3D INDUSTRIAL RECONSTRUCTION BY FITTING CSG MODELS TO A COMBINATION OF IMAGES AND POINT CLOUDS

Transcription

1 3D INDUSTRIAL RECONSTRUCTION BY FITTING CSG MODELS TO A COMBINATION OF IMAGES AND POINT CLOUDS Tahr Rabban, Frank van den Heuvel * Secton of Photogrammetry and Remote Sensng, Faculty of Aerospace Engneerng, Delft Unversty of Technology, Kluyverweg, 69 TS Delft, The Netherlands Emal: t.rabban@lr.tudelft.nl, F.A.vandenHeuvel@lr.tudelft.nl Commsson V, WG V/I KEY WORDS: Industral Reconstructon, Pont Cloud, Laser Scannng, Photogrammetry, Recognton, Regstraton, Modellng, Automaton ABSTRACT: We present a method for 3D reconstructon of ndustral stes usng a combnaton of mages and pont clouds wth a motvaton of achevng hgher levels of automaton, precson, and relablty. Recent advances n 3D scannng technologes have made possble rapd and cost-effectve acquston of dense pont clouds for 3D reconstructon. As the pont clouds provde explct 3D nformaton, they have a much hgher potental for the automaton of reconstructon. However, due to the measurement prncple employed by laser scanners and ther lmted pont densty, the nformaton on sharp edges s not very relable. It s precsely where mages have superorty over pont clouds. In addton mages are requred for vsual nterpretaton, texture mappng, and modellng parts not vsble n the pont clouds. Moreover, mage acquston s more flexble, and the cost and tme requred for t s much lower than that of laser scannng, makng ther combned use essental for a cost-effectve soluton. These reasons led us to develop a modellng strategy that uses both mages and pont clouds n combnaton wth a lbrary of CAD prmtves found n ndustral scenaros represented as CSG (Constructve Sold Geometry) objects. The modellng ppelne n our algorthm starts from pont clouds as the man data source for automaton. Frst of all we segment the pont cloud usng surface smoothness and detect smple objects lke planes and cylnders usng Hough Transform. Ths s followed by fttng of CSG objects to a combnaton of segments. These ftted CAD models are used as regstraton targets for addng more scans to the project. Addtonally, by fttng the projected edges to mage gradents we regster mages to pont clouds. Once we have a regstered data set, manual measurements are added to mages to model mssng parts and to ncrease the relablty of modellng for portons where laser data s known to be nosy. The fnal phase s smlar to bundle adjustment n tradtonal Photogrammetry as there we estmate pose and shape parameters of all CSG objects usng all mage measurements and ponts clouds smultaneously. We name ths fnal phase Integrated Adjustment as t ntegrates all avalable nformaton to determne the unknown parameters. The results of applyng ths method to data from an ndustral ste are presented showng the complementary nature of pont cloud and mage data. An analyss of mprovement n qualty of 3D reconstructon shows the benefts of the adopted approach.. INTRODUCTION As bult CAD models of ndustral stes are requred for many purposes lke mantenance, documentaton, and tranng. Moreover, current research s focusng on applyng Vrtual and Augmented Realty for provdng varous servces for tranng and operaton n ndustral envronments. The mplementaton of these technologes requres accurate 3D models of ndustral envronments for varous sub-processes lke trackng and algnment of vrtual and real objects. One such project on whch the Secton of Photogrammetry and Remote Sensng at TU Delft has been workng snce 00 focuses on usng Augmented Realty for provdng varous tranng servces to ndustral users (STAR, 003). In contrast to Vrtual Realty where everythng has to be modelled explctly, Augmented Realty s more flexble as t uses a mxture of real-tme vdeo and vrtual objects and humans. As a result more realstc scenaros and servces can be easly mplemented wthout requrng explct 3D models for all the objects. At the same tme the requrement for algnng the vdeo frames to 3D objects becomes more crtcal. Ths necesstates more accurate 3D geometrc models for the objects present n the surroundng envronment, whch are used as targets durng trackng and algnment. Tradtonal technques for modellng ndustral envronments use pont and lne Photogrammetry as t s much faster and convenent compared to manual surveys. An mproved approach sutable for Photogrammetrc modellng of ndustral envronments was presented by (Vosselman et al., 003) whch uses fttng of mage edge measurements to back-projected contours of the CSG object n the mage. Ths elmnates the measurement to CAD model converson stage, whch s requred for pont and lne Photogrammetry based approaches. Addtonally, ncluson of varous nternal and external geometrc and parametrc constrants greatly reduces the degrees of freedom. Thus the number of the requred manual measurements s also reduced. Stll, ths process requres a lot of manual work, whch s the major cost n any modellng project. The prospects of mplementng any automatc strategy for ndustral envronments usng only mages are very dm; there are three man reasons for that. Frstly, there s no explct 3D nformaton n mages; at least two mages havng good * Correspondng author.

