FBE IE-11/00-11 AUTOMATED TOLERANCE INSPECTION USING IMPICIT AND PARAMETRIC REPRESENTATIONS OF OBJECT BOUNDARIES CEM ÜNSALAN AYTÜL ERÇĐL

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1 FBE IE-/- AUTOMATED TOLERACE ISPECTIO USIG IMPICIT AD PARAMETRIC REPRESETATIOS OF OBJECT BOUDARIES CEM ÜSALA AYTÜL ERÇĐL Ağustos August Fe Bilimleri Estitüsü Istitute for Graduate Studies i Sciece ad Egieerig Boğaziçi Uiversity, Bebek, Istabul, Turkey Boğaziçi Araştırmaları deeme iteliğide olup, bilimsel tartışmaya katkı amacıyla yayıladıklarıda, yazar(ları yazılı izi olmaksızı kedilerie atıfta buluulamaz. Boğaziçi Research papers are of a prelimiary ature, circulated to promote scietific discussio ad ot to be quoted without writte permissio of the author(s.

2 Automated Tolerace Ispectio usig Implicit ad Parametric Represetatios Cem ÜSALA Aytül ERÇĐL Ohio State Uiversity, Dept. of Electrical Egieerig Boğaziçi Uiversity, Dept. of Idustrial Egieerig Abstract: I this paper, two approaches of Automated Tolerace Ispectio will be proposed ad evaluated to see whether they are reliable, cosistet, robust ad applicable i a idustrial eviromet. A compariso of the studied methods will be preseted both i terms of their capabilities, ad their time requiremets. Keywords: implicit polyomials, parametric modelig, tolerace ispectio.. Itroductio Ispectio i its broadest sese is the process of determiig if a product deviated from a give set of specificatios [7,9]. May differet sectors of idustry have developed automatic systems to detect o-coformace ad treds, ad exercise process cotrol over discrete part maufacture. Dimesioal ispectio is oe of the most importat areas of ispectio. May automated ispectio systems have bee preseted i the literature [7]. Chi ad Harlow have surveyed some of the early work i automated visual ispectio i [3]. Chi preseted a secod survey of papers published from 98 to 987 i [4]. ewma ad Jai have surveyed the automated visual ispectio systems ad techiques that have bee reported i the literature from 983 to 988 i [3]. Curretly, may automated ispectio tasks are performed usig cotact ispectio devices that require the part to be stopped, carefully positioed, ad the repositioed several times. Machie visio ca alleviate the eed for precise positioig ad lie stoppage, ad there is also a lower level risk of product damage durig ispectio [8]. For automated ispectio to be feasible, however, it must ru i real-time ad be cosistet, reliable, robust ad cost-effective. I this paper, two approaches of Automated Tolerace Ispectio are proposed to see whether they are reliable, cosistet, robust ad applicable i a idustrial eviromet. Implicit polyomial ad parametric represetatios are the two approaches used for complex free-form object modelig ad ispectio. With these approaches, objects i D images are described by their silhouettes ad the represeted by D implicit polyomial curves or parametric equatios; objects i 3D data are represeted by implicit polyomial or parametric surfaces. The ispectio is carried out usig these boudary represetatios of the objects. The layout of the paper is as follows: Mathematical prelimiaries are give i sectio two. The proposed methods for automated tolerace ispectio i D are give i sectio three. The results are geeralized to ispectio i 3D i sectio four. I sectio three, a

