Multiresolution Interactive Modeling with Efficient Visualization

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1 Multiresolution Interctive Modeling with Efficient Visuliztion Jen-Dniel Deschênes Ptrick Héert Philippe Lmert Jen-Nicols Ouellet Drgn Tuic Computer Vision nd Systems Lortory Lvl University, Quéec, Cnd Astrct 3D interctive modeling from rnge dt ims t simultneously producing nd visulizing the surfce model of n oject while dt is collected. The current reserch chllenge is producing the finl result in rel-time. Using recently proposed frmework, surfce model is uilt in volumetric structure encoding vector field in the neighorhood of the oject surfce. In this pper, it is shown tht the frmework llows one to loclly control the model resolution during cquisition. Using ry trcing, efficient visuliztion pproches of the multiresolution vector field re descried nd compred. More precisely, it is shown tht volume trversl cn e optimized while preventing holes nd reducing lising in the rendered imge. 1. Introduction After more thn 20 yers of developments in 3D rnge sensors nd 15 yers of significnt dvnces in modeling ojects, the current chllenge is producing the finl result in rel-time while dt is collected. This new field of 3D interctive modeling integrtes the processes of scnning the surfce, incrementl surfce reconstruction, lignment, fusion, s well s compression nd visuliztion. The whole process is performed in feedck loop where user or mechnism typiclly selects the sensor viewpoint nd positions the sensor sed on the current stte of the surfce model. This pper dels with two specific ojectives in 3D interctive modeling: 1) Providing 3D interctive modeling with vrying resolution of surfce reconstruction. By loclly controlling the level of detils (resolution), interesting fetures cn e reconstructed with higher resolution thn surrounding context. This llows fithful modeling without unnecessry storge spce nd scnning time overhed. To chieve such reconstruction, the modeling process must cope with vrying smpling density. Fetures re densely smpled nd reconstructed with high resolution while the surrounding context is sprsely ut rpidly smpled. 2) Improving the qulity nd speed of online model visuliztion. So fr the min gol in visuliztion ws to provide visul feedck in order to ensure full coverge of the oject s surfce. By improving the qulity of visuliztion during interctive modeling, it ecomes possile to ssess the qulity of the reconstructed model s well. Surfce representtion is the corner stone in interctive modeling. Among the vrious representtions including point clouds, polygons nd volumetric cells, only the ltter ppers dequte in the most generl sitution. However, it is worth mentioning tht in recent work [1], comintion of these representtions ws proposed where lists of points re ccumulted in volumetric neighorhoods in order to uild tringultion in rel-time. This frmework is interesting lthough limited for some steps such s lignment nd its cpility to loclly ccommodte vrile surfce smpling. In the lst decde, volumetric representtion hs evolved progressively to encompss the whole modeling chin. Hoppe et l. [5] ddressed the surfce reconstruction step from point cloud using volumetric grid. An implicit surfce function is generted nd input to mrching cues lgorithm to produce mesh tht is rendered. Hilton nd Illingworth [4] s well s Curless nd Levoy [2] pushed the frmework further in the field of 3D imging from rnge dt. Although the interest ws not rel-time modeling, it ws possile to esily merge dt in the form of rnge imges nd reconstruct surfce. Improving the representtion with surfce norml estimtes ws essentil to llow online pproximte rnge imge lignment (registrtion) nd visuliztion y Rusinkiewicz et l. [9]. Surfce reconstruction is completed off-line efore visulizing the finl result. Tuic et l. [11] hve recently proposed volumetric frmework where ll modeling steps re integrted nd their complexity remins liner with respect to the numer of dt. The frmework lso llows the integrtion of ny type of rnge dt including points, curves nd surfce sections [12] while enefiting from the structure, nmely tngents, of the input dt when it is ville. The surfce is represented s vector field encoding the surfce norml nd the closest point t ech voxel in the neighorhood of the surfce.