2 Doman specfc knowledge Data base of CAD prmtves ther Features and Geometrc constrants Range data Segmentaton D Image Object Recognton Object-based Regstraton Constrant detecton Integrated Adjustment Ft fnal CSG models Manual Semautomatc Automatc modes User Input Fgure. Flowchart of the modellng ppelne ntersecton are needed along wth the nformaton about correspondng ponts to calculate 3D coordnates. Secondly, the clutter whch s a unversal feature of most ndustral stes combned wth uncontrolled lghtng makes the automatc object detecton much more dffcult. Thrdly, the nformaton n mages s contaned manly on the edges, as there the contrast s usually much better, whle n the absence of some dstngushable marks the nformaton about the 3D geometry of the rest of the surface s at best mnmal. Ths becomes a major lmtaton as n ndustral envronments the curved objects are unversally present, and for these objects only edge-localzed nformaton s not enough for automatc detecton and fttng. All these lmtaton are successfully resolved by laser scannng technques, where we get drect 3D nformaton n the form of a pont cloud makng the job of object detecton and fttng much easer. Furthermore, recent advances n 3D scannng technologes have made possble hgh-speed acquston of dense and accurate pont clouds at moderate costs (Laser Scanner Survey, 003). The strengths of Laser Scannng do not mean that mages lose all ther utlty. Actually the fact that mages provde accurate nformaton on edges becomes a source of strength f both mage and pont cloud data are smultaneously used. Most of the currently avalable laser scanners are usng technques based on ether trangulaton or tme of flght. In pont clouds acqured usng ether of them the data on edges s noser compared to that on the surface of the scanned object. Ths has to do wth the angle between the surface normal and the laser beam, whch changes very rapdly near the edges of the object, makng precse pont cloud acquston very dffcult. Addtonally, n the case of tme of flght multple reflected pulses lead to averagng of range measurements. Ths s especally true for the step edges. These lmtatons of scannng technologes make mages a complementary source of nformaton, especally on the edges of the objects. Acquston of mages as a supportng data source s not a problem, as cameras are stll much faster and cheaper than currently avalable laser scanners. Addtonally, mages are much better for vsual nterpretaton and are requred for producng texture-mapped models for realstc vsualzaton. Based on the above observaton of the complementary nature of mages and pont clouds the modellng strategy that we have developed uses both data sources smultaneously and thus explots all avalable nformaton to acheve a more accurate estmaton as well as hgher levels of automaton. The rest of the paper s organsed as follows. In Secton, we gve a summary of the modellng ppelne. Secton 3 provdes detals of fttng CSG objects to pont clouds and to mage edges. We present fttng results on an ndustral scenaro n Secton 4, along wth a dscusson about the mprovement n estmaton accuracy usng two experments of fttng on sngle objects. Fnally, we conclude n Secton 5 and propose some drectons for future work.. MODELLING PIPELINE Our modellng ppelne s shown n Fg.. As t was noted n the ntroducton we are usng both mages and pont clouds as data sources. We start from an ntal approxmate scan-to-scan regstraton usng Iteratve Closest Pont method (Besl and McKay, 99). The regstraton obtaned from ths preprocessng stage s used untl objects have been recognzed and ftted. Then ths ntal regstraton s refned n the fnal Integrated Adjustment usng object-based