SIMULTANEOUS REGISTRATION OF MULTIPLE VIEWS OF A 3D OBJECT

Transcription

1 SIMULTANEOUS REGISTRATION OF MULTIPLE VIEWS OF A 3D OBJECT Helmut Pottmann a, Stefan Leopoldseder a, Mchael Hofer a a Insttute of Geometry, Venna Unversty of Technology, Wedner Hauptstr. 8 10, A 1040 Wen, Austra - pottmann@geometre.tuwen.ac.at, leopoldseder@geometre.tuwen.ac.at, hofer@geometre.tuwen.ac.at KEY WORDS: regstraton, matchng, nspecton, three-dmensonal data, CAD, geometry. ABSTRACT In the reconstructon process of geometrc objects from several three-dmensonal mages (clouds of measurement ponts) t s crucal to algn the pont sets of the dfferent vews, such that errors n the overlappng regons are mnmzed. We present an teratve algorthm whch smultaneously regsters all 3D mage vews. It can also be used for the soluton of related postonng problems such as the regstraton of one or several measurement pont clouds of an object to a CAD model of that object. Our method s based on a frst order knematcal analyss and on local quadratc approxmants of the squared dstance functon to geometrc objects. 1 INTRODUCTION In the reconstructon process of surfaces wth help of stereo photogrammetry one often obtans several pont clouds arsng from dfferent vews of the object. By evaluaton of the surface texture n the dfferent mages, correspondences between ponts of overlappng pont clouds can be determned wth some confdence value. Now, the dfferent pont clouds have to be combned nto one consstent representaton. There, we may get errors n the overlappng regons. Mnmzaton of those errors s the goal of the present algorthm. A more challengng task s the smultaneous regstraton of several movng systems where no pont-to-pont correspondences are known. One applcaton where only two systems are nvolved s the followng: Suppose that we are gven a large number of 3D data ponts that have been obtaned by some 3D measurement devce (laser scan, lght sectonng, :::) from the surface of a techncal object. Furthermore, let us assume that we also have got the CAD model of ths workpece. Ths CAD model shall descrbe the deal shape of the object and wll be avalable n a coordnate system that s dfferent to that of the 3D data pont set. For the goal of shape nspecton t s of nterest to fnd the optmal Eucldean moton (translaton and rotaton) that algns, or regsters, the pont cloud to the CAD model. Ths makes t possble to check the gven workpece for manufacturng errors and to classfy the devatons. Another applcaton wth more than two systems s the multple matchng of dfferent 3D laser scanner mages of some 3D object. The 3D pont sets of dfferent vews wll be gven n dfferent coordnate systems, ther poston n a common object coordnate system may be known only approxmately. Now the key task s to smultaneously match, or regster, the dfferent pont sets such that they optmally ft n ther overlappng regons. In the followng we show how to solve these optmzaton problems wth an teratve algorthm. In each teraton step all N systems are regstered smultaneously. An arbtrary number of systems (at least one) s kept fxed. The algorthm uses a knematcal analyss of frst order and solves a lnear system of equatons, whch comes from a least squares problem. In the case that we do not have correspondences, the algorthm s based on local quadratc approxmants of the squared dstance functon to geometrc objects. The novelty n our method concerns the followng apects: We use local quadratc approxmants of the squared dstance functon nstead of pure pont-pont or pont-plane dstances. By an approach whch reles on nstantaneous knematcs we just solve a lnear system n each teraton step, even n case of smultaneously regsterng more than two systems. In Sec. 2 we brefly revew contrbutons n the lterature whch are closely related to our algorthm. In Sec. 3 some basc facts of spatal knematcs are collected. Sec. 4 s devoted to the mathematcal descrpton of our algorthm whch smultaneously algns multple pont clouds n the case that pont-to-pont correspondences are known. In Sec. 5 we descrbe how to treat the more dffcult case where no correspondences are gven. Fnally, n Sec. 6 topcs of further research are addressed. 2 CURRENT REGISTRATION ALGORITHMS Let us frst focus on regstraton problems where only two systems are nvolved (N = 2). One system moves relatve to the second system whch s kept fxed. If pont-topont correspondences are known, the optmal moton that mnmzes the Eucldean dstances between correspondng ponts can be explctly gven. The use of quaternons for determnng ths moton can already be found n (Faugeras Hebert, 1986, Horn, 1987). In many applcatons, however, no pont-to-pont correspondences are gven. One example s the algnment of a sngle 3D pont cloud to a geometrc entty whch could be a CAD model or another 3D pont set. Here a well-known standard algorthm s the teratve closest pont (ICP) algorthm of Besl and McKay (Besl McKay, 1992). In Sec. 2.1 we wll brefly summarze ths algorthm whch establshes pont-to-pont correspondences n each teraton step and uses the representaton of 3D Eucldean motons by unt quaternons. For an overvew on the recent lterature on ths topc we refer to (Eggert et al., 1998). A summary wth new results on the acceleraton of the ICP algorthm has been gven by Rusnkewcz and Levoy (Rusnkewcz Levoy, 2001).

