A Range Image Refnement Technque for Mult-vew 3D Model Reconstructon Soon-Yong Park and Mural Subbarao Electrcal and Computer Engneerng State Unversty of New York at Stony Brook, USA E-mal: parksy@ece.sunysb.edu Abstract Ths paper presents a range mage refnement technque for generatng accurate 3D computer models of real objects. Range mages obtaned from a stereo-vson system typcally experence geometrc dstortons on reconstructed 3D surfaces due to the nherent stereo matchng problems such as occlusons or msmatchngs. Ths paper ntroduces a range mage refnement technque to correct such erroneous ranges by employng eppolar geometry of a multvew modelng system and the vsual hull of an object. After regsterng multple range mages nto a common coordnate system, we frst determne f a 3D pont n a range mage s erroneous, by measurng regstraton of the pont wth ts correspondences n other range mages. The correspondences are determned on 3D contours whch are nverseprojectons of eppolar lnes n other 2D slhouette mages. Then the range of the pont s refned onto the object s surface, f t s erroneous. We employ two technques to search the correspondences fast. In case that there s no correspondence for an erroneous pont, we refne the pont onto the vsual hull of the object. We show that refned range mages yeld better geometrc structures n reconstructed 3D models. 1. Introducton Generatng 3D computer models of real objects s of much nterest n Computer Vson and Computer Graphcs. One of the recent research nterests s generatng a complete and closed 3D model by mergng multple mages of an object. One common technque s employng a sngle or multple rangng sensors to acqure and merge mult-vew range mages [1, 4]. There s a varety of rangng technques to obtan range mages of a scene of nterest. In order to generate accurate 3D models, most nvestgatons employ actve rangng technques such as laser rangng, structured lght pattern, space-tme coded pattern, etc. In contrast, passve technques work only on naturally formed mages produced by reflected lght from the object. A common technque s Stereo Image Analyss (SIA). However, due to nherent stereo problems, relatvely fewer number of researchers have employed stereoscopc magng sensors to generate complete 3D models [3, 5, 14]. Range mages obtaned from a stereo rangng sensor typcally experence erroneous ponts on reconstructed 3D partal surfaces. In order to obtan accurate range mages usng SIA, some researchers project random dot patterns onto object s surfaces to enhance contrast on them [9]. Others employ mult-baselne or mult-resoluton [14] technque to reduce the number of stereo msmatchngs. Even usng these technques, obtanng accurate range mages s stll dffcult n some portons of the object s surfaces. One smple approach of removng erroneous ponts s employng a lnear low-pass flter or a non-lnear flter [4, 5, 12]. However, t s not easy for these approaches to remove some erroneous regons where the errors are domnant wthn a flterng mask. Another smple approach s obtanng a large number of range mages to average out the errors [4, 10, 13]. However, acqurng multple range mages usng SIA s a computatonally expensve task. Another approach s employng a range mage regstraton technque, such as Iteratve Closest Pont (ICP), to reduce noses n range mages [11]. Some researchers try to remove these errors usng the vsual hull of the object [7, 8]. However, ths technque can remove only those outsde the vsual hull but not nsde. In ths paper, we present a range mage refnement technque to reconstruct accurate 3D models of real objects. After regstraton of mult-vew range mages, we frst determne f a 3D pont n a range mage s erroneous, by measurng the regstraton of the pont wth ts conjugates n other range mages. The conjugate ponts are searched on 3D contours whch are nverse-projectons of eppolar lnes n other 2D slhouette mages. When the 3D pont s decded as erroneous or t s outsde the vsual hull of the object, we refne the range of the pont to the mean of the conjugates or