regstraton (Djkman and Heuvel, 003). For mage regstraton or exteror orentaton, mage edge to back-projected CSG model contour fttng s used, durng whch only mage exteror orentaton parameters are adjusted and object parameters of the modelled objects are kept fxed. For next stages of segmentaton and object recognton only pont cloud data s used, as n contrast to mages t provdes explct 3D nformaton, and thus has better chances of achevng automaton. Ths s especally true for the reconstructon of ndustral stes as due to ther man-made orgn presence of well-defned CAD prmtves can be expected. For example as reported by Nourse et al. (980) 85% of objects found n ndustral scenes can be approxmated by planes, spheres, cones and cylnders. Ths percentage rses to 95% f torodal surfaces are ncluded n the set of avalable prmtves (Requcha and Voelcker, 98; Pettjean, 00). Usng pont clouds we take a two-step approach, consstng of segmentaton followed by Hough transform based object detecton. In the frst step we use a smple regon growng based segmentaton usng what we call Smoothness Constrant. It s based on the assumpton that most of the surfaces n ndustral envronments can be expected to be smooth wth ther surface normals changng rapdly only on the object edges. Frst of all we estmate the surface normal for each pont n the pont cloud usng plane fttng to the ponts wthn a small neghbourhood. Ths s followed by the stage of regon growng n whch we keep on addng ponts to one regon untl the angle between normals exceeds a specfed threshold. Actually, segmentaton and object recognton are two related problems, because f we know the type and locaton of objects, segmentaton s reduced to selectng ponts havng a low dstance from the object surface; and smlarly f we have a perfect segmentaton, the object recognton s just a matter of surface fttng and fndng the surface whch gves mnmum error of ft. Most of the segmentaton approaches to date haven t been able to acheve a hgh success rate (Hoover et al. 996; Mn et al., 000). The segmentaton approach we use leads usually to undersegmented results, wth multple objects beng assgned to one segment. The followng object recognton stage detects the planes and cylnders present n the segments usng a Hough Transform. As presence of multple objects and outlers s not a problem for the Hough transform we are able to recover from the errors of the precedng stage of segmentaton. The object

3 detecton s currently lmted to cylnders and planes, but n many ndustral envronments they can account for more than 70% of the objects. Next s the Constrant Detecton stage where objects whch have been multply detected are combned together, the cylnders whch mght be connected are found, and the presence of curves between ppes n proxmty s hypotheszed, and then checked aganst the pont cloud. The process of combnng varous segments and assgnng them to CSG objects from the Object Catalogue s currently manual. But n future, we plan to make t automatc. The next stage of surface fttng assumes that the combnaton of precedng segmentaton and object recognton stages have resulted n correctly labelled ponts, and we know whch ponts belong to whch CSG model. Smlarly n mages the pont measurements (ether manual or automatc) are correctly assgned to ther correspondng CSG objects. The detals of fttng the selected CSG objects to the pont cloud and the mage measurements as well as ther combnaton are dscussed n the followng sectons. 