2 In this pper, this frmework is extended to etter exploit multiresolution reconstruction for quick cpture of the context. It is shown how the multiresolution structure my dpt loclly to llow integrtion nd fusion of mesurements. Intermedite levels re preserved to ensure continuity in resolution. The min chllenge is to visulize the surfce representtion stte with high qulity s fusion is performed. To do so, three different visuliztion strtegies re nlyzed; these re vrints of ry trcing in volume. Avoiding holes due to multiresolution discretiztion s well s lising for visile surfce contours re the min concerns. A fovel strtegy for improving visuliztion speed nd progressive disply is lso descried. The principles of surfce modeling in vector field re first reclled efore the multiresolution issue is discussed in section 3. From the computer grphics literture, relted work on surfce visuliztion is ddressed in section 4 long with the rendering strtegies dpted to interctive modeling. Although intermedite results re presented throughout the pper, dditionl results follow in section Why vector field surfce representtion? If only one step in the modeling process is not of liner computtionl complexity with respect to the numer of mesured points, the system slows down s the mount of rnge dt increses nd eventully, ecomes too slow to e usle. Most of the computtionl complexity issues hve proximity opertions, closest point serch for exmple, s common denomintor. This is not only the cse for registrtion which is often sed on the Itertive Closest Points (ICP) lgorithm ut for surfce reconstruction s well: surfce reconstruction cn e seen s the opertor tht computes points on the finl model from the nerest mesured points. It ws shown erlier [11][12] tht reching the liner complexity depends not only on lgorithms ut on dequte surfce representtion s well. A prticulrly well suited surfce representtion tht respects computtionl complexity constrint, is vector field surfce representtion. The vector field is n implicit representtion defined on regulr 3D lttice of points tht encodes direction nd distnce towrds the closest surfce point. This is equivlent to encoding the tngent plne t the closest surfce point (see Figure 1). Vector fields llow surfce reconstruction from rnge imges, rnge curves, unorgnized sets of points or ny comintion of these. Furthermore, vector fields llow efficient registrtion, visuliztion nd compression, ll of them with liner computtionl complexity. The only exception is visuliztion tht should hve complexity independent of the model size since snpshot computed from the whole model must e rendered t ny time. When the model is completed, tringultion cn e produced if necessry. Figure 1. Exmple of the vector field. ) The vector field is computed on regulr grid of points (voxel centers) so tht the vector t ech voxel center points towrds the closest surfce point. The field is defined only in the vicinity of the surfce, tht is in the volumetric envelope (region ounded y two iso-distnt surfces, colored in green). ) A cross section of the vector field. 2.1 How does it work? In the vector field frmework, surfce reconstruction mens reconstructing vector field representtion from the rnge dt. Regrdless of the type of rnge dt, vector field reconstruction lwys proceeds in the sme wy. The tngent plnes encoded y vector fields re incrementlly reconstructed using either reltive positions of points, locl surfce informtion contined in normls (rnge imges), or tngents (surfce curves). Exploiting locl surfce orienttion contined in the rnge dt, surfce tngents or normls mkes the surfce reconstruction nd registrtion less sensitive to sensor positioning errors [12]. Theoreticlly, the vector field cn e otined s the derivtive of the sclr field. Nevertheless, numericl differentition cn e voided y directly constructing the vector field from input dt. Incrementl reconstruction is thus chieved y updting covrince mtrix from tngents, normls nd reltive positions of points t ll voxels in n envelope within predefined distnce round the mesured rnge dt. Liner complexity with respect to the numer of mesured points is chieved y defining locl volumetric neighorhood, nmely the fundmentl cell [11] tht selects voxels to e updted. The fundmentl cell is different for ech element of the input dt: point, line segment or tringle. Since such cell contins only voxels ffected y corresponding dt elements nd since its shpe cn e computed independently for ech element, the computtionl complexity is liner with respect to the mount of input rnge dt. Mtching closest points, the most computtionlly expensive prt of registrtion, is solved trivilly using vec-