onto the surface of the vsual hull. In order to make searchng and refnement fast, we employ two technques, pontto-projecton search and orderng constrant. Expermental results on a ground truth object and real objects show that 3D models from refned range mages yeld better geometrc structures. 2. Range Error due to Stereo Msmatchng Range mages obtaned by a stereo camera could experence some erroneous ponts on ther surfaces due to stereo msmatchng. An example of erroneous range surface s shown n Fgure 1. Fgure 1 shows a pcture of an object and Fgure 1 shows a reconstructed mesh model of the front partal surface of the object, whch s seen from the top. As n the fgure, stereo msmatchng errors appear as sharp peaks on the partal shape. Shape dstortons due to stereo matchng errors Fgure 2. Error dstrbuton n 3D space. Top: A range mage wth erroneous surfaces. Bottom: The same mage wthout error. Most errors are dstrbuted along the Z c axs of the camera coordnate system 3. Refnement Methodology 3.1. Erroneous 3D ponts Z X Y Camera coordnate system Fgure 1. An example of stereo msmatchng An object A partal shape of the object shows some erroneous surfaces When we compute the depth of a 3D pont P, whch maps to an mage pont p, the 3D pont s on the ray of p. Suppose an object s placed consderably far from the camera, then the ray s almost parallel to the Z c axs (vewng drecton) of a camera coordnate system. Therefore, we can estmate that the coordnates of msmatchng errors, between a correct ponts set {P} and an erroneous ponts set {P }, dstrbute mostly on the Z c axs. An example of error dstrbuton between two partal surfaces n Fgure 2 s plotted n Fgure 2. In Fgure 2, the top range mage has some errors, and the bottom one has neglgble errors n contrast. As shown n Fgure 2, most errors are dstrbuted along the Z c axs, whch concdes wth the vewng vector of the range mage. As presented n the prevous secton, stereo msmatchng errors n a range mage occur mostly on such object s surfaces whose normal vectors are n a hgh angle wth respect to the vewng drecton of the mage. However, they may be seen better from another vewng drecton because the object s scanned from multple drectons. Therefore, t s possble to refne an erroneous 3D pont, f we know ts correspondences on other vew s range mages. Suppose there s a 3D pont P on an object s surface and two dfferent vews V and V j can see the pont. Then we can wrte two representatons of the pont as P and P j n the correspondng range mages R and R j, based on the vew pont of V. When the two ponts are regstered to a common coordnate system, ther coordnates should concde n an deal case. However, n a real case, they may not concde because of systematc or calbraton error of the vson system. If the two ponts do not concde n 3D space, a problem arses that how we can determne the correspondences between multple range mages. We assume that all correspondences occur on the ray of a 3D pont. Consderng R as a reference range mage and P as a reference pont, we fnd all ntersectons n the other range mages on the ray of the reference pont. Suppose there are two range mages R and R j as shown
n Fgure 3 and they are regstered to a common vew V. Assumng regstraton of the range mages s very accurate, we know that P s very close to another pont P j, whch s on the lne of sght of the pont P from V. If the orgn of the vew V s O, the correspondence of P on R j s on the ray = αp,α R. Therefore, we wrte P j = R j (1) and P j =0. (2) Therefore, the two matchng ponts P and P j n Fgure 3 are very close and there s only a small regstraton error ɛ between them such that ɛ = P P j 0. (3) et us consder another 3D pont P as shown n Fgure 3. Ths pont s consdered as an erroneous pont n the range mage R due to stereo msmatchng, whle ts conjugate P j on the R j s a correct surface pont. Then there should be sgnfcant regstraton error between two ponts as n the fgure, whch s larger than a threshold d T such that R j P j ɛ = P P j >d T. (4) R R j d T P j R In addton to the error measure, we also use the vsual hull of an object to decde f a 3D pont s erroneous. When the 3D pont s outsde the vsual hull of the object, we consder t s erroneous. Ths technque s the same as what the shape-from-slhouettes technque does. In summary, let VH(O) be the vsual hull of an object O, then the 3D pont P s also { true, f P erroneous / VH(O) not determned, f P VH(O). (6) We frst check f the pont s outsde the vsual hull. If the pont s outsde, we consder t s erroneous and refne t. In case that the pont s nsde the vsual hull, we use Equaton (5) to check f t s erroneous. 