3 implicit fuctio fittig method ad a theorem for ispectio i 3D is give. Experimetal results are give i sectio 5. Summary ad coclusios are give i sectio 6. I this study, aligmet is assumed to be doe prior to the ispectio operatio.. Mathematical Prelimiaries I this sectio, mathematical theorems ad tools to be used throughout the paper are itroduced. First the distace fuctios betwee two poits i -dimesioal real space are give. Depedig o these distace fuctios, a theorem to fid the distace betwee a poit ad a poit set which represets a shape i -dimesioal space is itroduced. Usig this theorem, a ew method for tolerace ispectio i -dimesioal space is give. Defiitio : Let dist be defied as the distace betwee two poits, ad dist dist β ( β,..., β R by give formulas: ( α, β max α k β k ( dist, β k k / ( α, β ( α k β k (3 k I geeral for p / p p (, β α k βk k α ( α,..., α R ( α α β ( dist p α (4 Defiitio: Let f ( x,..., x be a parametric or implicit represetatio of a - dimesioal shape. Defie two sets as: F x, y : f x,..., x ;( x,..., R {( ( x },..., α, d ( x,..., x : dist( ( x,..., x,( α,..., α { d;( x,..., x,( α,..., R } ( α α C I the followig figures, the poit set havig equal distace from origi for three differet distace fuctios are plotted. It is see from Figure that, for dist the set correspods to a square, for dist the set correspods to diamod ad for dist the set correspods to circle. k k Figure. a C (,, d, dist b C (,, d, dist c C (,, d, dist Defiitio 3: The distace betwee a poit ( α,..., α R ad F, deoted by d, is such that F C( α,..., α, d φ ad F C( α,..., α, d + ε φ as ε Ispectio theorems to be itroduced i the followig sectios will heavily deped o fidig real roots of a polyomial or a fuctio withi give rages. For this purpose, two methods from umerical aalysis are give i the followig theorems. First of these theorems is cocetrated o a iterative search withi a give rage. This theorem is applicable both for fuctios ad polyomial represetatios. The secod theorem is oly applicable for polyomials. It checks the existece of a real root withi give rages.

4 Theorem : (Iterval Halvig Method If f is a cotiuous fuctio o the iterval [ α, β ] ad if f ( α f ( β <, the f must have a zero i [ α, β ]. Proof: Sice ( α f ( β < α, β ad, therefore, it has at least oe zero i the iterval. More iformatio about this theorem ca be foud i the give refereces [,]. f, the fuctio f chages sig o the iterval [ ] Theorem : (Fourier-Buda Theorem Let p deote a real polyomial of degree, ad v(x deote the umber of sig chages i the sequece { p( x, p' ( x,..., p ( x }. For ay real α ad β such that β > α, p ( α p( β the umber of zeroes, z, i the iterval [ α, β ] (each zero couted with its proper multiplicity equals v( α v( β mius a eve o-egative iteger. z v( v( β k k,,,3.... α { } Proof: The proof of this theorem ca be foud i []. Corollary : If ( α v( β α, β. Proof: Let k, so that we obtai the upper boud for the umber of zeros, z, i the iterval [ α, β ]. Give that v( α v( β, z. This meas o zeros i the iterval [ α, β ]. I the followig Lemma, Fourier series approximatio of parametric represetatio of the shape will be give i a special form. The same procedure is applied to obtai the implicit form of star shaped objects [6]. v the o zero i the iterval [ ] Defiitio 4: Let C be a + dimesioal vector such that i i C ( i + ( a, C (i + ( b i,,,3 where s a x( k s k (5 a x( k cos( π b s k s k π s k x( ksi( s (7 s k i i ad D ( i + ( c, D (i + ( d i,,,3 where s c y( k s k (8 c y( kcos( d s k s k π s k y( ksi( s ( s k s s (6 (9 a, b, c, d correspod to Fourier series coefficiets of the parametric represetatio of a D shape f ( x( t, y( t t [, ]. s s is the total umber of samples of the fuctio x(k of the cotour of the D shape.

5 + + Defie to be a operator such that a i S ai S ( i i i ( Lemma : th degree Fourier series approximatio for x(t ad y(t ca be represeted as: ( α ± C y( t ( α ± α x( t α where α si(t Proof: th degree Fourier series approximatios for x(t ad y(t are: x D ( t ( a si( t + b cos( t ( y t ( c si( t + d cos( t ( (3 With the help of the operator give i Defiitio 4 ad expasios of sie ad cosie terms give i Appedix, equatios. ( ad (3 ca be writte as: x ( t ( si( t + cos( t C (4 y t ( si( t + cos( t Let The ( D (5 si(t α. From polar coordiate represetatios, si ( t + cos ( t t R cos( α t ±, α < So the above represetatios for x(t ad y(t are obtaied. I the previous results, the square root terms ca ot be avoided. To be able to use Theorem for fidig the roots, Taylor Series expasio ca be used to obtai a polyomial approximatio for fuctio forms obtaied. For this purpose the followig theorems are give here. Theorem 3: (Taylor s theorem i R, Form II Let U be a ope set i R ad let c be a poit of U. Let f be a real-valued fuctio o U that is (k+ times cotiuously differetiable o a eighborhood (c of c. For each x i (c there exists a poit z x i (c,x such that j K * f ( x, x ( x c + ( x c f ( c, c + Higher Order Terms j j! x x The above polyomial is called the k th Taylor polyomial of f at c. As a polyomial i * two variables, f has degree less tha or equal to k. Proof: The proof of this theorem ca be foud i the give referece []. Corollary: (Taylor s theorem i D Taylor polyomial i D aroud c is, K j d f ( x x c f c + Higher Order Terms j j ( dx ( (6! Proof: The proof of this theorem ca be foud i the give referece []. The Taylor series approximatio for x x x 5x 7x y ± d d 8d 6d 8d 56d y ± d x d x d + Higher Order Terms ca be give as:

