Multiresolution for Algebraic Curves and Surfaces using Wavelets

Transcription

1 EUROGRAPHIS 0x / N.N. ad N.N. (Guest Editors) Volume xxx, (200x), Number yyy Multiresolutio for Algebraic urves ad Surfaces usig Wavelets Jordi Esteve Pere Bruet Alvar Viacua Uiversitat Politècica de ataluya, Barceloa, Spai [jesteve, bruet, alvar]@lsi.upc.es Abstract This paper describes a multiresolutio method for implicit curves ad surfaces. The method is based o wavelets, ad is able to simplify the topology. The implicit curves ad surfaces are defied as the zero-valued piece-wise algebraic isosurface of a tesor-product uiform cubic B-splie. A wavelet multiresolutio method that deals with uiform cubic B-splies o bouded domais is proposed. I order to hadle arbitrary domais the proposed algorithm dyamically adds appropriate cotrol poits ad deletes them i the sythesis phase. Keywords: geometric modelig, algebraic surfaces, wavelets, multiresolutio, topological simplificatio, coversio algorithms. 1. Itroductio I this paper a plaar curve ad surface multiresolutio method that simplifies the topology is preseted. We will work with algebraic curves ad surfaces defied as the zerovalued algebraic isosurface of a tesor-product uiform cubic B-splie. Thus, the cubic B-splie has oe dimesio more tha the dimesio of objects to be represeted (bivariate B-splies for curves ad trivariate B-splies for surfaces). The B-splie approximatio will produce the simplificatio of the objects ad the chage of their topology i a simple way. A preprocess is required to covert the iitial object model i a way that B-splie algebraic curve/surfaces ca be used. After a voxelizatio of the iitial object (a polyhedral BRep, a surface model, etc.), the coversio method from voxel ad octree represetatios to B-splie algebraic surfaces described i Viacua et al. 1 ca be used to obtai the iitial data for the wavelet decompositio. There are a lot of efforts aroud the solid multiresolutio or simplificatio, due to the iterest of iteractig with simplified versios for complex models. However, because of the complexity of the topological relatios iside the classical models used i AD (triagular meshes, BReps, SGs) it is actually difficult to make chages directly i their topology. May algorithms simplify triagular meshes doig edge collapses, for example Hoppe et al. 2 ad Hoppe 3. Some also make topological chages joiig the earest vertices of differet triagles: Rossigac et al. 4, Rofard et al. 5, Popovic et al. 6, Garlad et al. 7. Although they do topological simplificatio, they produce o-regular surfaces (isolate edges ad vertices appear) or o-maifold objects (ot every surface poit has a eighbourhood topologically equivalet to a disk). The algorithms that get topological chages keepig the solid correctess are based o space decompositio models, usually voxels ad octrees. It is the case of He et al. 8 with voxels ad Adújar et al. 9 with octrees. Nevertheless, they have several drawbacks: they must do two coversio steps to ad from the space decompositio model, the secod oe is very hard if we wat to compress the large umber of geerated faces ad it is complicated to save iformatio eeded for the recostructio of the iitial object. I our multiresolutio algorithm we use a very powerful tool: the wavelets. The wavelet methods allow doig a fast decompositio that miimizes the error data (the detail lost i the decompositio) ad, later, we ca recostruct the origial solid. Furthermore, the decomposed solid plus the error data use the same storage space as the origial solid. Other curve ad surface multiresolutio methods based o

2 2 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio wavelets exist, but they do ot use algebraic curves/surfaces ad they caot simplify the topology either. Reissell 10 implemets a parametric curves/surfaces multiresolutio with a type of wavelets called coiflets. The method works over the co-ordiates of the surface poits ad the approximatios are oly liear iterpolatios of the scalig coefficiets. Frequetly, the wavelets multiresolutio methods have bee applied over B-splie surfaces (see Sectio 3). Usig the ideas of Lyche et al. 11, Kaziik et al. 12 presets a multiresolutio over a parametric surface defied as a tesor-product of o-uiform B-splies. Fikelstei et al. 13 does the same over tesor-products of ed-iterpolatig uiform cubic B- splies. I Lousbery et al. 14 the wavelets are applied over recursive subdivisio surfaces with ay topology, although they caot simplify this topology. I Viacua et al. 1, a coversio method from a voxel or octree represetatio to a algebraic surface defied as the