Efficient Compression of 3D Dynamic Mesh Sequences

Transcription

1 Effcent Compresson of 3D Dynamc Mesh Sequences Rachda Amoun and Wolfgang Straßer WSI / GRIS Unversty of Tübngen, Germany {amoun, strasser}@grs.un-tuebngen.de Fgure : Sample frames of the anmatons used for the analyss. From left to rght: dance wth 4, dolphn wth 9, chcken wth and cow wth 6 clusters. Each cluster s colored dfferently and encoded separately. ABSTRACT Ths paper presents a new compresson algorthm for 3D dynamc mesh sequences based on the local prncpal component analyss (LPCA). The algorthm clusters the vertces nto a number of clusters usng the local smlarty between the traectores n a coordnate system that s defned n each cluster, and thus transforms the orgnal vertex coordnates nto the local coordnate frame of ther cluster. Ths operaton leads to a strong clusterng behavor of vertces and makes each regon nvarant to any deformaton over tme. Then, each cluster s effcently encoded wth the prncpal component analyss. The approprate numbers of bass vectors to approxmate the clusters are optmally chosen usng the bt allocaton process. For further compresson, quantzaton and entropy encodng are used. Accordng to the expermental results, the proposed codng scheme provdes a sgnfcantly mprovement n compresson rato over exstng coders. Keywords: 3D anmaton, anmated mesh compresson, segmentaton, PCA, rate-dstorton optmzaton. INTRODUCTION Anmated meshes are commonly used n computer games, computer generated moves, and many scentfc applcatons. The anmatons n these applcatons are often complex, nonlnearly generated and contan large geometrc datasets. They often consst of many frames, each of whch stores an own mesh. Even f key frame anmatons are used, they are too volumnous to be stored. Often the meshes dffer only slghtly between neghborng frames, leadng to a large redundancy between frames and between neghborng vertces n the same frame. Therefore, t s mportant to develop compact representatons that sgnfcantly reduce the storage space of anmated models and facltate ther transmsson over networks. Moreover, we need compresson algorthms that allow for small compressed repre- Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, or republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. Copyrght UNION Agency Scence Press Plzen, Czech Republc sentatons that mantan good vsual fdelty. Many exstng compresson schemes are restrcted to anmated meshes that do not change the topology from frame to frame so that the topology can be compressed once and only the vertex postons need to be compressed for the ndvdual frames. Here, we dstngush between four methods: predctve based methods [JCS2, IR3], PCA based representatons [AM, KG4, SSK5], wavelet based technques [GK4, PA5] and clusterng-based approaches [Len99, ZO4]. In ths paper, we present a new PCA-based technque as extenson of the work [ASS6]. The advantage of usng PCA s that t captures the lnear correlatons present n the dataset. The set of vertces can be represented by very few components and coeffcents dependng on the user s desred vsual qualty. The PCA s a good compressor for rgd moton and provdes a more compact representaton for temporallynvarant meshes. In many applcatons, however, anmated meshes exhbt hghly nonlnear behavor, whch s globally dffcult to capture usng standard PCA. Locally, the neghborng vertces have a strong tendency to behave and to move n a smlar way. The nonlnear behavor can therefore be descrbed n a lnear fashon by groupng the vertces of smlar moton nto clus-

2 (a) (b) Fgure 2: The poston of sx dfferent vertces over tme (llustrated wth dfferent colors) are represented wth global coordnates (a) and local coordnates (b)(dance anmaton). ters or by segmentng the mesh nto meanngful parts. Then a PCA s performed n each group. The process to construct ths representaton s called Local Prncpal Component Analyss. On the other hand, ntroducng a local coordnate frame (LCF) n each cluster may lead to extra clusterng of the coordnates before performng the PCA. If the segmentaton or clusterng process s effcent then t would be hghly probable that these coordnates change very slghtly relatve to the coordnate