n Los Angeles, Calforna, August [11] Erc Stollntz, Tony DeRose, and Davd Salesn. Wavelets for Computer Graphcs: Theory and Applcatons. Morgan Kaufmann, 1996.

8 a) 3 seconds... e) hgh color detal ) full resoluton b) 17 seconds... f) equal color and geometry detal j) lazy wavelets c) 59 seconds... g) hgh geometry detal k) 0-dsk wavelets d) full resoluton h) texture mappng l) 2-dsk wavelets Color Plate 1