zero-valued algebraic isosurface of a tesor-product uiform cubic B-splie is described. The multiresolutio is based o a octree level selectio ad a subsequet coversio to the algebraic surface. However, this coversio has a importat cost. The goal of our implemetatio is a quick multiresolutio thaks to directly hadlig the B-splie cotrol poits that defie the algebraic surface. Furthermore, our algorithm produces better approximatios due to the use of wavelet methods. I the ext sectio we preset the formal defiitio of the algebraic curves ad surfaces used i the applicatio. I Sectio 3 we discuss how to fit the wavelet multiresolutio to decompose/recostruct models of arbitrary size voxel spaces (i.e. ot ecessary powers of 2). Sectio 4 exposes the particularities of our algorithm: how to set the ukow cotrol poits i the decompositio ad how we ca do the recostructio process. Also, it describes a coveiet data structure for the model. Sectio 5 icludes several examples i 2D ad 3D domais. Fially, we poit out some coclusios ad possible improvemets of our work. its ode s resolutio. Therefore, the resolutio of the Node- ollectio must be chose accordig to the size of the features that we are iterested i. I our case, a fuctioal B-splie has bee chose to defie the surface stabbig the Node-ollectio. The surface is the zero-valued algebraic isosurface of the fuctioal B- splie. To model a plaar curve, for example, we cosider a 3D B-splie fuctio defied o the rectagular grid (ow the odes are 2D). The itersectio of its graph with the zerovalued plae defies the curve (see Figure 1). The Node- ollectio is the set of the 2D odes o the rectagular grid stabbed by the curve (see Figure 2). To model a surface, we defie a B-splie fuctio o the 3D regular grid such that the itersectio of its graph with the zero-valued hyperplae (w 0) produces the surface. Figure 1: The itersectio of the 3D B-splie fuctio with the zero-valued plae defies the object boudary curve 2. B-Splie-Based Algebraic Surface Models Let be the voxel model of a object surface. We do ot eed all the data cotaied i the voxel model, it is eough to store the set of voxels (odes) stabbed by the object surface. This set of odes is called Node-ollectio. The Node-ollectio ca be obtaied easily from voxel ad octree represetatios. The Node-ollectio ca also be calculated from polyhedral BReps ad surface models through a voxelizatio or 3D sca coversio algorithm 15. Usig the Node-ollectio to represet the surface offers us flexibility. Ay surface stabbig all odes i the Node- ollectio is a acceptable represetatio of the iitial object. This allow us to get smoother surfaces. As always, this discretizatio comes at the price of igorig detail below Figure 2: The Node-ollectio is the set of shaded odes Let us ow defie G as the set of spatial idices of the odes of the Node-ollectio G i j N i j Node ollectio (1) where N i j represets a uit cube with the vertices i δ 1 j δ 2, with δ 1 δ 2 beig either 0 or 1. Observe that the cardiality of G grows i the same maer as the size of the object does (quadratically if we are modelig a surface). O the other had, let us defie I as the set of spatial idices of the odes i the immediate viciity of the Node- ollectio:

3 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio 3 I i j with i δ 1 j δ 2 G δ 1 δ (2) The cardiality of I is of the order of 4 times the cardiality of G. A similar expressio is easily obtaied i a 3D case. Let us ow cosider the uiform, tesor-product B-splie fuctio F defied o the regular grid. F is a itegral, fuctioal splie ad we are oly iterested i evaluatig it o the Node-ollectio, to fid its zero-isosurface. F x y w i j N i x N j y 0 (3) i j I F is a valid represetatio if its zero-isosurface stabs all odes i the Node-ollectio. Let us cosider that F 0 iside the object ad F 0 outside the object (oly i odes belogig to the Node-ollectio). Sice F is defied o a regular grid (iduced by the voxels), we choose to represet it as a tesor-product uiform cubic B-splie. Our multiresolutio algorithm cosists of two mai steps: 1. The B-splie algebraic surface is obtaied. The B-splie fuctioal F must be calculated from a iitial Node- ollectio. This is a iterative algorithm that tries to miimize the curvature of the fial B-splie algebraic surface. Our implemetatio is based o Viacua et al. 1. We iitially assig weights of 1 to the vertices i the set I (Equatio 2) that lie outside the object ad 1 to the iterior vertices i that set. We the filter these weights usig a discrete Laplacia istead of the more elaborated method described i A surface multiresolutio is calculated applyig a wavelet multiresolutio method to the B-splie fuctioal F. The object surface approximatios are the zeroisosurfaces of the wavelet approximatios of F. This paper focuses o this last poit. Before presetig a detailed discussio i Sectio 4 we iclude for completeess ad otatioal coveiece a itroductio to waveletsbased B-splie multiresolutio i the ext sectio. 