Mesh Connectivity Compression for Progressive-to-Lossless Transmission

Transcription

1 Mesh Coectvty Compresso for Progressve-to-Lossless Trasmsso Pegwe Hao, Yaup Paer ad Ala Pearma ISSN RR Jue 005 Departmet of Computer Scece

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3 Mesh Coectvty Compresso for Progressve-to-Lossless Trasmsso Pegwe Hao 1 Yaup Paer 1 Ala Pearma 1 Departmet of Computer Scece Departmet of Electroc Egeerg Quee Mary, Uversty of Lodo, UK 1 {phao, paer}@dcs.qmul.ac.u ala.pearma@elec.qmul.ac.u Abstract A progressve mesh coectvty compresso techque s proposed ths paper. Our method s based o the edge collapse ad vertex splttg techque for progressve mesh compressodecompresso. Ths s a effcet ad reversble method that ca be used for progressve-to-lossless trasmsso. We derve a theoretcal upper boud for the lossless coectvty compresso bt rate, whe the solated vertces are gored. Our expermets show that all the meshes we have used ca be compressed better tha the derved bt rate upper boud. Furthermore, our method ca esure that less tha 10% of the vertces geerate accdetal code. Accordg to our aalyss, the optmal boud for tragular mesh codg ca be 3, smaller tha some of the reported results recet lterature. 1. Itroducto 1.1 Purpose of our wor 3D mesh codg methods are mportat for effcet storage ad fast trasmsso of large models. Such polygo models come from a umber of sources cludg computer-aded desg, rage scaers, terra mappg ad so-surface extracto from volume data. As model szes cotue to grow, fdg effcet ad mproved compresso methods wll rema a mportat problem computer graphcs. I partcular f complex scees are bult 3D ad set over commucatos les ad eed to be amated real tme the compresso methods become crucal for the qualty of amato to be acheved. A partcular applcato where ths s true s the computer games played teractvely over the et ad shared mmersve worlds. I our case, a UK collaboratve project called Prometheus [1, 11] udertoo to develop a ed-to-ed 3D televso cha. I ths project vrtual scees are used as sets ad avatars represet actors a studo. Vrtual worlds requre trasmsso of massve amouts of tragulated 3D geometrc data over the etwor. For both effcet trasmsso ad real-tme rederg of models to ad wth the ed-user computers at varous levels of detal, there was a eed for a effcet techque of mesh smplfcato ad progressve compresso. The techque s based o three aspects of curret research: mesh smplfcato, sgle-resoluto mesh compresso, ad progressve mesh recostructo. Our dea has bee to fd a mesh compresso method for progressve-to-lossless trasmsso whch produces good qualty smplfed meshes wth low bt rates ad wth lossless results comparable to the sgle-resoluto compresso methods. 1. Summary of related wor The ey dea of progressve codg s that data are set a coarse-to-fe way. Cocretely, progressve compresso trasmts a coarse approxmato frst, ad uses the followg subsequet bts to refe the mesh progressvely for more ad more detals. As for the progressve ecoders, Hoppe troduces [7] a algorthm for progressve trasmsso, startg from a coarse mesh ad sertg vertces oe at a tme to refe the mesh. A method called Progressve Forest Splt compresso s proposed by Taub et al. [13], usg a base mesh ad a forest of vertex splts. Gado ad Devllers [5] uses the d-tree ecodg of the geometry to drve the coectvty codg process. Pajarola ad Rossgac [10] group vertex-splt operatos to batches, the traverse the mesh ad specfy splts by marg each vertex usg oe bt. Khodaovsy et al. [8] proposed a progressve compresso algorthm based o the wavelet trasform. Cohe-Or et al. [4] preseted the patch colorg

