CLOSE RANGE DIGITAL PHOTOGRAMMETRY AND SOFTWARE APPLICATION DEVELOPMENT FOR PLANAR PATTERNS COMPUTATION

Transcription

1 CLOSE RNGE DIGITL PHOTOGRMMETRY ND SOFTWRE PPLICTION DEVELOPMENT FOR PLNR PTTERNS COMPUTTION FOTOGRMETRÍ TERRESTRE DIGITL Y PLICCIÓN SOFTWRE PR EL DESRROLLO DE PTRONES PLNOS PEDRO RIS Dep. Nturl Resources nd Environmentl Engineering, University of Vigo, Spin, pris@uvigo.es JULI RMESTO Dep. Nturl Resources nd Environmentl Engineering, University of Vigo, Spin, juli@uvigo.es JVIER VLLEJO Dep. Engineering Design nd Technicl Projects. Superior School of Engineering.Bilbo, Spin, eppvmj@bi.ehu.es HENRIQUE LORENZO Dep. Nturl Resources nd Environmentl Engineering, University of Vigo, Spin, hlorenzo@uvigo.es Recibido pr revisr diciembre 29 de 28, ceptdo bril 13 de 29, versión finl myo 26 de 29 BSTRCT: This pper shows there processes for the prticulr cse of objects with complex surfces: the 3D modelling of the object nd its development to 2D model. The procedure to obtin ccurte 3D CD models of objects with complex surfces through close rnge digitl photogrmmetry is described; severl methodologicl considertions re presented tht llow obtining high ccurcy results nd minimize processing time. Furthermore softwre ppliction tht clcultes the corresponding plnr profiles is developed. The ppliction utomticlly outputs CD neutrl file with the definition of the optimized plnr ptterns needed for the NC cutting mchines. The computtion of the plnr ptterns is bsed on the solution of n optimiztion problem tht tkes into considertion the geodesic curvture of the surfce of the isoprmetric curves. KEYWORDS: 3D modelling, Close Rnge Digitl Photogrmmetry, complex surfces, cutting profiles. RESUMEN: Este rtículo muestr estos procesos pr el cso prticulr de objetos con superficies complejs: el modeldo 3D de objetos y su desrrollo modelos 2D. Se describe el procedimiento pr obtener modelos CD 3D precisos de objetos con superficies complejs trvés de fotogrmetrí terrestre digitl; se relizn vris considerciones metodológics que permitirán obtener resultdos con lt precisión y minimizdo de los tiempos de procesdo. demás fue progrmd un plicción softwre pr el desrrollo de superficies complejs en el plno. L plicción proporcion utomáticmente rchivos CD estándr, definiendo ptrones plnos optimizdos necesrios pr los sistems CNC. El cálculo de estos ptrones plnos está bsdo en l solución de un problem de optimizción que tiene en cuent l curvtur geodésic de ls superficies de curvs isoprmétrics. PLBRS CLVE: Modeldo 3D, Fotogrmetrí Terrestre Digitl, Superficies Complejs, cutting profiles. 1. INTRODUCTION The only wy Europen mnufcturing compnies cn succeed in globl mrketplce is by implementing newer mnufcturing processes tht enble them to respond quickly to mrket chnges nd provide high qulity, low cost products. Dyn, ño 76, Nro. 16, pp Medellín, Diciembre de 29. ISSN

2 8 ris et l utomtion is one mjor step towrds chieving the qulity, cost nd delivery trgets. Due to cost effectiveness nd qulity results, CD/CM nd CNC mchining re replcing the old mnul or semi-utomted mchinery in wide rnge of industries. Frequently, it is common tht the design drwings corresponding to n object or prototype to be mnufctured through CNC system re not vilble. In such cses, reverse engineering techniques must be used in order to generte 3D digitl model of the object. fterwrds, this model is developed to obtin the cutting plnr ptterns tht will llow us to progrm the CNC mchinery opertions. The 3D modelling nd its development into plnr profiles re key processes for the utomtion nd the relibility of this type of mnufcturing process when it ffects objects with complex surfces. The trouble of creting development profiles of 3D surfces with double curvture to be mnufctured t lter time by forming plnr pieces of metl sheets hve been wide treted in the design community of industril products since it is present in gret vriety of industril res: utomtion, ircrft, nvigtion, boilermking, inclusive in the textile nd the shoemking industry [1] [2]. The current CNC mnufcturing systems long with CD/CM systems re becoming one of the mjor expnsion nd renovtion res in this type of industries. But the implementtion of CNC cutting mchinery in the flt sheet mnufcturing systems brings serious problems when trying to mnufcture complex geometry objects, since defining the optiml form of the cutting profiles, which is essentil to progrm the cutting pth, rises serious difficulties. Furthermore, when originl design drwings of the objects or prototypes to be mnufctured re not vilble, it is necessry to crry out previous modelling phse with the purpose to get the precise knowledge of