2 There are two major restrctons of the ICP algorthm. Frst, t s mplctly assumed that one of the data sets s a subset of the other. The presence of ponts that have no correspondng pont n the other set leads to ncorrect assgnments. Several dfferent approaches to threshold outlers have been presented, see the lterature cted n (Eggert et al., 1998). Secondly, the ICP algorthm s a two set approach and s not drectly extendable to multple data sets. It s not suffcent to apply regstraton to consecutve pars of 3D pont sets, snce algnment errors accumulate and certan pont sets wll be very poorly adjusted. There have been several approaches to the smultaneous regstraton of all data sets, see e.g. the sprng force model of (Eggert et al., 1998). (Bergevn et al., 1996) apply ncremental transformatons to all the movng systems wth respect to a fxed system and use pont-to-tangent plane correspondences n the overlappng regons. 2.1 The ICP algorthm A pont set ( data shape) s rgdly moved (regstered, postoned) to be n best algnment wth the correspondng CAD model ( model shape) n the followng teratve way. In the frst step of each teraton, for each pont of the data shape, the closest pont n the model shape s computed. Ths s the most tme consumng part of the algorthm and can be mplemented effcently, e.g. by usng an octree data structure. As result of ths frst step one obtans a pont sequence Y =(y 1 ; y 2 ;:::) of closest model shape ponts to the data pont sequence =(x 1 ; x 2 ;:::). Each pont x corresponds to the pont y wth the same ndex. In the second step of each teraton the rgd moton M s computed such that the moved data ponts M (x ) are closest to ther correspondng ponts y, where the objectve functon to be mnmzed s ky M (x )k 2 : Ths least squares problem can be solved explctly, see e.g. (Besl McKay, 1992, Horn, 1987). The translatonal part of M brngs the center of mass of to the center of mass of Y. The rotatonal part of M can be obtaned as the unt egenvector that corresponds to the maxmum egenvalue of a symmetrc 4 4 matrx. The soluton egenvector s nothng but the unt quaternon descrpton of the rotatonal part of M. After ths second step the postons of the data ponts are updated va new = M ( old ). Now step 1 and step 2 are repeated, always usng the updated data ponts, as long as the change n the mean-square error falls below a preset threshold. The ICP algorthm always converges monotoncally to a local mnmum, snce the value of the objectve functon s decreasng both n steps 1 and 2. 3 KINEMATICS An mportant part of the algorthm uses nstantaneous knematcs. Thus, we brefly descrbe the most mportant facts before gong nto the detals of the algorthm. Consder a contnuous one-parameter moton of a rgd body n space. If x s a pont n Eucldean three-space, the symbol v(x) denotes the velocty vector of that pont of the movng body whch s at ths moment at poston x. Thus v(x) s a tme-dependent vector attached to the pont x. It s well known that at some nstant t, a smooth moton has a velocty vector feld of the form v(x) =c + c x; (1) wth vectors c; c 2 R 3. Thus the velocty vector feld (or the nfntesmal moton) at some nstant t s unquely determned by the par (c; c). Of specal nterest are the unform motons, whose velocty vector feld s constant over tme. It s well known that apart from the trval unform moton, where nothng moves at all and all veloctes are zero, there are the followng three cases: 1. Unform translatons have c = o, but c 6= o,.e., all velocty vectors equal c. The paths of a unform translaton are straght lnes parallel to c. 2. Unform rotatons wth nonzero angular velocty about a fxed axs. We have c c = 0, butc 6= o. The pont trajectores of a unform rotaton are crcles wth the rotatonal axs as common axs. pt A x 3 x 1 x 2 Fgure 1: t x(t) x(0) = x Helcal moton. 