4 algorthm for progressve mesh compresso based o vertex decmato. I Allez ad Desbru s valecedrve progressve method [], they use the mmal graularty for coectvty,.e., they remove (or sert) oly oe vertex at a tme durg the ecodg (or decodg respectvely) phase, ad ecode (or decode) the valeces of the removed vertces afterwards (or beforehad respectvely). A determstc coquest avods a explct trasmsso of order over the vertces. Of course, progressve compresso methods caot compete wth those sgle-resoluto oes [1, 14, 3] sce ther techques bascally crease the dsperso of valece due to the re-tragulato. So far, the performace of [] loos the best, but terms of the upper boud of bt rate, the method s ot so satsfactory for all meshes. I [6], Gotsma preseted some theoretcal upper bouds for valece-based mesh codg, 3.4 bpv for tragular meshes. However, accordg to our aalyss, the upper boud for a tragular mesh ca be eve smaller, 3 bpv. the patch. A gate s a edge alog the boudary of a patch used to traverse or coquer the patch. Those traversed patches ad the vertces that belog to the patch are cosdered as coquered. A vertex or a patch s called avalable f t has ot bee coquered. A cetral vertex of a patch s called solated f there s o adjacet edge that ca be selected to collapse the ecodg phase or there s o vertex splttg eeded the decodg phase. Therefore, ay patch of a smplfed mesh has a cetral vertex that s ether solated or collapsed ad s watg for splttg whe decodg. Isolated vertces geerate addtoal accdetal codes the ecoded bt stream, for whch the addtoal operatos are tae as splts [6] ad some other publcatos. The vertex at the frot tragle of a gate s amed the frot vertex, smlar to that []. As llustrated Fgure 1, the two gate vertces are the left vertex L ad the rght vertex R, where the frot vertex s A. The two gate vertces ca be coquered (a, flled blac vertces) or ot (b, hollow crcles). 1.3 Overvew of our wor Frot vertex Frot vertex Our wor has maly cocetrated o the effcet ad reversble mesh smplfcato methods. I ths paper, we maly summarze the coectvty compresso for progressve decompresso. Because the edge collapse method s better tha a vertex removal method wth respect to the smplfed mesh qualty, our techque s based o the edge collapse ad vertex splttg method. We use gates to locate patches, patches to cover the whole mesh, ad two dces couted from the gate to locate vertex splttg. We also gve a theoretcal upper boud estmato of the bt rate for recordg the dces. Our expermets show the effcecy of our method.. Progressve Smplfcato ad Recostructo I order to smplfy a mesh, we eed to select a edge of a vertex par to collapse. Dfferet strateges of vertex par selecto result dfferet smplfed meshes, whch also gve dfferet ecodg effceces. We use a strategy that coquers patches of collapsed edge ad sustag vertces to cover the whole mesh..1 Deftos A patch cossts of all adjacet tragles of oe vertex. The vertex s, therefore, the cetral vertex of (a) Gate vertces coquered (b) Gate vertces stll avalable Fgure 1. Defto of a gate ad ts frot vertex A gate to fd the edge to collapse s also a gate to the ew patch. Gates are used to locate patches the ecodg phase ad relocate patches the decodg phase. We tae levels-of-detal (LoDs) as a level of the smplfed mesh at whch we cover a mesh wth adjacet but u-overlapped patches.. Ecodg: Edge Collapse ad Patch Coquest The ecodg phase s to smplfy a mesh ad to record the ecessary formato for mesh recostructo the decodg phase. Startg from the frst gate the orgal mesh, we collapse a edge adjacet to the frot vertex ad tae the ewly retragulated patch as coquered, ad the push all the patch boudary vertces to a queue. The we pop up these vertces pars as gates to locate the ext adjacet patches to coquer. We cotue wth ths