the object geometry nd dimensions. The 3D modelling nd its development into plnr profiles re key processes for the utomtion nd the relibility of this type of mnufcturing process when it ffects objects with complex surfces [3]. In recent yers, reserch efforts led to object modelling with complex surfces in the industril ppliction fields hve been focused on lser scnning technology [4] [5]. However, surfce reconstruction from rndom scnned dt usully requires complex time consuming reconstruction process nd significnt mount of computtion [6]. These considertions hve led us to concentrte in evluting the fesibility of pplying digitl monoscopic photogrmmetry in modelling objects with complex geometry. Some interesting previous works cn be found in [7]- [11]. This pper is focused on the monoscopic digitl photogrmmetry since it hs severl dvntges such s: equipment cost, flexibility, simplicity nd clcultion volume dt processing [12]. In our opinion, these fctors mke it more vible in industril pplictions in terms of cost-efficiency blnce point of view thn the stereophotogrmmetric techniques or single imge bsed processes [13]. Severl methodologicl indictions re estblished tht llow us to optimize the time required for the dt processing nd the finl 3D model ccurcy. Furthermore, custom ppliction hs been developed tht implements n efficient lgorithm for the plnr development, for the CD model obtined from the photogrmmetric procedure. This ppliction llows us to obtin CD file tht hs the definition of the optimum cutting profiles of the curved surfces. The mthemticl principles of this ppliction re shown. The method is pplied over wooden sculpture tht hs simple nd complex curved surfces (see figure 1). Figure 1. The methodology ws pplied over wooden sculpture tht hs simple nd complex curved surfces

3 Dyn 16, METHOD 2.1 Photogrm Metric Procedure: 3D Modelling Close rnge photogrmmetry bsiclly involves the use of network of photogrphs of n object tken from different ngles. In our cse study, the photogrphy network ws designed considering the prmeters tht determine the strength of the geometry ccording to [14]. digitl clibrted cmer, Cnon EOS 1D of pixels CCD resolution ws used. The interior orienttion prmeters re showed in tble 1. Tble 1. Interior orienttion prmeters of the cmer used Focl Length f (mm) 2, Principl Point Rdil Distortion Tngentil Distortion x (mm) 11,16145 y (mm) 7,52451 R 1 R 2 P 1 P 2 2,26E-4-4.9E-7 4.E-5 8.E-6 The sculpture ws plced on supported bse covered by 1 cm grid (see figure 2). the pckge Photomodeler Pro 5. [15]. The system is used for the digitl orienttion nd restitution of photogrphs. It enbles us to work with pictures obtined either with digitl cmers. The results obtined cn be of two types: numeric for qulity control of results (in SCII formt) nd grphic (DXF, 3D Studio, RW, 3dm nd Iges). The cross points within the grid nd the sculpture vertex re tken s trget points for the cmer poses computtion. dditionlly, grid of dots set t 5 cm mong them ws projected over the complex curved surfces for inner point restitution. The photogrphs were tken ll round the object from distnce of 1 meter, keeping the convergence of rys t 4º to 9º. The imge mesurement ws mde s mnul process for the identifiction of common pixels in different imges in the externl orienttion, s well s in the line drwing (points nd lines). The process of imge mesurement ws utomtic from grid of dots set projected over the complex curved surfces. The inner points in this kind of surfces (nd due the texture of the model) were clculted using structure light system. The photogrmmetric system used hs implemented lgorithms to clculte Crtesin coordintes of the grid of dots set with shpe of circle or ellipse. The structure light system ws builded using little serchlight with grid of dots built-in. The problem of obtining digitl model tht properly defines the geometry of the object from digitl imges cn be stted s follows: (X i, Y i, Z i ) of set of n points within the object surfce must be clculted in common Crtesin system given the imge coordintes (x i, y i ); then surfce S might be found tht pproximtes or interpoltes the set of points. Figure 2. The wooden sculpture ws plced on supported bse covered by 1 cm grid s photogrmmetric system we hve used monoscopic digitl photogrmmetric softwre, There is perspective reltionship between the 3D co-ordintes in the object spce nd the 2D corresponding coordintes in the imge plne, since n imge point i (x i, y i, -f), the perspective centre O of the cmer (X o, Y o, Z o ) nd the object point i (X i, Y i, Z i ), re over the sme line i oi. This reltionship cn be written mthemticlly ccording to eqution (1).