3 tor fields. Since the field is computed on regulr grid it is esy to find the closest voxel for ny point. The closest voxel on the other hnd directly encodes the closest tngent plne. The corresponding point is the closest point on the tngent plne. When necessry, compressing the vector fields is lso conceptully very simple. Since vector fields encode tngent plnes, they re ctully constnt in plnr regions of the surfce. Unlike sclr distnce fields tht re never constnt due to the chnge of the distnce to the surfce, vector fields cn e compressed simply y loclly replcing groups of voxels tht encode the sme tngent plne with smller numer of lrger voxels. When it comes to the visuliztion, the complexity prolem is relted to the ever incresing numer of points during the cquisition of dt. If ll cquired dt is displyed using renderer, the complexity increses s the numer of points increses, which slows down the visuliztion process. It is therefore essentil to use model tht merges nd/or discrds points in order to limit the size of the dt. Being defined on finite nd regulr grid, vector fields llow direct ry trcing with worst-cse constnt complexity when the smllest voxel size is set. The principle is simple, the surfce is viewed where the ry intersects with the tngent encoded in voxels. There is no solute need for n intermedite (polygonl) representtion. Using the vector field representtion, one defines the whole volume size to include the oject s well s the mximum resolution level (minimum voxel size). It is then gurnteed tht the whole modeling loop will not degrde fter long (infinite) cquisition time. For ech step, the complexity is constnt (liner with time or the numer of input dt) nd the constnt depends on the envelope size. The size of the envelope is typiclly 1 or 2 voxels round surfce dt. It depends on relted fctors such s the sensor resolution, smpling density, the expected level of detils on the surfce s well s the registrtion error. Although it is possile to prmeterize the size of the envelope, this cn e ddressed y uilding multiresolution representtion where the envelope size (in numers of voxels) remins constnt. 3. A multiresolution modeling pproch Since the vector field is defined in the neighorhood of the oject s surfce, most of the voxels in the volumetric grid re unused. It is thus dvntgeous to void llocting those unused voxels y encoding the volume in hierrchicl structure such s n octree. In the octree structure, voxel size divides y two t every level. One cn set mximum resolution level, N, nd ccumulte surfce informtion in the covrince mtrices t the lef nodes. The octree is then dynmiclly defined for voxels in the neighorhood of rnge mesurement t level N. c e Figure 2. Multiresolution: context nd detil with vrile smpling density. ) Sprse smpling density t level 7 ) High smpling density t level 7 c) Sprse smpling t level 9 d) High smpling density t level 9 e) Comined view of dt t levels with the highest locl level displyed. Although vrying the level of detils cn e chieved using utomtic compression, it is lso importnt to control locl resolution nd smpling density. In this scenrio, detiled surfce section is represented in the context of lrger surrounding re. For this lrger re, it is not necessry to cpture every detil nd use lrge quntity of memory (voxels). It is preferle to loclly vry the resolution level of the vector field. This cn e done mnully while the oject s surfce is eing cptured. In typicl scenrio, lower resolution level, N, is set nd the context is cptured in the whole re surrounding nd overlpping the region of interest. A higher resolution level N+P is then set nd the region of interest is cptured t higher smpling density, for instnce y slowly moving the hnd-guided d

