Generalized, Basis-Independent Kinematic Surface Fitting

Transcription

1 Generalzed, ass-independent Knematc Surface Fttng James ndrews a, Carlo H. Séqun a a UC erkeley, Soda Hall, erkeley, C bstract Knematc surfaces form a general class of surfaces, ncludng surfaces of revoluton, helces, sprals, and more. Standard methods for fttng such surfaces are ether specalzed to a small subset of these surface types (ether focusng exclusvely on cylnders or exclusvely on surfaces of revoluton) or otherwse are bass-dependent (leadng to scale-dependent results). Prevous work has suggested re-scalng data to a fxed sze boundng box to avod the bass-dependence ssues. We show that ths method fals on some smple, common cases such as a box or a cone wth small nose. We propose nstead adaptng a well-studed approxmate maxmum-lkelhood method to the knematc surface fttng problem, whch solves the bass-dependence ssue. ecause ths technque s not desgned for a specfc type of knematc surface, t also opens the door to the possblty of new varants of knematc surfaces, such as affnely-scaled surfaces of revoluton. Keywords: knematc surface fttng, reverse engneerng, slppable surfaces, velocty felds, approxmate maxmum lkelhood, Taubn s method C Fgure 1: smple example of knematc surface fttng: () slppable moton (llustrated by red streamlnes) s found for some selecton of the data (n blue). () The data (n green) s projected to some shared sweep plane, where t can be ft by a generator curve. (C) dvectng the generator curve along the slppable moton feld generates the knematc surface. 1. Introducton fundamental sub-problem of reverse engneerng s to ft a prmtve surface to a set of ponts [1]. Knematc surfaces are a class of prmtve surfaces notable for ther general applcablty. They can be used to classfy a wde range of common surfaces: spheres, planes, cylnders, cones, surfaces of revoluton, logarthmc sprals, helces, and more [2]. y channg smple knematc surfaces, even more nterestng surfaces can be ft, ncludng profle and developable surfaces [3]. number of recent systems use these methods for reverse engneerng tasks [4, 5, 6, 7]. knematc surface s a surface that s tangent everywhere to some easly parameterzeable, lnear velocty feld Emal address: jma@eecs.berkeley.edu (James ndrews) Fgure 2: Vsualzaton of two knematc motons of a cylnder. The arrows show the drecton of the felds at ndvdual ponts, and the red lnes are streamlnes showng paths traced by followng the velocty felds. over space (such as those n Sec. 2.1). Gven such a feld and a curve n space (whch we refer to as the generator curve ), advectng the generator curve by the velocty feld wll generate a knematc surface (Fg. 1). Such tangent-everywhere velocty felds are called the slppable motons of a surface [8]. Slppablty of a pont p wth normal n, wth respect to a feld v(p), can be tested by checkng that the normal s orthogonal to the velocty feld: v(p) n = 0. The full bass of slppable motons of a surface can be used to classfy the surface type: for example, f a surface s slppable by both a pure rotaton and a pure translaton n the drecton of the rotaton axs (as n Fg. 2), then the surface s a cylnder. The knematc surface fttng problem conssts of several sub-problems: segmentaton of a surface nto subsets that can be ft by separate knematc surfaces [8, 2]; fttng a knematc moton to a gven set of data ponts; and fnally, fndng a generator curve [9]. We focus specfcally on the problem of fttng the knematc moton. We show that prevous methods for fttng general knematc moton can fal on some smple, common cases such as a box or a cone wth small nose. We then show how to fx these problems. November 16, 2012

