A Fast Non-Uniform Knots Placement Method for B-Spline Fitting

Transcription

1 2015 IEEE Inernaional Conference on Advanced Inelligen Mecharonics (AIM) July 7-11, Busan, Korea A Fas Non-Uniform Knos Placemen Mehod for B-Spline Fiing T. Tjahjowidodo, VT. Dung, and ML. Han Absrac A wo-sep fas non-uniform kno placemen algorihm applicable for noisy daa is presened in his paper. The algorihm is sared by evaluaing he second derivaive of he sampled daa and, subsequenly, by using he half-spli approach, he bes piecewise linear funcion is fied o he compued derivaive. In he second sep, he fied funcions are adjused o define he final kno for he B-spline fiing. The proposed mehod is subsequenly validaed on an experimen daa wih a known nominal surface. I is demonsraed ha he proposed mehod offers a fas compuaional ime ha allow for online surface esimaion. I. INTRODUCTION In daa fiing applicaions, a kno vecor mus be defined in advance while he conrol poins are idenified based on he minimizaion of a leas squared error beween he daa poins and he fied funcion. Commonly, knos are chosen in uniform space. However, his migh resul in an overshooing problem in he case of smooh inhomogeneous or disconinuous daa. In order o overcome he problem, deermining a non-uniform kno space seems o be he bes soluion. Unforunaely, defining he knos and heir respecive locaions in non-uniform space is sill a challenging problem as i is compuaional cosly ([1]). Selecion of he locaion of non-uniformly spaced knos is very criical on he resuling shape. Many approaches are found in he lieraure for opimizing he kno vecor. In shor, here are hree main classes of approach for he kno placemen algorihm. The firs one is based on a rial-anderror approach. The main idea of his approach is by creaing an iniial kno vecor and evaluaing he error of he fied curve. The error will be evaluaed wih a predefined sopping crierion. The compuaion procedure will be erminaed and he seleced kno vecor is reurned if he crierion is me, oherwise a new kno vecor will be creaed based on a cerain crierion. Alhough i usually offers good resuls, bu he compuaional ime in his approach is raher cosly. Defining he kno vecor can be proceeded in eiher ways: (i) minimum number of knos are iniially defined and afer quaniaive checking, if necessary, an inserion of more knos is inroduced o improve he fied curve ([2],[3]); or (ii) he algorihm is sared from a dense kno vecor, where cerain knos are eliminaed if he fied funcion responses wih small change ([4],[5]). An evoluional algorihm ([6],[7],[8]) and immune sysem ([9]) are also inroduced as Tegoeh Tjahjowidodo and Dung Van Than are wih School of Mechanical and Aerospace Engineering, Nanyang Technological Universiy, 50 Nanyang Avenue, Singapore ( egoeh@nu.edu.sg). Han Mingli is wih Advanced Remanufacuring and Technology Cenre (ARTC), 3 CleanTech Loop, #01/01, CleanTech Two, Singapore alernaives o he firs wo approaches in his rial-and-error mehod. The second class in non-uniform kno placemen algorihm is by uilizing an opimizaion process o find he opimal kno poins ([10][11][12]). This caegory is raher hard o solve opimally, which is due o he nonlineariy in is cos funcion and he search space ha migh resul ino local opimum soluion ([13]). The knos placemen in he hird class is defined based on he feaure(s) of he daa. This approach offers a faser processing ime bu he opimal resuls canno be guaraneed. He e al. ([14]) uilized he wavele ransform o locae he knos on he high frequency segmen of he sampled funcion (da. Wavele is a well-known ime-frequency-represenaion mehod and i is proven o be effecive in he areas of sysem idenificaion and filering ([15],[16],[17]) compared o oher filraion echniques (e.g. [18],[19]). However, he processing ime of he Wavele decomposiion is raher cosly, and he seleced kno vecor is no opimized in erms of he number of kno. An alernaive mehod was presened by Li e al. ([20]) ha is based on he discree curvaure of daa. In he firs sep, he discree curvaure of daa are compued, and hen