Efficient Reconstruction of Indoor Scenes with Color

Transcription

1 Effcent Reconstructon of Indoor Scenes wth Color Ru Wang, Davd Luebke Department of Computer Scence Unversty of Vrgna {rw2p, Abstract In ths paper we present an effcent and general approach to computng and ntegratng 3D dstance felds drectly from multple range mages. We compute normal and confdence values drectly from a 2D range mage. We then approxmate 3D Eucldean dstance by correctng the lneof-sght dstance. To ntegrate multple scans, we effcently transform voxels of the target dstance feld to each scan s local coordnate system, then update voxels wth computed dstance and confdence values. Fnally, we extract an sosurface from the weghted dstance feld usng the marchng cubes algorthm. We extend the same dea to the assgnment of weghted colors or texture coordnates to the reconstructed model. Experments show that our approach s fast, has reasonable storage requrements, and can produce hgh-qualty surfaces from multple range scans. 1. Introducton Recent developments n 3D scannng technology have made t feasble to acqure complex models of real scenes on nexpensve platforms. Acquston of hstorc models as n the Dgtal Mchelangelo project [22] s one nterestng applcaton of ths technology. Whle extensve work has been done on precse dgtzaton of statues or museum collectons, ndoor scene modelng has receved relatvely lttle attenton. Together wth colleagues from Unv. of North Carolna at Chapel Hll, we ntated the Scannng Montcello project, the goal of whch s to create an extremely accurate model of Montcello, Thomas Jefferson s Vrgna home and a promnent achevement of Amercan archtecture. Wth a 3rdTech DeltaSphere laser scanner, we collected tens of ggabytes of geometrc and photographc data. We am to transform ths data nto accurate 3D models to be used for dgtal dsplay, vrtual toursm, and scentfc study. The 3D scannng ppelne of our system conssts of range mage preprocessng, multvew regstraton, effcent multvew ntegraton, reconstructon and colored dsplay of 3D models. In ths paper we focus on multvew ntegraton and reconstructon. Several technques have been proposed to reconstruct 3D models by ntegratng volumetrc representatons 3D dstance felds of algned range data. When dealng wth large models, most exstng approaches requre sgnfcant amounts of computaton and storage. In ths paper we present a new approach to computng and ntegratng 3D dstance felds, based on estmaton of 3D Eucldean dstance drectly from range mages. The same dea s used to assgned weghted colors or texture coordnates to reconstructed 3D models. Snce most computatons are performed n the 2D range mage doman, the new approach sgnfcantly reduces computaton and storage requrements and s therefore sutable for reconstructon of bg models such as ndoor scenes. 2. Background and Prevous Work 2.1. Range Image and 3D Regstraton A range mage stores depth values of sampled 3D ponts. Rotatng range scanners such as the DeltaSphere 3000 scan ponts on a 2D lattce n sphercal coordnates, recordng the drecton angles θ and φ and lne-of-sght dstance r. As Whtaker et al. [15] pont out, ndoor scene scannng s subject to nose and hole fllng problems, whch make t more complcated than the close-up scannng of statues or museum collectons. Range scanner accuracy s low because objects are dstant; dfferent surface propertes may lead to nvald or naccurate dstance measurements. A sngle scan may produce many range dscontnutes, further reducng confdence n the scanned depth. As a result, care must be taken when dealng wth ndoor scenes. To obtan a complete model of the scene, multple scans have to be acqured, regstered, and merged. Par-wse 3D regstraton has been extensvely studed, both n the context of automatc regstraton [27, 18] and n the context of teratve methods startng from an ntal algnment [5, 9, 26, 25]. The ICP (Iteratve Closest Pont or Iteratve Correspondng Pont) algorthm s frequently used. Researchers have also