3. Wavelet Multiresolutio Wavelets are a mathematical tool with a wide variety of applicatios i several fields. A lot of B-splie wavelet methods exist. I Fikelstei et al. 13 a method for editerpolatig uiform cubic B-splies has bee preseted. hui 16 cotais a study of the uiform B-splie wavelets o ubouded domais. I Lyche et al. 11 ad, later, i Kaziik et al. 12 the use of wavelets with o-uiform B-splies is described. I our case, due to the kid of algebraic surface to decompose, we will use uiform cubic B-splies o bouded domais (their kot sequece is uiformly spaced, with all kots of multiplicity 1 ad the domai is fiite). We develop a wavelet multiresolutio method, similar to the editerpolatig B-splies i Fikelstei et al. 13, implemeted with pre-calculated bad matrices. This will allow us to perform the aalysis ad sythesis processes of a row of coefficiets with liear cost. Our method produces similar results as hui 16 does, but usig wavelets defied o bouded domais. I hui 16 the same filter is applied to all coefficiets i the aalysis/sythesis process (a movig average algorithm). The disadvatage of the wavelets i hui 16 is that the aalysis filters are ifiite ad must be trucated (causig precisio errors). Before describig our algorithm, we briefly review the mai ideas behid wavelet multiresolutio ad defie the otatio used. For a more complete descriptio ad implemetatio details see, for example, Fikelstei et al. 13 ad Esteve et al. 17. It is importat to remark that, i this paper, the vector ad matrix idices represet the umber of itervals where the B-splie fuctio is defied istead of a level couter i the multiresolutio process. A B-splie o a bouded domai with 2 1 itervals ca be represeted as a colum vector of cotrol poits 2 1 (made up of 2 1 degree cotrol poits). The goal is to approximate the B-splie with aother B-splie fuctio of half resolutio (its domai will have half umber of itervals). This fuctio ca be represeted as a colum vector of fewer cotrol poits 2. lassical wavelets use a matrix A 2 to compute 2 A (4) ad a accompayig matrix B 2 to capture detail D 2 B (5) These two matrices (called aalysis filters) have idepedet colums, ad the process ca be iverted usig the sythesis filters P 2 ad Q 2 : 2 1 P 2 2 Q 2 D 2 (6) without loss of iformatio. Furthermore, the space required to store 2 ad D 2 is the same as that required for 2 1. This process ca be iterated yieldig a multiresolutio scheme (with each level usig half as much detail), which ca be stored usig the same space as oe approximatio 1 ad a sequece of detail vectors D 1 D 2 D 4 D 2. We shall eed to apply this level of aalysis i cotexts where the umber of itervals of the B-splies ivolved is arbitrary, ot a power of two, ad we will thus itroduce modificatios to this scheme i the followig sectio. Before dealig with that aspect of our problem, we eed to borrow oe more igrediet of classical wavelets. The sythesis matrices P 2 ad Q 2 for our bouded-domai uiform B-splies are baded ad, therefore, yield to ecoomic

4 4 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio umerical hadlig. However, their iverses A 2 ad B 2 are full, which meas that the aalysis phase usig (4) ad (5) has a quadratic cost (i the umber of cotrol poits). Quak et al. 18 shows how to improve o this. Notig by Φ 2 the vector of B-splie basis fuctios (the scalig fuctios) ad by Ψ 2 the vector of basis fuctios of the orthogoal of the resolutio 2 B-splies i the space of resolutio 2 1 B-splies (the wavelet fuctios) oe ca form the matrices I 2 ad J 2 of ier products of these, which satisfy18 17 obtaied i the aalysis process ad keepig a liear time cost. We will also reject the cotrol poits that we have set i the decompositio process. Figure 3 shows the chages i the Node-ollectio durig the aalysis process. The area that is occupied by the Node-ollectio grows. We must take ito accout the ukow cotrol poits aroud these ew areas of the Node- ollectio. I 2 2 P 2 T I (7) J 2 D 2 Q 2 T I (8) ad are baded. Thus, the aalysis amouts to computig the right-had sides of these ad solvig two baded liear systems, which gives a liear cost 19. Our techical report 17 discusses this i more detail ad also iclude the computatio of the sythesis matrices P 2 ad Q 2 ad the ier product matrices I 2 ad J 2 for the uiform cubic B-splies defied o a bouded domai. Next, we move o to discuss our adaptatio of these algorithms to our problem, extedig i Sectio 4.1 their applicability to a arbitrary eve umber of itervals. 