The Evaluation and Correction of the Reconstructed NURBS Surface Smoothing and Accuracy

Transcription

1 Sensors & Transducers 2013 by IFSA The Evaluaton and Correcton of the Reconstructed NURBS Surface Smoothng and Accuracy Shgang WANG, Janzhou ZHANG, Yong YAN School of Mechatroncs Engneerng, Qqhar Unversty, Qqhar , Helongang, Chna Tel.: , fax: E-mal: Receved: 17 October 2013 /Accepted: 22 November 2013 /Publshed: 30 December 2013 Abstract: NURBS surface reconstructon s dscussed from the two aspects of smooth surface and precson. Gven the test whether surface s smooth. To the not smooth surface, by usng modfed pont, secton feature curve to mprove the surface smoothness. Surface reconstructon accuracy does not meet the requrements of the stuaton, usng the modfed method of control vertces to correct surface reconstructon error. Ths paper provdes a method for the reconstructon of surfaces effectve assessment and correcton. Copyrght 2013 IFSA. Keywords: Surface reconstructon, Smoothng assessment, Accuracy correcton, Reverse engneerng. 1. Introducton In the reverse engneerng technology, the reverse process of surface needs to pass through the pont cloud data acquston, data processng, surface reconstructon and product manufacturng process [1]. The evaluaton of the effectveness and accuracy of the free-form surface, the commonly used method s to compare the surface and the orgn of reverse cloud data. From the reverse process of the entre technology loop wll produce errors, especally errors n the manufacturng process s very complex. Therefore, the reconstructed surface characterstc evaluaton model s an mportant step n the reverse process before the producton. If the surface reconstructon s the most essental part of the words, then the qualty evaluaton of surface reconstructon n reverse engneerng s the most key lnk. Surface qualty manly ncludes two aspects of smoothness and accuracy, Smoothng s a very subectve concept, whch contact wth human subectve feelng great. Smooth product appearance beautful generous, beautful shape, gvng a pleasant feelng, and ncludes streamlned dynamc model of the surface, wth good aerodynamc performance. Due to the subectve factors s relatvely large, t s dffcult to gve a specfc defnton smoothng, resultng n a number of dfferent evaluaton crtera. As wth the processng of the product, surface reconstructon has the accuracy of 3-D reconstructon, the 3D reconstructon accuracy, refers to meet the degree between the reconstructed surface and reverse prototype. If the surface reconstructon s not up to the gven precson requrements, even f the surface s reconstructed, there s no actual applcaton value. Accuracy assessment has ncreasngly become an mportant part of reverse engneerng n the whole lfe cycle. In ths paper, n order to reflect the real surface characterstcs of pont cloud data for evaluaton bass, from two aspects of the curved surface smoothng and surface accuracy of the free-form surface reconstructon of Artcle number P_