5 process to coquer the adjacet patches, f avalable, utl the whole mesh s covered. Fgure gves a demostrato of a geeral case of edge collapse the ecodg phase. The specfc algorthm s as follows: 1. At each LoD, add the frst caddate gate to the gate caddate queue as the program talsato, ad repeat, followg the patch coquest steps, utl there s o vertex collapsed the terato.. Get a caddate gate from the queue. If the queue s empty, go to Step 1 to smplfy the mesh for the ext LoD. 3. Chec f both the gate vertces L ad R are avalable. If ether, chec the ext caddates the queue. 4. Cross the gate of vertex L ad vertex R ad fd the frot vertex A. (As the drecto of the dashed arrow Fg. a) 5. Startg from vertex L, go clocwse to vertex R to search all the adjacet vertces of A for a edge AB to collapse. (For demostrato, we search alog the sold arrows Fg. a ad we fd the edge AB Fg. b as our selected edge to collapse.) If there s o such edge to collapse, we tag the cetral vertex as solated, record ths formato the bt stream, ad go to Step. 6. Collapse the edge AB, sert a ew vertex A as the cetral vertex of a ew patch, ad rebuld the coectvty of the ew patch. (As from Fg b to Fg. c) 7. Cout clocwse sequetally from the gate vertex L to locate the frst vertex adjacet to the collapsed edge ad get oe dex. (We get vertex Fg. c) 8. Cout couter-clocwse from the gate vertex R to locate the other vertex adjacet to the collapsed edge, ad get the other dex. (We get 1 Fg. c) 9. Ecode the two dces of the last two steps by a etropy coder to record them the bt stream order to re-locate the vertex splttg coectvty for perfect recostructo the decodg phase. 10. Startg from the etrace gate vertex L, go clocwse to vertex R to add all the adjacet vertces to the queue as the caddate gate vertces to adjacet patches. (As Fg. d) 11. Tag vertex A ad all ts adjacet vertces of the patch as coquered (As Fg. d), ad go to Step. 1. Ecode the fal coarsest mesh. Varous crtero for the edge selecto ca be used. I our expermets, we use the maxmum valece to determe whch edge s selected to collapse. As metoed [], ths ca reduce the data dsperso ad the the fal bt rate. If there s o edge avalable to collapse, we just tae the cetral vertex as a solated vertex, but we eed to record ths formato the bt stream, whch obvously reduces ecodg effcecy. The bt streams at dfferet LoDs are reverse recorded the whole bt stream of mesh codg. The coarsest mesh s ecoded ad put at the head of the whole code bt stream so that the decoder ca recostruct the mesh. (a) (b) (c) (d) Fgure Edge Collapse.3 Decodg: Patch Coquest ad Vertex Splttg The decodg phase s just the reverse of the ecodg phase. Startg from the frst patch the coarsest mesh, we tag the patch as a coquered patch ad push all the patch boudary vertces to a queue, ad splt the cetral vertex to produce a fer mesh, ad the pop up these vertces by pars as gates to the ext adjacet patches. Afterwards, we coquer these adjacet patches f they are stll avalable. The specfc algorthm s as follows: 1. Decode the coarsest mesh.. At each LoD, add the frst caddate gate to the gate caddate queue as the program talsato, ad repeat followg the patch coquest steps utl there s o data the bt stream. 3. Get a caddate gate from the queue. If the queue s empty, go to Step to refe the mesh for the ext LoD.

6 4. Chec f both the gate vertces L ad R are avalable. If ether s avalable, chec the ext caddates the queue. 5. Cross the gate of vertex L ad vertex R ad fd the patch ad ts cetral vertex A. (As Fg. c & d) 6. Startg from the etrace gate vertex L, go clocwse to vertex R to add all the adjacet vertces to the queue as the caddate gate vertces to adjacet patches. (As Fg. d) 7. Tag vertex A ad all ts adjacet vertces of the patch as coquered (As Fg. d). 8. If the cetral vertex of the patch s solated, go to Step Get the dces for vertex splttg from the bt stream. 10. Startg from the gate vertex L, go clocwse ad startg from the gate vertex R go couter-clocwse, usg the two dces to re-locate the two adjacet vertces of the collapsed edge. (As Fg. c) 11. Splt the cetral vertex to two, ad re-tragulate the patch.. (As from Fg c to Fg. b) 1. Go to Step 3. From the gate derved from the queue, for each LoD, we locate all the patches the same order as the ecoder. If the decodg tme s lmted such as for perodc rederg of a amated scee, the decodg procedure ca be stopped at ay tme. I ths case, the decoded mesh s ot losslessly decompressed..4 The Exceptoal Frst Gate I order to exactly re-locate all the patches the decodg phase, the frst gate must be fxed so that the decoder ca fd t at ay LoD. Therefore, the mesh ecoder eeds to fx the postos of the frst two gate vertces ad these are ept the same for all LoDs. Actually, f the frst gate s fxed ad both of the gate vertces are forbdde to collapse, they stay uchaged through the etre smplfcato at all LoDs, eve to the fal coarsest mesh. Obvously, ths results worse mesh qualty. If we tae the frst gate vertces as the u-coquered vertces, ad allow the frot vertex to choose ether of them to collapse, the smplfed mesh should have better qualty. Thus, we have three cases of edge collapse for the frst gate, as show Fgure 3. Fgure 3a shows a gate to the frst patch. We search all the adjacet vertces for a edge to collapse alog the arrows. For the frst patch, all the adjacet vertces are avalable for edge collapse. If the vertex chose s ot ether of the gate vertces, t s the ormal edge collapse case (3b), just the same as the followg patch coquests as Fgure. If the collapse s wth the left vertex of the gate (3c), the re-locato dex s just from the fxed rght vertex. If the collapse happes wth the rght vertex of the gate (3d), the re-locato dex should be from the fxed left vertex. To let the decoder dstgush the three cases of the frst gate, we oly eed oe umber to detfy them, say 0, 1 ad. 3. Upper Boud Estmato of Bt-Rate The proof of upper boud estmato of bt rate whe a mesh s ecoded wth a valece-drve method s gve [3]. We use a smlar dea to gve a geeral formula to estmate the upper boud of smlar methods ad gve a estmate for our method. (a) (b) (c) (d) Fgure 3 Three cases of the frst gate for vertex par searchg Suppose the average value of a teger sequece s m, ad the mmum s. Let H stad for the etropy of the tegers, for = m, the upper boud of H s: max H = log + ( + 1) 1