4 1 ris et l x y r = f r r = f r ( X X ) + r12( Y Y) + r13( Z Z ) ( X X ) + r32( Y Y) + r33( Z Z ) ( X X ) + r22( Y Y) + r23( Z Z ) ( X X ) + r ( Y Y ) + r ( Z Z ) (1) X Y X Y = = ( Z Z ) ( Z Z ) r11x + r12 y + r13 f r x + r y + r f r21x + r22 y + r23 f r x + r y + r f (2) Where (x, y) re imge co-ordintes of trget points, f, the principl distnce; (X, Y, Z) re object spce co-ordintes of trget points; (X, Y, Z ) re object spce co-ordintes of the projection centre; r ij re the components of the rottion mtrix R, which is function of the three rottion ngles ω, κ, φ tht define the sptil rottion between the object spce system to the imge co-ordinte system. The photogrmmetric model genertion involves the previous computtion of the six orienttion prmeters corresponding to the position (spce co-ordintes) nd orienttion (rottion ngles) of the cmer projection centre in Crtesin coordinte system for ech imge plne. For this end, the trget points in the object spce re mtched with the corresponding ones in the imge plnes of the network. If number enough of trget points re used, the six unknowns for ech imge plne cn be found. Generlly more thn the necessry minimum re used nd the prmeters re estimted by the bundle djustment functionl model, which is bsed on lest squres estimtion (for further informtion see [16]). The bundle djustment procedure ws crried out using, 12 control points locted on the supported bse tht llowed to scle the 3D model nd to fix the reference system, 17 imges in digitl formt of the sculpture. fter the bundle djustment the photogrmmetric system used to guge project qulity the vlue clled σ (or more properly priori σ ), clculted like the vrinces of the residuls. The vlue of σ ws estimted in.85 pixels. The computtion of the sptil 3D co-ordintes of the object vertexes nd the projected dots within the doubly curved surfce is nlogue to the method described for the cmer poses clcultion, see eqution (2). Now the unknowns re the co-ordintes (X, Y, Z ) of the points, nd they re computed given the corresponding (x, y ) imge co-ordintes nd the co-ordintes nd rottion ngles of the cmer projection centre, which were obtined for ech imge plne in the prior stge. In order to chieve enough level of redundncy mesurements, which is essentil for obtining high ccurcy results ccording to [14], more thn three imges were used for the restitution of ech point. The system is lso solved by bundle djustment, in this lst process the photogrmmetric system llow us to djust together the co-ordintes of the grid of dots set with the exterior orienttion prmeters. The photogrmmetric softwre used is convergent monoscopic multisttion, this system is bsed in process of intersection of rys common for the clculte of co-ordintes. This process is used for the restitution of ll points, including the grid of dots. Reconstruction of flt surfces is performed by fitting flt surfce to sets of coplnr points nd edges, in such wy tht the norml distnce from the surfce to the vertexes is minimized. Constrictions concerning prllel nd perpendiculr conditions of edges re previously estblished in order to minimize errors in the points restitution process. For the reconstruction of the curved surfces Deluny tringultion is performed tht pproximtes them. The tringulr mesh is then smoothed in order to get better fitting with the originl surfce. 2.2 Surfce Flttening Method The procedure implemented here nd described in points nd 2.2.3, is bsed on the zridis nd sprgthos method (see [17], [18] for detils).