6 To observe the approximatio quality of the give polyomials, circle ad two approximatig polyomials for this circle are plotted i Figure. Oly the regio close to the horizotal axis has some deficiecy, which will decrease as the degree of the approximatio is icreased. 3. Ispectio i D Figure. Circle ad its polyomial approximatio of degree I most works i computer aided desig, a surface is represeted parametrically as a smooth vector fuctio as s: R R 3 where each coordiate of s is typically either a polyomial or ratio of polyomials. Implicit polyomials, o the other had, are very effective for ivariat object recogitio ad are amog the most effective represetatios for complex free-form object modelig. For the ispectio process, objects i D images are described by their silhouettes ad the represeted by D parametric/implicit curves. The ext step is to obtai the image of the object to be ispected ad to extract the edge of the object. By applyig the procedure described below, each poit belogig to the edge is tested to check whether it is iside tolerace values or ot. This check is performed by formig a circle of radius equal to the tolerace limits aroud each pixel. If these circles itersect with the ideal template, the object is withi tolerace limits. The procedure is based o two sets, oe defiig the ideal template, ad the other defiig the tolerace regio aroud each poit belogig to the edge of the image, represeted by F ad C(d respectively. Here d deotes the radius of the circles, hece the tolerace. 3. Tolerace Ispectio usig a implicit polyomial represetatio of the object boudary A implicit surface is the set of zeros of a smooth fuctio f: R 3 R of three variables: Z(f {(x,x,x 3 t : f(x,x,x 3 }. Similarly, a implicit -D curve is the set of zeros of a smooth fuctio f: R R of two variables: Z(f {(x,x : f(x,x } The curves or surfaces are algebraic if the fuctios are polyomials. Other commo surface represetatios such as quadric surfaces (e.g. coes, ellipsoids, hyperboloids, etc. admit both a parametric ad a implicit form. However, there are some algebraic surfaces (third order ad higher that ca oly be represeted implicitly. A essetial requiremet for practicality of implicit polyomial related techiques is to have robust ad cosistet implicit polyomial fits to data sets [5]. The problem has bee formulated through differet miimizatio criteria ad costraits. A variety of iterative ad o-iterative solutio techiques, perturbatio methods ad stoppig rules have bee proposed i the literature [,4,5,6]. I this paper, Fourier series expasio of polar represetatio of a object cotour is used to obtai robust ad fast implicit polyomial

7 fittigs for star shaped objects i D ad 3D. The method is based o approximatig the polar represetatio of the cotour by a Fourier series ad the fidig the correspodig implicit polyomial usig the polar/cartesia coversio formulas. Fittig is achieved by fidig coefficiets i this structure usig the data. [6]. Lemma : Give a tolerace bad ± d ad a object havig a implicit polyomial model f ( x, y to observe whether a poit ( x, y is iside toleraces, due to differet distace fuctios, it is sufficiet to check the existece of at least oe real root for oe of the followig fuctios, havig oe variable. i- For distace fuctio dist f x x ± d y x d (7 (, ( ± d x y y f y d (8, ii- For distace fuctio dist f x x ± d x y x d (9 (, ( ( ± ( d y x y y, f y d ( iii- For distace fuctio dist ( x x, ± d x y ( ± d y x y y f x d ( f,, y d ( Proof: For three differet distace fuctios, if there exists at least oe real root withi give rages, the the implicit polyomial f ( x, y ad the circle, or the shape used for differet distace fuctios, havig a ceter ( x, y ad radius d has at least oe commo poit. From Defiitio 3, it is kow that the distace betwee poit ( x, y ad f x, y is smaller tha or equal to d. So the poit x, is iside toleraces. ( ( y 3. Tolerace Ispectio usig a parametric represetatio of the object boudary Although implicit algebraic curves ad surfaces have may good properties that make them the atural choice for object recogitio ad positioig, parametric curves ad surfaces outperform them i a fudametal area. More stable or robust algorithms are kow to approximate sets of measured data poits by parametric curves ad surfaces tha by their implicit couterparts. []. Lemma 3: Give a tolerace bad ± d ad a object havig a parametric model f ( x( t, y( t, t [, s ], to observe whether a poit ( x, y is iside toleraces, due to differet distace fuctios, it is sufficiet to check whether there exists at least oe t [, s ] that satisfies the iequality, for the distace fuctio used, give below. Aother method is to write the followig fuctios i special form give i Lemma ad search for the existece of real root for obtaied polyomial. i- For distace fuctio dist