3.2. Correspondence on Eppolar Contour In order to refne a reference 3D pont n a reference range mage, we search all ts correspondences n the other range mages. Suppose agan there s a reference 3D pont P (or P )onr as shown n Fgure 4. The perspectve projecton of the 3D pont to the mage plane of V sa2d mage pont p. et us consder a problem to fnd the correspondences of P n other range mages, R j and R k n ths example. We employ the eppolar geometry of our system. If we know the fundamental matrx between V and V j,we can restrct searchng of ntersectons on the eppolar lne u j whch s u j = F j p, (7) where F j s the fundamental matrx between V and V j. O j P O O j P' O O k p k,e U k Object p k,s u k I k P U j Fgure 3. A typcal error on a partal shape due to stereo msmatchng. Two matchng 3D ponts are very close P s far from P j p p j,e u j p j,s If multple range mages are ntersected by the ray, t s possble to determne several correspondences n the range mages. Suppose there are K range mages R j for j =0,,K 1, whch are ntersected by. Then we determne f the pont P s O Fgure 4. Eppolar geometry of a mult-vew system I j O j { true, f P erroneous P k >d T, k, 0 k K 1 false, f P P k <d T, k, 0 k K 1. On the mage plane I j n the fgure, the eppolar lne u j (5) overlaps wth a slhouette of the object, whch s a bnary
representaton of the perspectve projecton of the object to V j. On the object s slhouette, we determne a startng pont p j,s and an endng pont p j,e on the eppolar lne. From the startng pont to the endng pont, we then nverse project all 2D mage ponts on the eppolar lne u j to correspondng 3D ponts from P j,s to P j,e on the surface of R j. All the 3D ponts on the surface then form a 3D contour {U j } Along the 3D contour {U j }, we search all ntersectons by the ray, whch satsfy Equaton (1) and (2). Then from these canddates we pck one of them as P j,c, the correspondence of the pont P. The ntersecton pont P j,c satsfes followng equatons. {U j } = {P P T j M 1 j {p j,s p j,e }} (8) P j,c P =0 and P j,c {U j }. (9), where M j s the perspectve transform matrx of the jth vew, and T j s the transformaton of coordnate system from the jth vew to the th vew. Because there could be several ntersectons on the 3D contour by the ray, we fnd all of them by checkng sgns of cross products of the vector of and other vectors to ponts on {U j }. More detals are n the next secton. 4. Range Image Refnement 4.1. Correspondence Search In order to search all ntersectons on the 3D contour P j,s P j,e,wefrst fnd all zero crossngs of the cross product P j P, where P j P j,s P j,e. Because the Y s axs of the turntable s coordnate system s almost parallel to the Y c axs of the camera coordnate system, we check sgn of the Y component of the product to search zero crossngs. If the sgn changes, we consder there happens a zero crossng. Fgure 5 shows an example of zero crossngs on a 3D contour. Of course there could be multple (N c ) ntersectons on the contour, for example n the fgure from P j,c,0 to P j,c,nc 1. However, from the pont of th vew pont, only one ntersecton s vsble, whch s the closest to P. In ths fgure therefore, we take P j,c,0 as the correspondence P j,c on jth vew s contour {U j }. To avod false correspondence, we test and remove some ntersectons whose normal vectors are n a hgh angle wth respect to ther orgnal vew ponts, whch are hdden from the reference vew pont, or whch are outsde the vsual hull of the object. O P j,s P P j,c,0 P j,h P j,c,1 U j P j,c,2 Oj P j,e P j,c,nc-1 Fgure 5. Intersectons on a 3D contour by a reference vector 4.2. Fast Searchng Technque In ths secton, we consder computaton tme of ntersecton searchng and ntroduce fast searchng technques. Suppose there are Q ponts on a 3D contour and computaton for ntersecton searchng takes O(Q) operatons. If there are K overlappng ranges from a reference pont, we need to search ntersectons on K contours, thus O(KQ) computatons. In order to refne a range mage, whose object area s equvalent to M X M Y mage sze, then we need to compute O(KQM X M Y ) operatons. To refne all N range mages, we fnally need O(NKQM X M Y ) operatons. These computatons are computatonally very expensve. In a typcal Pentum-4 1.8GHz computer, t takes more than 30 mnutes to refne 16 range mages wth 320 240 resoluton. In order to make the computaton tme fast, we employ two technques based on pont-to-projecton range searchng and orderng constrant. 