Two Approaches for Vectorizing Image Outlines

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1 Internatonal Journal of Machne Learnng and Computng, Vol., No. 3, June 0 Two Approaches for Vectorzng Image Outlnes Muhammad Sarfraz, Member, IACSIT Abstract Two approaches, based on lnear and conc splnes, are proposed here for vectorzng mage outlnes. Both of the approaches have varous phases ncludng extractng outlnes of mages, detectng corner ponts from the detected outlnes, and curve fttng. Interpolaton splnes are the bases of the two approached. Lnear splne approach s straght forward as t does not have a degree of freedom.n terms of some shape controller n ts descrpton. However, the dea of the soft computng approach, namely smulated annealng, has been utlzed for conc splnes. Ths dea has been ncorporated to optmze the shape parameters n the descrpton of the generalzed conc splne. Both of the lnear and conc approaches ultmately produce optmal results for the approxmate vectorzaton of the dgtal contours obtaned from the generc shapes. Demonstratons and a comparatve study also make the essental parts of the paper. Index Terms Vectorzaton, corner ponts, generc shapes, curve fttng. I. INTRODUCTION Vectorzng outlnes of mages s requred n varous real lfe problems ncludng font desgn, cartoonng, pattern recognton, etc. It s one of the mportant problems of computer graphcs, vson, and magng. Varous mathematcal and computatonal phases are nvolved n the whole process. Ths s usually done by computng a curve close to the data pont set. Computatonally economcal and optmally good soluton s an ultmate objectve to acheve the vectorzed outlnes of mages for planar objects. Desgnng and modelng some approprate curve scheme []-[3] s one of the mportant phases of capturng and vectorzng outlnes of mages. It plays a sgnfcant role n varous applcatons. The representaton of planar objects, n terms of curves, has many advantages. For example, scalng, shearng, translaton, rotaton and clppng operatons can be performed wthout any dffculty. Although a good amount of work has been done n the area [4]-[6], t s stll desred to proceed further to explore more advanced and nteractve strateges. Most of the up-to-date research has tackled ths knd of problem by curve subdvson or curve segmentaton. The proposed work s the extenson of the work n [7]. Ths work s a presentaton of two approaches usng lnear and conc nterpolatons. The lnear nterpolant approach s straght forward. However, the conc approach s nspred by an optmzaton algorthm based on smulated annealng (SA) by Krkpatrck et.al [8]. It motvates the author to an optmzaton technque proposed for the outlne capture of planar mages. In ths paper, the data pont set represents any Manuscrpt receved March 0, 0; revsed Aprl 6, 0. Ths work was supported by Kuwat Unversty, Research Grant No. [WI 05/0]. M. Sarfraz s wth the Department of Informaton Scence, Kuwat Unversty, Adalya Campus, Kuwat (e-mal: muhamad.sarfraz@ku.edu.ke; prof.m.sarfraz@gmal.com). generc shape whose outlne s requred to be captured. We present an teratve process to acheve our objectve. The algorthm comprses of varous phases to acheve the target. Frst of all, t fnds the contour [9] [] of the gray scaled btmap mages. Secondly, t uses the dea of corner ponts [3] [9] to detect corners. That s, t detects the corner ponts on the dgtal contour of the generc shape under consderaton. These phases are consdered as preprocessng steps. Lnear and conc nterpolants are then used to vectorze the outlne. The dea of smulated annealng (SA) s used to ft a conc splne whch passes through the corner ponts. It globally optmzes the shape parameters n the descrpton of the conc splnes to provde a good approxmaton to the dgtal curves. In case of poor approxmaton, the nsertons of ntermedate ponts are made as long as the desred approxmaton or ft s acheved. The organzaton of the paper s as follows, Secton dscusses about preprocessng steps whch nclude fndng the boundary of planar objects and detecton of corner ponts. Secton 3 s about the nterpolant forms of lnear and conc splne curves. Overall methodology of curve fttng s explaned n Secton 4, t ncludes the dea of knot nserton as well as the algorthm desgn for the proposed vectorzaton