a low pass filer is applied o ge a smooher curvaure and knos are subsequenly seleced by a heurisic rule. This mehod offers a fas processing ime, bu i migh resul in a lo of redundan knos. Therefore, his mehod is suggesed o be used o find he iniial guess for he kno vecors before furher opimizaion processes. Yuan e al. ([21]) proposed a muli-resoluion basis funcion se for idenifying kno locaion. This mehod is based on he fac ha denser knos will occur a larger curvaure posiions. Iniially, a subse of basis funcions is seleced from he pre-specified muli-resoluion basis se using Lasso opimizaion. Subsequenly, a vecor space is consruced based on he basis funcion and a concise kno vecor is idenified o fi he unknown funcion. Despie he effeciveness of he mehod, he uilizing basis funcions requires excessive compuaional ime. Coni e al. ([22]) offered an alernaive o selec he kno vecor for a cubic spline based on he hird derivaive. However, his mehod is only applicable for smooh splines wih predefined number of knos and i is no applicable for non-smoohing funcions. Kang e al. ([23]) proposed anoher mehod ha is also based on he hird derivaive of he funcion ha is referred o as he sparse opimizaion. Given an iniial dense se of knos, by comparing he lef and righ hird derivaives for a cubic spline from a cerain kno (which are consans), he kno will be eliminaed if he difference beween he wo hird derivaives is smaller han a cerain hreshold. Tha is o say ha he second derivaives essenially are referring o collinear lines. Execuing /15/$ IEEE 1490

2 repeiive process, he real knos will evenually survive afer he process. The kno vecor is also adjused o find he bes resul. However, his mehod is no very effecive for noisy daa and i is no very efficien o handle muliple knos. This paper presens a new way for calculaing he kno vecor of a cubic spline based on he second derivaive (for cubic spline) of he daa. In he firs sep, he half-spli mehod wih a cerain error band as a crierion is applied o he second derivaive of he sampled daa. Subsequenly, he bes fied linear piecewise funcions for he compued second derivaive is evaluaed. The break poins of he linear piecewise funcion are considered as candidaes of kno vecor. In he second sep, an opimizaion process is employed o adjus he locaion of knos o handle muliple knos, which is refleced on he second derivaive. As he knos is solved analyically, he proposed mehod allows for he evaluaion o nearly exac knos (boh he number of knos and heir locaions) if he daa is sampled from a specific spline funcion. A he same ime, naurally, he mehod can handle muliple inerior knos. This means, he mehod will give he bes fied for various ypes of B- spline (non-differeniable and disconinuous funcions). Anoher imporan feaure of his mehod is ha i offers fas processing ime ha makes he mehod o be poenial for real-ime applicaions. The paper consiss of four secions. Secion II presens he mehodology o solve he knos based on he half-spli mehod and a predefined hreshold. Opimizaion of he kno locaion and muliple of kno handling is discussed in secion III. Experimenal validaions are presened in secion IV o demonsrae he effeciveness of he proposed mehod and some respecive conclusions are drawn in secion V. II. HALF-SPLIT METHOD FOR KNOTS EVALUATION A cubic spline is a piecewise funcion ha is coninuous up o is second derivaive where each polynomial is conneced o he oher a a kno or a break-poin. Mahemaically, a p- n degree B-spline, S( ) N p, i ( ) Pi, is defined on sequenial i0 knos: p p1 m p1 m p m where he knos p 1, p2,..., m p 1 are called inerior knos, p m n 1, wih P i is i h poin conrol and Npi, () is he i h basis funcion wih degree of p ha is defined in a recursive procedure ([13]). N N 0, i 1, if i i 1 0, oherwise i j1 ( ) Ni 1, j1 ( ) i i, j Ni, j1 i j i i j1 i1 The second derivaive of a cubic spline appears as linear polynomial piecewise funcions, where he inersecion of wo adjoining piecewise funcions is represened as a kno as illusraed in Figure 1. This means, if a spline