2 studed multvew regstraton, whch seeks to evenly dstrbute regstraton errors among multple scans [24, 4] Multvew Integraton and Dstance Felds Extensve research has examned the problem of ntegratng multple scans nto a sngle coherent model. Turk and Levoy [28] propose a technque to dentfy and zpper overlappng surfaces among scans. Curless and Levoy [10] present a technque based on ntegratng volumetrc representatons 3D dstance felds of range surfaces. Ther volumetrc method works better than polygon zpperng for producng topologcally sound reconstructons of geometry. The dstance feld s a 3D voxel array that stores the sgned dstances from each voxel to the nearby surface. The metrc of dstances need not be Eucldean dstance, but Frsken and Perry [13] suggest several advantages of usng Eucldean dstance, whch we use n our system. The zero level set of the dstance feld gves the underlyng surface, whch can be extracted usng the marchng cubes [23] algorthm. A common problem wth marchng cubes s the loss of sharp features. Kobbelt et al. [21] recently presented an extenson to marchng cubes for sharp features preservaton. Other approaches have also been proposed [11, 19]. The voxel feld needs to be very dense for adequate samplng and alas-free reconstructon. Computng and storng dstance values for every voxel s very expensve n both tme and space. That problem can be addressed by restrctng dstance evaluaton to a thn shell of voxels around the surface and by usng compresson technques such as runlength encodng of the 3D grd [10]. Frsken and Perry [14] have also successfully appled herarchcal grds. One way to produce dstance felds s by evaluatng the pont-to-trangle dstance from every voxel to nearby areas of the surface. It s also possble to frst measure dstance values wthn a narrow band around the surface [30], then propagate the values to other voxels n the thn shell usng the fast dstance transform [7] or fast marchng [20] method. Researchers have examned dstance feld generaton from unorganzed ponts [17, 3, 8] n whch no assumptons are made about the underlyng geometry. However, snce range mage ponts are sampled contnuously, they already contan mpled surface connectvty, hence these technques are less effcent for range mage reconstructon. In Curless and Levoy s volumetrc method [10], a range surface s frst obtaned from the range mage, and the dstance feld s then bult by computng the lne-of-sght dstance from a voxel to the nearest surface. Each voxel s assgned a confdence as weght. Scans are then ntegrated by weghted averagng of the dstance feld voxels from multple scans. Hlton et al. [16] compute Eucldean dstances based on local reconstructon of surface topology. Wheeler et al. [29] compute sgned dstance to the surface by fndng Fgure 1. (a)-(c) 2D verson of Frsken s technque; (d) Data loss due to projecton Fgure 2. 2D verson of our new approach a consensus of locally coherent observatons of the surface. Most of these exstng technques take range surfaces as nput. However, our hgh resoluton range scans typcally result n 5-15 mllon trangles, makng t very dffcult to process. Frsken and Perry [13] recently presented a method for estmatng Eucldean dstances drectly from 2D range mages. Frst they project range dstances, or lne-of-sght dstances, along a prmary scannng drecton and obtan projected dstances. They resample the projected dstances unformly to obtan a new orthographc range mage. Gven any voxel, t s trval to determne ts projected dstance to a nearby surface, then correct the projected dstance along the surface normal to approxmate the Eucldean dstance. Assumng that the range surface s smooth and contnuous and that the voxel n evaluaton s close to the surface, ths estmated dstance s a good approxmaton of the true Eucldean dstance. In addton, they propose an effcent way to fll holes around range clffs, where the range surface

3 s dscontnuous. Most of these computatons are n the 2D range mage doman, and they use Adaptvely Sampled Dstance Felds (ADFs) [14] to acheve better storage and computatonal effcency. As a result, ther method s very fast and has reasonable memory requrements. Fgure 1(a)-(c) shows a 2D verson to ther technque. Unfortunately, Frsken and Perry s technque s not easly appled to scans of ndoor scenes. Snce such scans typcally span a large range of angles (> 180 ), t s nearly mpossble to decde a prmary scannng drecton wthout dvdng the scan nto several parts, whch ntroduces artfacts. Carelessly choosng such a drecton may cause a loss of data when ponts from separate parts project to the same poston, as shown n Fgure 1(d). In general, projectng dstances along a major drecton bases the results n that drecton. Frsken and Perry s approach also nvolves resamplng of the range data, whch we would lke to avod. Fnally, snce ndoor scene scannng produces many range dscontnutes, sometmes t s better to leave the bg holes unflled. In the next secton we present a new approach that addresses these problems. 