frame of ther cluster. Of course, the number of clusters/segments wll also affect the compresson. If the number of clusters s very small, then a cluster mght contan vertces that have dfferent behavors. To overcome ths problem one mght possble mprove on the present approach by automatzng the selecton of the number of clusters. Fgure 2 demonstrates the dea of usng local coordnate systems. Fgure 2 (a) shows the path of sx ponts of a dance anmaton n the world coordnates. Note the hghly nonlnear behavor of the traectores. Fgure 2 (b) shows the path of the ponts usng a local coordnates. Note the relatve small changes and the tendency of the traectory of ndvdual ponts to cluster. In our approach, we perform a PCA on the local coordnates rather than the world coordnates. The advantage of combnng PCA wth the LCF s now obvous: f the moton of a group of vertces s rgd n the world coordnates, the postons of the vertces are slghtly nvarant relatve to ther LCF. Therefore, performng a PCA n these nvarant groups of vertces leads to a more compact representaton than the orgnal data, and a large number of PCA coeffcents are close to zero.. Overvew We propose a new compresson algorthm for anmated meshes of fxed number of vertces based on LPCA. Our man contrbuton s to cluster the mesh vertces usng the local smlarty of traectores. The orgnal vertex coordnates are transformed nto several LCFs defned by seed trangles. One LCF (one seed trangle) s assocated wth each cluster. The vertces are then clustered dependng on the varaton of ther local coordnates n each LCF. Thus, each vertex s added to the cluster where the vertex coordnates have the smallest varaton over tme. Ths automatcally "transforms" the nonlnear behavor of the orgnal vertces nto the clusterng behavor whch s very well compressable. The vertex postons wll tend to cluster around the same poston over tme (see Fgure 2(b)). Thus, the clusters themselves are almost nvarant to any deformaton. A PCA s then performed on each cluster such that the local coordnates of the vertces are transformed nto another bass whch allows for very effcent compresson. Our clusterng process produces clusters of dfferent szes. If one chooses a fxed number of bass vectors for all clusters, then there may be too few egenvectors to recover the clustered vertces at a desred accuracy and eventually too many egenvectors for other clusters (whch we call underfttng and overfttng, respectvely). Moreover, the number of bts needed to encode the unnecessary bass vectors n overfttng cases may be better allocated for other clusters n underfttng cases. Therefore the selecton of the best number of bass vectors to be extracted from anmaton data s necessary to properly recover the orgnal data of each cluster wth a certan accuracy. We ntroduce a rate dstorton optmzaton that trades off between rate and the total dstorton. We call our approach Relatve Local Prncpal Component Analyss (RLPCA) compresson. We use the term Relatve as the LPCA s performed n local coordnates. Our Algorthm acheves an ncreased compresson performance, s computatonally nexpensve (compared to a PCA for the full mesh) and s well suted for progressve transmsson. 2 RELATED WORK Statc Meshes A large number of compresson technques have been developed for statc meshes. Deerng [Dee95] was the frst to publsh work on geometry compresson for trangle meshes. Then, a successon of effcent schemes were proposed for both connectvty and geometry compresson [TR98, GS98, TG98, IA2]. Progressve compresson technques [Hop96], whch enable a mesh to stream from a server to a clent have also been proposed. Recently some comprehensve surveys of the developed technques have been provded [Ros4, AG5, JPK5]. Anmated Meshes Recently, research has started to focus on anmated meshes wth fxed connectvty. Lengyel [Len99] ntroduced the frst work on anmated geometry compresson. He parttoned the mesh nto submeshes and descrbed the moton of the submeshes by rgd body transformatons. The rgd body transformaton of a submesh was thereby estmated to best match the traectores of ts vertces. Hs approach s very effectve when large parts of an anmated model can be descrbed well by rgd body transformatons.

