Time-Varying Volume Geometry Compression with 4D Lifting Wavelet Transform

Transcription

1 Tme-Varyng Volume Geometry Compresson wth 4D Lftng Wavelet Transform Yan Wang and Heba Hamza NSF Center for e-desgn, Unversty of Central Florda, Orlando, FL , USA {wangyan, Abstract. Geometry compresson s an effectve way to dstrbute hgh-volume geometry data wthn lmted bandwdth and storage capacty. In ths paper, a new tme-varyng 3D geometry compresson method based on 4D lfted wavelet transform s presented. In ths hybrd approach, geometrc nformaton and anmaton are compressed based volume grd values. Isosurfaces are reconstructed from decompressed grd values. Wth rescalng and nteger-to-nteger lftng, compresson rato s sgnfcantly ncreased wthout compromsng qualty of surfaces. 1 Introducton There has been extensve research on 3D statc geometry compresson n the past decade [1,2]. In general, there are two approaches, mesh orented and mage orented. In mesh orented approach (e.g., [3]), both geometry (vertex coordnates n Eucldean space) and topology (connectvty among vertces) nformaton s compressed. Accurate 3D meshes can be reconstructed, whch s deal for engneerng applcatons. However, connectvty of mesh cannot be changed, whch restrans t from general applcatons. In mage orented approaches (e.g., [4]), topology nformaton s not consdered. Volumetrc geometry nformaton s represented by voxels or ponts, and shapes can be compressed based on 2D mage ale methods. The dynamcs of topology can be captured easly, and the compresson methods are general. However, to render a vsually recognzable and appealng surface requres a large amount of data. A balanced approach consderng these two ends wll possbly ntroduce general soluton wth acceptable performance. Vsualzaton technologes tend to merge [5]. In the past three decades, dgtal sgnal processng has evolved from 1D to 3D data. However, only a few focus on dynamc geometry change over tme. Smlar to 2D vdeo that complements 2D mage, tme-dependent 3D geometry can be looed as 3D vdeo and has great potental n varous applcatons, ncludng communcaton, entertanment, scentfc and medcal computng, computer-aded desgn and engneerng, as well as smulaton and vsualzaton. One can expect that 3D vdeos wth compressed formats are standardzed n the future and used as commonly as today s audo and vdeo. In general, there are two approaches n 3D anmaton compresson. In the frst approach, topology s assumed to be statc, and there s no or small change n connectvty. Mesh compresson s acheved by clusterng and segmentaton (e.g., Lengyel [6], Ahn et al. [7]), prncpal component analyss (e.g., Alexa and M.-S. Km and K. Shmada (Eds.): GMP 2006, LNCS 4077, pp , Sprnger-Verlag Berln Hedelberg 2006

2 Tme-Varyng Volume Geometry Compresson wth 4D Lftng Wavelet Transform 671 Müller [8], Sattler et al. [9]), moton predcaton (e.g., Ibarra and Rossgnac [10], Karn and Gotsman [11]), parametrc codng (e.g., Brceño et al. [12], Gusov and Khodaovsy [13]). In the second approach, topology may change arbtrarly between frames. Geometrc data are encoded wth voxel (e.g., Ma et al. [14]) and 3D wavelet codng (e.g., Guthe and Strasser [15], Sohn et al. [16]). Dfferent from the above methods, we propose a 3D anmaton compresson scheme based on 4D lfted wavelet transform (LWT), whch consders spatal and temporal coherence smultaneously. Ths method focuses on dynamc volume geometry compresson wth sosurface constructon. Topology wll not be coded as n mesh compresson, as n many cases the connectvty exsts purely for renderng purpose and t s not as essental as geometry nformaton. Isosurface representaton can reduce the data sze for surface boundary reconstructon. Wth compressed geometry as well as related surface nformaton such as normal drectons and colors, t s suffcent to reconstruct surfaces for vsualzaton wth relatvely low densty volume nformaton. In the rest of the paper, Secton 2 gves the overvew of the compresson scheme. Sectons 4 and 5 compare the floatng pont and nteger lftng based on 4D LWT. Secton 6 descrbes moton compensaton ssues n ths scheme. 2 Overvew of Proposed Compresson Scheme The volume data to be compressed s assumed to be regularly sampled, whch s commonly used n scentfc and medcal vsualzaton. Isosurface constructon based on volume data can provde a realstc renderng and can also be appled n other vsualzaton envronments wth artfacts and natural obects. The framewor descrbed here s based on the combnaton of volumetrc data and sosurfaces, as llustrated n Fg. 1. 4D volumetrc data s dvded nto groups of frames (GOFs) wth no specal requrement on GOF boundary. Each GOF s decomposed and compressed wth 4D LWT at the server sde. When receved by clent, the compressed data s decoded and decompressed. Isosurface s constructed based on the decompressed 4D volume data. The advantages of wavelet-based approaches nclude scalablty and smultaneous redundancy reducton for spatal and tme domans. Wth the nherent scalable representaton, wavelet provdes mult-resoluton soluton wthout extra costs. 4D LWT captures coherence localty of spatal and temporal domans. There s no need to dvde ntraframe data nto blocs, as n MPEG-2 standards based on dscrete cosne transform (DCT). Rescalng & Quantzaton LWT Encodng Internet / Intranet Decodng ILWT Isosurface Constructon Server Clent Fg. 1. Volume anmaton scheme based on 4D lfted wavelet transform