8 x( t x d ad y( t y d (3 ii- For distace fuctio dist x( t x + y( t y d (4 iii- For distace fuctio dist x( t x + y( t y (5 ( ( d Proof: For three differet distace fuctios, if there exists at least oe t [, ] that s satisfies the iequalities give i eqs. (3 to (5, the the parametric form f ( x( t, y( t ad the circle, or the shape used for differet distace fuctios, havig a ceter ( x, y ad radius d has at least oe commo poit. From Defiitio 3, it is kow that the distace betwee poit ( x, y ad f ( x, y is smaller tha or equal to d. So the poit ( x, y is iside toleraces. If the parametric form is represeted as a polyomial by Lemma, the the methods derived for implicit represetatios ca be used. 4. Ispectio i 3D For the ispectio of 3D objects, the object boudaries are represeted by 3D parametric/implicit surfaces. The ispectio process is very similar to that described i previous sectios for the case of D objects. This check is performed by formig a sphere of radius equal to the tolerace limits aroud each pixel. If these spheres itersect with the ideal template, the object is withi tolerace limits. 4. Ispectio by implicit represetatios Lemma 4: Give a tolerace bad ± d ad a object havig a implicit polyomial model f ( x, y, z to observe whether a poit ( x, y, z is iside toleraces, it is sufficiet to check the existece of at least oe of the followig o-degeerate implicit fuctios havig two variables. i- For distace fuctio dist f x x y y, ± d z x d, y d (6 (, ( x x, ± d y, z z ( ± d x y y, z z f x d, z d (7 f y d, y d (8, ii- For distace fuctio dist f x x y y, ± d x y z x d, y d (9 (, ( ( x x, ± ( d x z y, z z ( ± ( d y z x y y, z z f x d, z d (3, f y d, z d (3 iii- For distace fuctio dist

9 ( x x, y y, ± d x y z ( x x, ± d x z y, z z ( ± d y z x, y y, z z f x d, y d (3 f x d, z d (33 f y d, z d (34 Proof: If at least oe of the implicit fuctios give i eqs. (6 to (34, due to differet distace fuctios, represet a o-degeerate implicit fuctio the the sphere havig a ceter ( x, y, z, radius d ad implicit fuctio f ( x, y, z have commo poits. So from Defiitio 3, the distace betwee poit ( x, y, z ad f ( x, y, z is smaller tha d. that is to say the poit is iside toleraces. To check whether a implicit fuctio/polyomial is degeerate or ot, it is sufficiet to covert it to parametric form ad check for at least oe positive real poit belogig to this parametric form. Coversio betwee implicit ad parametric forms ca be foud i the give referece [7]. Lemma 5: Give a tolerace bad ± d ad a object havig a implicit polyomial model f ( x, y, z to observe whether a ad a poit ( x, y, z is iside toleraces, it is sufficiet to check the existece of a space curve defied by f ( x, y, z ad implicit surface(s represetig the distace fuctio used. Proof: A space curve is represeted by the itersectio of two implicit surfaces. The first implicit surface is f ( x, y, z which represets the object. The secod implicit surface will be defied by distace fuctio used. If both of these surfaces itersect, the the poit to be ispected will be iside toleraces. I the same time, there will be a space curve formed by these two implicit surfaces. Checkig for the existece of a space curve ca be foud i the give referece [7]. 4. Ispectio by parametric represetatios: Lemma 6: Give a tolerace bad ± d ad a object havig a parametric patch model f ( X ( θ, ϕ, Y( θ, ϕ, Z( θ, ϕ, to observe whether a poit ( x, y, z is iside toleraces, it is sufficiet to check the existece of at least oe real root for oe of the followig fuctios havig two variables. i- For distace fuctio dist X ( θ, ϕ x d ad Y ( θ, ϕ y d ad Z( θ, ϕ z d (35 There exists at least oe θ, such that satisfies both iequalities. ( ϕ ii- For distace fuctio dist X ( θ, ϕ x + Y ( θ, ϕ y + Z ( θ, ϕ z d (36 ( θ, ϕ There exists at least oe such that satisfies the iequality. iii- For distace fuctio dist ( X ( θ, ϕ x + ( Y ( θ, ϕ y + ( Z( θ, ϕ z d (37 There exists at least oe θ, such that satisfies the iequality. ( ϕ