3. MODEL FITTING 3. Fttng of CSG model to pont clouds The problem we are addressng can be formulated as follows. We have a set of ponts, whch are sampled from some object that can be approxmated by the gven CSG model. We want to estmate those values of the parameters for the CSG object, whch mnmze the sum of the squares of the orthogonal dstance of the ponts from the surface of the model.e., N mn Ω [ p, Γ( τ, τ,, τm)] () = Ω defnes the shortest dstance of a gven pont p to the surface of the CSG model Γ whch has M shape and pose parameters gven by τ, τ,, τm. The pont cloud conssts of N ponts, p, p,, pn (Fg. (a)). To solve ths non-lnear least-squares problem we need a method to fnd the value of the dstance functon Ω n () and ts partal dervatves wth respect to the CSG parameters.e. Ω, Ω Ω,, () τ τ τ M The calculaton of Ω s a dffcult problem, as due to the bounded surfaces used by the CSG specfcaton t s not possble to have closed form analytcal expressons. For a comparson of dfferent numercal methods for ts computaton we refer the reader to Rabban and van den Heuvel (004). Here we use ACIS (004), whch s a commercal geometrc modellng engne to compute Ω. Smlarly, the partal dervatves are estmated numercally usng fnte dfferences. As noted by Denns and Schnabel (996), for suffcently small step-sze, the results obtaned from the fnte dfference approxmaton of the partal dervatves for the least-squares soluton are ndstngushable from the analytcal ones. For mnmzng the functon () wth respect to parameters of the CSG model we use Levenberg-Marquardt method (Björk, 996; Press et al., 996). Startng from an ntal estmate of CSG parameters Γ 0, at each teraton we get an adjustment gven by: (a) Fg. : Calculaton of dstances for fttng (a) Ω for a Pont Cloud, the model s shown n yellow, the red arrows from green ponts to model surface ndcate the dstance Ψ for an Image, the measurements are n green, the back-projected model s n yellow, and red arrows ndcate ther dstance n mage space. D =Ω( p, Γ0) (6) Ω s the dstance of the th pont from the CSG surface, and k τ s the T T Γ = ( JJ +λi) ( JD ) (3) Γ =Γ0 Γ (4) where J s the Jacoban matrx and D s the dstance vector Ω kth parameter of the CSG tree. In (3) above λ s the Levenberg-Marquardt parameter. When λ = 0 Newton step s taken whle for λ results n steepest descent step. We are usng quaternons (Shoemake 985) for the specfcaton of rotaton as they provde a sngularty free representaton. Ths means we have four rotaton parameters wth one constrant.e.: q + q + q + q = (7) 3 4 The constrant n (7) cannot be enforced durng the adjustment, as Levenberg Marquardt s an unconstraned optmzaton method. Ths means that we have an over-parametersaton and the resultng matrx of normal equatons can be sngular. To avod the resultng numercal problems we use Sngular Value Decomposton (Golub, 996) for nvertng the matrx n (3). Ths way f there s a rank defcency we take the column correspondng to mnmum sngular value out of the matrx system and thus get a mnmum norm soluton. 3. Fttng of CSG Model to mages J k = (5) τ k The use of CAD models for fttng to mages was poneered by Lowe (99). He estmated the pose and the shape parameters by mnmzng the dstance of the vsble edges from the hddenlne projecton of the estmated model. Vosselman et al. (003) extended and modfed ths approach for fttng CSG objects to mage gradents and pont measurements for ndustral reconstructon. They also used nternal and external geometrc constrants to reduce the number of degrees of freedom and thus the requred mage measurements. We follow ther fttng approach for mages, wth one excepton that we don t know a pror the correspondence between mage measurements and back projected edges of the CSG model. Due to ths mssng nformaton we follow an teratve procedure, where before each teraton for fttng, the measurements are assgned to the closest edge.

