Issue, Volume 7, 03 Reverse Engneerng of the Dgtal Curve Outlnes usng Genetc Algorthm Muhammad Sarfraz, Malk Zawwar Hussan, Msbah Irshad Abstract A scheme, whch conssts of an teratve approach for the recovery of dgtzed, h prnted electronc planar obects, s proposed. It vectorzes the generc shapes by recoverng ther outlnes. The ratonal quadratc functons are used for curve fttng a heurstc technque of genetc algorthm s appled to fnd optmal values of shape parameters n the descrpton of ratonal functons. The proposed scheme s fully automated vectorzes the outlnes of planar mages n a reverse engneerng way. Keywords Ratonal functon, reverse engneerng, genetc algorthm, mages. I I. INTRODUCTION N the last two decades reorentaton of tradtonal artfcal ntellgence methods has been notced toward the soft computng technques. Ths development allows us to solve dffcult problems related to robotcs, computer vson, speech recognton machne translaton. Accordng to Zadeh [3], soft computng technques are characterzed by tolerance of mprecson, uncertanty parallel truth to acheve tractablty, robustness low soluton cost. Soft computng technques such as fuzzy logc (FL), neural networks (NN), genetc algorthm (GA), smulated annealng (SA), ant colony optmzaton (ACO), partcle swarm optmzaton (PSO) have receved a lot of attenton of researchers due to ther potentals to deal wth hghly nonlnear, multdmensonal, ll-behaved complex engneerng problems [4]. Genetc Algorthm (GA) [8], an evolutonary technque, gves us a method to perform romzed global search n a soluton space. In ths space, a populaton of cdate solutons, called chromosomes, s evaluated by a ftness functon n terms of ts performance. The best cdates evolve pass some of ther genetc characterstcs to ther offsprngs, who form the next generaton of potental solutons. The process of reverse engneerng of planar obects comprses of the steps lke: extractng data from boundary of the shapes, fndng the corner ponts usng some technque fnally fttng curve to these corner ponts usng ratonal quadratc functons GA. Ths paper utlzes genetc algorthm M. Sarfraz s wth the Department of Informaton Scence, Kuwat Unversty, Adalya Campus, P.O. Box 5969, Safat 3060, KUWAIT (phone: +965 498 309, e-mal:prof.m.sarfraz@gmal.com). M. Z. Hussan s wth the Department of Mathematcs, Unversty of the Punab, Lahore, Pakstan (e-mal: malkzawwar.math @pu.edu.pk). M. Irshad s wth the Department of Mathematcs, Unversty of the Punab, Lahore, Pakstan (e-mal: msbah09@gmal.com). technque for recoverng the outlnes of planar mages n a reverse engneerng way. Reverse engneerng s qute a modern research feld whch deals wth dverse actvtes. Its scentfc perspectve s generally related to computer scence heren to computer aded geometrc desgn (CAGD). Reverse engneerng of shapes s the process of representng an exstng obect geometrcally n form of computer aded desgn (CAD) model. A good reverse engneerng system not only creates a CAD model of the obect, but t also helps explorng understatng the structure of the obect. Generatng computer aded desgn (CAD) model from scanned dgtal data s used n contour stylng whch needs to adopt some curve or surface approxmaton scheme. Reverse engneerng of planar obects s referred to the process of fttng an optmal curve to the data extracted from the boundary of the btmap mage [0,, 4, 5, 6]. Curve fttng s frequently used n reverse engneerng to reproduce curves from measured ponts. It s always essental to provde new curve-fttng algorthms to acqure curves that satsfy dfferent condtons. Fttng curves to the data extracted by generc planar shapes s the problem whch s greatly worked on durng last two decades. Stll there s a room for researchers n ths feld due to ts applcatons n dverse felds ts dem n the ndustry. There are several advantages of curved representaton of planar obects. For