2 2. ackground Prevous work on fttng the velocty felds of knematc surfaces has used a common fttng method (Sec. 2.2) and problem formulaton: Gven a set of n ponts wth ther surface normals {p, n }, and a velocty feld (Sec. 2.1) parameterzed by some vector m, fnd feld parameters that are most slppable wth respect to the data ponts Velocty Feld Types Three velocty felds have been proposed. In order of ncreasng generalty, they are: Frst, a constant feld, whch accepts only translatonal moton [9]: v(p) = c (1) Second, a helcal feld, whch adds optonal rotatonal moton for helces and surfaces of revoluton [10]: v(p) = r p + c (2) Fnally, a spral feld whch adds optonal scalng moton for cones and logarthmc sprals [11]: v(p) = r p + c+γp (3) 2.2. Common Fttng Method lmost all knematc surface fttng papers use a common drect fttng algorthm [10] to fnd the feld parameters. Frst, the slppablty of the feld wth respect to the data s expressed n terms of some symmetrc covarance matrx M, such that (v(p) n) 2 = m T Mm, where m s the vector of velocty feld parameters. For example, for the spral feld wth parameters m = r x, r y, r z, c x, c y, c z, γ, matrx M wll be: M := f(x )f(x ) T (4) where x := p x, p y, p z, n x, n y, n z and f(x) := (p n) x, (p n) y, (p n) z, n x, n y, n z, p n Second, a normalzaton s ntroduced to avod degenerate solutons such as the feld wth v(p) = 0 everywhere. The most straghtforward soluton would be to normalze by v(p) : v(p ) n v(p ) However, ths normalzaton has been largely avoded because t would make the feld a non-lnear functon of ts parameters, leadng to a non-lnear optmzaton problem [10]. Instead, prevous work has normalzed the cumulatve squared error by some quadratc functon q (Sec. 2.3) of the moton parameters m: argmn m 1 n n =1 (v(p ) n ) 2 q(m) (5) (6) 2 Note that ths normalzaton can alternatvely be vewed as a constrant on the soluton vector: mnmzng Eqn. 6 s equvalent to mnmzng the non-normalzed metrc under the constrant q(m) = 1. Fnally, ths quadratc functon q s expressed n terms of some (often sngular) symmetrc matrx N: q(m) = m T Nm. Usng the method of Lagrange multplers to solve the constraned mnmzaton problem, m must be a soluton to the generalzed egenvalue problem: (M λn)m = 0 (7) ll egenvectors correspondng to small egenvalues of the matrx pencl (M λn) are then slppable motons of the data ponts. Some systems augment ths core fttng method by teratve re-weghtng (for example to downweght outlers) [10, 2] or subsequent applcaton of a general, non-lnear fttng technque (whch requres a reasonable ntal estmate from the core fttng method to perform well) [4] Quadratc Normalzaton Functons few dfferent quadratc normalzatons have been proposed. The two commonly used normalzatons are: Frst, the unt constrant, n whch the full parameter vector s constraned to unt magntude [8, 2]. For the spral feld, ths becomes: ( r 2 + c 2 + γ 2 ) = 1 (8) Second, the rotaton constrant, whch just constrans the rotaton axs r [10]: r 2 = 1 (9) The constant feld s smple enough that the constrant c 2 = 1 solves the problem perfectly; however ths s only applcable to the constant feld. The choce of normalzaton fundamentally affects the resultng ft: The rotaton constrant requres the soluton to nclude a rotaton component of constant magntude, so t should only be used when t s known n advance that rotaton s ncluded n the desred soluton [10]. The unt constrant s bass-dependent: the best ft wll change f the data s scaled or translated. Prevous work usng the unt constrant suggested re-scalng the data to a fxed sze to address ths ssue [8, 2]. Ths mtgates but does not elmnate the problem: We show n Sec. 3 that there s no fxed scale that can elmnate the artfacts of bass dependence. 3. Falure Cases of Current Methods The knematc surface fttng methods descrbed n Sec. 2 work on many examples, shown n prevous work [10, 8, 2, 7]. However, we found that they also fal on some common, smple cases. In ths secton, we demonstrate and explan the cases that cause these methods to fal.