2 performance assessment, and n vew of the tem does not meet the techncal modfcaton measures are put forward. 2. Smooth Surface Characterstcs Evaluaton and Correcton Methods To calculate a group of planes of P surface obtaned by lnes of curvature. Curvature analyss s an mportant tool for surface analyss, often usng color mappng and drawng curvature contour lnes of curvature n two ways curvature of the surface analyss. Fg. 1 shows a freeform mean curvature color maps. Smooth surface characterstcs evaluaton s to assess whether the surface prmarly aesthetc, whether wth a fluent lne and smooth surface, namely on the farng of the assessment. Products of free-form surface generally requre surface smoothng Crtera and Methods of Evaluaton of Reconstructed Surface Surface smoothng crtera s relatvely complcated, t s accordng to the crtcal curve on the surface (such as: the parameters of the u and v drecton lne coordnate or surface and parallel to the plane of a seres of plane cross secton of the lne, etc.) whether smoothng and surface curvature (man curvature and gauss curvature, mean curvature) whether changes evenly to udge. Surface smoothng crtera [2]: 1) The crtcal curve farng; 2) Surface wthout extraneous nflecton pont (or ponts) and varable perturbaton; 3) The prncpal curvature n the nodes of the curvature ump s small enough; 4) Gaussan curvature changes unform. Reconstructon surface due to the lmtaton of computer dsplay, t s very dffcult to drectly determne the smoothness of the surface, so we must adopt the method of mathematcal analyss. There are four knds of surface farng analyss method: the method based on curvature and reflecton lne algorthm, hgh brghtness lne method and sophotes method, each method has certan characterstcs. Fg. 1. Mean curvature surfaces color map. Drawng contour lnes of curvature on the surface s to connect the dots wth the same curvature nto the lne. Its method wth the curvature of color mappng s the same n essence, only expressed n dfferent ways. Fg. 2 s a freeform mean curvature contour curvature graph Curvature Based Method Curvature based method s the curvature analyss to the surface, the surface havng the same curvature pont values together nto a certan order of the curve, the contour of partal curvature can be obtaned through the establshment of the equaton [3-4]. Defne a parametrc surface M and a scalar functon k, whch means that some knd of curvature has the followng formula kuv (, ) kmuv ( (, )) (1) Usng the above formula to create surfaces Puv (, ) Puvkuv (,, (, )) PuvkMuv (,, ( (, ))) (2) Fg. 2. Surface contour curvature graph. As shown n Fg. 3 by the dstrbuton of curvature comb whether smooth unform, can determne the surface smoothng of whether meet the requrements Reflecton Lne Algorthm Reflecton lne refers to a set of rays on the surface obtaned by proectng surface curve [5]. Gven a set of parallel rays L() t, they pass through the same vewpont A to the obect rradaton, whch 396

3 wll produce a set of parallel rays reflected by the optcal theorem physcs that, the angle of reflecton equal to the angle of ncdence, by observng the reflecton densty of the yarn and deformaton to determne surface smoothness. Reflecton pont Q satsfes Fg. 4 (a) can be seen not only along the reflected ray and spacng are dfferent, udgng the surface s not smooth, Fg. 4 (b) for the modfed surface, the reflected ray comparson rules, dstrbuton s more unform, smoothness mproved sgnfcantly. L Q ( A Q ) ( L Q ) 2 N ( N, L Q ) A Q (3) Hgh Brghtness Lne Method Ths s a specal reflectve lne method, f there s no vew; the reflected ray becomes a set of ponts on the surface. Then lght lnes Lt () can be represented by the followng formula: L() t A Bt (4) Fg. 3. Curvature comb. A s the source pont on the lne, B s the drecton vector. For any pont Q on the surface, N represents the surface normal vector n the drecton of the pont. The N drecton n a straght lne through the pont Q s the Q-pont surface normal vector of extenson cord, whch can be expressed as Es () Q Ns (5) Fg. 4 s the use of reflecton lne method for detectng surface. Fg. 5 s a hgh-brghtness lne method to check the surface qualty. Fg. 4 (a). Reflecton lne method. (a) Fg. 4 (b). Reflecton lne method. (b) Fg. 5. Hgh brghtness lne method. 397