10 Proof: If oe of the fuctios give i eqs. (35 to (37, due to differet distace fuctios, have at least oe real root withi give rages, the the parametric patch model f ( X ( θ, ϕ, Y( θ, ϕ, Z( θ, ϕ ad the circle, or the shape used for differet distace fuctios, havig a ceter ( x, y, z ad radius d has at least oe commo poit. From Defiitio 3, it is kow that the distace betwee poit ( x, y, z ad f ( X ( θ, ϕ, Y( θ, ϕ, Z( θ, ϕ is smaller tha or equal to d. So the poit ( x, y, z is iside toleraces. The existece of at least oe real root ca be checked by umerical methods by iterative root fidig i D [8]. 5. Experimets I order to test the theory explaied above agaist real pieces several parts have bee obtaied from maufacturig orgaizatios. Both withi ad out of tolerace pieces were obtaied for testig purposes. After preprocessig operatios were coducted i a lab eviromet to obtai boudary data, the best fits were foud. ote that the order of the fit is differet for the implicit polyomial ad the parametric method. I the followig sectios, the applicatio of the implicit ad parametric method are explaied i detail ad the results are preseted. 5. Implemetatio of the Implicit Method The first step i the implemetatio is fidig the best fit for each part. As ca be see from Figure, the fit improves with the icrease i order of the polyomial. Sice the Fourier series expasio of the boudary is used i estimatig the implicit polyomial, error the actual image ad the fitted curve is easily obtaied. I the examples preseted, order was specified such that the error would be at the thousadths level. Oce a appropriate fit is foud, the ispectio is doe usig the procedure outlied above. To demostrate the importace of tolerace specificatio, the results of two differet tolerace values are displayed. For the first tolerace value (tolerace set to pixels, defects ca be detected (Figure 3.a, whereas i the secod tolerace value (tolerace set to 6 pixels all poits remai withi the tolerace limit (Figure 3.b. The poits that are out of tolerace are show with a red star i Figure 3.c. (for tolerace pixels. Figures 4 ad 5 show the ispectio results for some other objects.. Out of tolerac Figure 3. (a tolerace pixels defected, (b tolerace6 fail to detect defects (c out of tolerace poits marked red

11 (a (b (c (d Figure 4. (a The template object ad the implicit fit (b The template object ad the faulty part with implicit fit (c implicit ispectio with tolerace3 pixels (d ispectio with tolerace pixels. Figure 5 a Data ad 57 th order implicit fit, b Ispectio with tolerace pixels c Out of tolerace poits marked red The implicit polyomial method was very successful for the give examples, however, with the described fittig techique, oly star-shaped items could be modeled. Figure 5 shows a usuccessful attempt to fit a implicit polyomial to a o-star-shaped object at a order of. The image is of a alumium profile. I order to model these parts iterpolatio ad further aalysis is required. Whe the error of fit is above the give threshold, carryig out the ispectio task is ot reasoable. Figure 6. A usuccessful th order implicit fit 5. Implemetatio of the Parametric Method The same images used i applicatio of the ew implicit method were used to experimet with the parametric fittig techique. The sum of squared deviatios betwee the actual image ad the fitted curve is utilized i determiig the best fit. Figures 6-8 show the results of ispectig a object usig parametric represetatio.