4. The Proposed Wavelet Algorithm We have adapted the wavelet multiresolutio to our applicatio, sice the B-splie fuctio is ot defied o the whole domai, but oly o the odes of the 3D grid that belog to the Node-ollectio. The mai cotributios of the proposed wavelet algorithm are ext summarized: 1. The family of aalysis ad sythesis matrices has bee exteded. This is ecessary, as the B-splie fuctios must be defied o a domai more geeral tha domais havig 2 itervals. We restrict the B-splies to have a eve umber of itervals. This geeralizes the usual aalysis/sythesis process. 2. At each iteratio, oly B-splie cotrol poits i the viciity of the zero-isosurface are stored. 3. We must set some ukow cotrol poits (cotrol poits whose idices do ot belog to the set I) before decomposig the B-splie. The reasos are: We eed that the B-splies have a eve umber of itervals to decompose them, as idicated above. Itervals which are close together ca be joied i the decompositio. I this case, we must set the cotrol poits i betwee. The decomposed B-splie plus the produced detail should use the least space possible (ideally the same as the origial B-splie). 4. I the recostructio we must recover the cotrol poits whose idices beloged to the set I usig the iformatio (a) Level 2 (b) Level Figure 3: hages i the Node-ollectio 4.1. Wavelet Multiresolutio o Domais of Eve Legth The sigle coditio we will impose o the multiresolutio is that the B-splie must have a eve umber of itervals where it is defied (the umber of cotrol poits must be odd sice we are usig cubic B-splies). I the decompositio process, the umber of itervals will be divided by 2. Thus, the decompositio of a B-splie defied o 2 itervals 2 i oe defied o itervals ad its detail D ad the iverse recostructio ca be writte (aalogously to Equatios (4), (5) ad (6)) as A 2 (9) D B 2 (10) 2 P Q D (11) I the iterative applicatio of the aalysis process we ca obtai a B-splie with a odd umber of itervals. The, we must set additioal cotrol poits at each ed of the B-splie to retur to a eve iterval B-splie ad, i this way, cotiue the decompositio. Subsectio 4.2 discusses how to set these additioal cotrol poits. Therefore, we eed to add a assigmet process betwee the decompositio steps. Also, durig the recostructio phase, we must reject the cotrol poits thus itroduced at each level:

5 % % Esteve, Bruet ad Viacua / Algebraic surface multiresolutio 5 Aalysis A 2 2m m Sythesis B #" set A m B m D #" D m D iitial Figure 5: 10th row decompositio 2 $ P $ re ject 2m $ P m m Q Q m D D m The sythesis matrices P ad Q ad the ier product matrices I ad J for the uiform cubic B-splies defied o a domai with a arbitrary eve umber of itervals are listed i Esteve et al Aalysis Process ad the Assigmet of Extra otrol Poits As previously discussed, before applyig the aalysis process we must set the value of some ukow cotrol poits. Hece, we must decide which cotrol poits must be assiged ad which will be their value. We have studied several alteratives about how to set these cotrol poits. I the followig examples we detail ad explai the decompositio of the 10th row (we work o uidimesioal examples that represet the decompositio/recostructio i oe specific directio of the tesor product) Figure 4: Horizotal Decompositio First (see Figure 5), the B-splie fuctioal iitial is defied o the itervals [2, 3] ad [19, 22]. We have to set cotrol poits with odd idices (i positios 5 ad 17) to get the B-splie 2 defied o two domais with eve idices ([2, 4] ad [18, 22]). The we ca apply the decompositio process described i the previous sectio to obtai ad D. However, there is a problem: whe workig with a tesor product of the basis fuctios (o a 2 or 3D grid), after several steps of decompositio artifacts do appear (empty space iside the solid ad/or solid iside the empty space). The reaso is that the basis fuctios of the uiform cubic B-splies o a bouded domai are similar except at the edpoits, which are trucated. To miimize the error produced, the wavelet decompositio plays with the edpoit basis fuctios, that have less eergy tha the others, givig them high weights ad sig chaged with respect to the eighbours basis fuctios. The cotrol poits at each ed i oe directio ca become iterior cotrol poits whe we chage the directio of the decompositio process. I short, the value of the cotrol poits are weights of basis fuctios with differet eergy depedig o the directio we are decomposig. To overcome