3 Rotaton constrant Taubn constrant Unt constrant, sze=4 Taubn constrant Fgure 3: spral feld (Eqn 3) (red streamlnes) s ft to a blue selecton of a helx wth randomly perturbed vertces (Gaussan nose wth σ = 0.4% of boundng box sze) usng () the rotaton constrant and () Taubn s constrant. Fgure 5: spral feld (Eqn 3) (red streamlnes) s ft to a cone wth randomly perturbed vertces (Gaussan nose wth σ = 0.4% of boundng box sze) usng () the unt constrant wth boundng box sze 4 and () the Taubn constrant. Unt constrant, sze=4 Taubn constrant Fgure 4: spral feld (Eqn 3) (red streamlnes) s ft to a blue selecton of a box wth randomly perturbed vertces (Gaussan nose wth σ = 0.2% of boundng box sze) usng () the unt constrant wth boundng box sze 4 and () Taubn s constrant. data pont, causng the rotaton constrant to be systematcally based aganst translatonal or scalng moton n the face of nose. To demonstrate ths, we ft a helx wth small nose usng the rotaton constrant n Fg. 3. The rotaton constrant underestmates the ptch (translatonal moton) of the helx. In contrast, a fttng method wthout ths bas (ntroduced n Sec. 4.1) recovers the expected ptch. In each falure case, the key problem s that the slppablty measured for each data pont s scaled by v(p) a quantty that s unrelated to the actual tangency of the feld to the surface at that pont. Whle the quadratc normalzatons of Sec. 2.3 avod the degenerate case of v(p) = 0 everywhere, they stll permt low velocty felds for whch v(p) s reduced at every data pont. The most slppable solutons under these constrants are therefore based towards such low velocty felds. For nosy data, where the expected slppable motons have some error, these bas-favored solutons can be erroneously chosen as the most slppable felds. We demonstrate the falure cases n practce on smple synthetc example meshes, for whch the deal solutons are readly apparent. Each example mesh s generated wth approxmately unform vertex samplng. We ntroduce a small amount of Gaussan nose (wth σ less than 0.5% of the boundng box sze and smaller than half the average edge length), and we recompute the normal for each sample pont by averagng face normals. These small-nose examples should not be challengng, but they cause the prevous methods to perform poorly due to ther systematc bases Rotaton Constrant Falure Cases The rotaton constrant, r = 1, requres a fxed magntude rotaton be part of the soluton feld, but does not specfy the constrants on the translatonal or scalng moton of the feld. ny translatonal or scalng moton n the feld wll therefore ncrease the magntude of v(p) everywhere, beyond what s mandated by the rotaton constrant. Ths addtonal velocty scales the error at each Unt Constrant Falure Cases The unt constrant s bass-dependent: fttng results depend on the scale and translaton of the data ponts. Therefore, those who use t frst center the data ponts around the orgn, and scale the boundng box to a fxed sze (e.g. so the longest edge of the box has unt length) [8, 2]. The bases of the unt constrant depend on the chosen sze. One source of bas n the unt constrant favors scalng and rotaton at small scales: Veloctes of lnear scalng and rotatonal motons are proportonal to the dstance from the center or axs of the moton, so as data ponts come closer together, the veloctes (and thus errors) from scalng and rotaton become smaller. To demonstrate ths bas, we ft four sdes of a box wth small nose usng the unt constrant n Fg. 4. t scales wth boundng box sze 4 or smaller, the resultng ft has a sgnfcant, erroneous rotatonal component. nother source of bas favors offsettng the rotaton axs from the orgn. s the constant parameter c ncreases to acheve the offset, the rotaton axs r must scale down proportonally (to satsfy the unt constrant). Scalng down the rotaton axs scales down the velocty (and thus error) of rotaton. To demonstrate ths bas, we ft a cone wth small nose usng the unt constrant n Fg. 5. t scales wth boundng box sze 4 or greater, the resultng ft erroneously offsets the rotaton axs. From these two examples, we see that the bas of the unt constrant can cause problems at small scales (szes 4) and large scales (szes 4) alke: no sngle scale works well for all cases.