example, transformatons lke scalng, shearng, translaton, rotaton clppng can be appled on the obects very easly. Varous outlne approxmaton technques can be found n the lterature n whch dfferent splne models have been used by the researchers lke Be zer splnes [6], B-splnes [9], Hermte nterpolaton [8] ratonal cubc nterpolaton [9]. There are several other outlne capturng technques [3,, 3, 0-3, 9, 6, 4, 5, 7, 8] avalable n the current lterature most of them are based on least-squares ft [, 3, 0] error mnmzaton [3, 9, ]. Sarfraz et al. [] n ther outlne capturng scheme, calculated the rato between two ntermedate control ponts used ths to estmate ther poston. Ths caused reducton of computaton n subsequent phases of approxmaton. Few other technques nclude use of control parameters [8], genetc algorthms [3], wavelets [9]. In ths work ratonal quadratc functons (concs) are used for curve fttng usng genetc algorthm. The paper s organzed n a way that the frst second step of the proposed scheme, outlne estmaton corner
Issue, Volume 7, 03 detecton method, are descrbed n Secton. Revew of ratonal cubc ratonal quadratc functons s gven n Secton 3. Secton 4 explans the proposed scheme whch s demonstrated wth examples n Secton 5. The paper s concluded n Secton 6. II. CONTOUR EXTRACTION AND SEGMENTATION Frst step n reverse engneerng of planar obects s to extract data from the boundary of the btmap mage or a generc shape, shown n Fg. (a). Capturng boundary or outlne representaton of an obect s a way to preserve the complete shape of an obect. The obects n an mage can also be represented by the nteror of shape. Chan codng for boundary approxmaton encodng was ntally proposed by Freeman [7], whch has drawn sgnfcant attenton over last three decades. Chan codes represent the drecton of the mage help to attan the geometrc data from outlne of the mage. Extracted boundary of the btmap mage gven n Fg. (a) s gven n Fg. (b). (a) (b) (c) Fg. (a) btmap mage of a plane, (b) detected boundary of the mage (c) detected corner ponts from the boundary detected boundary ponts for dfferent mages s gven n Table. Detected corners of the boundary shown n Fg. (b), can be seen n Fg. (c). III. RATIONAL QUADRATIC FUNCTIONS In ths secton pecewse ratonal quadratc functons are presented used for curve fttng whch s an alternate of the ratonal cubc presented n Secton 3.. The ratonal quadratc possesses C contnuty A. C Ratonal cubc functon A pecewse ratonal cubc parametrc functon P C [ t, t + ], wth shape parameters v 0, =,..., n, s used for curve fttng to the corner ponts detected from the boundary of the btmap mage, the ratonal cubc functon s defned for t [ t, t + ], =,..., n, as follows 3 3 F ( θ) + vv ( θ) θ + vw ( θ) θ + F + θ P( t) = P ( t) = 3 3 ( θ ) + v ( θ ) θ + v ( θ ) θ + θ () where F F + are two corner ponts (gven control ponts) of the th segment of the boundary wth h = t t,. V h D = F + + + v v + h D W = F () where D, =,..., n + are the frst dervatve values at the knots t, =,..., n +. Table. Detals of Dgtal Contours Corner ponts. Image Name # of contours # of contour ponts # of ntal corner ponts Fork 673 5 Plane 3 95+36+54 8 Fsh 975 3 Segmentaton of obect boundary before curve fttng s very mportant for two reasons. Frstly, t reduces boundary s complexty smplfes the fttng process. Secondly, each shape conssts of natural break ponts (lke four corners of a rectangle) qualty of approxmaton can be mproved f boundary s subdvded nto smaller peces at these ponts. These are normally the dscontnuous ponts to whch we do not want to apply any smoothng lke to capture them as such. These ponts can be determned by a sutable corner detector. Researchers have used varous corner detecton algorthms for outlne capturng [,, 5, 7, 30]. The method proposed n [5] s used n ths paper. Number of contour ponts Fg. Demonstraton of ratonal cubc functon () It s to be noted that v, =,..., n, are used to control the shape of the curve. Effect of these shape parameters on the curve s shown n Fg. Fg.. Moreover, for v = 3, =,..., n, () represents cubc Hermte nterpolaton t can be consdered as default case of ratonal cubc (). If