3 672 Y. Wang and H. Hamza 3 4D Lfted Wavelet Transform (LWT) Lftng scheme [17] s also called second generaton wavelet transform. It s a two-step flterng process: predcton and update. In the predcton step, even sequences are used to predct odd sequences. The predcton error forms the correspondng hgh-pass subband. In the update step, an approxmaton subband s obtaned by updatng even sequences wth the scaled hgh-subband samples, generally descrbed as h l u p h f 2( ) where f [x] s the nput data to be processed at poston x n frame, h and l are resultng hgh-pass and low-pass sequences respectvely, p and u are predcton and update coeffcents of flters respectvely. The man advantage of lftng s ts memory effcency n computaton. Dfferent from tradtonal traversal dscrete wavelet transform (DWT), wavelet coeffcent calculaton n lftng can be embedded n-place. (1) frame 1 frame 17 frame 33 frame 64 (a) Orgnal anmaton of the polymer example wth the fle sze of 108,153 KB (b) Decompressed sosurfaces wth a threshold of 0.11% wth fle sze of 14,924KB (c) Decompressed sosurfaces wth a threshold of 0.55% wth fle sze of 3,944KB Fg. 2. Polymer example wth the grd sze 64x64x64 n each frame

4 Tme-Varyng Volume Geometry Compresson wth 4D Lftng Wavelet Transform 673 Another advantage s that bacward transform s easy to fnd and has the same complexty as the forward transform. An mportant ssue assocated wth spato-temporal decomposton s the choce of flters. Dfferent flters exhbt vared sgnal characterstcs n terms of energy compactness n the transform doman and codng gan. Long flters tend to explore coherences of large regons or long perod of tme. However, they may blur boundary of occupaton or movement. A dyadc decomposton approach s taen n ths paper snce t consders the coherence relaton between tme and space smultaneously. An example s used to llustrate the vsual effect. Fg. 2(a) shows four frames of the orgnal 3D anmaton of polymer morphology, durng whch sgnfcant topology change s observed. A 2-level decomposton process s appled wth Haar lftng scheme. Compresson s acheved by settng the transformed coeffcents to zero wth the orgnal values less than a threshold. Followng the meta nformaton ncludng data type flags and dmensons of arrays, concatenated 4D arrays of coeffcents form compressed fles. No addtonal arthmetc codng s appled n the fle sze comparson. The hghest coeffcent magntude n ths example s M= If the threshold T s 0.11%, whch s 0.11% of M, the decompressed surfaces are shown n Fg. 2(b) wthout sgnfcant vsual dfference. The classc marchng cubes algorthm s appled n sosurface reconstructon. As T ncreases to 0.55%, the reconstructed surfaces are shown n Fg. 2(c). An nterestng voxel effect occurs because the grd neghborhood wth same sovalues expands. To avod the voxel effect, resoluton of grd needs to be reduced by ether decreasng the densty of grds at clent sde or tang advantage of nherent mult-resoluton of wavelet transform thus transmttng only low resoluton data from server sde, whch further ncreases compresson rato. 4 Rescalng and Integer-to-Integer Transform Compared to classcal wavelet transform, n whch transformed wavelet coeffcents are floatng pont numbers even f the orgnal data are ntegers, lftng scheme supports lossless nteger-to-nteger transform [18]. It transforms nteger data to nteger coeffcents. Wth nverse transform, orgnal nteger data can be reconstructed. Another feature of the volume-based sosurface compresson scheme s that sosurface constructon s not senstve to the number of bts used n codng f the range of grd values s large. As a result, floatng-pont grd values can be rounded to ntegers and nteger-to-nteger LWT can be used to ncrease the compresson rato wthout compromsng qualty. If the range of grd values s too small, the grd values can be rescaled before the roundng. A good rescalng strategy s to rescale the sosurface values to close to zero and the overall grd values to be evenly dstrbuted between postve and negatve sdes. Ths reduces the number of bts to code values. In the example of Fg. 2, the maxmum and mnmum of grd values are ± The grd values are multpled by 100 before roundng. To reduce dstorton, floor operaton that rounds towards negatve nfnty s used. The result of nteger lftng s depcted n Fg. 3, where the sze of the fle contanng nteger coeffcents s 6,318KB compared to the floatng coeffcents of 61,900KB n Fg. 2(a) wth full reconstructon. The qualty of the surfaces s very close to the orgnal ones.