this problem, the B-splie fuctio has bee exteded at each ed before applyig the decompositio process. I this way, the trucated basis fuctios have o ifluece i the regio of iterest. Settig 3 additioal cotrol poits at each side is eough because the uiform cubic B- splies o bouded domais have 3 trucated basis fuctios at each ed. Our implemetatio, however, sets 4 additioal cotrol poits at each side which yields simpler algorithms. For example, Figure 6 shows a piece of the 10th row decompositio. First, as we described before, the cotrol poits with odd idices are assiged (the cotrol poit i positio 5) to get 2. The, 4 additioal cotrol poits at both sides are assiged ( set ). We shall call these accessory cotrol poits. Ad fially 2 set is decomposed i 4 ad D 4. Obviously, this proposal produces 8 additioal cotrol poits i 4 plus D 4 tha those i 2. To overcome this drawback, first observe that oly the cotrol poits of 4 ad D 4 eeded to recostruct the iitial B-splie 2 must be stored. Sice P is a bad matrix such that its middle rows have 2 or 3 o-zero coefficiets, all the 2 cotrol

6 6 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio iitial set +4 D O O O O O O O O Figure 6: Piece of 10th row decompositio. 4 additioal cotrol poits at both sides are assiged to 2 poits (i positios 1, 2, 3, 4 ad 5) are calculated from the cotrol poits i positios 0, 2, 4 ad 6 (for example the positio is idicated by subidices 1 2 is recovered form 0 ad 2, 2 2 is recovered form 0, 2 ad 4, etc). Therefore we ca reject 2 cotrol poits at both sides of each iterval of 4. We caot do the same with D 4 because Q has the middle rows wider tha P (Q has 5 or 6 ozero coefficiets i the middle rows). I this way, each iterval decompositio produces 4 additioal cotrol poits i plus D 4 tha those i 2. Next observe that if i the recostructio process we kow the values assiged to the accessory cotrol poits of 2 durig the decompositio process we may do without these extra 4 cotrol poits. I the preset work we set the accessory cotrol poits of 2 i the followig way: they ca be K or K (K is a model costat whose value is i the same order of magitude as the cotrol poits) ad, further, all correlative cotrol poits have the same sig ad they coicide i sig with the fuctioal B-splie at the edpoits of the domai where it is defied. This is cosistet with the sematics of our model: egative values correspod to the iterior whereas positive values correspod to the exterior. I this way, we do t eed keep so may cotrol poits of D 4 i the decompositio, because we take advatage of kowig several fragmets of 2. The computatioal procedure to take advatage of this iformatio is discussed i Sectio 4.3. As ca be see i Figure 7, we achieve that the umber of cotrol poits stored i plus D is the same as the iitial umber i 2. We have tested other ways to set the accessory cotrol poits. For example, settig the accessory cotrol poits as the average value of the closest kow cotrol poit at the left side ad the closest kow cotrol poit at the right side. Or settig the accessory cotrol poits as the average value O D +D set O O O O O O O O O O O O O O O O rej O rej O O O O O O O O O O O O O O Figure 7: 10th row decompositio. 4 additioal cotrol poits at both sides are assiged to 2 ad, later, 2 cotrol poits at both sides are rejected from ad D of the fuctioal B-splie evaluated at the eds of the Node- ollectio at the left side ad right side. We have also tested the assigmet of accessory cotrol poits that miimizes the error data i the wavelet decompositio. Noe of them ca take advatage of kowig several fragmets of 2 to reduce the umber of cotrol poits stored i the decompositio. Ad oe of them has produced so good results as the assigmet of K or K we have previously described. This assigmet produces B-splie fuctios with strog slopes aroud their zeroes i the odes of the Node- ollectio, leadig to better approximatios. Note that, i the aalysis process, if two itervals of kow cotrol poits of 2 are very close, they ca be coverted ito oe iterval i. This happes whe the distace of the earest edpoit cotrol poits of two itervals (situated o odd positios) is lower or equal tha 4 (see Figure 8). I these cases we caot use the ecoomies discussed above, ad our scheme will eed a few more cotrol poits to represet the multiresolutio (see Sectio 5 for figures measurig the impact of this i several examples). 2 + D +D set O O O O O O O O O O O O rej O O rej O O O O O O Figure 8: 4th row decompositio. Joiig of 3 close itervals 4.3. Sythesis Process ad the Recovery of otrol Poits Let us see how to do the recostructio 2 P Q D uder these coditios. We start by kowig the previous