3. Algorthms and Implementaton 3.1. Overvew of the Algorthm Gven a voxel P n space, the drecton from the orgn (scanner center) to P defnes the vew drecton v. If H s the ntersecton pont between v and the range surface, the range value at H can easly be determned by nterpolatng between values n the range mage. The sgned dstance between H and P s the lne-of-sght dstance of P to a nearby surface. To estmate the Eucldean dstance, we project the lne-of-sght dstance along the surface normal n at ht pont H. Fgure 2 llustrates a 2D verson of ths approach. If the underlyng surface s smooth and contnuous and P s wthn a small dstance (called the ramp wdth [10]) of the surface, ths estmated dstance s a good approxmaton of the true Eucldean dstance. We then assgn the confdence value of H to P as a weght. Close to range edges where the range surface s dscontnuous, however, the estmated dstance s no longer a good approxmaton of the Eucldean dstance. We address ths problem n Secton 3.7. Lke Frsken and Perry, we estmate dstance values drectly from the 2D range mages. However, we perform all computatons n sphercal coordnates. Snce ths s the orgnal format of these range mages, we do not have to convert range dstances to projected dstances or resample the range data. Ths makes our approach both effcent and more sutable for our ndoor scene scans Range Data In the followng, we use r(θ, φ j ) to represent a range mage. Here θ and φ j are the sphercal rotaton angles. We assume the range mage s regularly sampled, such that the samples at each (θ, φ j ) are algned on a unform 2D lattce. In practce, sample ponts produced by our Delta- Sphere scanner are slghtly msalgned, so we need to frst realgn our range mages onto a unform 2D grd. The vsual effect of ths realgnment s neglgble. After range scannng, we take color mages wth a dgtal camera fxed at the same poston as the range scanner. We then algn color mages to range scans usng a pre-computed camera calbraton. As a result, each range mage pont s assgned a color value. Later n the ppelne, each sample pont wll also be assgned surface normal and confdence values. These values are stored together wth range values n the range mage. Gven any drecton (θ, φ), we can ndex nto the range mage and obtan a range value r along ths drecton usng blnear nterpolaton. Ths s the dstance to an magnary ht pont along the gven drecton. A negatve value of r means no pont s ht n ths drecton. Smlarly, we can obtan the color s, surface normal n and confdence c wth blnear nterpolaton. Range mages of ndoor scenes are usually very nosy due to errors n the range values 1. In a preprocess, we apply ansotropc dffuson [6] on range mages to reduce range noses whle preservng sharp features. Fgure 3(a) shows the surface of a scanned char before applyng ansotropc dffuson. Fgure 3(b) shows the surface after eght teratons of ansotropc dffuson. The result s much cleaner than the orgnal data. Notce that a few sharp features are blurred. In Secton 4 we wll dscuss the tradeoffs n applyng ansotropc dffuson Computng Normal and Confdence Values Because surface normals at range mage ponts are needed to correct lne-of-sght dstances, we precompute and store these values n the range mage. Surface normals can be computed drectly from the range mage. Gven a set of sample ponts P = [x, y, z] that compose a range surface, each surface normal s computed as n = P θ P φ ( s cross product). We see from the followng steps that we only need the dot product of surface normal wth vew drecton (v n), so we need not store the surface normal n tself. Here the vew drecton s the vector from a sample pont to the orgn: v = P = [x, y, z]. By substtutng 1 The range accuracy of the DeltaSphere 3000 laser scanner s ±0.3 n. at 40 ft. dstance, and the angular accuracy s ±0.015 degree.