4 (a) The problem of mnmzng Ψ s also a non-lnear least squares problem and s smlar to that of mnmzng Ω dscussed n the above secton. We need partal dervatves wth respect to CSG parameters.e., Ψ, Ψ Ψ,,. τ τ τ M Although analytc expressons for the estmaton of the partal dervatves for some of the CSG objects are gven by Ermes et al (999) here we estmate them numercally usng fnte dfferences. The fnal estmaton uses Levenberg-Marquardt n combnaton wth Sngular Value Decomposton. The detals are smlar to the ones dscussed n Secton FITTING EXPERIMENTS (c) (d) Fg. 3: Cylnder fttng experment (a) Pont Cloud (b-d) Images wth back-projected model n yellow, pont measurements n red, and sub-sampled pont cloud n whte Each pont measurement n the mage gves us a ray n 3D. Gven a set of mages wth measured ponts we want to estmate those values for CSG parameters that result n the mnmum dstance between all these rays and the estmated CSG model. Alternately, the ray to body dstance can be calculated n mage space. There we have to compute the dstance n pxels between an mage measurement and the closest back-projected contour of the CSG model. The back projecton must have a mechansm for hdden-lne removal, so that the effects of self and external occlusons are taken nto account. We follow the second approach, and use ACIS (004), whch a commercal geometrc modellng engne to compute the hdden lne projecton of the model n the mage. For an example see Fg.. Thus the fttng problem reduces to the estmaton of those values of the CSG parameters, whch mnmze the sum of the squares of the orthogonal dstance of the mage measurements from the back-projected edges of the model n the mage.e., N mn Ψ [ m, Η( τ, τ,, τm)] (8) = Where Ψ defnes the shortest dstance of a gven measurement m n an mage to the closest edge of the back projected CSG model Η, whch has M shape and pose parameters gven by τ, τ,, τm. There are N mage measurements gven by Parame Image ter 3 m, m,, mn. Pont Cloud Both X Y Z t E-.0E0.8E-3.0E-3 5 t E-.E 3.96E-3.9E-3 6 t Length Radus Table : Standard devaton for Cylnder fttng experment As t was sad n the ntroducton, mages and pont clouds provde complementary sources of nformaton, and by ther combnaton we can expect better estmaton accuracy. Edges of the object where laser scanner usually provde nosy data are captured best n the mages. Addtonally, whle fttng bounded objects pont clouds do not contan enough nformaton about determnng the bounds, whereas by provdng the full edge outlne mages fx the bounds. For example n the case of a cylnder usually the closng lds on both sdes are not scanned ether because they are not vsble due to the connectons wth other surroundng objects, or because t s not convenent to place the scanner n a poston where the lds are vsble. As a result we expect the length of the cylnder to be poorly determned by such a pont cloud. In contrast the measurements n the mage provde ponts on the edges and thus help mprove the precson of the length estmate. To demonstrate the complementary nature of the nformaton comng from mages and pont clouds we wll do some fttng experments on two test objects. Each object wll be ftted three tmes, frst usng only pont cloud, then usng only mage measurements and fnally a combnaton of both. The pont clouds we wll use were captured usng a Cyrax scanner. We assume standard devaton of 5mm for each pont. The mages were captured usng a Nkon CoolPx camera havng a resoluton of 5 mega pxels and usng a fxed focal length of 7.34 mm. The standard devaton for mage measurements s taken to be pxel. 4. Cylnder fttng The arrangement we used for the frst experment s shown n Fg. 3. A cylnder s scanned from the front, and mages are taken from three dfferent postons. We see back-projected hdden lnes n yellow, ponts measured on edges n red, whle the sub-sampled pont cloud s shown n whte. A cylnder s represented by 8 parameters, 3 for the poston, 3 for the axs, one for the radus and one for the length. In Table we see the standard devatons obtaned for dfferent parameters by dong fttng to pont clouds, mages and to a combnaton of both. For mages we dd fttng separately usng one, two and three mages, whle n case of both all of the three mages were used. As expected n the case of usng only pont cloud the length of cylnder s not determned because n the absence of ponts on upper and lower lds there s not enough nformaton n the pont cloud for ts determnaton. Because we use sngular value decomposton the length parameter s taken out of the estmaton durng matrx nverson and thus ts value remans fxed on the ntal startng pont ths results n standard devaton of for length.