4 4. Improved Fttng Methods The problems we have dentfed n knematc surface fttng methods are smlar to those faced by early methods for algebrac surface fttng [12]. pproxmate maxmum lkelhood (ML) methods are a general class of methods for fttng algebrac surfaces and general parametrc models, whch have been appled to many other problems [13, 14, 15]. In ths secton, we show how to apply ML methods to the knematc surface fttng problem to create an mproved knematc surface fttng method. ML methods can apply to any parametrc model that takes the form m f(x) = 0. For our knematc equatons (Sec. 2.1) ths holds n the case of a spral feld, for example, m would be the parameter vector r x, r y, r z, c x, c y, c z, γ, and f(x) would be the transformaton defned n Eqn. 4. The maxmum lkelhood (ML) method seeks to mnmze the squared dstance from each data pont x to the nearest correspondng pont on the model surface x ; n other words, to mnmze: x x 2, subject to m f( x ) = 0 ecause x s the zero of m f(x) that s closest to x, the dstance to ths root can be approxmated to frst order by the magntude of one step of Newton s method. Ths gves the ML dstance: (m f(x )) 2 x (m f(x )) 2 (10) Rewrtten n terms of knematc surface fttng, the ML method then becomes: (v(p ) n ) 2 p (v(p ) n ) 2 + v(p ) 2 (11) Note that the ML and ML methods are both scale dependent, because the closest element x can be dfferent from x n both poston and normal. If the ponts are scaled up, dfferences n the normal are unchanged, but dfferences n the poston ncrease. We can make ths tradeoff explct: scale the data ponts to a fxed sze boundng box (we scale t so the longest axs has length 1), then ntroduce a weght parameter w p that scales the contrbuton of the poston-based term: (v(p ) n ) 2 w p p (v(p ) n ) 2 + v(p ) 2 (12) Note that as w p goes to zero, the ML fttng equaton becomes the non-lnear mnmzaton (Eqn 5) prevously proposed for knematc surface fttng [10]. Therefore, n theory, all results on ML fttng apply drectly to ths fttng problem. However, ths orgnal non-lnear mnmzaton has numercal ssues around sngulartes where v(p) goes to zero: at these ponts, that slppablty metrc s undefned. small, non-zero value for w p gves a more stable metrc wth no undefned ponts Drect ML Method: Taubn s Constrant The ML dstance metrc s non-lnear, requrng teratve methods or approxmaton to fnd a soluton. One popular approxmaton s Taubn s method [13], whch approxmates the non-lnear ML metrc (Eqn. 10) by summng all numerator and denomnator elements separately: (m f(x ))2 x(m f(x )) 2 Lke the prevous knematc surface fttng methods (Sec. 2), ths s a drect method solveable by a small generalzed egenvalue problem. Lke those methods, t rescales the cumulatve squared error (as n Eqn. 6), so ndvdual ponts are stll scaled by local velocty. However, when w p := 0, ths constrant ensures that the overall average squared veloctes have a fxed magntude Taubn s constrant becomes: 1 n n v(p ) 2 = 1 (13) =1 ecause the average squared magntude velocty s drectly constraned, we can t cheat the Taubn-constraned error metrc by choosng a feld that globally reduces the velocty at all data ponts. Ths prevents falure cases of the varety descrbed n Sec. 3. Note that lettng w p := 0 ensures that Taubn s method s bass ndependent, and does not cause stablty ssues: degenerate ponts where v(p ) goes to zero smply do not contrbute to ether the numerator or denomnator sums. To mplement Taubn s method, we express the normalzaton n the form m T Nm requred by the standard fttng algorthm (Sec. 2.2). The matrx N = ( x f(x ))( x f(x )) T ; for the spral feld (Eqn 3) ths s: N = n =1 n +w p =1 [p ] T [p ] [p ] 0 [p ] T I p 0 T p p p [n ] T [n ] n n wth m = r x, r y, r z, c x, c y, c z, γ (14) For generalty, we have ncluded the w p terms here; when applyng the Taubn constrant ths weght should be zero, but n a non-lnear, teratve method (Sec. 4.2) t can be non-zero. To demonstrate the Taubn constrant n practce, we show a number of practcal test cases of the Taubn constrant n Fgs. 3-6 and Whle the Taubn constrant works well n practce, t remans a based approxmaton t systematcally places less weght than deal on ponts where the velocty feld s small, and more where the velocty feld s large. To address ths, we turn to teratve, non-lnear ML methods.

5 wp := 10 5 C wp := 10 3 Fgure 7: spral feld (Eqn 3) s ft to a cone usng the HEIV method wth varyng values for wp. The HEIV method never converges for wp := 10 5 ; we show the state after 101 teratons. The method converges n 2 teratons for wp := D E F Fgure 6: spral velocty feld ft to a number of dfferent selectons (n blue) usng Taubn s constrant. Streamlnes tracng the best fttng feld are shown for each mage Iteratve ML Methods: HEIV and Reduced To mnmze the true non-lnear error term (ether Eqn. 5 or Eqn. 12), we must use an teratve method. Fortunately, a number of teratve ML methods have been developed [15, 16, 17] all of whch have been shown to converge very quckly n theory for data wth small nose [15]. ll of these methods are based on solvng a smlar egenvalue problem to the drect methods (Sec. 2.2) but teratvely adjusted