Volumetric Textures. Esteban W. Gonzalez Clua 1 Marcelo Dreux 2

Transcription

1 Volumetc Textues Esteban W. Gonzalez Clua 1 Macelo Deux 2 1 PUC-Ro - Depatment of Compute Scence Rua Maquês de São Vcente, 255, , Ro de Janeo, RJ, Bazl 2 PUC-RIO Mechancal Engneeng Depatment Rua Maquês de São Vcente, 255, , Ro de Janeo, RJ, Bazl Abstact Thee ae some types of natue elements that ae adequately epesented n Compute Gaphcs only though volumes. In ode to vsualze scenes wth volumes, togethe wth geometcal objects, t s necessay to make use of a hybd endeng algothm. Howeve, the pesence of volumetc elements heavly nceases the vsualzaton pocessng tme, ndependently of the technque beng used. Ths atcle pesents a method that educes the volumetc endeng to a pojecton of a specal textue on a geometc suface that nvolves the volume. Ths technque s patculaly effcent to volumes that epesent natue elements, such as gases, clouds, smoke and leaves. It s possble, howeve, to extend the method to othe types of volumetc objects. Keywods: Volume Rendeng, Natue Elements, Textues. 1. Intoducton Tadtonal Volume Rendeng technques pefom a densty/tanspaency evaluaton, along each vsted voxel, elatve to each pxel that contans pat of the volumetc object. Consde, fo nstance, a volumetc object n a scene wth 1000x1000x1000 voxels. In ths case, fo each pxel of the mage that contans pat of the pojected object, on aveage 1000 voxels wll be vsted. Ths numbe vaes accodng to the angle and poston of the ay that comes fom the obseve and eaches the volume. If Ray-tacng s beng used, such as t s descbed by Sobeajsk et al 9, the stuaton s even wose snce, due to the ecusons, the volume s tavesed moe than once. A numbe of volume vsualzaton optmzatons have been poposed 3, 7, 8. These methods, howeve, ty to mnmze the numbe of vsted voxels along the ay path, nsde the volume, by avodng unnecessay voxels o by goupng voxels wth low densty contbuton. Even wth those optmzatons, t s necessay, n many stuatons, to tavese a long voxel lst. Gadne 4 makes use of pocedual textues n ode to dstbute densty onto an object suface, when dealng wth natue elements. Howeve, n that method, the textue s only appled on the suface, and thus nfomaton fom the nteo of the volume s not stoed. They ae not eal volumes such as the ones modeled by Ebet 1, 2. Volumetc Textues, as they ae called n ths atcle, ty to model a volume though a set of specal textues appled to smple geometcal sufaces. Those textues ae bult pe-pocessng the volume, to avod tavesng ts nteo dung the vsualzaton. That appoach enomously acceleates the endeng pocess. It becomes a smple textue mappng wth a value that

2 E. Clua and M. Deux / Volumetc Textues coesponds to the densty accumulaton along the path tavesed by the ay nsde the volume. 2. Textue Evaluaton The volumetc textue pe-pocessng conssts bascally on stong the densty accumulaton of a ay that eaches the volume at a cetan pont, consdeng a vaety of possble angles. Theefoe, volumetc textues ae matces, whee each element maps a pat of the geometc suface that nvolves the volume. In ths wok, these elements ae called nput polygons. Each nput polygon has an assocated table that stoes the esults obtaned by a ay ntesectng the volume at that pont wth dffeent nput angles. So, the esoluton of the volumetc textue object depends on the nput polygon quantty and on the esoluton of the table of nput angles (see fgue 1). Whee s the face to be mapped by the volumetc textue VX, U p s face up vecto, N s face nomal and and N. S s a vecto pependcula to both U p As the face s squae the thee vectos fom a local coodnate system to face (see fgue 2). These base vectos must be nomalzed n ode to smplfy futue evaluatons and the ogn of ths local coodnate system wll be located at the cente of the top left nput polygon of the face. It s also necessay to set up a base change matx T, that tansfoms a pont fom ths local coodnate system to the global coodnate system. Each polygon s, ntally, a squae of sze m. The locaton of the cente of each nput polygon s detemned by (j.m, k.m, 0) and J.m and K.m ae the hozontal and vetcal face szes, espectvely. Fgue 1 The volume s epesented by a geometc object (a box n ths fgue),whch s dvded n nput polygons. Each of those polygons have a table wth dffeent nput angles. It stoes the densty accumulaton along the ay path elatve to each nput angle. Consde the volume epesented by a box defned by the vetces v1, v2,..., v8 and by the faces F1, F2,..., F6. In ths case thee wll be 6 matces, one fo each face. Each volumetc textue (VX ) wll also need to stoe the followng vectos, n ode to speed up the vsualzaton pocess: vk v j Up = vk v j And S = Up N (1) Fgue 2 Each face has a local coodnate system, fomed by the vectos ( S, Up, N ). Each face must have ts own base tansfomaton matx T, whch convets a local coodnate to the global coodnate system. The pe-pocessng, then, conssts n evaluatng the densty to a set of dffeent nput angles to each face polygon. That set of angles s a samplng of all possble nput angles. Each polygon table of angles s a matx addessed by the pa of angles α e β fomed between