5 674 Y. Wang and H. Hamza frame 1 frame 17 frame 33 frame 64 (a) Fully decompressed sosurfaces from nteger lftng scheme wth fle sze of 6,318KB (b) 1-level decompressed sosurfaces from nteger lftng scheme wth fle sze of 667KB (c) 0-level decompressed sosurfaces from nteger lftng scheme wth fle sze of 91KB Fg. 3. The Polymer example wth nteger-to-nteger 4D LWT 5 Moton Compensaton Moton compensaton (MC) s to remove temporal redundancy of vdeo sgnal further n 4D subband codng. Bloc-based moton models are predomnantly used n tradtonal moton-compensated MPEG codng. They can accurately represent very smooth moton felds but not complex ones. In contrast, deformable mesh moton model can mprove moton compensaton by tracng expansons and contractons whle sustanng a contnuous moton feld. Wth the notaton n (1), h l u p h f 2( ) [ MC [ MC ( x)] ( x)] s the MC scheme n spatal doman, where MC denotes moton-nduced transformaton from frame to frame. In tradtonal 2D vdeo, MC ncreases sgnalto-nose rato thus vdeo qualty wth the reducton of energy n hgh-pass temporal subbands. Yet t may ntroduce dstorton because of moton nverson errors. (2)

6 Tme-Varyng Volume Geometry Compresson wth 4D Lftng Wavelet Transform 675 In the context of our approach, moton of surfaces s represented by value change of dscrete grds, whch s more reslent than drect pxel or voxel representaton. We develop a Control Grd MC model for lftng transform to study the effectveness on our 4D LWT scheme. In each frame, the 3D space s dvded nto small cells. The average value of wthn each cell s taen to be elements of moton vectors. For example, the Haar lftng wth MC s shown n Fg. 4. Odd frames are predcted by even frames wth moton vectors before the lftng process where hgh-pass temporal subbands are generated. -pass temporal subbands are generated wth nverse moton vectors and lftng. Fnal hgh-pass and low-pass subbands are normalzed. x 2 x 2+1 M M 1/ / 2 2 l h Fg. 4. MC n Haar lftng scheme A smple ball example s used to test the Control Grd MC scheme. As shown n Fg. 5, shape dstorton besdes moton dstorton s observed. As the cell sze decreases, the dstorton s reduced. The extreme case s that only one grd pont s n each cell, whch s ndeed regular LWT. Compared to nteger LWT, no compresson gan s obtaned wth the moton compensaton. The result ndcates that bloc or grd based MC n our 3D vdeo approach, n whch grds are not as dense as pxel or voxel representaton n 2D vdeo, does not show sgnfcant advantages. frame 1 frame 32 frame 64 Fg. 5. Moton compensated Ball example shows dstorton (MC cell sze of 4x4x4, compressed fle sze s 6,238KB wth moton vector ncluded) 6 Concludng Remars and Future Wor In ths paper, a new tme-varyng 3D geometry compresson scheme based on 4D lfted wavelet transform s presented. It s demonstrated that a hybrd approach wth volume grd values and sosurfaces s feasble for 3D anmaton compresson. Geometrc nformaton and anmaton are compressed based on volume grd values. Surfaces are