5 Dyn 16, ppliction plug-in development The computtion of the plnr profiles of the developed surfces strts from the 3D model definition obtined s result of the photogrmmetric process. The fces of the model defined with developble surfces (Gussin curvture = ) re esily developed using the functionlity of commercil CD ppliction. However, this is not possible for the fces with doubly curved geometry tht cnnot be flttened into mp without stretching, tering, or compressing itself. When Gussin curvture is not very high, n optimiztion procedure could be proposed tht provides fir pproximtion to the shpe of the plnr pttern minimizing the deformtion of the plte. The mthemtics of this method tkes into ccount the isometric development of the isoprmetric curves with geodesic curvture preservtion. Mnipultion nd complex dt computtion of model geometry ws required therefore the ppliction for plnr ptterns computtion ws conceived s custom plug-in to CD ppliction. plug-in is softwre module tht extends the functionlity dding new commnds or fetures to the ppliction. The ppliction plug-in ws developed with the Rhino 3. SDK, n PI tht llows the ccess to the functions of Rhinoceros 3. CD ppliction. This plug-in is build with Microsoft Visul C++ 6. s Windows DLL (Dynmic Link Librry) tht could be executed from Rhino s commnd line. neutrl file of the model is generted in order to mnge its geometry into the CD ppliction. Rhino s SDK provides ll the functionlity needed to select the surfce ptches nd ccess the dt structure of the model, computing position, norml nd tngent vectors, needed for the mthemticl procedure The resulting plnr pttern obtined with the plug-in is geometricl entity of the CD ppliction nd cn be exported in different formts of CD stndrd neutrl files Mesh definition The mesh development procedure tht is pplied is bsed on the development of one of the fmilies of isoprmetric curves. Therefore, the surfce is ccurtely pproximted by set of flt tringulr elements tht re homogeneously distributed. This mens tht their vertexes re plced t the intersection of u-isoprmetric curve nd v-isoprmetric curve of the x= f(u, v) surfce. Ech strip of tringles is defined between two u- or v-isoprmetric curves tht re pproximted by the set of tringle sides with vertices on the sme curve Isometric mpping of surfce curve with geodesic curvture preservtion The second stge consists of n initil estimtion of the mesh development. One set of isoprmetric curves is developed preserving the length of the curve between ech two points (isometric mp) nd the geodesic curvture. This criterion is essentil since it gurntees the uniqueness of the resulting curve. The development procedure of curve C lying on surfce requires the pproximtion by finite set of points C i tken on the curve. The tngent plne T i nd the norml vector N i to the surfce hs to be known t ech of these points. The curve C is isometricly mpped onto plnr curve C* mking the chord lengths of C* the sme s the distnce between the corresponding points of C. The geodesic curvture of C is preserved t ech intermedite point C i if we keep the ngle θ i between two successive chords (C i-1 *, C i *, C i+1 *) of the plnr curve C* the sme s the ngle defined by the projections onto the tngent plne T i of the segments (C i-1, C i, C i+1 ). In the tringulr mesh of the surfce, the sides of the tringles tht join the vertexes lying on these curves pproximte the u- nd v-isoprmetric curves. The first estimtion of the developed mesh is obtined pplying n isometric mp to these lines with geodesic curvture preservtion. By crrying out this procedure, developble

6 12 ris et l surfces re flttened without distortions, while distorted flttened mesh is obtined in the cse of doubly curved surfces Reduction of the distortions: n energy bsed method The development of the surfce pproximte tringultion my be expressed by the sme procedure pplied to the mpping of one unique curve. Isometric mpping requires the preservtion of the distnces between pirs of nodes mesured on the 3D mesh. It should be tken into ccount tht ech node of the tringultion lies on the crossing point of two u- nd v-isoprmetric surfce curves. Preservtion of the geodesic curvture t ech mesh node requires the ngle to be preserved between the projected segments of ech one of the isoprmetric curves nd lso the reltive ngle between them. If the surfce is doubly curved, it is not possible to preserve the geodesic curvture nd the isometric mp, therefore the plnr development of the mesh will hve n error. To estimte this error n energy function is proposed. This function combines two min error components. The first error term estimtes the error due to the distnce difference between the nodes in the 3D mesh nd the corresponding nodes in the plnr development. The second term estimtes the ngulr distortion between the isoprmetric curves t ech node. 3. RESULTS 3.1 3D Modelling: Photogrmmetric Procedure The 3D surfces modeling ws obtined with 74 restituted points (including the projected grid of dots). The bundle djustment provide the results show in the tble 2, for ll points (including the grid of dots) used to generte the 3D model. This tble shows informtion bout: stndrd devition of the cloud of points (verge vlues in X, Y, Z xes nd qudrtic composition of the verges); the men qudrtic error, the men error nd the mximum error observed in the prllx (m); Root Men Squre (RMS) error nd mximum residul error (pixels); ngle mximum nd minimum between imges (deg). The use of flt grid s support bse for the object simplifies the processing stge, since the point identifiction on 2D surfce is significntly less ffected by visul perception errors thn the identifiction of 3D points. The genertion of the 3D surfce model through digitl monoscopic photogrmmetry hs shown to be strightforwrd, fst nd relible method when developble surfces re surveyed (see figure 3). In the cse of complex surfces, the use of the projected grid of dots hs shown to be simple nd ffordble system tht provides high ccurcy results (see tble 2). Tble 2. 3D model ccurcy ssessment: stndrd devition (m) of the cloud of points (verge vlues in X, Y, Z xes nd qudrtic composition); the men squre error, the men error nd the mximum error observed in the prllx (m); Root Men Squre error nd mximum residul error (pixels); ngle mximum nd minimum between imges (deg) Stndrd Devition (m) X Y Z verge,34,32,29 Qudrtic compos,55 Prllx (m) Pixels Imges (deg) verge,73 3,18 RMS,85 Mximum error,198 12,87 Mximum ngle 89,9537 Minimum ngle 39,8627 Nevertheless we must point out tht the effectiveness of this method depends on the size of the object nd the complexity of the surfce, since the lrger nd/or the more complex it is, the higher number of dots might be projected nd processed to wrrntee the mtch between the 3D digitl model nd the prototype.