to correct the weghts of the data ponts. ecause the weghts can only be corrected wth respect to one feld, these teratve methods focus on fndng a sngle best soluton rather than the full bass of solutons provded by drect methods. Prevously the reduced method [16] has been suggested for use on the knematc surface fttng problem [18], although wthout evaluaton. Ths method smply teratvely re-weghts the unt-constrant method. Specfcally, t repeatedly solves the egenvalue problem descrbed n Sec. 2.2, wth normalzaton matrx N = I and error matrx M re-computed at the (j + 1)th teraton as: Mj+1 := X f (x )f (x )T x (mj f (x )) 2 HEIV, wp := 10 3 T aubn Fgure 8: spral feld (Eqn 3) s ft to a cone wth Gaussan nose appled to the base (σ = 1% boundng box sze) usng the HEIV method (converged n 3 teratons) and the Taubn method. when evaluated n the context of algebrac curve fttng [15] replcatng the robustness under nose of the Taubn method, but wth lower error. Intutvely ths may be expected because ths method can be seen as an teratve reweghtng of Taubn s method. t each teraton, t solves a generalzed egenvalue problem wth M reweghted as n Eqn. 15 above, and N reweghted as: Nj+1 := (15) X Where mj s the parameter vector at teraton j; m0 can be ntalzed by solvng wth any drect method. Unfortunately, ths fals n the same way as the non-teratve unt constrant method: the nherent bases of that method are not addressed by re-weghtng, and smlar results to those of Fgs. 4 and 5 occur. Ths result s consstent wth the poor performance observed for the reduced method on algebrac curve fttng problems under mld nose [15]. In contrast, the heteroscedastc errors-n-varables (HEIV) method [17] performed much more successfully (mj f (x ))2 x (mj f (x )) 4 ( x f (x ))( x f (x ))T Ths method performs smlarly to the Taubn method on most examples we tested. Some care must be taken to choose the wp large enough for stablty; we found wp.001 worked consstently, whle smaller values could be unstable and thus fal to converge as shown n Fg. 7. We therefore set wp :=.001 for our tests. dvantages of HEIV over Taubn become evdent when nose s dstrbuted unevenly over the surface, n areas whch Taubn wll systematcally over-weght, as shown n Fg. 8. 5

6 C D Fgure 9: spral feld (Eqn 3) s ft to a scanned model of a drll bt, on whch outlers are concentrated at one end. In ()-(D), the bestfttng vector feld s vsualzed by red stream lnes. () and (C) are ft wthout usng RNSC; n () the outlers at the end have been omtted manually by not selectng that porton of the mesh, whle n (C) the outlers are ncluded, sgnfcantly affectng the result. (D) s ft usng RNSC, and t gves a result smlar to manually avodng the outlers. The Taubn method s used for each ft. 5. Robustness to Outlers Lke any least-squares fttng method, these methods are all senstve to outlers. Many methods can be, and have been, used to reduce the mpact of such outler ponts; n partcular, M-estmators [19] and RNSC [20] have both been suggested [10, 2]. The RNSC approach s explaned n detal, n the context of knematc surface fttng, by [2]. We recommend ths procedure, albet usng the new fttng technques and normalzed dstances presented here nstead of Eqn. 8. n example demonstratng the effectveness of ths approach s shown n Fg. 9. For ths example we assumed that outlers have an error (computed by Eqn. 5) greater than.1, and that 90% of ponts are not outlers. 6. New Velocty Felds Prevous methods for knematc surface fttng dd not generalze well beyond the few feld types lsted n Sec. 2.1: the unt constrant s problems only become worse wth more complex felds, and t s unclear how to apply the rotaton constrant unless the feld promnently features a rotaton axs parameter. The Taubn and HEIV methods, n contrast, apply to any velocty feld lnear n ts parameters m that s, any velocty feld that can be expressed n the form v(p) := m f (p) where m are the elements of the parameter vector m, and the functons f (p) can be any functons from postons to vectors. Therefore these methods can be used to ft new, more general velocty felds. For sensble results, the class of felds chosen should also be closed under the Le bracket operator n other words, composng motons of multple velocty felds n a class should result n motons whch are also n that class. 