7 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio 7 accessory cotrol poits of 2 (with value K or K) ad those stored i ad D. From the recostructio poit of view the kow cotrol poits of 2, ad D of the example i Figure 7 are marked i Figure set O O O O O O O O O O O O O O O O D Figure 9: 10th row recostructio. Kow cotrol poits of 2, ad D Splittig each colum vector i two vectors with the kow k ad ukow u cotrol poits (, k ad u have the same dimesio. k is filled with zeroes i the etries where there are ukow cotrol poits. u is filled with zeroes i the etries where there are kow cotrol poits.) ad applyig the sythesis filter: 2 2 k 2 u (12) k u (13) D D k D u (14) I 2 k 2 u P k u Q D k D u (15) I 2 k P k Q D k P u Q D u I 2 u (16) The left-had side i the last equatio ca be evaluated: it is the idepedet term V of the liear equatio system. The right-had side ca be expressed with a sigle matrix ad a sigle vector of ukow cotrol poits if we elimiate the matrix colums that are multiplied by zero (the vectors ad matrices simplified are marked with ) V P& ' Q& ' I& & ' u D& ' u & 2 T u (17) Reorderig the colums of matrix ' ' P& Q& I&, we trasform it ito a bad matrix. So, we have a baded liear system that ca be solved i liear time doig a LU decompositio 19. I the recostructio process we must reject the accessory cotrol poits of 2 that have bee assiged. So, we must store additioal iformatio about which cotrol poits are assiged durig the aalysis process (i the preset implemetatio we store the itervals where there are defied cotrol poits, see ext sectio) Data Structure Because our model oly works with the cotrol poits with spatial idices cotaied i I, a set usually much smaller tha the possible cotrol poits that ca be defied iside the domai s grid, we thik it is suitable to store the cotrol poits i a hash table. We also keep some additioal data about the itervals where the cotrol poits are defied i oe directio (a vector of iterval lists i 2D ad a matrix of iterval lists i 3D). The hashig table eables the storage of the kow cotrol poits i a compact way (the storage space is proportioal to the kow cotrol poits, ot to the domais s grid) ad to cosult them i a almost costat time. The iterval iformatio about where the defied cotrol poits are, avoids searchig for ukow cotrol poits i the hashig table. Thus, it will be feasible to do fast computatios i the directio of the iterval data i the aalysis ad sythesis process. Furthermore, this iformatio will be very useful i the recostructio process sice it will allow us, i oe multiresolutio level, to distiguish betwee the kow ad the accessory cotrol poits. Due to the alteratio of the directio of the evaluatios i both the aalysis ad the sythesis processes we must calculate the iterval iformatio i oe directio from the same iformatio i other directio. This costs O i 2D ad O 2 i 3D (it eeds to visit oly oce each elemet i the iterval vector/matrix: we assume the same dimesio i all directios of the data domai). This cost has the same order of magitude as each decompositio ad recostructio step if we cosider that the cotrol poits i the Node- ollectio are sparse i relatio to the domai s grid ad, therefore, the aalysis ad sythesis process of each row of few cotrol poits has a costat cost. This is oly true at high resolutio. It is there, however, that the questio of cost if meaigful. 5. Examples ad Discussio We have tested the proposed algorithm over several curves ad surfaces. Figures 10 through 13 show examples of curves, whereas Figures 14 ad 15 show two 3D solids. Figure 10 shows a multiresolutio of the Africa cotiet, Figure 11 of a camera, Figure 12 of a skeleto ad Figure 13 of a tiger. The images correspod to the resolutio levels after doig the decompositio process i the two directios. Therefore, there is a itermediate resolutio betwee two cosecutive images. The iitial curve is obtaied from a bitmap image. We have developed a simple algorithm for that coversio. Table 1 lists the dimesios of the domai, the umber of cotrol poits stored (spatial idices of I) ad the umber of defied odes (same order of magitude as the Node-ollectio G) for each resolutio level. The umber of coefficiets i the wavelet trasform of Africa (Africa i the lowest resolutio level plus all the detail data produced i the decompositio process) is 27116, 22.4% more tha the origial curve. The wavelet trasform of tiger has