3 Encoder Input: M, M 2,, M F Geometry data Segmentaton Decoder Stream PCA detals + resduals Connectvty WCS to LCS Transformaton Local coordnates reconstructon Statc compresson PCA, PCA2 PCAN LCS to WCS Transformaton LCF resduals computaton Quantzaton Stream Connectvty Output: M ~, M ~,, M ~ 2 F Fgure 3: Overvew of the compresson / decompresson ppelne. Jnghua et al. [ZO4] used an octree to spatally cluster the vertces and to represent ther moton from the prevous frame to the current frame wth a very few number of moton vectors. The algorthm predcts the moton of the vertces enclosed n each cell by tr-lnear nterpolaton n the form of weghted sum of eght moton vectors assocated wth the cell corners. The octree approach s later used by K. Mueller et al. [MSK + 5] to cluster the dfference vectors between the predcted and the orgnal postons. Alexa et al. [AM] used PCA to acheve a compact representaton of anmaton sequences. The PCA coeffcents were shown to be well compressable. Karn and Gotsman [KG4] mproved ths method by applyng second-order Lnear Predcton Codng (LPC) to the PCA coeffcents such that the large temporal coherence present n the sequence s further exploted. Sattler et al. [SSK5] ntroduced the clustered PCA. The mesh s segmented nto meanngful clusters whch are then compressed ndependently usng a few PCA components only. Predcton technques can also be used to effcently compress anmated meshes. Assumng that the connectvty of the meshes doesn t change, the neghborhood n the current and prevous frame(s) of the compressed vertex s exploted to predct ts locaton or ts dsplacement [JCS2, IR3]. The resduals are compressed up to a user-defned error. Guskov et al. [GK4] used wavelets for a multresoluton analyss and exploted the parametrc coherence n anmated sequences. The wavelet detal coeffcents are progressvely encoded. Payan et al. [PA5] ntroduced the lftng scheme to explot the temporal coherence. The wavelet coeffcents are thereby optmally quantzed. Segmentaton Mesh segmentaton has recently become useful for many applcatons n geometry processng. In the context of compresson, segmentaton s often used to decrease the computatonal costs as well as to preserve the global shape of the mesh because some compresson algorthms (e.g. PCA for a full mesh) can destroy mportant features of the mesh. To fnd the vertces that have smlar moton, Lengyel [Len99] proposed that one select a set of seed trangles randomly and compared ther traectores. Trangles wth a smlar moton are combned. Then the vertces are assocated wth the trangle whose traectory best ft thers. Sattler et al. [SSK5] proposed that one cluster the traectores of vertces usng Lloyd s algorthm n combnaton wth PCA. In the both cases, the segmentaton s computatonally expensve. 3 ANIMATION COMPRESSION In ths secton, we descrbe n detal the core of our compresson algorthm for the moton of vertces of anmated trangle meshes. An overvew of compresson and decompresson ppelne s llustrated n Fgure 3. Gven a sequence of trangle meshes M, =,..,F of constant connectvty wth V vertces and F frames (meshes), we frst construct N LCFs n each frame, then group the mesh vertces nto N clusters, where each cluster contans V, =,..,N, vertces. 3. Local Coordnate System Expressng the vertex locatons n a LCF s an optmal way of exhbtng clusterng behavor. It makes the clusters qute nvarant over tme to any rotaton and/or translaton. Ths representaton can be very compressable wth the PCA. Ths s the key feature of our algorthm. Fgure 4 llustrates the LCF that we use n our algorthm durng and after segmentaton. We consder that each cluster s ntalzed wth seed trangle (p,p 2,p 3 ). Each cluster C has ts own LCF defned on the seed trangle. The orgn o s the center of one of ts three edges (typcally (p,p 2 )), the x-axs (red arrow) ponts down the edge (p,p 2 ), the y-axs (green arrow) s orthogonal to the x-axs n the plane of the seed trangle and the z-axs s orthogonal to the x- and y-axs. The transformaton of a pont p to ts local coordnate system q can be accomplshed by an affne transformaton wth a translaton o and a lnear transformaton T (orthonormal matrx): q = T(p o) In our algorthm, for each frame f ( f F) and for each frame cluster G f C ( N), we computed {T f,of } from the ponts of the seed trangle (p, f,p, f 2,p, f 3 ).