schemes. Demonstraton of the proposed schemes as well as comparatve study s presented n Secton 5. Fnally, the paper s concluded n Secton 6. II. PREPROCESSING The proposed schemes start wth fndng the boundary of the generc shape and then usng the output to fnd the corner ponts. The mage of the generc shapes can be acqured ether by scannng or by some other mean. The am of boundary detecton s to produce an object s shape n graphcal or non-scalar representaton. Chan codes [], n ths paper, have been used for ths purpose. Demonstraton of the method can be seen n Fg. whch s the contour of the btmap mage shown n Fg.. Corners, n dgtal mages, gve mportant clues for the shape representaton and analyss. These are the ponts that partton the boundary nto varous segments. The strategy of gettng these ponts s based on the method proposed n [3]. The demonstraton of the algorthm s made on Fg.. The corner ponts of the mage are shown n Fg.. III. CURVE FITTING AND SPLINE The motve of fndng the corner ponts, n Secton, was to dvde the contours nto peces. Each pece contans the data ponts n between two subsequent corners nclusve. Ths means that f there are m corner ponts cp, cp,, cp m then there wll be m peces p, p,, p m. We treat each pece separately and ft the splne to t. In general, the th pece 30

2 Internatonal Journal of Machne Learnng and Computng, Vol., No. 3, June 0 contans all the data ponts between cp and cp + nclusve. After breakng the contour of the mage nto dfferent peces, we ft the splne curve to each pece. To construct the parametrc splne nterpolant on the nterval t, t ], we have m [ 0 n F R, = 0,,..., n, as nterpolaton data, at knots t, = 0,,..., n. Obvously, the parameters u 's, when equal to, provde the specal case of quadratc splne. Otherwse, these parameters can be used to loose or tghten the curve. Ths paper proposes an evolutonary technque, namely smulated annealng (SA), to optmze these parameters so that the curve ftted s optmal. For the detals of SA approach, the reader s referred to [30]. Fg.. Pre-processng steps: Orgnal mage, Outlne of the mage, Corner ponts acheved. A. Lnear Splne The curve ftted by a lnear splne s a canddate of best ft, but t may not be a desred ft. Ths leads to the need of ntroducng some extra treatment n the methodology. Ths secton deals wth a form of lnear splne. It ntroduces parameters t s n the descrpton of lnear splne defned as follows: P t) P ( θ ) + P θ () ( = + ( t t ) θ [ t t ( t), ) =, h + = t + t, h and P and P + are corner ponts of the th pece. B. Fnal Stage The curve ftted by a conc splne s a canddate of best ft, but t may not be a desred ft. Ths leads to the need of ntroducng some shape parameters n the descrpton of the conc splne. Ths secton deals wth a form of conc splne. It ntroduces shape parameters u s n the descrpton of conc splne defned as follows: P ( θ ) + uu θ ( θ ) + P + θ = ( θ ) + uθ ( θ ) + θ P ( t), () U ( V + W ) h D =, V = P +, u = h D. + W P + u D and D + are the correspondng tangents at corner ponts P and P + of the th pece. The tangent vectors are calculated as follows: ( P P0 ) D0 = ( P P0 ) D = a ( P P ) + ( a )( P + P ), (3) ( P n Pn ) Dn = ( Pn Pn ) P + P a =. P P + P P + IV. SIMULATED ANNEALING (SA) Smulated annealng (SA) [9], [30] s a global optmzaton method based on the Monte Carlo method. It works on the analogy of the energy n an n-body system the materal s cooled to lower temperatures gradually to result n a perfect crystal structure. The perfect crystal structure s attaned by havng mnmum energy n the materal. Ths analogy translates to the optmzaton done n smulated annealng n fndng a soluton that has the lowest objectve functon value. The soluton space s all the possble solutons. The current soluton s the present state of the materal. The algorthm teratvely tres to change the state of the materal and check whether t has mproved. The materal s state s changed slghtly to fnd a neghborng state.e. a close soluton value n the soluton space. It s possble that all neghborng states of current states are worse solutons. The algorthm allows gong to a worse state wth a certan probablty. Ths probablty decreases as the algorthm teratons proceed. Fnally, t only allows a change n state f t s strctly better than the current