7 Dyn 16, Surfce Flttening Method custom Rhino plug-in ppliction tht obtins the plnr development of doubly curved surfces strting from the CD definition of the surfces hs been developed nd implemented. Figure 3. Wire-frme 3D model of the wooden sculpture The ppliction genertes CD neutrl file with the shpe of the cutting profile tht is needed for the mchining processes of the metl sheets (see figure 4). The mthemtics of this method is bsed on the solution of n optimiztion problem tht tkes into ccount the isometric development of the isoprmetric curves with geodesic curvture preservtion. The surfce is first pproximted with tringulr mesh. Distnce deformtions nd ngulr distortions re mesured between ech pir of mesh nodes nd ech pir of crossing isoprmetric curves t the nodes respectively, re observed s result of the mesh flttening process. n optimiztion procedure is pplied to function bsed on the vlue of these deformtions. Tble 3 shows the evolution of these vlues in successive itertions, where the vlues of distnce deformtions nd ngulr distortions re expressed s percentge of the ctul distnces nd ngles. The medium nd the mximum vlues re computed ech itertion. Figure 4. 3D model of the grid surfce with doubly curved surfce (left), plnnr development of the grid (up right) nd cutting profile (down right) Tble 3. Results of the minimiztion development method used on double curved surfces Itertion Distnce deformtions (%) ngulr distortions (%) Medium Mximum Medium Mximum Totl error Initil vlue (mm)

8 14 ris et l 4. CONCLUSIONS In this pper n pproch to the ppliction of digitl monoscopic photogrmmetry technique in the 3D modelling of doubly curved objects is presented, surfce flttening method, bsed on isometric mpping nd geodesic curvture preservtion, is implemented s CD ppliction plug-in, nd finlly, n itertive energy bsed optimiztion procedure is pplied tht minimises distortions in the finl plnr profiles. The use of Photogrmmetric procedures in this kind of industril pplictions hs shown some dvntges from other clssicl methodologies, could be ppointed: - Esy methodology nd with fst results. For people without previous knowledge bout photogrmmetric procedures. - Low cost nd simple equipment. Infrstructures or complex devices re not necessry. - The results cn get it with ccurcy better thn millimeters in the finl model. For other side, we cn see sme spects or chrcteristics would hve to be more developed, for exmple: - The grid of dots projected on complex surfces is essentil to get qulity results. - When the complex surfces cn be observed in the element. The use of grid of dots mke necessry the development nd building of industril structure light systems, like this, the device will llow the utomtic process of the grid of dots projected on the surfce. - There is n importnt influence in the finl results for the kind of softwre CD used, to process the cloud of points nd to djust there to surfces models cn be development lter, to get nd extrct plnr ptches to mnufcture process. Independently of the procedure or system photogrmmetric used. In conclusion, two min contributions should be highlighted. Digitl photogrmmetry hs shown to be fesible nd ffordble tool in the ccurte 3D model genertion of complex objects; it is relevnt tht both of the spects of fesibility nd ffordbility re decisive fctors in the ccomplishment of utomted mnufcturing processes. Then the 3D model is sved to neutrl file formt tht could be processed with CD ppliction. The surfce flttening method is implemented s customized plug-in. The proposed development nd optimiztion methods hve been proven to be effective nd highly efficient in the cse study. Finlly, the resulting plnr pttern is then sved to neutrl file tht could be ccessible from wide number of different mchine tool progrmming pplictions. 5. CKNOWLEDGMENTS This rticle is prt of reserch project finnced by Ministry Eduction nd Science Utilizcion de tecnics geomtics pr el nlisis dimensionl-estructurl y conservcion de puentes historicos singulres (BI ). REFERENCES [1] KRUNKRN, K.P., GRWL, S., VENGURLEKR,P.D., SHSRBUDHE, O.S., PUSHP,.V. ND ELY, R. Segmented object mnufcturing. IEEE Trnsctions, 39, , 25. [2] RIS, P., RMESTO, J., LORENZO, H. ND ORDÓÑEZ, C. 3D terrestril lser technology in sporting crft 3D modelling. Interntionl Journl of Simultion Modelling, 6 (2), 65-72, 27. [3] CURTS, E ND PRIETO, F. Reconstrucción de objetos con superficies Tipo 2-vriedd diferencible. Dyn, 156 (75), 7-18, 28.