4 the LCF to the cluster C and store ts local coordnates for the next step (compresson). The teraton stops f no more canddate vertces exst. When a vertex s added to a cluster, t s marked as vsted. We end up wth N clusters that have V vertces each. Fgure 4: Illustraton of the local coordnate frame 3.2 Segmentaton based on Clusterng Our segmentaton algorthm starts wth several seed trangles upon whch the LCFs are constructed. Then the clusterng s obtaned by assgnng the vertces to the seed trangle n whose local coordnate frame they have mnmal coordnates varaton across the F frames. The clusterng process conssts of the followng steps: Intalzaton: Intalzes the N cluster S, =,...,N, to be empty. All vertces are unvsted. Seed Selecton: Selects N seeds usng the far dstance approach [YKK]. The frst seed s selected as the vertex correspondng to the largest eucldan dstance from the geometrcal center of all vertces n the frst frame. The next seeds are selected sequentally untl all N seeds are selected. Each seed s selected to be the vertex wth the farthest dstance from the set of already selected seeds. We assocate wth each seed one of ts ncdent trangles and call ths trangle the seed trangle. The regons are ntalzed wth ther three ncdent vertces denoted as (p, f,p, f 2,p, f 3 ) the three vertces of seed trangle of -th cluster n the f -th frame. Local Frame Constructon: A local coordnate frame s constructed for each seed trangle (see secton 3.). Vertex clusterng: Gven an unvsted vertex p f k, we do the followng: Transform ts world coordnates nto the N local coordnate frames constructed n each frame f, so: {q, f k,q 2, f k,...,q N, f k }, ( f =,...,F), compute the total devaton (moton) of the vertex between each two adacent frames f and f n eucldan space: θ k, = F q, f k f =, f qk 2 θ k, represents the total moton of the vertex k n the LCF assocated wth the cluster. A small value means that the vertex poston has moton that s smlar to C. Thus the vertex should belong to the cluster for whch the devaton s very small, note mn : mn := argmn N {θ k, } We terate over all vertces, addng the unvsted vertex whose local coordnates are almost nvarant n Our algorthm provdes a smple and effectve way of effcently clusterng mesh vertces. The results of the segmentaton technque can be seen n fgure. 3.3 Compresson Once the mesh vertces are clustered, ther coordnate systems need to be encoded usng PCA. In order to be able to transform back to the world coordnates durng the decodng step, we also have to encode the world coordnate of the ponts of seed trangles (used to construct the transformatons). The affne transformaton should then be correctly computed (at decodng) wthout loss of nformaton. Therefore, we propose the seed trangle ponts be encoded separately wth delta codng. Delta codng Gven the sequence of the seed trangle ponts (p, f,p, f 2,p, f 3 ), we frst encode ther world coordnates n the frst frame. Then, the dfferences between each two adacent frames n the sequence are computed. To avod error accumulaton durng anmaton, these resduals are computed between the coordnates of the pont p, f n the current frame and ther recovered coordnates p, f n the prevous frame: δ, f = p, f, f p, ( =,2,3) where =,...,N and f =,...,F. Prncpal Component Analyss Prncpal Component Analyss (PCA) s a statstcal technque that can reduce the dmensonalty of a dataset. It determnes lnear combnatons of the orgnal dataset whch contan maxmal varaton and represents them n an orthogonal bass. PCA reconstructs the orgnal dataset optmally n the mean square-error sense. If we have F frames of 3V dmenson each, PCA produces a reduced number L F of prncpal components that represent the orgnal dataset. We now consder how a cluster evolves over the frames of the anmaton. Let G f be the -th cluster n the f -th frame, =,...,N and f =,...,F. A sngle cluster C thus conssts of F clusters (one for each frame) C = {G,G 2,...,G F } where G f represents the vector wth the geometry of the cluster n frame f G f = (q, f 4,q, f 5,..,q, f V ) t, whose elements are the local coordnates of correspondng vertces (except the coordnate of the seed trangle). All these vectors G f have the same length 3(V 3), and construct a geometrc matrx A wth 3V 9 rows and F columns. A = [ G G 2...G F ] A sngular value decomposton on A s A = U D V t