3 the nput vecto and the vectos S e U p, espectvely (fgue 3). The nput vecto can be expessed, n the local coodnate system of face, by: R = (j.m, k.m, 0) (j.m + tg -1 α, k.m + tg -1 β, 1) (2) Howeve, as ths vecto wll tavese the whole volume, whch s descbed n the global coodnate system, t s necessay to apply the tansfomaton matx T to obtan the coect nput vecto (eq. 3). R = T.(j.m, k.m, 0) When α o β ae 90 o o 90 o, t s not necessay to evaluate o to stoe the accumulated densty value snce, fo a vecto paallel to the face, the accumulated densty value s 0. The ays n that stuaton ae tangent to the face and, theefoe, do not penetate the volume. Note that the fp ntegal can be evaluated based on the densty functon descpton. Theefoe, the volume to be vsualzed does not need to exst at any tme. Ths fact can bng, n some cases, an enomous optmzaton. Howeve, f only the volume s known, and not ts densty functon, then the ntegal evaluaton s analogous to the ncemental densty accumulaton of the volume endeng algothm 6. T.(j.m + tg -1 α, k.m + tg -1 β, 1) (3) The followng algothm bulds the volumetc textue fo the face F, usng egula ncements to obtan the conjunct of nput angles. The accumulated densty fo each ay passng by the volume s stoed n VX, j, k (VX s the volumetc textue of F and j, k ae the nput polygon coodnates of the pont of the volume beng evaluated). Fo each nput polygon of face F do α = 180 / vetcal angula esoluton β = 180 / hozontal angula esoluton Fo α = (-90 + α ) to (90 - α ) do Fo β = -90 to (90 - β ) do R = T.(j.m, k.m, 0) T.( j.m.tg -1 α, k.m.tg -1 β, 1) Evaluate the ponts t n and t out,. They ae the ntesecton ponts between vecto R and the box. out VX, j, k [α, β] = f ρ( t) dt β = β + β α = α + α t tn Whee s one of the 6 faces, (j, k) s the coodnate of the cuent polygon, gven n the local system of face and fp s the functon that descbes the densty nsde the volume as a functon of space. Fgue 3 In ode to obtan the angles whee the accumulated densty s evaluated t s necessay to apply egula ncements to α and β. 3. Vsualzaton The volume vsualzaton conssts n evaluatng, fo each nput polygon, whch ae the α and β angles fomed between the obseve vecto O (vecto fom the obseve to the nput polygon beng vsualzed) and the local coodnate system of the face the polygon belongs to. So, α s the angle fomed between the pojecton of O onto the plane fomed by the vectos N and S ( P α ) and β s the angle fomed between the pojecton of O onto the plane fomed by the vectos N and U p ( P β ) (fgue 4).