4 mage. The way we assgn confdence values s smlar to [2]. Three factors can decrease our confdence n a range pont: 1. Dstance weght the pont s dstant from the scanner. 2. Vew weght the pont has a large gazng angle, as determned by the dot product of vew drecton wth surface normal (v n). 3. Boundary falloff the pont s a small number of hops away from a range edge. To produce smooth transtons between merged surfaces, we use a quadratc boundary falloff (correspondng to α b = 2 n [2]). (a) 3.4. Estmatng 3D Eucldean Dstance In the followng we defne functons that convert between Cartesan coordnates (x, y, z) and sphercal coordnates (r, θ, φ) as: r = x 2 + y 2 + z 2 θ = arctan 2(y, x) φ = arcsn(z/r) (a) x = r cos φ cos θ y = r cos φ sn θ z = r sn φ (b) (1) As descrbed n Secton 3.1, the algorthm for estmatng 3D Eucldean dstance s as follows: 1. Traverse dstance feld voxels n order. For each voxel, convert ts poston (x, y, z) nto sphercal coordnates (r, θ, φ) wth Equaton 1(a), where (θ, φ) s the vew drecton v. (b) Fgure 3. Char model before and after eght teratons of ansotropc dffuson Equaton 1(b), we get r cos φ (v n) = cos 2 φ(r 2 + rφ 2) + r2 θ where r θ = r θ and r φ = r φ, whch can be estmated from the range mage usng central dfferencng: r θ (θ, φ j ) = r(θ+1,φj) r(θ 1,φj) 2 θ r φ (θ, φ j ) = r(θ,φj+1) r(θ,φj 1) 2 φ where θ and φ are the angular dstances between ponts n the range mage. For weghted averagng of range values, we also need to precompute confdence values and store them n the range 2. Use (θ, φ) to ndex nto the range mage and obtan an nterpolated range value r. If r s negatve, skp processng the current voxel. Smlarly, obtan the nterpolated normal n and confdence value c. 3. Estmate 3D Eucldean dstance by projectng lne-ofsght dstance (r r ) onto normal n, as shown n Fgure 2. Snce the lne-of-sght dstance s along the vew drecton v, the corrected dstance value d = (r r )(v n ). 4. If the absolute value of d s wthn a user-defned ramp wdth [10], update the current voxel wth d and the confdence value c ; f not, the voxel s outsde a thn shell (defned by the ramp wdth) of the underlyng surface, hence wll not be updated. 5. Repeat steps 1 through 4 untl all voxels are traversed. If the above algorthm were run on the entre dstance feld, a substantal amount of computaton would be wasted on voxels that are dstant from the underlyng surface. To reduce the amount of unneeded computaton, we frst construct a 3D bt array correspondng to the dstance feld.