has piecewise linear funcions as is second derivaive, he wo adjoining sraigh lines will be conneced hrough a break-poin. Given a daa se of an unknown funcion (eiher clean or noisy), we can define is second derivaive from he given daa. Approximaing he analyzed second derivaive by linear piecewise funcions, he inersecion of each wo adjoining sraigh lines and each parameric,, as he inerior kno of he cubic spline can be defined. A. Daa Spliing In his approach, a subse of he second derivaive sample poins will be approximaed as a sraigh line. The firs key poin of his approach is o spli he daa ino a se of piecewise linear funcion. In order o perform i, we employ he half-spli mehod o subdivide he daa (in ascending order) as illusraed in Figure 2. A he firs sep, all he daa is assumed o be a par of a single sraigh line wih kno vecor is T 1 = ( 0, 1 ). Subsequenly, we examine wheher all daa, indeed, belongs o a single sraigh line. If i is no, we spli he subse T 1 = ( 0, 1 ) ino wo sub-inervals by inroducing an inermediae kno ha will urn he kno vecor ino T 2 = ( 0, 1, 2 ). This kno vecor, subsequenly, will be re-examined wheher he wo subses obained from sep 2 are originaed from wo lines. If no, anoher inermediae kno will be inroduced. A his poin, le us assume ha a sep 3 he kno vecor now is T 3 = ( 0, 1, 2, 3, 4 ). Based on a sopping crierion/hreshold (ha will be discussed in secion IIB), he subinerval [ 0, 1 ) is defined as a piecewise linear funcion and he hree remaining inervals are no (noe: in he figure, 1 indicaes he daa subse ha does no saisfy he sopping crierion and 0 denoes he daa subse ha saisfies he crierion as a sraigh line). Sep 1 Sep 2 Sep Figure 1. Typical B-spline curve 0: error band error band hreshold 1: error band > error band hreshold lim lim lim Sep lim Sep lim Figure 2. Half spli mehod of daa spliing 1491

3 y y1 ax b1 y ax b As shown in Figure 3, he error band AB of a daa subse can be expressed as AB = max(y i -ax i )-min(y i -ax i ), where index i represens he i h sample of he subse daa. A(0, b ) 1 C(0, b) O B(0, b ) 2 H ( x, y ) Figure 3. Error band of a daa subse h h y2 ax b b) c) d) Figure 4. Half spli mehod wih error bands o define he opimal kno vecor As he inerval [ 0, 1 ) is defined as a linear funcion, i will be kep and he remaining hree inervals are proceeded furher for spliing based on he predeermined crierion. As a final resul, a sep 5, he kno vecor is defined as T 5 = ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ) and he spliing process is erminaed based on wo condiions, i.e. all segmens are linearly defined and/or he remaining inerval is now lesshan-or-equal o he minimum value. Typically, his value is equal o he sample space or minimum inerval. For convenience, he sampling order as a subjec for he half-spli mehod will be used insead of he parameer,, For example, if we have 101 samples in an ascending order from 0 o 100. A he firs sep, he kno vecor is [0,100] and second sep is [0, 50, 100] and so on, while he half-spli procedure sops when lim = 1. B. Sopping Crierion A se of daa (x i,y i ) belongs o a sraigh line y = ax+b if he daa lay wihin a cerain band > 0 such y i -(ax i +b) <. In order o examine wheher a cerain subse belongs o a single piecewise linear funcion, we evaluae an error band AB wih respec o he regressed line on he subse. Error band widh is defined as a disance in y direcion beween wo sraigh lines ha are boh parallel o he fied line where all sample daa say wihin he corresponding error band. x Figure 5. Joining subses from redundan knos b) The error band, AB, will be compared o a conrol hreshold,, ha serves as a conrol facor and has o be properly chosen. If i is much smaller han he noise level, oo many knos would be seleced and he curve will be overfied. On he conrary, if he error band is oo large compared o he noise level, he number of knos is insufficien and he curve will be under-fied. Based on a predefined conrol hreshold,, he knos ha define he daa subse are idenified. Figure 4 illusraes he use of he half spli