4 E. Clua and M. Deux / Volumetc Textues P and α fomulas: P = ( T P β can be evaluated by the followng v ( O) S ) S + ( T ( O) N ) N 1 1 α (4) P β and = ( T 1 v ( O ) S ) S + ( T 1 ( O ) U p ) U p (5) Fgue 4 Evaluaton of angles α and β whch ae fomed between vecto O and the local coodnate system of the face the cuent polygon belongs to. It s necessay to apply the nvese of the tansfomaton matx T, ove vecto O, snce t s descbed n tems of the global coodnate system and the othe vectos ae n tems of the face coodnate system. It s convenent to evaluate T -1 dung the pepocessng phase and to stoe t wth the volumetc textue. Once the α and β angles have been evaluated, t should be only necessay to seach the volumetc textue VX, j, k [α, β] n ode to obtan the accumulated densty value of polygon j, k fom volume face. Howeve, α and β ae eal values whle the table of angles has a dscete angle samplng. It s necessay to pefom a lnea ntepolaton of the densty values to avod an undesable flckeng effect dung the vsualzaton. That ntepolaton slghtly nceases the computatonal pocessng tme, howeve t geneates mages wth geate qualty, when compaed to mages wth no ntepolaton. Volumes wth few geometc detals, such as gases, clouds and leaves, do not suffe a sgnfcant loss wth the ntepolaton. When dealng wth models wth fne detals, such as n medcal mages, the ntepolaton only woks f the table of angles has a hgh esoluton. 4. Optmzaton The sze of the volumetc textues s popotonal to the table of angles esoluton and to the sze of the nput polygons. Fo nstance, f the table consdes vaaton of 30 o to both α and β, thee wll be 25 densty values to each nput polygon. Consdeng a 1000 x 1000 x 1000 volume esoluton, then the sze of each matx wll be , n case each nput polygon s the sze of a voxel. As t s necessay to use 6 matces to epesent a complete volume, then the total sze wll be , whle the volume, not consdeng possble compessons, has 10 9 voxels. In that case, the volumetc textues not only heavly acceleate the vsualzaton pocess, but also eque less stoage space, when compaed to volume endeng. If a quadtee 5 s appled to the table of angles epesentaton t s possble to obtan a moe accuate and pecse esult. The matx s, then, sub-dvded each tme the densty vaaton fom one angle to ts neghbo s hgh. Ths method could also be used to pefom table compesson. The matx could be bult by jonng pats of the table wth smla values. Anothe possble optmzaton s to make use of gaphcal acceleaton (OpenGL, fo example). Fo ths case t s possble to plot an ente nput polygon by applyng the appopate tanspaency, whch was obtaned by ts volumetc textue. 5. Concluson and Futhe Reseach The poposed technque s extemely effcent to vsualze volumetc objects wth a dffuse densty dstbuton, as n the case of natue elements. Moeove, the technque allows the pesence of volumes n scenes wth suface endeng (ay-tacng, scan-lne, etc.), snce t s not necessay to pefom the tadtonal volume endeng. The volume pocessng s teated as a knd of textue mappng and the eal volume does not have to be pesent n the scene. Ths wok appled the textues n volumes beng epesented by 3D boxes. A possble extenson could be the ceaton of othe types of boundng volumes, such as sphees, cylndes o mplct sufaces, snce fo othe

5 volumetc objects those boundng volumes would be moe appopate. Othe compesson data stuctues could also be studed, snce n cetan stuatons the use of quadtees could pove to be not so effcent. Ths wok only stoes, fo each nput angle, the accumulated densty fo a ay n cetan dectons. Howeve, to geneate a moe ealstc mage, t could be necessay to stoe exta data, such as the gadent o nomal of the object nsde the volume fo each one of ths nput angles. The volumetc textues stll have lmtatons and t needs a thoough nvestgaton to pusue possble solutons. The man dawback concens the volumes wth hgh detals that could not be well epesented, as mentoned befoe. Othe lmtaton s that the obseve cannot penetate the volume, because the textues stoe accumulated densty fom the nput pont untl the end of the volume. Futhemoe, f thee s an alteaton of a voxel densty the volumetc textues wll have to be econstucted, whch s an expensve task. Acknowledgement The fst autho s gateful to CAPES, a bazlan govenment eseach councl, and to TeCGaf, Technologcal Goup n Compute Gaphcs of PUC- Ro, whch sponsoed ths eseach. Refeences 1. Davd Ebet. Pocedual Volumetc Modelng and Textung, SIGGRAPH 97, couse 14, notes, chapte 6, August John Danskn and Pat Hanahan. Fast Algothms fo Volume Ray-tacng. Poceedngs of the 1992 Wokshop on Volume Vsualzaton, The Assocaton fo Computng Machney, Boston, MA, pp Specal ssue of Compute Gaphcs, ACM SIGGRAPH, New Yok. 4. Geoffey Y. Gadne. Smulaton of natual scenes usng textued quadc sufaces. In Hank Chstansen, edto, Compute Gaphcs (SIGGRAPH 84 Poceedngs), volume 18, pp 11-20, July C. L. Jacklns and S. L. Tanmoto. Octees and the use n epesentng thee-dmensonal objects. Compute Gaphcs and Image Pocessng, 14, pp , Mac Levoy. Dsplay of Sufaces fom Volume Data. IEEE Compute Gaphcs and Aplcatons 8 (5) pp May Mac Levoy. Volume Rendeng by Adaptve Refnement. The Vsual Compute 6 (1), pp Renben Shu and Alan Lu. A Fast Ray Castng Algothm Usng Adaptve Isotlnea Subdvson. Poceedngs Vsualzaton, IEEE Compute Socety Pess, pp , L. M. Sobeajsk and A. E. Kaufman. Volumetc Ray Tacng. Poceedngs of the Symposum on Volume Vsualzaton, pp.11-18, Davd Ebet. Pocedual Volumetc Cloud Modelng and Anmaton. SIGGRAPH 99, couse notes. August 1999.