3. Unform helcal motons are the superposton of a unform rotaton and a unform translaton parallel to the rotaton s axs. They are characterzed by c c 6= 0. If the pont x s stuated on the axs, ts path concdes wth the axs. The trajectores of the other ponts are helces. If! s the angular velocty of the rotaton, and v the velocty of the translaton, then p = v=! s called the ptch of the helcal moton. We use the conventon that! s nonnegatve, that p > 0 for rght-handed helcal motons, and that p<0for left-handed ones. To further clarfy the concept of a unform helcal moton, we assume the x 3 -axs of a Cartesan system (x 1 ;x 2 ;x 3 ) to be the helcal axs (see Fg. 1). A unform helcal moton may then be wrtten as x 7! x(t) wth x(t) = cos t sn t 0 sn t cos t A 0 0 p t 1 A ; (2) p =0means a unform rotaton, and for p!1the moton tends to a unform translaton.

3 Snce all possble pars (c; c) actually occur, we can use these three cases to classfy the type of velocty vector feld at one nstant of an arbtrary smooth moton: Infntesmal translatons are characterzed by c = o, and nfntesmal rotatons by c c =0. The remanng velocty vector felds are sad to belong to nfntesmal helcal motons. At all nstants, f the velocty vector feld of a smooth moton s nonzero, t belongs to one of the three cases. If (c; c) represents the velocty vector feld of a unform rotaton or helcal moton, then the Plücker coordnates (g; g) of the axs A, the ptch p and the angular velocty! are reconstructed by (g; g) =(c; c pc); p = c c=c 2 ;! = kck; (3) see e.g. (Pottmann Wallner, 2001). The Plücker coordnates (g; g) of a straght lne A consst of a drecton vector g and the moment vector g about the orgn. From the moment vector, we can easly compute a pont p of the lne A, snce for all ponts p on A we have the relaton g = p g. The above results about nfntesmal motons are a lmt case of the followng fundamental result of 3-dmensonal knematcs: Any two postons of a rgd body n 3-space can be transformed onto each other by a (dscrete) helcal moton (consstng of a rotaton about an axs and a translaton along that axs), ncludng the specal cases of a pure rotaton and a pure translaton. Our algorthm actually teratvely computes the velocty vector feld of a dscrete helcal moton. Those underlyng helcal motons are then used for the dsplacement. 4 SIMULTANEOUS REGISTRATION WITH KNOWN CORRESPONDENCES The frst applcaton we have n mnd s the smultaneous regstraton of N pont clouds whch have been obtaned by stereo photogrammetry. The pont clouds partally overlap, and n these regons correspondences (plus confdence values) between ponts of dfferent pont clouds are known from surface texture analyss. The N pont clouds can be vewed as rgd systems and are denoted by ±. An arbtrary number (at least one) of the gven systems remans fxed. The others shall be moved such that after applcaton of the motons the dstances of correspondng ponts, weghted wth ther confdence value, are as small as possble. Snce n our case we have N > 2, only an teratve procedure s possble. We use a geometrc method that nvolves nstantaneous knematcs, and thus t s smlar to the approach n (Bourdet Clément, 1988). Only those ponts n a cloud are used for the algnment process whch belong to an overlappng regon wth a neghborng pont cloud. For such a gven data pont par (x ; y ) we know the ndex j of the system ± j to whch x belongs, Fgure 2: Example of multple regstraton wth known correspondences: All gven 30 pont clouds (top) and detal of 4 clouds showng the overlappng areas (bottom). Data by courtesy of Gerhard Paar, Joanneum Research. and the ndex k ndcatng the system ± k of the pont y. The pont pars have found to be n correspondence wth a confdence value w 2 (0; 1). Our goal s to move each system ± l by a moton M l n a way, such that after applcaton of all these motons M l, the new postons of correspondng ponts are as close as possble to each other n a least squares sense. Thereby we have to keep n mnd the confdence values of correspondences. 