mehod and he conrol hreshold o define he kno vecor wih 18 samples indexed from 0 o 17. In he firs ieraion, (, he kno vecor is [0, 17]. When he error band is shown o be larger han he conrol hreshold, he daa subse is spli and, subsequenly, he kno vecor is half-spli (b) ino [0, 9, 17]. The process is repeaed unil in he las sep, (d), no error band is greaer han he conrol hreshold and he repeiive process is erminaed. As an oucome he kno vecor now becomes [0, 5, 8, 9, 12, 14, 17]. III. KNOTS OPTIMIZATION Previous secion presened he half-spli mehod o define he kno vecors ha is bes fied o he second derivaive of he sampled daa. However, in some scenarios, he mehod canno be implemened direcly. One case is illusraed in Figure 4c when redundan knos are idenified afer halfspliing process. The problem occurs when he oal number of knos creaed from he mehod is larger han he acual knos, resuling in o redundan knos ([5, 9 and 14] in Figure 4c). In his case, cerain acions have o be aken o eliminae he redundan knos, o join linear funcions ha belong o a same line and o generae new relevan knos o he joined lines. A. Joining Collinear Lines To avoid redundan knos, every wo adjoining lines have o be examined wheher boh lines can be merged as a single line. The wo lines are considered as a single subse if boh error bands are smaller han a same conrol hreshold as well as he merged line. Figure 5 illusraes he eliminaion of wo redundan knos and joining wo pairs of subse [0, 5] o [5, 8] and [5, 8] o [8, 9]. B. Knos Adjusmen In real cases, i is less likely ha he idenified knos coincide wih he sample poins. There are wo possible cases where adjusmen of he physical kno poin o he analyzed knos locaions is required. The firs case occurs 1492

4 when he inersecing poin of he wo adjoining lines (analyzed kno) is close o a real physical kno and he neighboring physical knos [he (i-1) h and/or he (i+1) h ] is more han one sample away from he corresponding physical kno. The case is illusraed in Figure 6a. In his case, he idenified physical kno a i h is replaced wih he new analyzed kno and hen, for regressing he line, he kno vecor convenion is convered from index number o real parameric values,. The righ panel of Figure 6 illusraes he second case when he inersecing poin of he wo adjoining piecewise linear lines lies beween wo knos [i h, (i+1) h ]. Similar o he previous case, he wo idenified physical knos are replaced wih he new kno idenified from he inersecing poin. For regressing purpose, he kno vecor will also be represened in real parameric values. In his paricular case, as he wo physical knos will be replaced by a new defined kno, he oal number of kno will be one number less han ha in he original condiion. C. Muliple Knos A unique feaure of he B-spline echnique is is abiliy o capure smooh and non-smooh funcions. In paricular, hree disinc non-smooh funcions, a funcion wih a kink on is firs derivaive, a funcion wih a kink, and a disconinuous funcion. A p-degree B-spline is coninuous up o is (p-1) h derivaive a knos. A a special kno case, when here are k-mulipliciy of he knos hen he B-spline funcion will only be coninuous up o (p-k) h derivaive. Double knos For a cubic spline wih double knos hen a he repeiive knos posiion he B-spline funcion will only be coninuous up o firs derivaive y. This means he second derivaive of he funcion a he corresponding poin will be disconinuous. The case is illusraed in Figure 7. There are four possible cases of double knos In he firs case, he double knos are idenified close o he (i+1) h sample and he wo physical knos are idenified nex o he (i+1) h sample, i.e. a i h and (i+2) h sample poins, while he inersecing poin of he wo adjoined lines lies ouside he [i, i+2] inerval. In his case, he wo physical kno locaions a i h and (i+2) h will be subsiued by a double kno poin a (i+1) h. The case is illusraed in Figure 8a. The second case (Figure 8b) occurs when wo small inervals wih a physical kno a (i+1) h is idenified. A double knos will be defined a (i+1) h and he same reamen