8 8 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio coefficiets, 8.3% more tha the origial curve ( coefficiets). This icrease is due to set the accessory cotrol poits i the decompositio process. Figure 10: Africa multiresolutio Figure 12: Skeleto multiresolutio (a image of the voxelizatio of a skull from a computerized tomography). Fially, Figure 15 shows a multiresolutio of a mechaical part s boudary. The iitial surface is obtaied from a octree ad usig the coversio algorithm preseted i Viacua et al. 1. The images correspod to the resolutio levels after doig the decompositio process i the three directios. Therefore, there are two itermediate resolutios betwee two cosecutive images. Tables 2 ad 3 list the same parameters as the previous table, but here for the surface models. The wavelet trasform of the skull has coefficiets, 14.4% more tha the origial surface. The wavelet trasform of the mechaical part has coefficiets, 17.2% more tha the origial. Because of the small thickess of the skull, the surface topology becomes more complicated (holes ad solid fragmetatio) i the itermediate resolutios. Figure 11: amera multiresolutio Figure 14 shows a multiresolutio of a medical model Figure 16 shows the same mechaical surface as Figure 15, but ow the simplificatio is achieved pruig oe or more levels of the octree structure ad usig the coversio algorithm of Viacua et al. 1. From the images we realize that our wavelet multiresolutio produces results closer to the iitial object tha pruig the octree, surely because of

9 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio 9 Table 1: urve multiresolutio: Africa Resolutio Domai #defied #defied level dimesio coeff. I odes ( G 1 367x a 173x x a 88x x a 46x x a 25x x a 14x Figure 13: Tiger multiresolutio 6 15x a 9x x a 6x x a 5x x a 4x x sitio step for the mechaical part are 6.067, 1.571, 0.465, 0.111, 0.037, 0.019, secods respectively. Each decompositio step cosists i doig three decompositio substeps ad chagig the directio, Y, Z of decompositio. These times have bee obtaied o a persoal computer with a 350Mhz AMD-K6-2 PU ad 64 Mb of mai memory. 6. oclusios ad Future Work Figure 14: Skull multiresolutio the error miimizatio property of wavelet aalysis ad the good approximatio properties of B-splies. Note that i the first simplificatio steps we get objects very similar to the origials alog with a importat data reductio (if the detail data is throw out). I fact, we could use the wavelet method as a compressio algorithm if we oly preserve the most simplified solid alog with the sigificat detail coefficiets. Due to the liear time cost i 2D ad the quadratic cost i 3D, the multiresolutio is calculated quickly. For example, the ru-times of each decompo- We have preseted a multiresolutio method to simplify the geometry ad also the topology of curves ad surfaces. The proposed scheme uses wavelet decompositio ad obtais a set of multiresolutio algebraic surface models. By raisig the problem s dimesio ad viewig the object as a level curve/surface of a fuctio i oe more dimesio, we allow topological chages i the object whe we aalyse this fuctio. The proposed algorithm calculates the simplificatio ad recostructio directly over the kow cotrol poits of the fuctio usig wavelet multiresolutio methods. This allows obtaiig a object decompositio (the lowest resolutio versio plus several detail at differet levels) that eeds almost the same space as the iitial object. Further, the detailed data (error) of the B-splie fuc-

10 * 10 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio Table 2: Surface multiresolutio: Skull Resolutio Domai #defied #defied level dimesio coeff. I odes ( G 1 97x97x a 97x37x b 37x50x x50x a 50x20x b 20x27x x27x a 27x12x b 12x15x x15x a 15x8x b 8x9x Figure 15: Mechaical part multiresolutio 5 9x9x a 9x6x b 6x6x x6x a 6x5x b 5x5x x5x a 5x4x b 4x4x x4x Figure 16: Mechaical part multiresolutio pruig the octree levels tio that defies the algebraic curve/surface is miimized accordig to the ier fuctio product f u g u ) f u g u du. It is ot easy to describe how miimizig the B-splie error affects the B-splie zeroes that defie the algebraic curve/surface. For example, we ca get better approximatios usig B-splie fuctios with strog slopes aroud their zeroes, at least i the first simplificatio steps. The algorithm uses wavelets o geeral domais (ot restricted to 2 ). A specific wavelet scheme for aalysis/sythesis o geeral eve domais has bee developed, icludig strategies for additio ad rejectio of cotrol poits. At each iteratio, oly B-splie cotrol poits i the viciity of the zero isosurface are stored. As a cosequece, the umber of defied cotrol poits i geeral for the 3D case is proportioal to 2 - surface area of the object - ad ot to the dimesio 3 of the complete voxel domai. The wavelet algorithms preset liear cost i 2D ad quadratic cost i 3D, ad 2D ad 3D cases are based o repeated applicatio of oe-dimesioal aalysis ad sythesis. The complexity of oe simplificatio step is of the order of the total surface area of the object. Workig with algebraic isosurfaces of cubic B-splies is well suitable for applicatios that deal with smooth curves ad surfaces. The mai drawback of these algebraic isosurfaces is the iability to represet sharp features. To improve the results ad avoid shrikig/dilatig the solid we are explorig the possibility to chage the isosurface value or, o the other had, to start with a iitial ob-