7 676 Y. Wang and H. Hamza reconstructed from grd values and sovalues. Rescalng and nteger-to-nteger LWT shows sgnfcant mprovement on compresson rato wthout compromsng qualty of surfaces. Bloced based moton compensaton appears unnecessary. The proposed 4D geometry compresson can be used n general applcatons such as scentfc computng and vsualzaton, collaboratve engneerng, modelng and smulaton, teleconferencng, and entertanment. To enable applcaton of 4D LWT to general 3D vdeos, future wor may nclude volume data edtng methods to support multple sosurface constructon, content-based moton compensaton, and assocated moton error, as well as flter selecton and comparson. References 1. Allez, P. and Gotsman, C.: Recent advances n compresson of 3D meshes. In: Proc. Symp. on Mutresoluton n Geometrc Modelng Peng, J., Km, C.-S. and Kuo, C.-C. J.: Technologes for 3D mesh compresson: A survey. J. Vsual Communcaton & Image Representaton. 16 (2005) Taubn, G., Horn, W., Lazarus, F. and Rossgnac, J.: Geometry codng and VRML. In: Proc. the IEEE. 96 (1998) Mura, S.: Volume data and wavelet transforms. IEEE CG&A. 13 (1993) Nelson, G.M., Brunet, P., Gross, M., Hagen, H., Klmeno, S.V.: Research ssues n data modelng for scentfc vsualzaton. IEEE CG&A. 14 (1994) Lengyel, J.: Compresson of tme dependent geometry. In: Proc ACM Symp. on Interactve 3D Graphcs. (1999) Ahn, J.-H., Km, C.-S., Kuo, C.-C. J., and Ho, Y.-S.: Moton-compensated compresson of 3D anmaton models. IEE Elec. Lett., 37 (2001) Alexa, M. and Müller, W.: Representng anmatons by prncple components. Computer Graphcs Forum. 19 (2000) Sattler, M., Sarlette, R., and Klen, R.: Smple and effcent compresson of anmaton sequences. In: Proc. EUROGRAPHICS (2005) Ibarra, L. and Rossgnac, J.: Dynapac: space-tme compresson of the 3D anmatons of trangle meshes wth fxed connectvty. In: Proc. EUROGRAPHICS (2003) Karn, Z. and Gotsman, C.: Compresson of soft-body anmaton sequences. Computers & Graphcs. 28 (2004) Brceño, H.M., Sander, P.V., McMllan, L., Gortler, S., and Hoppe, H.: Geometry vdeos: A new representaton for 3D anmatons. In: Proc. EUROGRAPHICS (2003) Gusov, I. and Khodaovsy, A.: Wavelet compresson of parametrcally coherent mesh sequences. In: Proc. EUROGRAPHICS (2004) Ma, K.-L., Smth, D., Shh, M.-Y., and Shen, H.-W.: Effcent encodng and renderng of tme-varyng volume data. NASA/CR ICASE Report No (1998) 15. Guthe, S. and Strasser, W.: Real-tme decompresson and vsualzaton of anmated volume data. In: Proc. IEEE Vsualzaton 2001 (2001) Sohn, B.-S., Baa, C., and Sddavanahall, V.: Volumetrc vdeo compresson for nteractve playbac. Comp. Vs. & Image Understandng. 96 (2004) Sweldens, W.: The lftng scheme: A custom-desgn constructon of borthogonal wavelets. Appl. & Comp. Harmonc Analyss. 3 (1996) Calderban, A.R., Daubeches, I., Sweldens, W., and Yeo, B.-L.: Lossless mage compresson usng nteger to nteger wavelet transforms. In: Proc. IEEE Int. Conf. Image Processng. (1997)