4.2.1 Pont-to-Projecton Searchng Instead of searchng ntersectons on a 3D contour, we use a pont-to-projecton technque to solve the correspondence problem as shown n Fgure 6. To fnd the jthe vew s correspondence from a reference pont P, we project t to the mage plane of V j to a 2D pont p j, nversely project the 2D pont to jth range mage R j to get P j. In other words, P j = T j R j(m j T j P ), (10) where R j (p) s the jth vew s range of a 2D pont p. By searchng all nverse-projecton ponts P k, k = 0 K 1 n other range mages, we use Equaton (5) to determne f the reference pont s erroneous or not. Inverseprojecton s not exactly an ntersecton pont by the ray. However, f two range surfaces are very closely regstered, we can approxmate the pont as an ntersecton. Once the
R j Object P P j R P U j O j O Fgure 6. Pont-to-Projecton searchng I p (x,y ) p (x +1,y ) ; p' p j,s p j,s p j (x j,y j ) j,e p' j,e u j I j O reference pont s determned as erroneous, we then fnd the real ntersectons on the ray of usng the technque presented n Secton 3.2. Usng ths searchng technque, we refne all range mages n about 5 or 10 mnutes dependng on objects. 4.2.2 Orderng Constrant The next fast searchng technque s usng orderng constrant n a par of mult-vew mages. Once a reference pont P s determned as erroneous, we fnd ts conjugates on a 3D contour {U j }, whch s the nverse projecton of a 2D segment on u j. In Secton 3.2, we search the conjugates along u j startng from p j,s to p j,e. However, nstead of all ponts on the eppolar lne we use the orderng constrant between a par of mult-vew mages. et us consder an example n Fgure 7. As n the fgure, suppose a 2D reference pxel p (x,y ) on I matches wth ts conjugate p j (x j,y j ) on I j, where (x,y ) and (x j,y j ) are ther coordnates on I and I j respectvely. In other words, ther 3D conjugates P and P j match each other. Now consder another reference pont p (x +1,y ) on I, whch s the next pxel of p (x,y ). Then we know that ts matchng pxel on I j s always on the rght-sde of p j (x j,y j ) accordng to the orderng constrant. Therefore we search an ntersecton of the ray on a 3D contour along {U j }, whch s the nverse projecton of a lne segment p j,s p j,e rather than p j,sp j,e. Based on the constrant, the x coordnate of the startng pont could be the same wth that of p j (x j,y j ), that s x j. However, n order to fnd a zero crossng on the contour, we gve a margn n the left drecton. As a result, we set the range of ntersecton searchng along the x axs from x j δ to x j + δ +.By assumng there s no sudden depth change on the object s surface, we set δ to 2 and δ + to 10 pxels. 4.3. Refnement onto Vsual Hull Some erroneous range ponts are outsde the vsual hull (VH(O)) of an object. If a 3D pont s outsde of VH(O), Fgure 7. Reducng search range usng orderng constrant we have to refne t to a correct 3D pont on the object s surface. If there are enough matchng ponts n other range mages, we can refne t usng the presented technques. However, a problem arses f there s not enough or no matchng pont. In ths case, we decde to refne t onto the vsual hull of the object. As shown n Fgure 5, a vsual hull pont P j,h can be found by the ntersecton of two rays, and P j,s O j. Intersecton of two non-ntersectng rays can be solved usng a smple lnear equaton. Because there are multple vews, we acqure multple vsual hull ponts P k,h, 0 k K 1. Then the problem s now decdng whch one s on VH(O) of the object. et us consder an example n Fgure 8. On the ray from a vew orgn O, we acqure multple vsual hull ponts from P 0,h to P 3,h. To determne whch one s on VH(O), we smply project every pont to mage planes of all multple vews and check f t s on VH(O). In ths fgure, P 3,h s on the vsual hull. Then we move P to P 3,h. On a lne segment p j,s p j,e on an eppolar lne u j n Fgure 4, we get two vsual hull ponts usng two rays P j,s O j and P j,e O j. However, we take only one of them from P j,s O j, because the pont P can see only the startng pont P j,s. After acqurng all vsual hull ponts from K overlappng vews, we sort them accordng to dstance from the reference pont and check f they are nsde VH(O). 