4.1 Dsplacement estmaton va nstantaneous knematcs Snce the expected motons are small dsplacements anyway, we replace them by nstantaneous motons. The nstantaneous moton of system ± l aganst one fxed system (called ± 0 henceforth) possesses a velocty vector feld. It s characterzed by two vectors c l ; c l 2 R 3, and analogously to Eq. 1, the velocty vector v l0 of a pont x 2 ± l s then gven by v l0 (x )=c l + c l x : (4) For a par of correspondng ponts (x ; y ) we would lke to estmate ther dstance after the motons have been appled to ther systems ± j and ± k, respectvely. In frst order, we

4 can use the velocty vectors, and thus the squared dstance of the dsplaced ponts s gven by Q 1 (x ; y )=(x + v j0 (x ) y v k0 (y )) 2 = (x y +(c j + c j x ) (c k + c k y )) 2 : Ths term Q 1 (x ; y ) s a quadratc functon n the unknowns c j ; c j ; c k ; c k of the nstantaneous motons appled to the nvolved systems ± j and ± k. There s an alternatve to Eq. (5): Instead of lnearzng the moton of ± j aganst ± 0 and the moton of ± k aganst ± 0, one can lnearze the relatve moton of ± j aganst ± k. The velocty vector v jk of a pont x 2 ± j for ths relatve moton s gven by (5) v jk (x )=v j0 (x ) v k0 (x ): (6) Here, the dstance of nterest s between the pont x + v jk (x ) and y (.e., x s nterpreted to be movng wth system ± j relatve to the pont y n system ± k ). The squared dstance of these two ponts of nterest s gven by Q 2 (x ; y )=(x + v jk (x ) y ) 2 = (x y +(c j + c j x ) (c k + c k x )) 2 : The term Q 2 (x ; y ) s agan a quadratc functon n the unknowns c j ; c j ; c k ; c k. We see that any par of correspondng ponts gves rse to such a quadratc term Q 1 or Q 2 n the nvolved unknown moton parameters. That term s a frst order estmate of the squared dstance (error) after applcaton of the motons. Hence, to perform the error mnmzaton, we wll mnmze the followng weghted sum F = (7) w Q 2 (x ; y ): (8) The weght w s the known confdence value of the par (x ; y ). Note that snce pont-to-pont correspondences are known, both Q 1 and Q 2 can be used. Wthout known correspondences, however, t s necessary to use the velocty vectors v jk for the relatve moton of ± j aganst ± k, see Sec. 5. The mnmzaton of F s mathematcally smple, because F s a quadratc functon n the unknown moton parameters c l ; c l. Collectng all unknowns n the vector C = (c 1 ; c 1 ; c 2 ; c 2 ;:::;c N ; c N ) T, we may wrte F n the form F = C T B C+2A C+ w (x y ) 2 : (9) Hence, the mnmzer C of F solves the followng lnear system, B C+ A T =0; (10) where B s a 6 6N matrx and A s a 1N 6N matrx. Note that t s very easy to fx more than one system. Fxng ± l just requres to set both vectors c l and c l equal to zero. 4.2 Computng the actual dsplacements from veloctes In the prevous subsecton we have estmated the dsplacement vector of a pont (.e., the vector pontng from the old to the new poston) wth help of the velocty vector of an nstantaneous moton. However, dsplacng ponts n ths way would result n an affne mappng of the correspondng system ± l and not n a rgd body moton. Although such affne transformatons are actually used n the lterature (Bourdet Clément, 1988), we prefer to compute exact rgd body motons n the