o he wo adjacen physical knos as in he previous case will be aken. The hird case happens when he double kno is locaed near o (i+1) h or (i+2) h. In his case, as shown in Figure 8c, he half-spli process will divide he inerval [i, i+3] ino 3 sub-inerval, namely [i, i+1], [i+1, i+2], and [i+2, i+3]. In his case, he double kno is defined a he middle of subinerval [i+1, i+2], denoed by (i+1.5) o subsiue he four physical knos a (i), (i+1), (i+2) and (i+3). The las case (Figure 8d) is similar o he hird case, excep ha only one segmen wih 3 space inerval (or wo physical knos) is idenified and he inersecion of wo adjoining piecewise linear funcions lies ouside [i, i+3]. Triple knos A B-spline funcion wih a kink will have disconinuiy a is firs and second derivaives as illusraed in Figure 9. When wo small inerval wih a cener a (i+1) h and wo physical knos adjacen o he cener a i h and (i+2) h poins, he riple knos locaion will be chosen a he middle poin of inerval [i, i+2] o replace hree physical knos locaed a (i), (i+1) and (i+2) wih he riple knos a he middle poin, (i+1). The case is illusraed in Figure 10a. In case when he riple knos is idenified inside [i+1, i+2], as shown in Figure 10b, he half-spli process will resul in hree single-space sub-inerval [i, i+1], [i+1, i+2] and [i+2, i+3], wih a very sharp spike. The riple knos will be defined a he middle of [i+1, i+2] inerval o replace o four respecive physical knos. Fourfold knos Fourfold kno will occur when we have disconinuous daa as illusraed in Figure 11. The locaion of fourfold kno will be defined a he middle poin of inerval [i, i+3], denoed by (i+1.5). The fourfold kno, subsequenly, will replace he four physical knos a (i), (i+1), (i+2) and (i+3). Inersecion poin i h Inersecion poin i h i h (i+1) h b) Figure 6. Two cases of kno adjusmen Figure 7. Double kno case i h (i+2) h b) i h (i+1) h (i+1) h (i+2) h (i+3) h c) d) Figure 8. Typical double kno cases i h (i+2) h (i+3) h 1493

5 profile a he middle par han ha a he wo ends and, in urn, i will cause more noise on is second derivaive. i h Figure 9. B-spline wih riple knos h (i+2)h (i+1) i h (i+1) h (i+2) h(i+3)h b) Figure 10. Two cases of riple knos The more irregular he profile, he higher he knos will be idenified. The case occurs in he middle par of he propeller daa poins. Because of he rougher profile on he measuremen daa a he middle par, denser knos are idenified a he corresponding region as indicaed by riangles ( ) a he boom par of he righ panel in Figure 13. Therefore, as he enire fiing process is carried ou wih a uniform hreshold,, he denser knos will resul in an overfied curve. Figures 14 shows he measuremen resuls and he fied curve of he compuer mouse wih irregular clay addiives. As shown on he lef panel, he es sample feaures a few sharp ridges as indicaed approximaely a posiion 4mm and 27mm. These feaures correspond o he second derivaive plo on he righ panel of he same figure, where i shows wo high spikes a he respecive sharp spike locaions. The proposed mehod is shown o be able o capure he feaures wih higher number of knos (denoed by ) wih no overshoo. This demonsraes he abiliy of he mehod o handle muliple kno problem. Figure 11. B-spline wih fourfold knos IV. EXPERIMENTAL VALIDATION Validaion of he proposed mehod is carried ou on experimenal daa ha were aken on a free form surface. In his case, we consider a wised surface in a mini-uav propeller and a compuer mouse wih some clay addiives o simulae produc defecs. For he laer case, wo differen scenarios are considered, i.e. (i) wih arbirary-shaped addiive, and (ii) wih conrolled-shape addiive. The esing apparauses ogeher wih he sampling planes are shown in Figure 12, where he lef panel shows he propeller and he righ one illusraes he compuer mouse wih arbirary addiive shape. Keyence LJ-V7080 profilomeer is used wih uniform sampling space of 0.05mm o acquire he sampled poins. The lef panel of Figure 13 shows he sampled profile of he propeller ogeher wih he b-spline fied curved, while he righ one depics he