4.4. Refnement Steps Range mage refnement steps are as follows. For every 3D pont n a range mage, we frst determne f t s erroneous or not usng two methods. One s to determne f t s outsde of the vsual hull and the other s to determne f t s erroneous usng the pont-to-projecton searchng technque presented n Secton 4.2.1. If the pont s erroneous, we O j
Vsual Hull to R + N (K = N/2). We set the error threshold d T to 3 4 mm and the varance threshold σt 2 to 2 mm. O 0 Object 5.2. Error Analyss O 1 Cones of multple vews O 2 P 1,h O 3 Fgure 8. Refnement of P onto a vsual hull pont P 3,h search all ntersectons on surfaces of other range mages. If there are at least two ntersectons P j,c, 0 j<k 1, we change the coordnates of the pont to the mean P m of the all ntersectons f σp 2 <σt 2. Here P m and σp 2 are mean and varance of the ponts j P j,c and σ2 T s a threshold for the varance. When there are not enough ntersectons or σ p 2 >σ2 T, but f the reference pont s outsde vsual hull, we change the coordnates of P to P j,h as presented n Secton 4.3. In ths case P j,h should be also nsde the vsual hull of the object. 5. Expermental Results 5.1. Range Image Acquston P 0,h P O P 3,h We generate 3D models of a ground truth object to analyze and compare reconstructon errors of before and after refnement. Fgure 9 shows a pcture of a cylndrcal object for error analyss. Fgure 9 s the dmenson of the object. We acqure range mages of the object from 8 or 16 vew ponts and regster and ntegrate them to a 3D mesh model. Then we measure dmensonal and regstraton errors between the 3D model and a ground truth model. To generate the ground truth model, we reconstruct a 3D model of dense pont clouds. We synthesze the object s surface for every 1 mm 2 to produce about 50000 ponts to represent ts outer surface. R = 37.59 mm H = 105.7 mm Fgure 9. Cylnder object for error analyss Pcture of the object Dmenson of the object To generate mult-vew range mages, we use a stereobased range mage acquston system, whch conssts of a stereo camera, a turntable stage, and a personal computer. A range mage s obtaned from a par of stereo mages. In order to fnd correspondence n the mage par, we use a multresoluton stereo matchng technque [2, 8]. The stereo par s resampled to form a Gaussan mage pyramd and a smple SSD-based (Sum of Squared Dfference) correlaton technque s performed at each pyramd level. Mult-vew range mages are obtaned by rotatng the turntable wth a fxed rotaton angle accordng to number of vews. We acqure range mages of an object from 8 or 16 (N =8, 16) dfferent vew drectons. We use a Pentum-4 1.8GHz personal computer for our vson system. A range mage resoluton for each vew s 320 240 n row and column drectons. For every range mage R,werefne t pxel by pxel (or pont-by-pont). In order to refne an erroneous pont, we search ts correspondences n range mages from R N 4 The two 3D models, the reconstructed model and the ground truth model, are regstered nto a common coordnate system usng ICP regstraton technque. After regstraton, we measure mean and varance of the radus of the reconstructed model. Table 1 shows results of mean and varance of the radus of the object. We compare errors of two 3D models whch are generated by mergng orgnal range mages and refned range mages. We show the errors when the models are merged from 8 vews and 16 vews. To show the results wth dfferent sze of voxel, we reconstruct dfferent 3D models by changng voxel sze of 2 mm, 2.5 mm, and 3 mm. We also measure RMS and maxmum errors of closest dstance between a 3D pont on the reconstructed model and the ground truth model. Table 2 shows results of these errors. As shown n both tables, 3D models generated from refned range mages have better geometrc structures than those from orgnals.