followng way. It s suffcent to explan ths for one movng system, whch we denote by ±, and whose nstantaneous dsplacement s gven by the vectors c; c. In the unlkely case that there s no rotatonal part,.e., c = 0, we are done, snce then we have a translaton wth the vector c, whch of course s a rgd body moton. Otherwse we note that the velocty feld of the nstantaneous moton s unquely assocated wth a unform helcal moton. Its axs A and ptch p can be computed wth formula (3). The dea now s to move ponts va that helcal moton approxmately as far as ndcated by the velocty vectors (ponts are now moved along helcal paths of that moton). Note that kck gves the angular velocty of the rotatonal part. We apply a moton to ± whch s the superposton of a rotaton about the axs A through an angle of ff = arctan(kck) and a translaton parallel to A by the dstance of p ff. A rotaton through an angle of ff about an axs (wth unt drecton vector a = (a x ;a y ;a z )) through the orgn s known to be gven by x 0 = R x wth orthogonal matrx R = m 00 m 11 2(b 1 b 2 + b 0 b 3 ) 2(b 1 b 3 b 0 b 2 ) 2(b 1 b 2 b 0 b 3 ) m 22 2(b 2 b 3 + b 0 b 1 ) 2(b 1 b 3 + b 0 b 2 ) 2(b 2 b 3 b 0 b 1 ) m 33 m 00 = b b2 1 + b2 2 + b2 3 ;m 11 = b b2 1 b2 2 b2 3 ; ; m 22 = b 2 0 b b2 2 b 2 3;m 33 = b 2 0 b 2 1 b b2 3; (11) where b 0 = cos(ff=2), b 1 = a x sn(ff=2), b 2 = a y sn(ff=2), b 3 = a z sn(ff=2). The superposton of the rotaton about the axs A wth Plücker coordnates (a; a), cf. Eq. (3), through an angle ff and the translaton parallel to A by p ff s then gven by x 0 = R(x p) +(p ff)a + p; (12) where R s the matrx gven above and p s an arbtrary pont on the helcal axs (e.g. p = a a). 4.3 Iteraton and termnaton crtera Wth the methods from 4.1 the algorthm teratvely computes nstantaneous motons of the movng systems, whose actual dsplacements are then computed as n 4.2. Ths teratve procedure s termnated f one of the two followng condtons s satsfed.

5 1. The mprovement n the objectve functon (8), after some teraton step, s below a chosen value. 2. The number of teratons exceeds a chosen constant. 5 SIMULTANEOUS REGISTRATION WITHOUT CORRESPONDENCES Gven are N clouds of 3D data ponts (systems ± 1 ;:::;± N ) that partally overlap, but now we have nether correspondences between pont pars, nor confdence values, as n Sec. 4. But we stll assume that a good ntal poston of these N 3D pont sets n a global coordnate system s known. Furthermore we know whch of the pars of systems ± j, ± k actually overlap. In Sec. 4 each data pont x 2 ± j that corresponds to y 2 ± k contrbutes to the functonal F n Eq. (8) wth the term Q 2 (x ; y ) whch s quadratc n the unknowns v jk (x )= (c j c k )+(c j c k ) x. Now we also want to set up such a quadratc functonal for each data pont x n an overlappng regon. We present a strategy based on a quadratc approxmaton of the squared dstance functon, cf. (Pottmann Hofer, 2002). If two systems ± j, ± k overlap we trangulate the pont cloud n ± k and refer to the trangulated pont cloud as T (± k ). Frst we determne those ponts x 2 ± j that actually le n the overlappng regon,.e., ther dstance to T (± k ) s below a certan threshold. For lterature on these thresholdng technques, see e.g. (Blas Levne, 1995, Eggert et al., 1998, Zhang, 1994). Now, the goal s to brng the ponts x closer to the geometrc shape T (± k ),.e., to move the ponts x to lower levels of the squared dstance functon to the trangulated surface T (± k ). In (Pottmann Hofer, 2002) t s descrbed how to compute for any pont x 2 R 3 a local quadratc approxmant F d;x =: F d to a smooth surface. For our applcatons ths has to be a nonnegatve quadratc functon, F d (x) 0; 8x 2 R3. Here we do