5 Each bt ndcates whether a gven voxel s close to the surface and hence computaton should proceed at step 1. To buld the bt array, we go through every range mage pont, fnd ts poston n space, and mark a surroundng volume centered at the pont as true n the bt array. The volume marked s a cube of sde length at least twce the ramp wdth. If the range ponts are dense enough, the bt array s a good estmaton of a thn shell around the surface. Whle our dstance feld structure s accessed sequentally and can therefore be run-length encoded [10] effectvely, the bt array s not run-length encoded. Ths allows us to mantan access speed but can be a problem for hgh resoluton dstance felds. To save storage and setup tme, we buld the bt array at a lower resoluton than the actual dstance feld. For example, gven a dstance feld of sze , the twcedownsampled bt array s of sze ; each bt represents a 4 3 volume of voxels and s set to true f any of the 4 3 voxels s close to the surface Effcent Multvew Integraton In ths secton we descrbe an extenson to the above algorthm that allows for effcent multvew ntegraton. Suppose we have N range scans and ther correspondng regstraton matrces M, whch we precompute usng Stanford Unversty s Scanalyze software package [1]. Applyng regstraton matrx M to scan places the scan n a common (target) coordnate system. Smlarly, applyng the nverse matrx M 1 on a pont n the target coordnate system transforms t back nto scan s coordnate system. The sze of the dstance feld we wll use s determned by both the boundng box of all scans n the target coordnate system and a user-defned voxel sze. We apply the transformaton to every dstance feld voxel p = [x, y, z] T. The transformed voxel, p = M 1 1 p, s then used at step 1 for M 1 dstance estmaton. When updatng voxels at step 4, we do an ncremental weghted average as descrbed n [10]. Unfortunately, transformng every dstance feld voxel nto each scan s coordnate system s an expensve operaton. We can mprove the speed, however, by observng that voxel traversal s ncremental and voxel transformaton s a lnear operaton. Suppose m 1, m2, m3 are the frst three column vectors of M 1 such that M 1 = [m 1, m2, m3, t]; applyng the transformaton on voxel p = [x, y, z] T produces p = [x, y, z ] T = M 1 p; consder the next voxel traversed n x drecton p x+ = [x+ x, y, z] T, applyng the transformaton produces p x+ = M 1 p x+ = p +m 1 x, whch s smply the prevous result p plus an ncrement m 1 x. Here ( x, y, z) s the user-defned voxel sze. Hence we can pre-compute and store the three ncrements m 1 x, m2 y and m3 z n the x, y, and z drectons, respectvely. Now the only transformaton needed s on the startng voxel; for each subsequent voxel traversed n the x, y, or z drecton, we smply add the correspondng ncrement to get transformed voxel. After all scans have been ncorporated, we run the marchng cubes algorthm on the target dstance feld to reconstruct the merged 3D model Weghted Colors and Texture Coordnates Wth lttle modfcaton, we can extend the above algorthm to assgn weghted colors to vertces n the reconstructed 3D model. The steps are: 1. Startng from the frst scan, convert each vertex of the reconstructed 3D model to (r, θ, φ) coordnates. 2. Use the computed (θ, φ) to ndex nto the range mage and get the nterpolated range value r, confdence c, and color s. 3. If r r s wthn an acceptable error metrc, update the vertex color wth s and c. Otherwse, the vertex s probably occluded n ths scan, hence ts color s not updated. 4. Repeat the above steps for all scans. An alternatve approach would be to use the color data from each range mage as a texture map and apply color to the reconstructed model at runtme usng weghted texture mappng. Ths would requre that we assgn a set of texture coordnates to each vertex for every texture map. The confdence values would be stored along wth the color data so that the texture maps could be weghted approprately. We could also use vew-dependent projectve texture mappng [12], where the vertex color seen by the vewer s dynamcally blended from all the texture maps based on the vewer poston and the vew drecton Dstance Correcton around Range Edges Range edges are places where range values are dscontnuous (.e., the scanned surface has a large gazng angle). Holes may be left n the range surface due to these dscontnutes. Snce our algorthm assumes a smooth and contnuous range surface formed by the range ponts, dstance estmaton s subject to error around edges. As demonstrated by Fgure 4, the computed dstance at voxel P s no longer a good approxmaton of the true dstance. Frsken and Perry [13] propose a method for fllng holes around range edges. In our case, however, fllng holes s not always a good dea. Range edges are common n ndoor scene scannng, so sometmes fllng holes may produce undesrable results. For example, the edge of a char should not be connected wth the wall behnd t. Therefore, rather than fllng such holes, we rely on scans wth better vews to fll n the