5 4. Box Fttng (a) The second example s that of a box, wth only two of ts faces fully scanned. Addtonally, three mages are taken from dfferent postons (Fg.4). The box has 0 parameters, 3 for the poston, 4 for the rotaton, and 3 for the szes. Agan, smlar to the example of cylnder dscussed above, we have an overparametersaton for rotaton, as we use 4 nstead of requred 3 parameters, and cannot enforce the constrant. Agan we fnd a very hgh correlaton between dfferent q parameters that lowers the confdence n the otherwse low standard devaton values. For example the correlaton between q0 and q s 0.5. (c) (d) Fg. 4: Box fttng experment (a) Pont Cloud (b-d) Images wth back-projected model n yellow, pont measurements n red, and sub-sampled pont cloud n whte Para meter X Y Z q0 q q q3 X sze Y sze Z sze Images e-.4e-3 7.8e-4 4.3e e- 5.0e-4 3.0e-4.6e e-.3e-4 6.0e-5 3.4e Pont Cloud Both e-.40e-4 5.0E E e-.0e-4 5.0e-5.0e Table : Standard devaton for Box fttng experment As the z-axs s algned wth the length of the cylnder there s a very hgh correlaton between both of them. As a result the estmaton of z-poston s also very weak compared to the estmaton of x and y poston. But f we combne the pont cloud wth measurements from the mages (Table, column Both ) the stuaton mproves dramatcally as the edges n the mages provde enough nformaton about the length and the resultng standard devatons are much lower, ndcatng much better estmaton precson. Cylnder axs can be specfed usng two parameters, but as we are usng 3 wthout enforcng the constrant there s an overparametersaton. Although the standard devaton of axs parameters look qute good, but due to over parametersaton ther correlaton s very hgh. For example the correlaton between t and t s 0.5, whch ndcates that the low values of standard devatons are due to some numercal effects. As expected as we use more mages the standard devaton of parameter estmaton goes down. It also shows that even a sngle mage n combnaton wth a good scan can lead to sgnfcant mprovement n the estmaton of those parameters whch are not well-determned from the pont cloud. In the absence of ponts on all faces of the box, t s not possble to relably determne the sze parameters of the box. That s what we see n the standard devaton resultng from fttng usng only pont clouds (Table ), where the standard devaton for y and z szes s meanng that they could not be estmated. The value of standard devaton for x sze s low only because of the coordnate system chosen for the box, whch has ts orgn n the left corner. Ths fxes the poston of rght sde and thus the x sze s also determned. Due to hgh correlaton between zposton and z-sze, ts estmaton s also bad. Once agan, we see from the last column of the Table that the ncluson of mage measurements leads to a much better estmaton of sze and poston parameters. Both of these examples prove our thess, that although pont clouds contan drect 3D nformaton, whch s very useful for automatc object recognton, the fnal adjustment must use a combnaton of both data sources to account for mssng or nosy nformaton n pont clouds. 4.3 Modellng of an ndustral ste We appled the presented methodology for makng 3D model of an ndustral ste shown n Fg. 5. Seven scans were made usng a Cyrax laser scanner. Each scan conssted of one mllon ponts wth a standard devaton of 5mm. Addtonally about 60 mages were taken from dfferent postons. Followng the modellng ppelne dscussed n Secton we started wth approxmate regstraton usng ICP. The approxmately regstered scan was segmented usng Smoothness constrant based regon growng. Cylnders and planes were automatcally detected usng the Hough transform, and then used to refne the scan-to-scan regstraton. For mages the orentaton was approxmated usng vanshng ponts. Ths was followed by scan-to-mage regstraton usng a few mage measurements and keepng all object parameters fxed, whle estmatng only exteror orentaton parameters of the mages. The process of combnng automatcally detected cylnders and planes to full CSG objects as well as the process of addng measurements to mages was done manually. Once we have mage measurements as well as segmented ponts clouds, we proceed wth the Integrated adjustment usng both data sources smultaneously. Ths ntegrated adjustment mnmzes the sum of square of the dstances of pont cloud from the model surface and sum of square of the mage measurement dstance from the back projected edges of the model, whle estmatng the pose and shape parameters of the CSG object as well as the regstraton parameters of the ndvdual scans and exteror orentaton of the mages. Ths process s an extenson of the dea of bundle adjustment n tradtonal Photogrammetry but s