5 where U s a (3V 9) F column-orthogonal matrx that forms an orthogonal bass and contans the egenvectors of the A A t. D s a dagonal matrx whose nonzero elements represent the sngular values and are sorted n decreasng order. Thus D = dag{λ,λ 2,...,λ F }. V s a F F orthogonal matrx. To reduce the dataset, we pck only the frst L egenvectors (L s a user specfed number). So, U = {u,l,l =,...,L} contans the most mportant prncpal components u that correspond to the largest egenvalues λ,...,λ L. Then each cluster G f s proected nto the new bass U to get a new matrx of coeffcents C of sze L F. C = U t A After performng the PCA for all N clusters C, we get N new sets {U,U 2,...,U N } and coeffcent matrces {C,C 2,...,C N } wth dfferent szes. Quantzaton and Arthmetc Coder For further compresson, the floatng-pont values (32 or 64 bts) are often quantzed to a user specfed number of bts per coordnate relatve to the maxmum extend of the boundng box of the model. The quantzed values are encoded wth an arthmetc coder [WNC87]. In the case of an anmaton, the quantzaton s often performed accordng ether to the tght axs-algned boundng box for each frame or to the largest boundng box for all frames. Snce we have to encode the bass vector values and the coeffcents rather than the vertex coordnates, we use two dfferent encodng contexts. The frst concerns the matrces and the second the delta vectors. The bass matrx U and the coeffcent matrx C of each cluster C are truncated usng a fxed number of bts q u and q c respectvely (typcally q u = q c ). We frst compute the mnmum and the maxmum values (u mn,,u max, ), (c mn,,c max, ) of U and C respectvely. Then nteger values are straghtforwardly derved accordng to u q (m, ) = u (m, )/u max, u mn, 2 q u + /2 c q (, f ) = c (, f )/c max, c mn, 2 q c + /2 where m 3V 9, L and f F. The resultng sgned nteger values of the matrces are encoded wth an adaptve arthmetc coder and sent wth the extreme numbers. For delta vectors, the coordnates are encoded accordng to the boundng box of each frame. Usng a fxed number of bts q, the coordnates of the delta vectors are mapped nto ntegers whch are then encoded separate from PCA detals wth an arthmetc coder. We assume that the quantzaton errors of PCA detals are neglgble up to 2 bts quantzaton. Note that the total number of bts needed for storng delta vectors s very small. It ranges between. and bt per vertex per frame when the quantzaton ranges between 2 and 6 bts dependng on the number of egenvectors, the level of quantzaton, and the number of clusters. 3.4 Rate-Dstorton Optmzaton In LPCA-based technques often PCA s performed usng a fxed number of components per cluster, neglectng the fact that whole mesh sequences are often not rgd and the dfferent parts can have dfferent behavor (.e. ther moton s not smlar). Thus, usng a fxed number of components per cluster may results n an nsuffcent number to represent a gven cluster at the desred accuracy whle havng too many for the representaton of other clusters. To mprove the PCA based compresson and avod ths overfttng and underfttng, we ntroduce the Rate- Dstorton Optmzaton (RDO) whch s also called the bt allocaton. The obectve s to fnd the best tradeoff between the btrate and the dstorton of coordnates of the vertces. Gven N clusters C, that we have to encode separately, and a set of egenvectors I = {l,l,...,l L }. For each cluster C, let (R l,dl,) denote the rate-dstorton pont for each number l I, (typcally l =,...,4 components). The rate R l represents the number of bts requred to encode the bass vector values and the coeffcents. The dstorton D l s the root square error between the orgnal and the reconstructed coordnates of all vertces n the cluster. Let R target be the gven total bt rate for all clusters. Then the optmzaton problem s to fnd the best number of components l for the cluster, ( =,...,N) that mnmze D = N = Dl subect to the constrant N = Rl R target. In our codng, we ntroduce an R-D optmzaton whch s based on an ncremental computaton of the convex hull [WS]. For smplcty, and snce the number of bts ncreases wth the sze of the bass vectors, we defne the rate R as the number of bass vectors rather than the number of bts. Brefly, we defne the optmzaton algorthm n the followng:. For each cluster C we compute: The number of components l that corresponds to the smallest rate; The number of components k that corresponds to the next RD pont on the lower convex hull; The slope λ between the ponts (R l,dl ) and (R k,dk ). 2. We compute the total rate R t = N = Rl 3. As long as ( N = Rl R target ) s verfed, we: Select the cluster S n whose λ n s mnmal; Update R t Modfy l by k ;