6 Fgure 4. Dstance estmaton s problematc at range dscontnutes 4. Results and Dscusson We tested our algorthm on sx range mages acqured nsde the Montcello lbrary room. For each scan, we apply eght teratons of ansotropc dffuson durng preprocessng. We expermented wth several dfferent voxel szes. Smaller voxel szes result n hgher resoluton dstance felds, hence the reconstructed models are more fnely tesselated. Ths also ncreases executon tme, however. We use ramp wdths of 0.6 to 0.8 nches to defne the thn shell around the surface. Performance s measured on a 1.53GHz AMD Athlon processor wth 1GB of memory. Fgure 6 shows one of the sx range scans we used n reconstructon. Fgure 7 shows the model as reconstructed wth a voxel sze of 0.2 nches and a dstance feld of sze Fgure 8 shows the same model wth each vertex assgned a weghted color Performance Fgure 5. Before and after dstance correcton at range dscontnutes. Notce the artfacts n the top mage mssng geometry. Small dscontnutes, on the other hand, should be connected, because ndoor scene scannng often leaves many small holes, whch are not always possble to fx by scannng from more vewponts. To address ths problem, we use a hybrd scheme whch computes the true dstance values around edges. The followng s performed for each scan after the dstance estmaton s completed: 1) Mark range ponts that are close to edges. Recall we already have edge hops when computng the confdence values, and they can be drectly used here. 2) Form trangles by connectng neghbor ponts around the marked ponts. Remove those trangles that have sde lengths over a pre-defned threshold these are the holes we wll not fll. 3) In the dstance feld, wthn a thn shell of the remanng trangles, compute pont-totrangle dstances as true dstance values, and use them to overwrte the estmated dstance values. The overhead due to ths computaton s neglgble snce only a small porton of the trangles s used. Fgure 5 llustrates the reconstructed surface before and after dstance correcton. For performance analyss we consder two parts: Ansotropc dffuson, normal, and confdence values are computed n the 2D range mage doman. For these calculatons, the executon tme grows lnearly wth the range mage resoluton. Gven a range mage resoluton of n 2, we have executon tme t 1 = O(n 2 ). In dstance estmaton, wth the the auxlary bt array, expensve computatons are restrcted to voxels wthn the thn shell around the target surface. If we assume the surface s a 2D manfold, the processng tme s roughly quadratc wth respect to the wdth of dstance feld. Notce that the wdth of a dstance feld s nversely proportonal to the voxel sze. The cost of buldng the bt array depends on the number of range mage ponts and the number of voxels wthn the ramp wdth. Despte some overhead, the bt array helps ncrease overall executon speed by a factor of two to three. Gven a dstance feld resoluton of N 3, we have executon tme t 2 = O(N 2 ) and a total executon tme T = t 1 + t 2 = O(n 2 + N 2 ). Table 1 shows the executon tme for computatons performed n the range mage doman. On average t takes 6 to 8 seconds for a range mage. Table 2 shows the executon tme for calculatng the dstance felds when mergng all sx scans. It takes rougly 10 to 15 seconds to calculate a dstance feld usng a sngle scan. Curless and Levoy [10] report a reconstructon tme of 197 mnutes wth hole fllng for 71 range mages of the dragon model on a volume of sze on a 250MHz MIPS R4400 processor. Ths equates to roughly 200 seconds of processng for one scan on a dstance feld. Frsken and Perry report two to sx seconds of reconstructon tme for several models on dstance felds; our approach s a lttle slower but s more general. Wth respect to speed and generalty, our method compares favorably to exstng technques.