not have a smooth surface but a trangulated pont cloud T (± k ). We can apply the results of (Pottmann Hofer, 2002), f we are able to locally approxmate the trangulated surface by a smooth surface. For ths we can use local quadrc fts to T (± k ), see e.g. (Yang Lee, 1999). In case that the trangulaton s too coarse, we may frst apply mesh refnement technques (e.g. nterpolatory subdvson) to get a suffcently dense trangulaton (Dyn et al., 1990, Zorn, 1997). Now the regstraton of two such overlappng pont clouds s found by teratvely mnmzng the functonal F d (x + v jk (x )); (13) whch s quadratc n the unknowns c j ; c j ; c k ; c k. If we do not consder ponts x from one overlappng regon (± j ; ± k ) only, but ponts from all overlappng regons smultaneously, then we have to mnmze the functonal (13) for the unknowns c 1 ; c 1 ;:::;c N ; c N. In order to fx certan systems ± l, one smply has to set the vectors c l ; c l equal to zero. As a smple example for a quadratc approxmant F d of the squared dstance functon of T (± k ) at the pont x,we would lke to present a squared pont-tangent plane dstance: Let y denote the closest pont of T (± k ) to x.we estmate the unt normal vector n to T (± k ) n y, e.g. by computng a regresson plane usng the ponts n a certan neghborhood of y. In the followng we refer to the plane wth normal vector n through y as the tangent plane of T (± k ) n y. Let d denote the orented dstance of x to ths plane. If x s already close to T (± k ) t s better to frst refne the trangulaton and then compute the nearest pont and proceed as mentoned above. We want to mnmze the squared dstance of the dsplaced pont x Λ = x + v jk (x ) to the tangent plane at y. The dstance s gven by d + n v jk (x ) and thus we get the functonal Q 3 (x )=F d (xλ)=(d + v jk (x ) n ) 2 =(d +(c j c k ) n + det(c j c k ; x ; n )) 2 (14) ; whch s quadratc n the unknowns c j ; c j ; c k ; c k. In ths way we can compute a quadratc term Q 3 (x ) for all ponts x that have been found to le n an overlappng regon. Hence, to smultaneously regster all N pont clouds (where at least one s fxed) we have to mnmze the followng functonal F = Q 3 (x ): (15) As mentoned n Sec. 4 one has to observe the fact that the map x 7! x + v j0 (x ) s no Eucldean rgd body moton. See Sec. 4.2 for the computaton of the rgd moton whch brngs x close to tps of the vectors x + v j0 (x ). The squared dstance functon to the tangent plane has already been used n several varants to the ICP algorthm, cf. (Bergevn et al., 1996, Chen Medon, 1992). Our approach of a local quadratc approxmant of the squared dstance functon s more general and ncludes the squared dstance functon to the tangent plane as a specal case. Fg. 3 gves an example for our algorthm n the case N =2, namely the regstraton of a pont cloud to a CAD model. In Fg. 4 the mean squared error of the data ponts to the CAD surface after each teraton s gven, both for our algorthm and for the standard ICP algorthm. Our algorthm converges much faster than ICP, and wth respect to the number of teratons t s comparable to Chen and Medon s method. However, we solve just a lnear system n each teraton step, whereas Chen and Medon n each step run a numercal optmzaton algorthm. We are currently workng on an effcent organzaton of the local quadratc approxmants n a spatal data structure n order to speed up the algorthm. Then, ndustral nspecton tasks are expected to come close to real tme performance, snce the computatons on the approxmaton of the squared dstance functon to the CAD model can be done n a preprocessng step. Further deas for acceleraton and stablzaton of the algorthm can be taken from (Rusnkewcz Levoy, 2001).