soluton. Detals of SA theory can be found n [9], [30] and an example applet can be seen n [7]. A detaled descrpton of the mappng of the SA technque on the proposed problem s gven n the next secton. V. PROPOSED APPROACH The proposed approach to the curve problem s descrbed here n detal. It ncludes the phases of problem matchng wth SA usng conc splnes, descrpton of parameters used for SA, curve fttng, and the overall desgns of algorthms A. Problem Mappng Ths secton descrbes about the SA formulaton of the current problem n detal. Our nterest s to optmze the values of conc curve parameters u such that the defned curve fts as close to the orgnal contour segments as possble. We use SA for the optmzaton of these parameters for the ftted curves. Hence the dmensonalty of the soluton space s for conc curves. Each state n the SA soluton space represents a value of u for concs. We start wth an ntal state that s a gven value of u for concs. A startng temperature s also chosen arbtrarly. Ths temperature s an nherent nternal parameter of SA and has no sgnfcance or mappng on our problem. The algorthm mantans a record of the best state ever reached throughout the algorthm run. Ths s the value of u for concs that has gven the best curve fttng so far. Ths best soluton gets updated whenever the algorthm fnds a better soluton. The algorthm teratvely looks for neghborng states that may or may not be better than the current one. These neghborng 30

3 Internatonal Journal of Machne Learnng and Computng, Vol., No. 3, June 0 states are values of u for concs that are slghtly dfferent from the values of u for concs. The coolng rate n SA s the factor affectng the lkelhood of selectng neghborng values of u for concs that gve a curve fttng worse than the values of u for concs. Note that we apply SA ndependently for each segment of a contour that we have dentfed usng corner ponts. SA s appled sequentally on each of the segments, generatng an optmzed ftted curve for each segment. The algorthm s run untl the maxmum allowed tme s reached, or an optmal curve fttng s attaned. B. Intalzaton Once we have the btmap mage of a generc shape, the boundary 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 as explaned n Secton. Ths corner detecton technque assgns a measure of corner strength to each of the ponts on the boundary of the mage. Ths step helps to dvde the boundary of the mage nto n segments. Each of these segments s then approxmated by nterpolatng splne descrbed n Sectons 3.. The ntal soluton of splne parameters u values for concs are randomly selected wthn the range [-, ]. C. Curve Fttng After an ntal approxmaton for the segment s obtaned, better approxmatons are obtaned through SA to reach the optmal soluton. We experment wth our system by approxmatng each segment of the boundary usng the conc splnes of Secton 3.. The conc splne method s a varaton of the quadratc splne. It provdes greater control on the shape of the curve and also effcent to compute. The tangents, n the descrpton of the splne, are computed usng the arthmetc mean method descrbed n Eqn. (3). Each boundary segment s approxmated by the splne. The shape parameter u, n the conc splne, provdes greater flexblty over the shape of the curve. These parameters are adjusted usng SA to get the optmal ft. Snce, the objectve of the paper s to come up wth optmal technques whch can provde decent curve ft to the dgtal data. Therefore, the nterest would be to compute the curve n such a way that the sum square error of the computed curve wth the actual curve (dgtzed contour) s mnmzed. Mathematcally, the sum squared dstance s gven by: S [ P u ) P ], t [ t, t ], = 0,,..., n m = (, j, j, j + j=, (4) P,j = (x,j, y,j ), j =,,,m, (5) are the data ponts of the th segment on the dgtzed contour. The parameterzaton over t's s n accordance wth the chord length parameterzaton. Thus the curve ftted n ths way wll be a canddate of best ft. Once an ntal ft for a partcular segment s obtaned, the parameters of the ftted curve u s for concs are adjusted to get better ft. Here, we try to mnmze the sum squared error of Eqn. (4). Usng SA, we try to obtan the optmal values of the curve parameters. We choose ths technque because t s powerful, yet smple to mplement and as shown