6 Second Ed., Phladelpha. Djkman, S.T., Heuvel, F.A. van den, 00. Sem automatc regstraton of laser scanner data. Internatonal Archves of Photogrammetry and Remote Sensng, 34(5), pp - 7 Ermes, J.P.A.M., Heuvel, F.A. van den, Vosselman, G., 999. A Photogrammetre Measurement Method usng CSG Models. Internatonal Conference proceedng on Measurement, Object modellng and documentaton n archtecture and ndustry, Thessalonk, , IAPRS, 3(5), pp (a) Golub, G.H., Van Loan, C.F., 996. Matrx Computatons, 3rd Ed. The Johns Hopkns Unversty Press. Hoover, A., Jean-Baptste, G., Jang, X., Flynn, P.J., Bunke, H., Goldgof, D.B., Bowyer, K., Eggert, D.W., Ftzgbbon, A., and Fsher, R.B., 996. An expermental comparson of range mage segmentaton algorthms. IEEE Transactons on Pattern Analyss and Machne Intellgence, 8 (7), (c) (d) Fg. 5: Modellng an ndustral ste (a) Pont Cloud (b-c) Images (d) Fnal Model much more general, as t uses both pont cloud and mage measurements n bg adjustment. Ths ntegrated adjustment was appled to the test scenaro shown n Fg. 5. Only cylnders, boxes and tor were used from the catalogue of CSG objects. The results of the fttng are shown as a 3D Model n Fg. 5(d). 5. CONCLUSIONS We have presented a modellng technque for fttng CAD models descrbed as CSG prmtves to measurements n mages and pont clouds. Whle the pont clouds are excellent for automatc object recognton, the comparson of mprovement n the standard devaton of the estmated parameters clearly shows that mages have a complementary role as they provde more nformaton on the edges and help fx the bounds of models where pont clouds fal to do so. In future we plan to extend the fttng procedure from pont measurements to edge and curve measurements n mages. Smlarly dfferent strateges for automatc object recognton n ndustral envronments usng a combnaton of magery and pont clouds wll also be nvestgated. REFERENCES: ACIS, 004. The 3D ACIS Modeler (ACIS) (Last accessed Aprl 9, 004) Besl, P., and McKay, N., 99. A method for Regstraton of 3D shapes. In Pattern Analyss and Machne Intellgence, 4(), Björk, Å., 996. Numercal Methods for Least Squares Problems. Socety for Industral and Appled Mathematcs, Phladelpha. Denns, J.E. and Schnabel, R.B., 996 Numercal Methods for Unconstraned Optmzaton and Nonlnear Equatons, SIAM, Laser Scanner Survey, 003 (Conducted by POB Magazne). (Accessed March, 004). Lowe, D. G., 99. Fttng parametrzed three-dmensonal models to mages. PAMI, 3(5), Mn, J., Powell, M.W., Bowyer, K.W., 000. Progress n automated evaluaton of curved-surface range mage segmentaton. In Internatonal Conference on Pattern Recognton (ICPR), Barcelona, Span, pp Nourse, B., Hakala, D., Hllyard, R., Malrason,P., 980. Natural quadrcs n mechancal desgn.in Proc. of Autofact West, Anahem, CA., Pettjean, S., 00. A survey of methods for recoverng quadrcs n trangle meshes. In ACM Computng Surveys, 34(), pp. -6. Press, W.H., Teukolsky S.A., Vetterlng W.T., Flannery B.P., 996. Numercal Recpes n C: The Art of Scentfc Computng, nd ed. Cambrdge Unversty Press, Cambrdge, U.K. Rabban, T., Heuvel, F.A. van den, 004. Methods For Fttng CSG Models to Pont Clouds and Ther Comparson. The 7th IASTED Internatonal Conference on Computer Graphcs And Imagng, August 7-9, 004, Kaua, Hawa, USA. Requcha, A., Voelcker, H., 98. Sold modelng: a hstorcal summary and contemporary assessment. IEEE Comput. Graph. Appl. (), 9-4. Shoemake, Ken, 985. Anmatng rotaton wth quaternon curves. Proceedngs of the th annual conference on Computer STAR-Servces and Tranng through Augmented Realty. (Accessed Nov, 003) Vosselman, G., Tangelder, J.W.H., Ermes, P., Heuvel van den, F. A., 003. CAD-Based Photogrammetry for Reverse Engneerng of Industral Installaton. Computer-Aded Cvl and Infrastructure Engneerng, Vol. 8 pp