6 C Fgure 10: The general lnear feld (llustrated wth red streamlnes) s ft to a number of selectons (n blue) on varous objects. The space of possble classes of velocty felds s enormous, and not easy to understand ntutvely. ut t opens doors to fttng some new, nterestng prmtves and some smple prmtves that were notably mssng from the past repertore of knematc surfaces. For example: although spheres are handled by the more tradtonal knematc surfaces, ellpsods and general quadrcs are not, because there s no support for rotaton combned wth some scalng. Ths would correspond to a non-lnear feld wth some scalng matrx S as a new parameter: v(p) := S 1 (r (Sp)) + c (16) Just by addng ths scale factor, knematc surfaces would now nclude all quadrc surfaces as a subtype they could handle. Ths scaled equaton s no longer lnear n the parameters, but f we multply out (lettng = S 1 [r] S, where [r] s the matrx form of a cross product by r) we see that ts felds are a subset of a class of general lnear felds: v(p) := p + c (17) where s an arbtrary 3 3 matrx. Ths general lnear feld can be used to ft felds wth rotaton combned wth some scalng, as we demonstrate n Fg. 10. From ths example t s clear that the more general felds do nclude at least one addtonal, useful shape prmtve, and thus seem worthy of further nvestgaton. 7. Dscusson and Future Work We dentfed a weakness n the standard algorthm for fttng knematc surfaces, gvng detaled examples of how ths causes problems n practce as well as an explanaton of the underlyng source of these problems. We then presented a soluton that s general, bass-ndependent, and D

7 correctly handles the falure cases of the prevous methods. Ths soluton should mprove the robustness of algorthms based on knematc surface fttng. Its generalty also makes t easy for us to ntroduce a new type of feld. The newly ntroduced feld s a proof of concept that the generalty of our method could apply to even broader applcatons. It also presents new challenges: wth more complex velocty felds, a useful nterpretaton of the resultng parameters becomes more dffcult. Developng a complete surface reconstructon ppelne that explots the full range of possble knematc surfaces wll lkely requre exploraton of addtonal new feld types and new algorthms to ft and nterpret those feld types. [15] N. Chernov, On the convergence of fttng algorthms n computer vson, J. Math. Imagng Vs. 27 (3) (2007) [16] K. Kanatan, Further mprovng geometrc fttng, n: Proc. 5th Int. Conf. 3-D Dgtal Imagng and Modelng, 2005, pp [17] Y. Leedan, P. Meer, Heteroscedastc regresson n computer vson: Problems wth blnear constrant, Int. J. Comput. Vson 37 (2) (2000) [18] H. Pottmann, J. Wallner, Computatonal Lne Geometry, Sprnger-Verlag New York, Inc., Secaucus, NJ, US, [19] P. Huber, Robust Statstcs, John Wley and Sons, New York, [20] M.. Fschler, R. C. olles, Random sample consensus: a paradgm for model fttng wth applcatons to mage analyss and automated cartography, Commun. CM 24 (6) (1981) cknowledgements Ths work was supported n part by the Natonal Scence Foundaton (NSF award #CMMI (EDI)) and by dobe Systems. Thanks to the Image-based 3D Models rchve, Tlcom Pars, and to the Stanford Computer Graphcs Laboratory for some of the test meshes used. References [1] T. Varady, R. Martn, J. Cox, Reverse engneerng of geometrc models - an ntroducton, Computer ded Desgn 29 (1997) [2] M. Hofer,. Odehnal, H. Pottmann, T. Stener, J. Wallner, 3d shape recognton and reconstructon based on lne element geometry, n: Tenth IEEE Internatonal Conference on Computer Vson, [3] H. Pottmann, H.-Y. Chen, I. K. Lee, pproxmaton by profle surfaces, The Mathematcs of Surfaces VIII (1998) [4] Y. Lu, H. Pottmann, W. Wang, Constraned 3d shape reconstructon usng a combnaton of surface fttng and regstraton, Computer-ded Desgn 38 (2006) [5] G. Harary,. Tal, The natural 3d spral., Computer Graphcs Forum 30 (2) (2011) [6]. Gfrerrer, J. Lang,. Harrch, M. Hrz, J. Mayr, Car sde wndow knematcs, Computer-ded Desgn 43 (4) (2011) [7] J. ndrews, H. Jn, C. H. Séqun, Interactve nverse 3d modelng, Computer-ded Desgn and pplcatons 9 (6) (2012) [8] N. Gelfand, L. Gubas, Shape segmentaton usng local slppage analyss, n: Eurographcs Sympoum on Geometry Processng, [9] T. Randrup, pproxmaton by cylnder surfaces, Computer ded Desgn 30 (1998) [10] H. Pottmann, T. Randrup, Rotatonal and helcal surface approxmaton for reverse engneerng, Computng 60 (1998) [11] H. Pottmann, I.-K. Lee, T. Randrup, Reconstructon of knematc surface from scattered data, Proceedngs of Symposum on Geodesy for Geotechncal and Structural Engneerng (1998) [12] V. Pratt, Drect least-squares fttng of algebrac surfaces, SIG- GRPH Comput. Graph. 21 (4) (1987) [13] G. Taubn, Estmaton of planar curves, surfaces and nonplanar space curves defned by mplct equatons, wth applcatons to edge and range mage segmentaton, IEEE Transactons on Pattern nalyss and Machne Intellgence 13 (1991) [14] K. Kanatan, N. Ohta, Comparng optmal three-dmensonal reconstructon for fnte moton and optcal flow., J. Electronc Imagng 12 (3) (2003)