9 Dyn 16, [4] RBBNI, T., DIJKMN, S., VN DEN HEUVEL, F. ND VOSSELMN, G. n integrted pproch for modelling nd globl registrtion of point clouds. ISPRS Journl of Photogrmmetry nd Remote Sensing, 61 (6), , 27. [5] ROC-PRDIÑS, J., LORENZO, H., RIS, P. ND RMESTO, J. From lser point clouds to surfces: Sttisticl nonprmetric methods for three-dimensionl reconstruction. Computer ided Design, 4 (5), , 28. [6] YU, H.T., LEE, R.K., CHUNG, C.M. ND HSU, C.Y. NC Tool-Pth Genertion Bsed on Surfel Models Constructed from Rndom Scnned Dt. Computer-ided Design & pplictions, 2 (1-4), , 25. [7] Y. ZHNG. Mesurement of industril sheetmetl prts with CD, designed dt nd nonmetric imge sequence. Proceedings of the XXth ISPRS Congress Commission 5, Istnbul, Turkey, 24. [8] TNGELDER, J.W.H., ERMES, P., VOSSELMN, G. ND VN DEN HEUVEL, F.. CD-bsed photogrmmetry for reverse engineering of industril instlltions. Computer- ided Civil nd Infrstructure Engineering, 18, , 23. [9] BOSEMNN, W. dvnces in photogrmmetric mesurement solutions. Computers in Industry, 56 (8-9), , 25. [1] ERGUN, B. n expert mesurement system for photogrmmetric industril ppliction. Mesurement, 39 (5), , 26. [12] RIS, P., CMÑO, J.C., LORENZO, H. ND RMESTO, J. 3D Modeling nd Section Properties of ncient Irregulr Timber Structures by Mens of Digitl Photogrmmetry. Computer-ided Civil nd Infrstructure Engineering, 22 (8), , 27. [13] SLZR,., SÁNCHEZ, L.G. ND PRIETO, F. Sistem de dquisición de imágenes de rngo con bse en estéreo-ctivo. Dyn, 153 (74), , 27. [14] MSON, S.O. Conceptul Model of the convergent multisttion network configurtion tsk. Photogrmmetric Record, 15 (86), , [15] EOS SYSTEMS INC. Photomodeler Pro 5.. User mnul. Vncouver, Cnd, 23. [16] COOPER, M. ND ROBSON, S.. Theory of Close Rnge Photogrmmetry, In: Close Rnge Photogrmmetry nd Mchine Vision (Ed. K.B.tkinson), Whittles Publishing, 9-51, 21. [17] ZRIDIS,P. ND SPRGTHOS, N. Design of plne ptterns of doubly curved surfces. Computer-ided Design, 29 (1), , [18] ZRIDIS,P. ND SPRGTHOS, N. Geodesic curvture preservtion in surfce flttening through constrined globl optimiztion. Computer-ided Design, 33, , 21. [11] D'NGELO, P., WOHLER, C. Imgebsed 3D surfce reconstruction by combintion of photometric, geometric, nd rel-perture methods. ISPRS Journl of Photogrmmetry nd Remote Sensing, 63 (3), , 28.