4 Hgh brghtness method assocated wth the vector method of curved surface, and to the drecton of the vector change stck don't senstve parameters for a gven surface Suv, (, ) the method of one pont vector u, v can be used n both drectons of a dervatve, namely tangent vector to defne. If the frst dervatve s dscrete, the hghlghted lne n whch the showed rregular, so senstve method of hgh brghtness can be locally s not only sutable for testng surface. As shown n Fg. 5(a) that lne s not smooth, you can determne the surface smoothness s poor, Fg. 5(b) flow smoothly and evenly spacng change, whch means that surface smoothng good Isophote Method The same defnton wth the curvature lne, have the same llumnance pont accordng to the order form of the curve s the sophote [6-8]. The sophotes can be defned by type ( N( u, v), L) c const In the formula, X ( uv, ) s the parametrc surface, L as the drecton vector parallel lght; Nuv (, ) represents the normal pont of surface. Smooth sophote represents a surface have good qualty; rregular dstorted sophotes dsplay defects n one or two dervatve surface; contnuous and unform dstrbuton of the sophotes sad surface smoothness hgh degree. From Fg. 6 we can see the sophote lnes smooth, and space evenly, thereby determnng the surfaces have good smoothness. curvature value between adacent patches A and B contnuty. Ths method s very senstve to small changes n surface shape. Illumnaton model essentally reflects the changes of the surface normal are based, can help us to determne the surface patches A and B contnuty. The renderng surface, we can make the surface shape has a very ntutve understandng, but can not udge C contnuty between surfaces. Contour map can help us to understand the shape of surface, fnd the peak, valley, maxmum, mnmum, the contour smoothness help us udge the farness of the surface. In actual use, we can choose accordng to the specfc stuaton of a surface analyss method, the ntegrated use of multple methods can also be Not Smooth Surface Modfcaton There are many knds of the correcton method of curved surface shape characterstcs; there are local modfcatons and overall correcton. Overall correcton effect s good, but computatonally ntensve and slow. Partal correcton method s manly through dfferent means of nspecton, found only on the surface along the secton, and then uses some method to adust some control surface vertces to acheve smooth surfaces. Correcton method s usually caused by local surface shape characterstc evaluaton method to decde what correctve acton. In vew of the commonly used evaluaton method based on curvature and contour, the most smple and calculaton of curved surface shape characterstcs of the mnmum amount of correcton measures are characterstc curve usng the modfed cross-secton the way to correct t. Let the detected sequence of ponts P ( 0,1,, n), Pt () s the nterpolaton curve of PP, k s the relatve curvature at the pont P. When k 1k 0 and kk 1 0, the pont P s dead pxels. Make PQ P Q 1 1 max 1 1 /2, Let dead pxel s P Pt Q P P, the frst amendment pont s. Fg. 6. Isophote method. The above methods have certan characterstcs, overall: the curvature based method allows us to easly observe the dstrbuton of surface curvature. Curvature color mappng, curvature lne method can help us to recognze the surface fluctuaton, concave and convex regons, fnd out the surface maxmum Wth P of the adacent pont where P and 1 P 1 vector and the tangental vector nformaton construct curve Ct (), D C P, when D ( s the maxmum number of modfcatons), Modfy the P to the C ; Otherwse, modfy the P to the D P. D 398

5 Re-fttng the modfed data ponts, get the new cross secton characterstc curve, and then rebuld the surface, realze the correcton on the curved surface shape characterstcs. 3. Surface Accuracy Assessment and Correcton 3.1. Surface Accuracy Assessment In reverse engneerng, the physcal prototype has been dgtzed, hgh-densty measurement pont cloud contans a large amount of data, fully expressng the measured surface characterstcs, t can be used the same as a real surface, therefore, the error between the physcal prototype and model surface, can be expressed by the samplng error between the pont and the model surface. Comparson of the model wth the physcal problem nto a calculated pont-tosurface dstance, usng accuracy of mnmum dstance from a pont to surface to characterze the surface reconstructon model [9], and the fnshed surface models from pont cloud data analyss whch, f the mnmum dstance s n the error range you can assess the reconstructon model of qualfed over the range of error, settng that surface accuracy s not qualfed, as shown n Fg. 7 between the surface model and the orgnal pont cloud through dstance analyss, ths can be used to evaluate the model accuracy ndex. general, spatal arbtrary pont on the surface of proecton can be expressed as S QP d S u u S S v v, (6) where P s the proecton of Q on the surface; d s the shortest dstance between Q to the surface S(u,v); Su, v Su, v Su, Sv s the partal dervatve u v of parametrc surfaces. For complex surfaces, the equaton (6) s a hgh-order nonlnear equatons, Newton's method can often be used to solve, but the Newton method for ntal demandng at the boundares may lead to dvergence calculaton; whle Newton's method s tme-consumng, practce n order to avod solvng lnear equatons, often usng geometrc segmentaton method, the surface plane slces nto a seres of dscrete polyhedra, and then pont to calculate the dstance to each flat pece, take the mnmum value for the shortest dstance Error Correcton Measures In practce, once the error exceeds the allowable value of the surface, the surface should be corrected. Correcton method also has many, there by ncreasng or decreasng the number of control ponts to mprove, by modfyng the surface wth a method of control vertces. Ths paper used to modfy the surface control vertces surface error correcton method to mprove the surface accuracy [10]. Let h k tmes gven pont NURBS surfaces p p( u, v), u ( u, u h 1), v ( v, v k ) 1 Change the control vertces d to d, so that the surface pont p moves along the c drecton gven * dstance s to pont p. That s p p acrh,, k,( u, v), (7) where Rh,, k,( u, v ) s the ratonal bass functon. Then Fg. 7. Pont cloud to surface dstance. The key ssue here s to calculate the samplng pont to the surface of the shortest dstance, set pont P s the nearest pont to the surface Suv (, ) of pont Q, and then the vector (P-Q) must be the same as the surface normal at pont P. Therefore, the pont to the surface of the problem can be transformed nto calculaton ponts n the parameter surface S(u,v) of the proecton, the proecton drecton of the surface normal vector, n a cr s uv h,, k,, Thus, you can get new control vertces are d d ac, to reconstruct the surface wth new control vertces, untl t meets the accuracy requrement. 4. Conclusons Ths paper summarzes the surface smoothng performance testng standards and methods, summarzes the varous characterstcs of curved 399