Issue, Volume 7, 03 v, then the ratonal cubc functon () converges to lnear nterpolant gven by L ( t) = ( ) F + F (3) θ θ + whch means that the ncrease n v pulls the curve towards F F + n the nterval [ t, t + ] the nterpolant s lnear as shown n Fg.. For v 0 Equaton () can be wrtten n the form P ( t; v ) = R ( θ; v ) F + R ( θ; v ) V + R ( θ; v ) W + R ( θ ; v ) F + (4) where V W are gven n Equaton () R ( θ ; v ), = 0,,,3 are ratonal Bernsten-Bezer weght functons such that C 0 3 3 = 0 R ( θ; v ) = B. Ratonal quadratc functon Consder the general ratonal quadratc, gven as Fg. 3. V ˆ ˆ ˆ ˆ ( θ) + rz ( θ ) θ + W θ P( t) = P ( t) = ˆ ˆ ˆ ˆ ( θ) + r ( θ ) θ + θ where V, Z such that conc passes through (5) W are the control ponts for th segment V Z affects the shape of the conc. for th segment. W the pont r s the shape parameter F ( θ ) + rv ( θ) θ + Z θ P( t) = P ( t) = ( θ ) + ( θ ) θ + θ The other conc whch passes through les n the convex hull of F +, W r Z Z s gven by Z ( θ ) + rw ( θ ) θ + F θ (6) F + t + ( ) = P ( t) = ( θ ) + r ( θ ) θ + θ P t (7) (a) (b) (c) Fg. 3 Demonstraton of ratonal quadratc functon In order to have an alternate quadratc representaton for cubc defned n Secton 3., each segment of pecewse ratonal cubc curve should be dealt ndvdually so that t could be represented by two segments of ratonal quadratc. Ths process wll be done n a way that one conc passes through F Z. Smlarly, the other conc nterpolates Z F +. Consder the frst conc whch passes through t les n the convex hull of F, V Z. F Z Fg. 4 (a) demonstraton of conc (6), (b) demonstraton of conc (7), (c) combned vew of both the concs (6) (7). Followng propertes should be satsfed by concs (6) (7) to be C. h D V = F + r h D + W = F + r V + W F + F h Z D D + = = + + 4r ( ) 3
Issue, Volume 7, 03 It s to be noted that, for r =, =,..., n, denomnators for ratonal quadratc (6) (7) become () ratonal quadratc can be wrtten as: P( t) = P ( t) = F ( θ ) + V ( θ ) θ + Z θ (8) P ( t) = P ( t) = Z ( θ ) + W ( θ ) θ + F θ (9) + Ths s the default case of ratonal quadratc. Both the concs (6) (7) are demonstrated n Fg. 4(a) Fg. 4(b) respectvely whereas Fg. 4(c) depcts combne vew of both the concs (6) (7). Fg. 5 represents combned vew of ratonal cubc () the concs (6), (7) for dfferent values of r v, t can be noted that the ratonal 3r cubc () concdes wth concs (6) (7) f v =. Fgs. 5(a) 5(b) show default vew of ratonal quadratc ratonal cubc respectvely. Fg. 5(c) gves the combne vew of both the default ratonal functons. Fg. 5(d), (e) (f) represent combned vew of concs for r = 4 ratonal cubc for v = 6, combned vew of concs for r = 6 ratonal cubc for v = 9 combned vew of concs for r = 8 ratonal cubc for v = respectvely. (a) (b) (c) (d) (e) (f) Fg. 5 (a) default vew of cons, (b) default vew of ratonal cubc, (c) combned vew of concs ratonal cubc for default cases, (d) combned vew of concs for r = 4 ratonal cubc for v = 6, (e) combned vew of concs for r = 6 ratonal cubc for v = 9, (f) combned vew of concs for r = 8 ratonal cubc for v = A. Parameterzaton Number of parameterzaton technques can be found n lterature for nstance unform parameterzaton, lnear or chord length parameterzaton, parabolc parameterzaton cubc parameterzaton. In ths paper, chord length parameterzaton s used to estmate the parametrc value t assocated wth each pont. It can be observed that θ s n normalzed form vares from 0 to. Consequently, n our case, h s always equal to. B. Estmaton of Tangent Vectors A dstance based choce of tangent vectors s defned as: For open curves: D ' s at F ' s 4