calculaed second derivaive of he sampled daa. As shown in he lef panel of he figure, he proposed mehod is proven o be able o fi he propeller shape curve. One hing can be highlighed from he second derivaive plo is ha he middle pars (beween 15-30mm on he plo) indicae more noise compared o he daa from he oher end pars. This is due o numerical differeniaion of he sampled daa. The noise is aribued o he roughness surface on he propeller apparaus. During he measuremen, he es sample is horizonally fixed o is normal axis. The line of sigh of he nonconac profilomeer is perpendicular o he normal surface of he middle par of he es sample. This orienaion brings higher sensiiviy o he roughness profile a he middle par of he es sample during he measuremen. Consequenly, he measuremen daa indicaes rougher Figure 12. Tes objecs: a propeller and a compuer mouse Figure 13. Fiing of he propeller profile Figure 14. Fiing of he compuer mouse surface wih arbirary-shaped clay addiive Figure 15. Fiing of he compuer mouse surface wih regular-shaped clay addiive 1494

6 TABLE I. Daa Size of daa RESULT SUMMARY OF THREE CASE STUDY Processing ime (s) i Inerior knos RMSE (mm) Profile e-3 Profile e-3 Profile e-2 To validae he capabiliy in handling he muliple kno case, clay addiives wih a regular profile is appended o he compuer mouse. The clay is molded and finished carefully o ge smooh surfaces and sharp edges, where he profile is shown in he lef panel of Figure 15. As expeced, wo spikes are idenified on is second derivaive daa ha correspond o he abrup changes of he surfaces, where denser knos are idenified on he corresponding locaions. The fied curve is also shown on he lef panel of he figure and no indicaion of overshooing implies he abiliy of he mehod o deal wih a muliple kno problem. Quaniaive performances of he fied curves are abulaed in Table 1 ogeher wih quaniaive measures in erms of roo-mean-square error (RMSE). V. CONCLUSION A fas mehod for knos calculaion in a B-spline fiing based on he second derivaive is proposed. The working principle of he mehod is based on employing he half-spli mehod and conrolling i wih a predeermined hreshold o find he bes fied piecewise linear funcion on he compued second derivaive and o idenify he knos. Considering he effec of muliple knos on he second derivaive, our proposed mehod can naurally handle muliple knos. This mehod is proven o be able o reconsruc a B-spline from sample daa, if he second derivaive of he given daa can be compued wih permissible error. One paricular aenion has o be highlighed ha he performance of he proposed mehod depends on he accuracy of he second derivaive compuaion. Neverheless, his can be easily miigaed by uilizing denser daa poin o improve he accuracy of he knos idenificaion. As he mehod is shown o be superior in he processing ime, i is very poenial in he applicaion of online nominal surface reconsrucion ha is commonly required in remanufacuring processes for defec idenificaion and feaure removal. The resuls of he surface reconsrucion uilizing he proposed mehod are communicaed and presened in anoher paper. ACKNOWLEDGMENT The auhors acknowledge he suppor of Nanyang Technological Universiy and he Advanced Remanufacuring and Technology Cenre (ARTC) in his research work. REFERENCES [1] M. Unser, A. Aldroubi, and M. Eden, B-spline signal processing. II. Efficiency design and applicaions, IEEE Trans. on Signal Processing, vol. 41, pp , [2] H. Park and J.-H. Lee, B-spline curve fiing based on adapive curve refinemen using dominan poins, Compuer-Aided Design, vol. 39, pp , [3] L. A. Piegl and W. Tiller, Surface approximaion o scanned daa, The Visual Compuer, vol. 16, pp , [4] T. Lyche and K. Mørken, Kno removal for parameric B-spline curves and surfaces, Compuer Aided Geomeric Design, vol. 4, pp , [5] T. Lyche and K. Mørken, A daa-reducion sraegy for splines wih applicaions o he approximaion of funcions and daa, IMA Journal of Numerical Analysis, vol. 8, pp , [6] S. Miyaa and X. 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