6 surface smoothng method and accordng to the surface phenomenon, and by amendng secton gves the characterstc curves for surface partal correcton method to fx t, whch can mprove the surface smoothng performance very well. Accordng to the precson of the reconstructed surface problem, to reconstruct surface ntersects the measured pont nstead of the normal and the dstance for error estmaton based on surface, has realzed the error of the reconstructed surface, to the surface accuracy can not meet the actual requrements, adopt the method of modfed control vertex to error correcton surface, effectvely ensure the accuracy of the surface performance meet the requrements. Acknowledgements Ths work was supported by the Natural Scence Foundaton of Helongang Provnce, Chna (Grant No. E201106) and the Proect of Department of Educaton of Helongang Provnce, Chna (Grant No ). References [1]. Weun Wang, Yuwen Sun, Reverse engneerng prncples, methods and applcatons, Machnery Industry Press, [2]. Yuan L, Kafu Zhang, Janfeng Yu, Technology and applcaton of computer aded geometrc desgn, Northwestern Polytechncal Unversty Press, [3]. Gang Le, Changpng Zou, Janmng Huang, Research progress of rapd reverse engneerng, Mechancal Engneer, Vol. 10, 2008, pp [4]. Zhongy Ca, C contnuous curved surface reconstructon and smoothng of the fnte element method, Journal of Engneerng Graphcs, Vol. 4, 2006, pp [5]. Syuan Wu, Study on the method of free-form surface dgtzng based on ultrasonc dstance measurement, Chna Mechancal Engneerng, Vol. 17, No. 22, 2006, pp [6]. S. Kabacher, G. Husler, A new approach for modelng and smoothng of scattered 3D data, n Proceedngs of IEEE Conference on Vsualzaton, 2008, pp [7]. L. Peel, W. Tller, Surface sknnng revsted, The Vsual Computer, Vol. 18, 2002, pp [8]. W. Ueng, J. Y. La, J. Doong, Sweep-surface reconstructon from three-dmensonal measured data, Computer-Aded Desgn, Vol. 30, No. 10, 1998, pp [9]. Sad Motavall, Rafe Shamsaasef, Obect-orented modelng of a feature-based reverse engneerng system, Internatonal Journal of Computer Integrated Manufacturng, Vol. 9, No. 3, 1996, pp [10]. Hong Lu, Jawu Fu, Yongchang Peng, Reconstructon of freeform feature evaluaton and correcton method, Machne Tool and Hydraulcs, Vol. 40, No. 20, 2012, pp Copyrght, Internatonal Frequency Sensor Assocaton (IFSA). All rghts reserved. ( 400