7 For the mathematcal dervato, please refer to the Appedx of ths paper. For a mesh wthout boudares, f the umber of faces, edges ad vertces are F, E, ad V respectvely, we have 3F = E ; F + V E =. The, we have E = 3V 6 ad the total valece of all vertces s E = 6V 1. So the average valece for arbtrary mesh s approxmately 6. If we tae the average valece m= p = 6, where the mmum valece s = 3. = 3 The, we have = m = 3. Therefore, the correspodg etropy upper boud of valeces s the same as gve [3]: 4 4 max H = log = For our method, the mmum dces 1 ad are 0, ad the average value of the sum of two dces ( m,m ) ad should be less tha the average valece 1 of a smple mesh, 6. Therefore, we have 1 = = 0, ad we tae m1+ m+ = 6 1 or m1+ m = 3. Suppose 1 = m1 1, = m, we have 1+ = 3 ad 0 1, 3. I the case of the uform dstrbuto, we suppose 1 = = 3. The, we ca get the correspodg upper boud of bt rate, whch s the addto of the bt rate upper bouds of two dces: ( 1 1) ( + 1) max H = log + log 1 =.47 = If we use a tragle-remove method for progressve mesh compresso, wth the upper boud of a teger sequece wth the average costrat as show the Appedx of ths paper ad based o our aalyss Secto 3, we ca easly fd the bt rate upper boud of the three dces. To use a ew vertex to replace three vertces of a tragle, there are totally two vertces removed after each operato ad three dces are eeded for decodg. I such a case, we tae 1 = = 3 = 0, ad m1+ m+ m3+ = 6 1 or m1+ m + m3 = 3. Thus, we have = 3. The worst case s 1 = = 3 = = 1, ad we obta the upper boud: + ( + 1) 1 max H = 3log = 3 / = 3 (bpv) However, above aalyss, the addtoal operatos for solated vertces (as splts are eeded [6]) are ot cosdered yet. 4. Expermets We used e models studed the lterature to mae the expermets ad test ther performace. The ecodg bt rates of the complete mesh (for lossless recostructo) are lsted Table 1. They are as good as we expected. The umbers of the solated vertces are all less tha 10%, ad the bt rates of coectvty are all less tha our estmated upper boud, The tme for decodg s very close to that for ecodg the same model. I comparso wth the best results by [], Allez & Desbru, 001, we lst ther results the last colum. From the commo models used, our results are ot always better tha thers, but t s terestg to meto that the case of ther worst examples ours perform better. For models fads ad horse, ther bt rates are 4.99 ad 4.61 bpv, respectvely, ad above tha ther upper boud. I comparso, our results are 4.06 ad 4.47 bpv, respectvely, ad the bt rates for all our tested meshes are below our upper boud. The upper boud of ther vertex removal method s 3.45 bpv, but ther worst case s 4.99 bpv, 1.75 more tha that. We beleve that ther method has less fluece o the umber of the solated vertces tha ours, whch results cases that exceed the upper boud. I our experece, therefore, our method resultg less tha 10% solated vertces performs better. Therefore, our method gves more flexblty by selectg a edge to collapse ad the locatg the patch tha a techque where a patch s located ad the the cetral vertex s removed. 5. Coclusos I cotrast to the vertex removal method [], the effcecy results from a lower umber of solated vertces, ad ths maes our estmated upper boud tght ad reasoable. Edge collapse strategy has the flexblty to choose oe of the adjacet vertces to collapse, whch s obvously produces a lower umber of solated vertces (less tha 10%) ad also better smplfes mesh qualty. For our future wor, we eed to mplemet the correspodg geometrc data compresso. The lterature [9] provdes us some clues for effcet ecodg whch should be useful for our further research. For our theoretcal aalyss of our ew bt rate upper boud wth face removal method, the upper boud should be verfed by expermets future.