4 sensor within this re. The system is then enled to uild the octree in rel-time t higher levels. In Figure 2, the result fter pplying this procedure is shown. Figure 2 displys view of the low resolution section nd its surroundings mesured t sprse smpling density. In Figure 2, surfce smpling is higher ut the resolution of the model, N, remins the sme. There is no significnt difference. Figures 2c nd 2d show these two viewpoints uilt from the sme corresponding dt, ut where the resolution of the model is set to N+P. It cn e seen tht mny holes re present in the reconstructed section with lower smpling density since the envelope is lso two voxels t the finer resolution. Since the high resolution reconstruction requires high smpling density, low density regions cnnot e reconstructed dequtely, leving lrge numer of holes. The difference thus clerly ppers. Figure 2e shows the hyrid representtion exploiting oth resolutions. Ech ry intersects with the volume t its loclly higher ville resolution level. This cuses holes to e filled in the surrounding section while detils re visile in the centrl section. During interctive modeling, one cn lso set ck the resolution level to N nd only these nodes t level N will e updted. If one chnges the resolution level from N to N+P (with P>1), ll intermedite levels re uilt (independently). In this wy, it is possile to mintin the continuity etween levels while vrying the level of resolution during cquisition. Actully, since the surfce envelopes re two voxels with size corresponding to specific levels, the envelope of the vector field t level K will generlly overlp with the tngent plne of the vector field t level K-1. The sme surfce cn thus e trcked in the octree y progressing through the resolution levels. However, this is not strictly gurnteed for ny resolution nd noise level. In the next section, it will e explined tht overlpping through levels is lso necessry property for coherent visuliztion. Figure 3 shows 2D cross section of the octree. The overhed nd memory requirement for mintining intermedite levels is not limittion since the sum of voxels including ll levels etween N nd N+P-1 is elow 1/7 of the size t level N+P. One importnt concern when uilding nd visulizing models is the continuity of the reconstructed surfce. Simultneously displying t lest two levels of resolution my led to discontinuities t the interfce. It is importnt to understnd tht the structure encodes the surfce t oth levels nd tht there re two models where the sections overlp, s opposed to mixture of models. It is thus not n ojective to hide possile discontinuities since the lower resolution section only presents the context to disply the reconstructed detil of interest. Moreover, discontinuities etween levels will e minimized when sections overlp over smooth sections. In these res, strong discontinuity indictes mislignment of the dt. Figure 3. Cross section of the vector field depicting the finest locl resolution levels. 4. Interctive visuliztion Gret efforts hve een devoted to ccelerting high qulity rendering. Pushed y the gme industry, the disply (rsteriztion) of tringulr meshes, now ssisted y texture mps, norml mps nd displcement mps to nme few, is good exmple. High performnce grphics hrdwre is now widely ville. Spltting [14] nd its recent versions [10][15] is nother worthy method tht chieved oth fst nd fine qulity rendering when dt cn e preprocessed. Finlly, pioneering work on efficient ry trcing of volumetric dt [6], lso followed y recent progress [13], proposed nother lterntive tht cn provide renders of the highest qulity. Except for qulity, ry trcing hs not chieved the performnce of the former methods especilly in the context of interctive modeling. Nevertheless, it solely depends on the numer of pixels nd octree depth, property tht spltting does not shre. When the displyed surfce re is smll compred with the ctul surfce re of the model, volumetric ry trcing will e especilly indicted. The primry representtion for integrting mesurements is volumetric, not n explicit surfce. Moreover, ry trcing is perfectly tilored to prlleliztion. For these resons, our reserch ims t developing this venue in the context of interctive modeling, despite current performnces of PCs. 4.1 High qulity rendering Integrting rendering in the modeling loop llows one to oserve the evolution of surfce coverge nd its qulity. It is lso possile to oserve flws tht re cused y the sensing device. After descriing the mechnism of ry trcing in the multiresolution vector field, three methods re compred. They differ in their simplicity (rpidity) nd cpility to limit discretiztion nd consequently, lising long the contour curves of n oject.