6 ACKNOWLEDGEMENTS Ths work has been carred out wthn the K plus Competence Center Advanced Computer Vson and was funded from the K plus program. REFERENCES Fgure 3: Regstraton of a pont cloud to a CAD model: Intal and fnal poston of the pont cloud MS algnment error Mnmzaton wth quadratc approxmants Mnmzaton of pont to pont dstances (ICP) Iteratons Fgure 4: Comparson of the convergence rate for our method vs. mnmzaton of pont-to-pont dstances n each step (ICP). 6 ETENSIONS AND FUTURE RESEARCH Here we dd not deal wth those systematc and random errors n the 3D pont clouds that arse n the data capturng process. Our algorthm for the smultaneous regstraton of multple pont clouds teratvely mnmzes a functon F = P! Q k (x );k =1; 2; 3. The weghts! can be used to successvely downweght outlers. Approprate weghtng schemes may be found n (Rousseeuw Leroy, 1987). In our contrbuton we have assumed that a rough, ntal algnment of the pont clouds s gven and that small dsplacements of the pont clouds are suffcent to brng them n optmal algnment. A very ambtous task for future research s to derve a stable algorthm to fnd these rough ntal algnments. Such an algorthm should be applcable to multple pont clouds and should explot as much nformaton on the nvolved geometrc enttes as possble. Fnally, for specal geometres lke 3D objects composed of smple surfaces one may have addtonal or more precse nformaton on the squared dstance functon (Kverh Leonards, 2002). Addtonal work has to be done to nclude such nformaton n the basc algorthm. Bergevn R., Laurendeau D., Poussart, D., Regsterng range vews of multpart objects. Comput. Vson Image Understandng, 61, pp Besl, P. J., McKay, N. D., A method for regstraton of 3D shapes. IEEE Trans. Pattern Anal. and Mach. Intell., 14, pp Blas, G., Levne, D., Regsterng multvew range data to create 3D computer objects. IEEE Trans. Pattern Anal. Mach. Intell., 17, pp Bourdet, P., Clément, A., A study of optmal-crtera dentfcaton based on the small-dsplacement screw model. Annals of the CIRP, 37, pp Chen, Y., Medon, G., Object modellng by regstraton of multple range mages. Image and Vson Computng, 10, pp Dyn, N., Levn, D., Gregory, J. A., A butterfly subdvson scheme for surface nterpolaton wth tenson control. ACM Transactons on Graphcs, 9(2), pp Eggert, D. W., Ftzgbbon, A. W., Fsher, R. B., Smulaneous regstraton of multple range vews for use n reverse engneerng of CAD models. Computer Vson and Image Understandng, 69, pp Faugeras, O. D., Hebert, M., The representaton, recognton, and locatng of 3-D objects. Int. J. Robotc Res., 5, pp Horn, B. K. P., Closed-form soluton of absolute orentaton usng unt quaternons. J. Opt. Soc. Am. A, 4, pp Kvehr, B., Leonards, A., A new refnement method for regstraton of range mages based on segmented data. Computng, 68(1), pp Nkolads, N., Ptas, I., D Image Processng Algorthms. Wley. Pottmann, H., Hofer, M., Geometry of the squared dstance functon to curves and surfaces. Proceedngs Mathematcs and Vsualzaton, Sprnger, to appear. Pottmann, H., Wallner, J., Computatonal Lne Geometry, Sprnger-Verlag. Rousseeuw, P. J., Leroy, A. M., Robust Regresson and Outler Detecton, Wley. Rusnkewcz, S., Levoy, M., Effcent varants of the ICP algorthm. In: Proc. 3rd Int. Conf. on 3D Dgtal Imagng and Modelng, Quebec. Yang, M., Lee, E., Segmentaton of measured pont data usng a parametrc quadrc surface approxmaton. Computer-Aded Desgn, 31, pp Zhang, Z., Iteratve pont matchng for regstraton of smooth surfaces usng dfferental propertes. In: Proceedngs of the 3rd European Conference on Computer Vson, Stockholm, pp Zorn, D., Schröder, P., Sweldens, W., Interactve multresoluton mesh edtng. In: SIGGRAPH 97 Conference Proceedngs, pp