Issue, Volume 7, 03 ( ) ( ) ( ) ( ) ( ) ( )( ) D0 = F F0 F F0 Dn = Fn Fn Fn Fn D = a F F a F + F, =,,..., n For close curves: F = Fn, Fn + = F D = a ( F F ) ( a )( F + F ), = 0,,..., n where F + F a =, = 0,,..., n. F F + F F + (a) (b) (c) (d) (e) (f) (g) (h) () Fg. 6 Demonstraton of proposed scheme (a) btmap mage of plane, (b) detected boundary of the mage n (a), (c) corners detected from the boundary, (d) default concs ftted to the corners along wth boundary, (e) st teraton of concs ftted through GA, (f) nd teraton of concs ftted through GA, (g) 3 rd teraton of concs ftted through GA, (h) 4 th teraton of concs ftted through GA, () 5 th teraton of concs ftted through GA. IV. OPTIMAL RATIONAL QUADRATIC FUNCTIONS Ths Secton descrbes the process of evaluatng optmal quadratc functons usng GA [8]. Genetc Algorthm formulaton of the curve fttng problem dscussed n ths paper s also elaborated here. Suppose, for =,..., n, the data segments (, ) P = x y, =,,..., m be the gven data segments.,,, Then the squared sums S ' s of dstance between P, ' s ther correspondng parametrc ponts P( t )' s on the curve are m determned as S = P ( u, ) P,, =,..., n where u s = are parameterzed n reference to chord length parameterzaton. Conc : When conc represented by the ratonal quadratc (6) s consdered, the squared sum S would be defned as m S = P( u ) P, =,..., n,, = 5
Issue, Volume 7, 03 where P( u, ) s defned as n (6). Conc : Smlarly for conc represented by the ratonal quadratc (7), the squared sum S would be defned as: where m =,, = S P ( u ) P, =,..., n P ( u ) s defned as n (7)., (a) (b) (c) (d) (e) (f) (g) (h) () Fg. 7 Demonstraton of proposed scheme (a) btmap mage of fork, (b) detected boundary of the mage n (a), (c) corners detected from the boundary, (d) default concs ftted to the corners along wth boundary, (e) st teraton of concs ftted through GA, (f) nd teraton of concs ftted through GA, (g) 3 rd teraton of concs ftted through GA, (h) 4 th teraton of concs ftted through GA, () 6 th teraton of concs ftted through GA. Table. Number of corner ponts together wth number of ntermedate ponts for teratons, 3 of GA. Image Name # of ntal corner ponts # of ntermedate ponts n cubc nterpolaton wth threshold value 3 Fnal Itr. Itr. Itr.3 tr. Total tme elapsed Fork 5 0 0 9 35 6.075 sec Plane 8 0 9 30 39 6.7 sec Fsh 3 0 7 9 35 9.395 sec Now to fnd optmal curve to gven data, such values of parameters r s, are requred so that the sums S ' s are mnmal. Genetc Algorthm s used to optmze ths value for the ftted curve. Romly chosen values of r are needed to ntalze the process. Successve applcaton of search operatons lke selecton, crossover mutaton to ths populaton gves optmal values of r s. A. Intalzton Once we have the btmap mage of a character, the boundary 6
Issue, Volume 7, 03 of the mage can be extracted usng the method descrbed n Secton. After the boundary ponts of the mage are found, the next step s to detect corner ponts to dvde the boundary of the mage nto n segments as explaned n Secton. Each of these segments s then approxmated by nterpolatng quadratc functons (6) (7) descrbed n Secton 3.. (a) (b) (c) (d) (e) (f) (g) (h) () Fg. 8 Demonstraton of proposed scheme (a) btmap mage of fsh, (b) detected boundary of the mage n (a), (c) corners detected from the boundary, (d) default concs ftted to the corners along wth boundary, (e) st teraton of concs ftted through GA, (f) nd teraton of concs ftted through GA, (g) 3 rd teraton of concs ftted through GA, (h) 4 th teraton of concs ftted through GA, () 5 th teraton of concs ftted through GA. B. Curve Fttng After an ntal approxmaton for the segment s obtaned, Genetc Algorthm helps to obtan better approxmatons to acheve optmal soluton. The tangent vectors at knots are estmated by the method descrbed n Secton 3.4. C. Breakng Segment For some segments, the