5 Three vrints of the sic procedure re descried; these methods re nmed first ctive voxel, first tngent intersection, nd first entry respectively. A C B The first ctive voxel method detects hit s soon s the ry penetrtes into voxel within which there is tngent plne. In prctice, this trnsltes to voxel with defined norml whose length is less thn hlf the digonl of the voxel, tht is 3 2 times the voxel size. Figure 5 illustrtes the principle for rys A nd B. This pproch is rpid nd does not led to hole rtefcts (tunnels) due to visuliztion. The compromise is the stronger discretiz Figure 4. Importnce of preserving intermedite levels. ) The tngent plne t level N does not overlp with the envelope t level N+3 t its intersection with the ry pth. ) Cross section view: the low resolution level hides the higher surfce resolution representtion since it is outside the envelope. The principle of ry trcing is simple. For ech pixel in the rendered imge, ry is cst into the volume. When the ry hits the surfce, the pixel color (or grey level) is computed from the ry direction, the surfce norml orienttion nd the simulted source direction if different from the ry direction. The depth could lso e returned to provide 2.5D mp. The surfce is hit t the zero-crossing of the vector field norm tht occurs etween two neighoring voxels whose vectors hve opposite directions within the volume. The octree is trversed efficiently using the strtegy proposed in [7]. However, since the model is multiresolution nd the lef nodes re not ll t the sme level, we dopt the strtegy of displying the finest locl level where ville. The ry thus penetrtes into the structure t the finest level. This strtegy requires tht the vector field e defined t ech intermedite level etween the lower nd higher resolution levels. Here gin, the envelope of ech resolution level, K, must overlp with the tngent plne t its djcent level K-1. This cn e seen in Figure 4 where the envelope t the higher level does not progressively overlp with the surfce representtion t the lower level. In this sitution, the corser model occludes the finer level (N+3) since there is no ctive resolution voxel t level N+1 nd the octree is not explored further. The rendered pixel would e displyed sed on the corse level N. It is possile to disply the pproprite intersection ut one would hve to explore the su-tree exhustively, losing the multiresolution interest. Since the resolution progression is power of two nd the envelope is t lest two voxels wide, the sme surfce will overlp with its djcent level. Building the intermedite levels mkes it possile for ech voxel to hve correspondent t corser levels through continuous envelope progression. Figure 5. Method First ctive voxel. ) Principle ) Effect on contours nd surfce. A C B Figure 6. Method First tngent intersection. ) Principle ) Effect on contours nd surfce. A C B Figure 7. Method First entry. ) Principle ) Effect on contours nd surfce.

6 tion effect leding to lising tht is esily oserved on occluding contours. The reson for this is clerly seen in Figure 5 where ry A intersects with the voxel ut not with the tngent plne of the surfce model. One cn improve the qulity of the rendered imge y limiting discretiztion rtefcts. This is ccomplished y further imposing tht the ry intersects with the tngent plne. In prctice, the tngent plne must e ounded to descrie locl neighorhood. To do so, it is defined on circle whose rdius length is 3 2 times the voxel size. Figures 6 nd show the principle s well s the qulity improvement despite slight increse in the computtionl cost. Nevertheless, this vrint introduces new rtefcts tht cn e seen on the nose of the mnnequin hed t the lower right prt of Figure 6. These rtefcts re cused y rys entering flse tunnels into the surfce. Ry C in Figure 6 illustrtes this ll too common sitution. Although incresing the circle rdius would reduce the prolem, corser surfce of the modeled field would e displyed. A etter solution consists in tking dvntge of the volumetric representtion where the surfce is defined within the envelope s the vector field. This comines the dvntges of the two preceding pproches. Actully, ech voxel defining the surfce cn e tgged s inside (sign -), outside (sign +) the surfce or oth. The two signs re present within voxels split y tngent plne. The position of the sensor is used to set the signs when the field is uilt. The ry trcing strtegy then consists of serching for sign trnsition given the sign sttus of the oserver. The interior/exterior oundry my thus e reched 1) when the tngent plne is crossed over or 2) when the ry enters new voxel. This pproch prevents the ppernce of tunnels while limiting lising. Figure 7 illustrtes the method s well s its resulting effect in. This lst method leds to the est qulity results. Nevertheless for the tests we rn, the First ctive voxel method ws on verge 40% fster while the First tngent intersection ws out 20% fster. 