7 Image Sze n, c Dffuson Total (t 1 ) s 13.07s 13.73s s 13.17s 13.81s s 10.34s 10.93s s 18.84s 19.63s s 10.01s 10.58s s 10.97s 11.58s Table 1. Executon tmes for processng n the range mage doman Voxel D.F. Res. Bt Array Dst. Est. 0.8 n s 20.20s 0.5 n s 54.53s 0.3 n s s 0.2 n s s Table 2. Executon tmes for producng dstance felds 4.2. Storage Requrements The storage requrements of our algorthm are reasonable. Each range mage pont s composed of floatng-pont range, normal, and confdence values (recall from Secton 3.3 that we only store a scalar value for normal) and one nteger color value, for a total of 16 bytes. Each dstance feld voxel stores one floatng-pont dstance value and one float weght. The dstance feld s run-length encoded. The auxlary bt array s twce downsampled from the dstance feld s resoluton, so t requres only 2MB memory for a dstance feld Accuracy It s partcularly challengng to analyze the accuracy of our model. Snce we approxmate Eucldean dstance, t s theoretcally more accurate than lne-of-sght dstance or other dstance metrcs. The accuracy may be affected durng nose reducton. As mentoned n Secton 3, ansotropc dffuson effectvely reduces range nose and mproves overall accuracy of our scans. However, due to the smoothng process, detals are lost n hgh curvature regons. Ths can be seen, for example, n the bookshelf n Fgure 3. Proper choce of dffuson parameters, ncludng the edge preservaton functon and the number of teratons, may help balance nose reducton wth detal preservaton. Alternatvely, we could consder usng more aggressvely smoothed data for normal estmaton and use slghtly smoothed data for dstance estmaton. 5 Conclusons and Future Work In ths paper we present an effcent approach to computng and ntegratng dstance felds drectly from multple range mages. Our approach s based on the estmaton of Eucldean dstance by correctng the lne-of-sght dstance. Normal and confdence values are also effcently computed drectly from range mages. We also propose a hybrd approach that corrects dstance values around range dscontnutes. Snce most computatons are n the 2D, our approach s effcent n both computaton and storage. We hope to mplement a herarchcal structure for our dstance feld such as the adaptvely sampled dstance feld to further reduce computaton and storage requrements. We also seek to apply better sosurface extracton algorthms to reduce the sharp feature loss caused by marchng cubes. Fnally, we would lke to ntegrate vew-dependent projectve texture mappng nto our system to mprove nteractve vewng of these models. References [1] Scanalyze software package. stanford.edu/software/scanalyze/. [2] Vrppack user s gude. stanford.edu/software/vrp/gude/. [3] C. Bajaj, F. Bernardn, and G. Xu. Automatc reconstructon of surfaces and scalar felds from 3d scans. In Proc. of SIGGRAPH 1995, [4] R. Bergevn, M. Soucy, H. Gagnon, and D. Laurendeau. Towards a general mult-vew regstraton technque. IEEE Trans. on PAMI, 18(8): , [5] P. Besl and N. McKay. A method for regstraton of 3d shapes. IEEE Trans. on PAMI, 14(2): , February [6] M. Black, G. Sapro, D. Marmont, and D. Heeger. Robust ansotropc dffuson. IEEE Trans. on Image Processng, 7(3): , [7] D. E. Breen, S. Mauch, and R. T. Whtaker. 3d scan converson of csg models nto dstance volumes. In Proc. of the IEEE VolVs 1998, pages 7 14, [8] J. C. Carr, R. K. Beatson, J. B. Cherre, T. J. Mtchell, W. R. Frght, B. C. McCallum, and T. R. Evans. Reconstructon and representaton of 3d objects wth radal bass functons. In Proc. of SIGGRAPH 2001, [9] C. Chen and G. Medon. Object modelng by regstraton of multple range mages. In Proc. IEEE Conf. on Robotcs and Automaton, [10] B. Curless and M. Levoy. A volumetrc method for buldng complex models from range mages. In Proc. of SIGGRAPH 96, [11] P. de Brun, F. Vos, S. Frsken-Gbson, F. Post, and A. Vossepoel. Improvng trangle mesh qualty wth surfacenets. In MICCAI, pages , [12] P. Debevec, Y. Yu, and G. Boshokov. Effcent vewdependent mage-based renderng wth projectve texturemappng. In 9th Eurographcs Renderng Workshop, 1998.

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