best ft obtaned through teratve mprovement, may not be satsfactory. In that case, we subdvde the segment nto smaller segments at ponts where the dstance between the boundary parametrc curve exceeds some predefned threshold. Such ponts are termed as ntermedate ponts. A new parametrc curve s ftted for each new segment as shown n Fgs. 6(f-h), Fgs. 7(f-h) Fgs. 8(f-h). Table gves detals of number of ntermedate ponts acheved durng dfferent teraton of Genetc Algorthm appled n process of curve fttng. D. Algorthm All the steps of computng the desred outlne curve manpulaton can be summarzed nto the followng algorthm: Step. Input the data ponts. Step. Subdvde the data, by detectng corner ponts usng the method mentoned n Secton. Step 3. Compute the dervatve values at the corner ponts by usng formula gven n Secton 3.4. Step 4. Ft the ratonal quadratc functons, of Secton 3, to the corner ponts found n Step. Step 5. If the curve, acheved n Step 4, s optmal then go to Step 7, else fnd the approprate break/ntermedate ponts (ponts wth hghest devaton) n the undesred curve peces. Compute 7
Issue, Volume 7, 03 the best optmal values of the shape parameters r s. Ft ratonal quadratc functons n Secton 3 to these ntermedate ponts. Step 6. If the curve, acheved n Step 5, s optmal then go to Step 7, else add more ntermedate ponts (ponts wth hghest devaton) go to Step 3. Step 7. Stop. Algorthm together wth corner ponts boundary, (f), (g), (h) () depct ftted outlne for nd, 3 rd, 4 th fnal (5 th ) teratons respectvely usng Genetc Algorthm together wth corner ponts, breakponts boundary. Smlarly, the automatc algorthm has been mplemented on the Fork Fg. 7(a) to produce Fgs. 7(b-) n a smlar manner as those n Fg. 6. However, the last teraton n Fg.7() has appeared to be the 6 th one n ths case. The Fsh mage, n Fg, 8(a), has also been attempted for the algorthm mplementaton. The output appeared, n Fgs. 8(b-), n a smlar way as n Fgs. 6(b-). Fg. 9 Graph of mnmum cost for plane showng mxed behavour. Fg. Graph of mnmum cost for plane showng mx behavour. Fg. 0 Graph of mnmum cost for plane showng decreasng behavour. V. DEMONSTRATION Curve fttng scheme, proposed n Secton 4, has been mplemented on dfferent mages of a plane (Fg. 6(a)), a fork (Fg. 7(a)), a Fsh (Fg. 8(a)). In Fg. 6 ((a) represents orgnal mage, (b) shows outlne of the mage, (c) demonstrates corner ponts, (d) presents ftted Hermte curve to the corners along wth boundary of the mage, (e) gves ftted outlne to the corners for st teraton usng Genetc Fg. Stoppng cretera met by GA for mage of plane. Some analytcal study has also been made for the performance of the devsed algorthm. Fg. 9, Fg. 0 Fg. represent mnmum cost durng dfferent generatons of GA for the mage of plane. Fgs. -3 gve the percentage of stoppng crtera met by GA for the mages of plane fsh 8
Issue, Volume 7, 03 respectvely the parameters used whle applyng GA are gven n Table 3. Fg. 3 Stoppng cretera met by GA for mage of fsh. Table 3. Parameters of GA. Sr. No. Name Values Populaton sze 5 Genome length 5 3 Selecton rate 0.5 4 Mutaton rate 0.0 VI. CONCLUSION The reverse engneerng technque of planar obects s presented whch uses concs for curve fttng genetc algorthm to fnd the optmal values of parameters n the descrpton of concs. Two ratonal quadratc functons are mplemented n the replacement of a ratonal cubc. Intal rom populaton of parameters s requred for the proposed scheme to get started then the algorthm assures the values of parameters whch provde the optmal ft to the boundary of the btmap mages of planar shapes. The authors are nterested to proceed further extend the scheme to vectorze 3D shapes. Ths work s n progress wth the authors. REFERENCES [] G. Avraham, V. Pratt, Sub-pxel edge detecton n character dgtsaton, Raster Imagng Dgtal Typography II, 99, pp. 54 64. [] H. 