8 Table 1 Expermets wth patch coquest strategy of vertex par searchg Models Vertces Average Idces Isolated coectvty Coectvty LoDs Idex1 Idex vertces (%) (ours, bts/v) ([], bts/v) fads > horse > maequ > torus > dosaur > N/A eght 766 -> N/A fele > N/A sampleavatar 190 -> N/A Appedx: Upper boud of a teger sequece wth costrats Ths proof follows that gve [3], after the dervato of the Lagrage multplers s corrected. Cosder a seres of teger umbers,, dstrbuted from to + (the mmum umber s ), the dstrbuto probablty of umber s p, ts correspodg etropy of the umber sequece s H plog p. Ad, we also have = p = 1 as our = = oe costrat. Suppose the average value of the sequece s m, we have the other costrat, p = m. = Usg Lagrage multplers to fd the upper boud of H for the teger sequece: f( p, λµ, ) = p log p + λ ( p 1) + µ ( p m) = = = Dervatves of f wth respect to each of p must be 0: log p = λ loge+ µ Let λ log e+ µ λ loge µ ( ) p = = = α β, we have β p = α β = α = 1 = = 1 β ad β ( 1) β ( 1) β p = α β = α = = m = = 1 β 1 β 1 β Therefore, the etropy ca reach the upper boud whe 1 ( m + 1) α =, m β = ( m ) m + 1 The etropy upper boud the s ( α β ) max H = α β log = log = = = α log α β α β β = logα m log β = ( m + 1)log m + 1 ( m )log m = log ( ) ( ) m + 1 ( m + 1) m ( m ) Let = m, we obta the bt-rate upper boud of the teger sequece: + ( + 1) 1 max H = log (bts/umber) Refereces [1] [] Perre Allez, Matheu Desbru, Progressve Compresso for Lossless Trasmsso of Tragle Meshes, I Proceedgs of ACM SIGGRAPH, pp , 001. [3] Perre Allez, Matheu Desbru, Valece-Drve Coectvty Ecodg of 3D Meshes, I Proceedgs of EUROGRAPHICS, v.0,.3, pp , 001. [4] D. Cohe-Or, D. Lev, ad O. Remez, Progressve compresso of arbtrary tragular meshes, IEEE Vsualzato, pp. 67 7, [5] P.-M. Gado ad O. Devllers, Progressve lossless compresso of arbtrary smplcal complexes, ACM Trasactos o Graphcs, v.1.3, July 00, pp (Proceedgs of ACM SIGGRAPH 00). [6] C. Gotsma, O the Optmalty of Valece-based Coectvty Codg, Computer Graphcs Forum, v.,.1, pp , 003. [7] H. Hoppe. Progressve meshes, I Proceedgs of ACM SIGGRAPH, pp , [8] A. Khodaovsy, P. Schroder, ad W. Sweldes, Progressve geometry compresso, SIGGRAPH Proceedgs, pp , 000.

9 [9] H. Lee, P. Allez ad M. Desbru, Agle-Aalyzer: A Tragle-Quad Mesh Codec, I Proceedgs of EUROGRAPHICS, v. 1,. 3, pp , 00. [10] R. Pajarola ad J. Rossgac. Compressed Progressve Meshes. IEEE Trasactos o Vsualzato ad Computer Graphcs, v.6,.1, pp , 000. [11] M. Prce, et al, Real-tme producto ad delvery of 3D meda, I Proceedgs of Iteratoal Broadcastg Coveto, Amsterdam, Netherlads, Sept 00. [1] J. Rossgac. EdgeBreaer: Coectvty Compresso for Tragle Meshes, IEEE Trasactos o Vsualzato ad Computer Graphcs, pp , [13] G. Taub, A. Guezec. W. Hor, ad F. Lazarus. Progressve Forest Splt Compresso, I Proceedgs of ACM SIGGRAPH, pp , [14] C. Touma ad C. Gotsma. Tragle Mesh Compresso, I Proceedgs of Graphcs Iterface, pp. 6 34, 1998.