4.2 Improving speed Although visuliztion of the multiresolution representtion sed on ry trcing is of constnt complexity order, it is still the most demnding process in the modeling loop. Depending on the mchine technology, complete visuliztion cn e requested t lower rte thn other modeling processes. This leds to prtition of the visuliztion process where smples of the imge to e rendered re displyed t ech step. Generlly, dynmic susmpling strtegy is necessry for high resolution imges with higher cquisition rte. Nevertheless, when the user stops cquiring dt to concentrte on the disply, we should e le to otin ll of the detils. This is the well known principle of progressive displying. c d Figure 8. Fovel progression ) Resolution res from pixels. In -c-d, shded pixels re those updted t the corresponding step. Dots, X nd dimonds re reference pixels to select the grey level of the re (2x2 or 1x1 t the finest level). The re displyed in -c-d is t the upper corner of n re trnsition. In this work, we hve dopted fovel smpling strtegy tht is well suited to hnd-guided sensors. For these sensors, the viewpoint cn dpt to the cmer viewpoint or sty fixed. In oth cses, quick disply of detils will e prioritized in the center of the imge, which is nturlly the re of interest. The progressive disply is split into 5 itertions. Figure 8 illustrtes the progression. In Figure 8, 512x512 imge is seprted into three concentric res of 128x128, 256x256 nd 512x512. In Figures 8, c, nd d, the progression is shown t the upper left corner of trnsition etween two resolution res. In ech figure, the grey shded res show the pixels tht re updted t given itertion. The white res re left t their previous vlues. The signs (,, ) mrk the pixels tht re used to set the uniform grey level in squre lock of incresing resolution. One cn see tht the centrl re fills more rpidly thn its surroundings. Using this scheme, it tkes 5 itertions to completely disply the 512 x 512 imge. 5. Additionl results The dditionl results presented in this section s well s results presented in the preceding sections were otined using hnd-held rnge sensor whose principle ws

7 c d c d Figure 10. Fovel progression: imges. In --c-d, views fter steps 1, 2, 3 nd 5 re displyed. e Figure 9. Multiresolution model. In --c-d, the mximum displyed resolution level is limited to nd 11 respectively. e) Sme view s in d with 3 colors comprising the corse, ll intermedite nd the finest resolution levels. descried in [3]. The sensor projects crosshir lser pttern. A more recent version of this sensor extrcts nerly points/sec. t 15Hz. The recent version sensor cn lso exploit fixed trgets for self-referencing on lrge res. Figure 9 shows the levels of the multiresolution representtion. The context ws cptured y setting the voxel size to 5 mm in cuic volume of 1280 mm for the whole oject ( lrge vse). The 5 mm vlue corresponds to level 8 in the octree. The oject shpe ws cptured quite rpidly without pying ttention to dense smpling of the surfce. Then, the resolution level ws incresed to voxel size of mm which corresponds to the level 11. The centrl detil displyed in the figure ws scnned more crefully. To visulize the multiresolution representtion, we hve imposed constrint on the mximum resolution level to disply using the First entry method. In Figure 9,, c, nd d, the mximum displyed resolution levels re set to 8, 9, 10 nd 11 respectively. We cn oserve the progression in the detils. Figure 9e displys ll resolution levels such s in d ut with distinct colors for the sis level 8 (pink), the intermedite levels 9 nd 10 (lue) nd the higher resolution level 11 (eige). It is interesting to oserve the lue re tht ppers t the trnsition etween high nd low resolution smpled res. One property of the system is to cpture nd disply the relevnt detils dpted to the smpling density. If the smpling density is incresed y moving ck with the sensor to the sme re, the resolution will progressively switch to the finer level. Figure 