Table 1. Mean (m) and varance (v) error of the radus of Cylnder object n mm Orgnal Range Refned Range N voxel 2.0 2.5 3.0 2.0 2.5 3.0 8 m 36.32 36.32 36.33 36.35 36.35 36.36 v 0.416 0.438 0.40 0.316 0.307 0.313 16 m 36.40 36.39 36.41 36.41 36.41 36.41 v 0.280 0.296 0.317 0.232 0.253 0.252 Comparson of two 3D models s shown n Fgure 12 and. In ths fgure, the sze of a voxel grd s 2.5 mm. The shaded surface models show the back surface of the object. The refned 3D model n Fgure 12 shows better surface structure than the orgnal n Fgure 12. Fgure 13 shows reconstructon results of another object Soccerball. Fgure 13 s the 3D model from orgnal range mages, and Fgure 13(c) s from refned mages. The fgure also shows that refned range mages yeld better surface structure. Table 2. RMS and MAX error of the radus of Cylnder object n mm Orgnal Range Refned Range N voxel 2.0 2.5 3.0 2.0 2.5 3.0 8 rms 1.33 1.33 1.32 1.27 1.27 1.26 max 3.56 3.60 3.37 3.58 3.14 3.32 16 rms 1.21 1.22 1.21 1.19 1.20 1.20 max 2.63 2.54 2.60 2.50 2.25 2.79 500 500 480 480 5.3. Real Objects The proposed technque s also tested on real objects. Fgure 10 shows an example of a sngle lne refnement of Potatohead object shown n Fgure 1. Fgure 10 shows the orgnal front vew (V 0 ) of the object and a 3D contour whch corresponds to a sngle horzontal lne on ts 2D plane. In Fgure 10, we plot all correspondng profles from V N N to V 4 + N on the XZ plane of the camera 4 coordnates. On the plot of V 0, there are some erroneous ponts as shown the contour n Fgure 10. A refned range mage and all profles for the same horzontal scan lnes are shown n Fgure 10 and (d). Some of orgnal and refned range mages of the object are shown n Fgure 11 and (c). Fgure 11 shows there are some artfacts n range mages due to stereo msmatchng. Some brght and dark portons on the object area represent shape dstortons on surface. The refned range mages n Fgure 11(c) show that ther surface structures are refned compared to ther orgnals. Processng tme of range refnement and number of range pxels on all 16 mages are shown n Table 3. Table 3. Processng tme of refnng range mages Object Potatohead Soccerball Number of vertces 264608 156662 Refnement tme (sec) 510.0 122.0 Z axs 460 440 vew12 vew13 420 vew14 vew15 400 vew0 vew1 380 vew2 vew3 vew4 360 0 10 20 30 40 50 60 70 80 90 X axs (c) Z axs 460 440 vew12 vew13 420 vew14 vew15 400 vew0 vew1 380 vew2 vew3 vew4 360 0 10 20 30 40 50 60 70 80 90 X axs Fgure 10. An example of range refnement. A horzontal scan lne s plotted on a orgnal partal shape. The same lne s plotted on a refned shape (c) Profles of all overlappng range contours before refnement (d) Profles of all overlappng range contours after refnement 6. Conclusons We present a range mage refnement technque to generate accurate 3D computer models. The proposed technque employs the eppolar geometry of our vson system and the vsual hull of an object to refne erroneous 3D ponts. After mult-vew range mages are regstered to a common coordnate system, we frst determne f a reference 3D pont n a range mage s erroneous by measurng regstraton error of the pont to ts correspondences n the other overlappng range mages. The correspondences are ntersectons by the ray of the reference pont. Then the range of an erroneous pont s refned onto the object s surface. Vsual hull of the object s also employed to decde f the pont s erroneous (d)
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