6 Fgure 5: Reconstructed chcken. Top raw: Frame 34. Bottom raw: Frame 4. From left to rght: Orgnal, optmzed RLPCA, RLPCA, and LPCA performed n world coordnates ( clusters; components). Determne k that corresponds to the next RD pont on the lower convex hull; Compute λ n. 3.5 Compresson Parameters The compresson parameters defne the desred amount of compresson. In our approach, there are three parameters that govern the compresson rato: The number of bass vectors/rate L: If ths number s fxed for all clusters, then the user defnes t (dependng on the desred accuracy). The larger ths number s, the better reconstructon wll be (at the expense of less compresson). If the RDO s used, then we wll need only to specfy the amount of compresson (rate) or the maxmum number of bass vectors that are to be used to approxmate each cluster as we do n our codng. The number of vectors n each cluster s then optmally selected such that the total rate s below the gven user-specfed rate (or the total number of vectors s below the gven user-specfed maxmum number of vectors). The number of clusters N: If ths number s very small, then the cluster may contan vertces of dfferent behavor and ther local coordnates wll have a large varaton over tme. However, t s dffcult to fnd a lnear space that effcently represents these coordnates usng PCA. In the future, we want to automatze the selecton of ths number. Typcally, n moton capture based anmaton the number of clusters should be equal to the number of onts. The reconstructon error: Ths error presents the devaton of the reconstructed postons from the orgnal one. It s measured usng L2-norm or the metrc whch we call KG error [KG4]. Moreover, t controls the compresson durng the RDO. Ths number should ncrease wth decreases n the number of clusters or the number of egenvectors. 4 DECOMPRESSION Fgure 3 llustrates the decodng process. After recevng the sequences of the PCA detals and the delta vectors, we decode and undo quantzaton of delta vectors, we reconstruct the ponts of the seed trangles of each cluster n each frame, then reconstruct the LCFs. In the second stage, we undo the quantzaton of all bass vector values and coeffcents, we reconstruct the local coordnates of all vertces n each cluster, and transform them back to world coordnates. Fnally, we collect all clusters to reconstruct the sequence of meshes. 5 RESULTS In order to see the performance of our scheme, RLPCA, we measured the number of bts per vertex per frame (bpvf), and as most other recently proposed methods for anmated geometry codng, we used the KG error metrc to measure the dstorton n the reconstructon anmaton wth regard to the orgnal anmaton. We also computed the dstorton per frame usng the L2 norm of all reconstructed vertex postons relatve to the orgnal postons of each frame. We compare the compresson performance of our algorthm aganst AWC [GK4], TLS [PA5], PCA [AM], KG [KG4] and CPCA [SSK5]. RLPCA vs. LPCA We want to fnd the nfluence of the segmentaton and the local coordnates on the rate and on the reconstructon of anmaton. We performed LPCA n the world coordnate system as well as n the local coordnate systems for a gven numbers of clusters, components and bts of quantzaton N, L and q c respectvely. Furthermore, we compared LPCA wth the standard PCA. Fgure 6 (a) shows the reconstructon results relatve the orgnal frame usng q c = q u = 2 and L = when

7 (a) (b) (c) (d) (e) (f) Fgure 6: Rate dstorton curves for the cow (b), dolphn (c), chcken (e) and dance (f) sequences usng KG error. The error plot for the chcken sequence: (a) usng LPCA n the world and the local coordnates and usng the RD-optmzaton ( clusters, components) and (d) usng, and 2 clusters and 2, and 2 components. the LPCA s performed n the world coordnates (green) and n the local coordnates (blue) and when the R-D optmzaton s ntroduced (red) at the same number of bt per vertex per frame. We can see that the local coordnates are more compressable than the the orgnal coordnates. Fgure 6 (d) and Fgure 6 (f) shows the effect of the number of clusters and the component on the frame reconstructon for the chcken anmaton usng (N,L) ={(2,); (2,2); (,)} and on the