12 Fig 7. a Imb-4 Data ad 4 th order b Ispectio with c Out of tolerace parametric fit tolerace pixels poits marked red Fig 8. a Imb4- Data ad 5 th order b Ispectio with c Out of tolerace parametric fit tolerace pixels poits marked red Fig 9. a Imb6- Data ad 5 th order b Ispectio with c Out of tolerace parametric fit tolerace pixels poits marked red It is importat to ote the modelig capability of parametric equatios. Although the implicit method was limited with star-shaped objects, it is possible to fid a fit for very complicated, o-star-shaped objects usig the parametric method. To give a example, the alumium profile that could ot be modeled usig implicit polyomials is preseted here with its parametric fit ad the ispectio output (Figure. Figure. a- imb8-4 th order parametric fit b- tolerace pixels (ote lower right edge c- out of tolerace poits marked red 5.3 Compariso of the Two methods A compariso of the implicit polyomial ad parametric represetatio methods is preseted i Table. Table : Compariso of the implicit ad the parametric method

13 Image # of poits Order of fit T. Fittig (sec T. Ispectio (sec Implicit parametric implicit parametric implicit Parametric Imb_ Imb_ Imb_ Imb_ Imb3_ Imb3_ Imb3_ Imb3_ Imb4_ Imb7_ Imb8_ The compariso is performed for both the fittig ad the ispectio procedures. The images icluded are show together with the umber of poits they have ad the order of the model their fit was achieved at the give tolerace level. T. Fittig is the time that is required to calculate the coefficiets of the implicit polyomial that is fitted. T. Ispectio is the time it takes for the ispectio algorithm to check whether object poits are withi the tolerace limits aroud the fit. Both the fittig time ad the ispectio times are foud to be icreasig with umber of poits ad degree of order. The time values give are i MATLAB, hece do ot represet real ispectio times. C implemetatio of the code would speed up the give times by at least a factor of. I Figure, the degrees of order for both methods are displayed. It is see that there is o clear relatio betwee method ad order of fit. Therefore it ca be cocluded that the difficulty to achieve a fit for a object is ot related to the method employed. As for the fittig time requiremets Figure shows that fittig time is cosiderably shorter for the parametric method. The ispectio time requiremets for the two methods are compared i Figure 3. Order of Fit 6 4 Imb_ Imb_ Imb_3 Imb_4 Imb3_ Imb3_3 Imb3_4 Imb3_6 Imb4_ Imb7_ Imb8_ Implicit Parametric Figure. Order of fit i the two methods for ivestigated images

14 Fittig Time (sec.5.5 Implicit Parametric Imb_ Imb_ Imb_3 Imb_4 Imb3_ Imb3_3 Imb3_4 Imb3_6 Imb4_ Imb7_ Imb8_ Figure. Fittig time requiremets i the two methods for ivestigated images T.Isp(d (dist (sec Imb_ Imb_ Imb_3 Imb_4 Imb3_ Imb3_3 Imb3_4 Imb3_6 Imb4_ Imb7_ Imb8_ Implicit Parametric Figure 3. Ispectio time requiremets i the two methods for ivestigated images I this example, ispectio of a 3D object by itersectio of two implicit surfaces is give. I ispectio, the first surface correspods to object to be ispected ad the secod surface correspods to the set created to ispect the poit, (,,, at the ceter. Tolerace value is take as d. Both of these implicit surfaces are plotted i Figure 4. Space curve is formed sice the poit to be ispected is iside toleraces. This formed space curve is plotted i Figure 5. For the object: x + y + z The ispectio surface: x + y + ( z. 3 Their itersectio set is obtaied as: x + y ad correspodig z-level is z Figure 4. Object ad data set for ispectio Figure 5. Space curve formed