11 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio 11 Table 3: Surface multiresolutio: Mechaical part Resolutio Domai #defied #defied level dimesio coeff. I odes ( G 1 129x129x a 129x66x b 66x66x x66x a 66x35x b 35x35x x35x a 35x19x b 19x19x x19x a 19x11x b 11x11x x11x a 11x7x b 7x7x x7x a 7x5x b 5x5x x5x a 5x4x b 4x4x x4x ject that has bee modified with the same eergy where the B-splie fuctio is positive (solid) as where the B-splie fuctio is egative (empty space). It is iterestig i complex models to save memory space. Thus, we are comparig several compressio techiques to reduce the amout of detail or error. We are also studyig other ways to set the accessory cotrol poits that, geeratig correct simplificatios, could reduce the total error. Also, we are workig o directly computig a B-splie algebraic surface from a sufficietly dese set of poits i 3D space. I the future, we will work o the multiresolutio editig of the curves ad surfaces defied by this model. This is a very useful tool i multiresolutio eviromets: we ca edit i oe multiresolutio level ad, later, add the other levels to get the object with local or global chages. To make this editig more useful, we are ivestigatig ways to localize the simplificatio. Refereces 1. A. Viacua, I. Navazo, ad P. Bruet. Octtrees meet splies. I G. Fari, H. Bieri, G. Bruett, ad T. DeRose, editors, Geometric Modellig, omputig [Suppl], volume 13, pages Spriger-Verlag, H. Hoppe, T. DeRose, T. Duchamp, J. McDoald, ad W. Stuetzle. Mesh optimizatio. SIGGRAPH. omputer Graphics Proceedigs, Aual oferece Series, pages 19 26, H. Hoppe. Progressive meshes. SIGGRAPH. omputer Graphics Proceedigs, Aual oferece Series, pages , J. Rossigac ad P. Borrel. Multi-resolutios 3d approximatios for rederig complex scees. I B. Falciedo ad T. L. Kuii, editors, Modellig i omputer Graphics, pages Spriger-Verlag, R. Rofard ad J. Rossigac. Full-rage approximatio of triagulated polyhedra. omputer Graphics Forum (Proceedigs of Eurographics), 15(3):67 76, J. Popović ad H. Hoppe. Progressive simplicial complexes. SIGGRAPH. omputer Graphics Proceedigs, Aual oferece Series, pages , M. Garlad ad P. S. Heckbert. Surface simplificatio usig quadric error metrics. SIGGRAPH. omputer Graphics Proceedigs, Aual oferece Series, pages , T. He, L. Hog, A. Kaufma, A. Varshey, ad S. Wag. Voxel based object simpliicatio. I G. M. Nielso ad D. Silver, editors, Visualizatio 95, pages , Atlata, GA, Adújar, D. Ayala, P. Bruet, R. Joa-Ariyo, ad J. Solé. Automatic geeratio of multiresolutio boudary represetatios. omputer Graphics Forum, 15(3), L.-M. Reissell. Wavelet multiresolutio represetatio of curves ad surfaces. Graphical Models ad Image Processig, 58(3): , T. Lyche ad K. Mørke. Splie-wavelets of miimal support. Numerical Methods of Approximatio Theory, 9: , R. Kaziik ad G. Elber. Orthogoal decompositio of o-uiform bsplie spaces usig wavelets. omputer Graphics Forum (Proceedigs of Eurographics), 16(3):27 38, A. Fikelstei ad D. H. Salesi. Multiresolutio curves. SIGGRAPH. omputer Graphics Proceedigs, Aual oferece Series, pages , M. Lousbery, T. DeRose, ad J. Warre. Multiresolutio aalysis for surfaces of arbritary topological type.

12 12 Esteve, Bruet ad Viacua / Algebraic surface multiresolutio Techical Report b, Dept. of omputer Sciece ad Egieerig. Uiversity of Washigto, M. Sramek ad A. E. Kaufma. Alias-free voxelizatio of geometric objects. IEEE trasactios o visualizatio ad computer graphics, 5(3): , harles K. hui. A itroductio to Wavelets. Academic Press, J. Esteve, P. Bruet, ad A. Viacua. Multiresolutio for algebraic curves ad surfaces usig wavelets. Techical Report LSI R, Dept. L.S.I., Uiversitat Politècica de ataluya, E. Quak ad N. Weyrich. Decompositio ad recostructio algorithms for splie wavelets o a bouded iterval. Techical Report AT 294, eter for Approximatio Theory, Texas A&M Uiversity, W. H. Press, B. P. Flaery, S. A. Teukolsky, ad W. T. Fetterlig. Numerical Recipes. ambridge Uiversity Press, ambridge, secod editio editio, 1992.