10 illustrtes the fovel disply. The result fter the first three steps is displyed in,, nd c respectively. The imge in d corresponds to the lst step 5. One cn clerly see the vrition of progression etween the centrl nd the peripherl res. For this oject with fovel disply, time sttistics re presented for ech of the 5 steps in Figure 11. The curves illustrte the time progression when the higher resolution level vries from 8 to 11. The system ws run on Pentium IV, 3 GHz. Although these specific curves were otined for the vse oject with one view depicted in the upper figures, it clerly illustrtes tht the progression does not follow the progression in the numer of voxels. In theory the

8 numer of surfce voxels qudruple t ech resolution level since it follows the surfce re of the oject. Time progression vs octree depth 6. Conclusion Interctive modeling integrtes concurrent surfce reconstruction nd visuliztion processes. We hve exposed the dvntges of using vector fields nd proposed multiresolution extension suited for interctive modeling. In this hierrchicl representtion, lef nodes re creted t different levels, determined during cquisition. This wy, it is possile to cpture nd control the resolution dpted to the level of detils nd surfce smpling. In this frmework, the surfce model is loclly ville for ll intermedite levels etween the finer nd the corser levels. This ensures etter resolution continuity tht is exploited for more efficient visuliztion. Among the three compred methods for visuliztion, the First entry method provides the est qulity results. Future work will concentrte on improving rendering performnces tht were not highly optimized prt from fovel progression. In ddition, the multiresolution representtion is good tool for exploring more efficient registrtion pproches. References Time(ms) highest resolution level Step 1 Step 2 Step 3 Step 4 Step 5 Totl Figure 11. Evolution of rendering time with respect to the highest octree level, for ech step of the fovel rendering. [1] T. Bodenmueller nd G. Hirzinger, Online Surfce Reconstruction from Unorgnized 3D-Points for the DLR Hndguided Scnner System, in Proc. of the 2nd Interntionl Symposium on 3D Dt Processing, Visuliztion, nd Trnsmission (3DPVT'04), Thessloniki, Greece, Sept. 2004, pp [2] B. Curless nd M. Levoy. A Volumetric Method for Building Complex Models from Rnge Imge, in SIGGRAPH '96 Proceedings, 1996, pp [3] P. Héert, A Self-Referenced Hnd-Held Rnge Sensor, in Proc. of the Third Interntionl Conference on 3D Digitl Imging nd Modeling, Queec, Cnd, My 2001, pp [4] A. Hilton nd J. Illingworth. Geometric Fusion for Hnd- Held 3D Sensor, Mchine vision nd pplictions, 12:44-51, [5] H. Hoppe, T. DeRose, T. Duchmp, J. McDonld, nd W. Stuetzle, Surfce reconstruction from unorgnized points, In SIGGRAPH '92 Proceedings, volume 26, July 1992, pp [6] Levoy M., Efficient Ry Trcing of Volume Dt, ACM Trnsctions on Grphics (TOG), Volume 9, Issue 3, July 1990, pp [7] J. Revelles, C. Ureñ, nd M. Lstr, An Efficient Prmetric Algorithm for Octree Trversl, in Proc. of the 8th Interntionl Conference on Computer Grphics nd Visuliztion'2000, Plzen (Czech Repulic), Ferury [8] G. Roth nd E. Wiowo. An Efficient Volumetric Method for Building Closed Tringulr Meshes from 3-D Imge nd Point Dt. In Grphics Interfce, My 1997, pp [9] S. Rusinkiewicz, O. Hll-Holt nd M. Levoy, Rel-Time 3D Model Acquisition, in ACM Trnsctions on Grphics, 21(3), July 2002, pp [10] S. Rusinkiewicz nd M. Levoy, Qsplt: A Multiresolution Point Rendering System for Lrge Meshes, in Proc. of the 27th nnul conference on Computer grphics nd interctive techniques, 2000, pp [11] D. Tuic, P. Héert, nd D. Lurendeu. A Volumetric Approch for Interctive 3D Modeling. Computer Vision nd Imge Understnding, 92:2003, pp [12] D. Tuic, P. Héert, J.-D. Deschênes, nd D. Lurendeu, A Unified Representtion for Interctive 3D Modeling, in Proc. of the 2nd Interntionl Symposium on 3D Dt Processing, Visuliztion, nd Trnsmission (3DPVT'04), Thessloniki, Greece, Sept. 2004, pp [13] I. Wld, P. Slusllek, Stte of the Art in Interctive Ry Trcing, in Proc. of EUROGRAPHICS 2001, Mnchester, United Kingdom, Sept. 2001, pp [14] L. Westover, Interctive Volume Rendering, in Proc. of the 1989 Chpel Hill Workshop on Volume Visuliztion, Chpel Hill, USA, 1989, pp [15] M. Zwicker, H. Pfister, J. vn Br, M. Gross, Surfce Spltting, in Proc. of the 28th nnul Conference on Computer Grphics nd Interctive Techniques, 2001, pp