15 6. Summary ad Coclusios I this paper, two approaches of Automated Tolerace Ispectio were itroduced ad their properties were studied. The ispectio is carried out usig implicit/parametric boudary represetatios of the objects. Fourier series expasio of polar represetatio of a object cotour is used to obtai robust ad fast implicit polyomial/parametric fuctio fittig of the object boudary. The order of the fit was specified such that the error would be at the thousadths level. As expected, better fits are achieved as the order of the model is icreased at the expese of icreased computatioal time. The processig time seems to icrease liearly with the order of the polyomial fit ad with the umber of poits o the boudary of the object. After the itroductio of two ew methods with implicit polyomial ad parametric represetatio the studied methods were compared o basis of their capabilities ad time requiremets. The studied implicit method works oly for star-shaped objects whereas the parametric method ca obtai fits for all free form objects allowig more flexibility i the implemetatio. I terms of time requiremets, the parametric method is faster tha implicit polyomial represetatio i both fittig ad ispectio. dist, which is the Euclidea distace, gives fastest result for both methods. Experimets doe i the previous sectio show clearly the practical applicatio power of the itroduced ispectio methods. The ispectio methods itroduced are suitable for parallel processig, hece ca be implemeted i real time. Ackowledgemets: This work has bee partially supported by TUBITAK MISAG 6, SF 465, EUREKA 77 ad BU Research Fud 99A3 projects. Refereces. M. M. Blae, Z. Lei, H. Civi ad D. B. Cooper The 3L Algorithm for Fittig Implicit Polyomial Curves ad Surfaces to Data IEEE PAMI,. R.L. Burde, J.D. Faires ad A.C. Reyolds, umerical Aalysis, Pridle, Weber & Scmidt, R.T. Chi ad C.A. Harlow, Automated Visual Ispectio: A Survey IEEE Tras. PAMI, vol. 6, pp , R.T. Chi, Automated Visual Ispectio: 98 to 987 Computer Visio Graphics ad Image Processig, vol. 4, pp , S.A. Douglas, Itroductio to Mathematical Aalysis, Addiso-Wesley, M. Hebert, J. Poce, T. Boult ad A. Gross eds. Object Represetatio i Computer Visio. Spriger Lecture otes i Computer Sciece Series. Spriger-Verlag, K. Hedegre, Methodology for Automatic Image-Based Ispectio of Idustrial Objects, i J. Saz (ed., Advaces i Machie Visio, pp. 6-9, Spriger Verlag, ew York, P. Herici, Computatioal Complex Aalysis vol., Joh Wiley ad Sos, ew York, C.W. Keedy, E.G. Hoffma ad S.D. Bod, Ispectio ad Gagig, Sixth Editio, Idustrial Press Ic., ew York, D. Kicaid ad W. Cheey, umerical Aalysis, d editio, Brooks Cale, Califoria, 996.

16 . Z. Lei, H. Civi ad D.B. Cooper, Free form Object Modelig ad Ispectio Proceedigs, Automated Optical Ispectio for Idustry, SPIE s Photoics Chia 96, Beijig, Chia, ovember 996. J.H. Mathews, umerical Methods for Mathematics Sciece ad Egieerig, d editio, Pretice Hall, ew Jersey, T.S. ewma ad A.K. Jai A Survey of Automated Visual Ispectio Computer Visio ad Image Uderstadig, vol. 6, pp. 3-6, March G. Taubi, Estimatio of plaar curves, surfaces ad o-plaar space curves defied by implicit equatios, with applicatios to edge ad rage image segmetatio. IEEE Trasactios o Patter Aalysis ad Machie Itelligece, ovember G. Taubi, F. Cukierma, S. Sulliva, J. Poce ad D.J. Kriegma, Parametrized family of polyomials for bouded algebraic curve ad surface fittig IEEE Trasactios o Patter Aalysis ad Machie Itelligece, March C. Usala ad A. Ercil, A ew robust ad fast implicit polyomial fittig techique, Proceedigs of MVIP 99, Akara, Turkey, September 999, 7. C. Usala ad A. Ercil, Coversio betwee parametric ad implicit forms for computer graphics ad visio, Proceedigs of ISCIS 99, pp , October 999, Kusadasi, Turkey 8. C. Usala ad A. Ercil, Automated Tolerace Ispectio By Implicit Polyomials, Proceedigs of ICIAP '99 th Iteratioal Coferece o Image Aalysis ad Processig, Image-based Emergig Applicatios, p. 8-5, September 999, Veice, Italy

17 Appedix We ca represet cos θ ad si θ for ay iteger, from recursive calculatios startig from cos θ ad si θ as: cos θ cos θ si θ si θ si θ cosθ 3 3 cos 3θ cos θ 3si θ cosθ si 3θ 3siθ cos θ si θ cos 4θ cos θ 6si θ cos θ + si θ si 4θ 4siθ cos θ 4 cosθ si θ