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Khan, An automatc algorthm for approxmatng boundary of btmap characters, Future Generaton Computer Systems, vol. 0, 004, pp. 37 336. [] M. Sarfraz, M. R. Asm, A. Masood, Capturng outlnes usng cubc Be zer curves, Proceedngs of the IEEE nternatonal conference on nformaton communcaton technologes: from theory to applcaton, Aprl 9-3, 004, pp. 539 540. [3] M. Sarfraz, S. A. Raza, Capturng outlne of fonts usng genetc algorthm splnes, The proceedngs of IEEE nternatonal conference on nformaton vsualzaton-iv-uk, July 5-7, 00, pp. 738 743. [4] M. Sarfraz, M. Ryazuddn, M. H. Bag, Capturng planar shapes by approxmatng ther outlnes, Journal of Computatonal Appled Mathematcs, vol. 89, 005, pp. 494 5. [5] M. Sarfraz, S. A. Raza, Vsualzaton of data wth splne fttng: a tool wth a genetc approach, Proceedngs of nternatonal conference on magng scence, systems, technology, Las Vegas, Nevada, USA, June 4-7, 00, p. 99 05. [6] M. Sarfraz, A. Masood, M. R. Asm, A new approach to corner detecton, Computer vson graphcs, vol. 3, 006, p. 58 533. [7] P. J. Schneder, An algorthm for automatcally fttng dgtzed curves, Graphcs Gems, 990, pp. 6 66. 9
Issue, Volume 7, 03 [8] F. A. Sohel, G. C. Karmakar, L. S. Dooley, Arknstall J. Enhanced Be zer curve models ncorporatng local nformaton, Proceedngs of IEEE Internatonal Conference on Acoustcs, Speech, Sgnal Processng, USA, March 8-3, 005, pp. 53-56. [9] Y. Y. Tang, F. Yang, J. Lu, Basc processes of chnese character based on cubc B-splne wavelet transform, IEEE Transactons on Pattern Analyss Machne Intellgence, vol. 3, no., 00, p. 443-448. [30] C. H. Teh, R. T. Chn, On the detecton of domnant ponts on dgtal curves, IEEE Transactons on Pattern Analyss Machne Intellgence, vol., no. 8, 989, pp. 859 873. [3] L. A. Zadeh, Fuzzy Logc Soft Computng: Issues, Contentons Perspectves, Internatonal Conference on Fuzzy Logc, Neural Nets Soft Computng, Izuka, Japan, 994, pp. -. M. Sarfraz s a Professor n Kuwat Unversty, Kuwat. He receved hs Ph.D. from Brunel Unversty, UK, n 990. Hs research nterests nclude Computer Graphcs, CAD/CAM, Pattern Recognton, Computer Vson, Image Processng, Soft Computng. He s currently workng on varous proects related to academa ndustry. He has been keynote/nvted speaker at varous platforms around the globe. He has advsed/supervsed more than 50 students for ther MSc PhD theses. He s the Char of the Informaton Scence Department, Kuwat Unversty. He has publshed more than 50 publcatons n the form of varous Books, Book Chapters, ournal papers conference papers. He s member of varous professonal socetes ncludng IEEE, ACM, IVS, IACSIT, ISOSS. He s a Char, member of the Internatonal Advsory Commttees Organzng Commttees of varous nternatonal conferences, Symposums Workshops. He s the revewer, for many nternatonal Journals, Conferences, meetngs, workshops around the world. He s the Edtor-n-Chef of the Internatonal Journal of Computer Vson Image Processng. He s also Edtor/Guest Edtor of varous Internatonal Conference Proceedngs, Books, Journals. He has acheved varous awards n educaton, research, admnstratve servces. M. Z. Hussan s a Professor n Punab Unversty, Lahore, Pakstan. He receved hs Ph.D. from Punab Unversty, Pakstan, n 00. Hs research nterests nclude Computatonal Mathematcs, Computer Graphcs, CAGD, CAD/CAM, Image Processng Soft Computng. He has advsed/supervsed more than 30 students for ther M.Phl. PhD theses. He has publshed more than 50 publcatons n the form of ournal conference papers. M. Irshad s a Ph.D. student n Punab Unversty, Lahore, Pakstan. She receved her M.Phl. from Punab Unversty, Pakstan, n 006. Her research nterests nclude Computatonal Mathematcs, Computer Graphcs, CAGD, CAD/CAM, Image Processng, Soft Computng. She has publshed more than 6 publcatons n the form of ournal conference papers. 0