ratedstorton curves for the dance anmaton usng, 2 and 3 clusters. In Fgure 6 (a), the mprovement n the second curve (blue) s due to the transformaton of the orgnal coordnates nto local coordnates whch forces the coordnates of a vertex to cluster around one pont (see Fgure 2). Ths mprovement ncreases (red) when the optmzaton were ntroduced. Fgures 5 shows the reconstructed two frames n the chcken sequence when the world and the local coordnates are used and when the optmzaton s ntroduced usng components and clusters. Comparson to other coders Fgure 6 also llustrates the comparson to other methods as rate-dstorton curves for the cow (b), dolphn (c) and chcken (e) anmatons. At frst glance, we can see that our approach acheves a better rate dstorton performance than the standard PCA, LPC and TG for the three models. Ths result s obvous snce the anmaton codng based on statc technques only explot the spatal coherence and the lnear predcton codng only uses the temporal coherence. Furthermore, the standard PCA only approxmates the global lnearty and s less effectve for nonlnear anmaton. For the CPCA and AWC algorthms, we acheve better or smlar results. Fgure 6 (b) shows that for the cow anmaton our method s sgnfcantly better than the method of Karn and Gostman and than the CPCA. And t comes close to AWC. For the dolphn and the chcken sequences our method performs better than all the above methods. Ths mprovement s due to the segmentaton of the model nto meanngful parts (whose vertces move qut smlarly) as well as to the use of local coordnates rather than world coordnates. On the other hand, the RLPCA performs well for the models of large number of vertces n contrast to KG. Therefore, by combnng RLPCA wth LPC, we mght acheve a better compresson rato. Fgure 6 also demonstrates that the rate dstorton optmzaton we ntroduce n our algorthm (ORLPCA) s mportant for achevng better compresson performances especally when the number of vertces s large and the anmaton s complex. From the computatonal vewpont, PCA s computatonal expensve but n combnaton wth LPC [KG4], t gves a better compresson performance, partcularly for a long sequence of ust a few number of vertces. CPCA [SSK5] outperforms both methods snce they explore a robust segmentaton whch s based on a data analyss technque but remans expensve. In contrast, our RLPCA uses a smple clusterng and transformatons and acheves a better compresson rato.

8 Table : Comparson compresson and decompresson tmngs wth CPCA. CPCA Models vertces trangles frames bpvf d KG t(sec) enc RLPCA t(sec) FPS bpvf d KG N L t(sec) enc chcken cow dolphn t dec (sec) Tmngs: Table shows the tmngs n seconds of the codng (t enc ) and decodng (t dec ) processes (wthout optmzaton) for the three anmatons wth a comparson to CPCA (t FPS for dsplay whle decodng). We observe that for the chcken and cow anmatons, our coder s much faster and performs better than CPCA. Our tmng results are measured on Pentum 4 wth 2.53 GHz and CPCA on AMD Athlon64 XP CONCLUSION We ntroduced a new compresson technque for the anmated meshes whch s based on LPCA. The mesh vertces are clustered usng the moton n the LCF. Then, the world coordnates of each cluster are transformed nto local coordnates. Ths step enables the algorthm to compress an anmated mesh effcently. It explots the "local" behavor of the local coordnates. Fnally, an LPCA s performed n each cluster wth the rate dstorton optmzaton. Our approach s smple, fast and acheves a better performance than other current exstng compresson technques. It s applcable to meshes and pont-based models. It performs well for anmatons wth a large number of vertces. For very long sequences, we suspect that the moton of a local coordnates also becomes complex and non-lnear. Therefore, we want to combne our method n the future wth LPC whch s good for long sequences or splt the sequences nto small clps. Furthermore, we plan to develop an adaptve segmentaton over tme and encode the clusters wth dfferent quantzaton levels. The number of clusters can also be chosen automatcally. Acknowledgements We would lke to thank Zach Karn and Hector Brceño for provdng us the anmated meshes and Mrko Sattler, Igor Guskov and Frédérc Payan for the results of ther methods. The Chcken sequence s property of Mcrosoft Inc. REFERENCES [AG5] P. Allez and C. Gotsman. 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