n Secton 6, performs well for our purpose. D. Segmentaton Usng Intermedate 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 the dstance between the boundary and parametrc curve exceeds some predefned threshold. Such ponts are termed as ntermedate ponts. A new parametrc curve s ftted for each new segment. VI. OVERALL APPROACHES AND ALGORITHMS We can summarze all the phases from dgtzaton to optmzaton dscussed n the prevous sectons. The algorthms of the proposed schemes are contaned on varous steps. Algorthm of Secton 6. explans the mechansm for the computaton of lnear curve. Curve manpulaton methodology wth concs, usng SA, has been lad down n Algorthm of Secton 6.. A more detaled descrpton for concs, descrbng the whole system wth step by step flow, s shown n the flowchart demonstrated n Fg.. A. Algorthm for Lnear Interpolant The summary of the algorthm, desgned for optmal curve desgn usng lnear nterpolant, s as follows: Step AG.: Input the mage. Step AG.: Extract the contours from the mage n Step AG.. Step AG.3: Compute the corner ponts from the contour ponts n Step AG. usng the method n Secton. Step AG.4: Ft the lnear splne curve method of Secton 3. to the corner ponts acheved n Step AG.3. Step AG.5: IF the curve, acheved n Step AG.4, s optmal then GO To Step AG.8, ELSE locate the approprate ntermedate ponts (ponts wth hghest devaton) n the undesred curve peces. Step AG.6: Enhance and order the lst of corner and ntermedate ponts acheved n Step AG.3 and AG.5. Step AG.7: GO TO Step AG.4. Step AG.8: STOP. B. Algorthm for Conc Interpolant The summary of the algorthm, desgned for optmal curve desgn usng conc nterpolant, s as follows: Step AG.: Input the mage. Step AG.: Extract the contours from the mage n Step AG.. Step AG.3: Compute the corner ponts from the contour ponts n Step AG. usng the method n Secton 3. for concs. Step AG.4: Compute the dervatve values at the corner / ntermedate ponts. Step AG.5: Compute the best optmal values of the shape parameters u s for concs usng SA. Step AG.6: Ft the splne curve method of Secton 3. for concs to the corner / ntermedate ponts acheved n Step AG.. Step AG.7: IF the curve, acheved n Step AG.6, s optmal then GO To Step AG.0, ELSE locate the approprate ntermedate ponts (ponts wth hghest 303

4 Internatonal Journal of Machne Learnng and Computng, Vol., No. 3, June 0 devaton) n the undesred curve peces. Step AG.8: Enhance and order the lst of the corner / ntermedate ponts acheved n Step AG.3 and AG.7. Step AG.9: GO TO Step AG.4. Step AG.0: STOP. Start of SA to get Optmzed values for u for ftted curve on gven segment Compute Tangent Vectors for segment usng Arthmetc Mean Method Fg.. Flowchart of the system computaton of conc curves usng SA. 304

5 Internatonal Journal of Machne Learnng and Computng, Vol., No. 3, June 0 Fg. 3. Results of lnear scheme: Ftted outlne of the mage, Ftted outlne of the mage wth ntermedate ponts. Results of conc scheme: Ftted outlne of the mage, (d) Ftted outlne of the mage wth ntermedate ponts. (d) Fg. 6. Pre-processng steps for curve fttng Image of a plane, Extracted outlne Intal corner ponts. Fg. 4. Pre-processng Steps: Orgnal Image, Outlne of the mage, Corner ponts acheved, (d) Ftted Outlne of the mage. Fg. 7. Results of lnear scheme : Ftted outlne of the mage, Ftted outlne of the mage wth ntermedate ponts. Results of conc scheme: Ftted outlne of the mage, (d) Ftted outlne of the mage wth ntermedate ponts. The above mentoned schemes and the algorthms have been mplemented and tested for varous mages. Reasonably qute elegant results have been observed as can be seen n the followng Secton of demonstratons. (d) Fg. 5. Results of curve fttng. lnear curve fttng: Ftted outlne of the mage, Ftted outlne of the mage wth ntermedate ponts. Conc curve fttng: Ftted outlne of the mage, (d) Ftted outlne of the mage wth ntermedate ponts (d) VII. DEMONSTRATIONS The proposed curve scheme has been mplemented successfully n ths secton. We evaluate the performance of the system by fttng parametrc curves to dfferent bnary mages. Fg. 3 shows the mplementaton results of the two algorthms for the mage Lllah of Fg.. Fgs. 3 and 3 are the results for the lnear scheme, respectvely, wthout and wth nserton of ntermedate ponts. Smlarly, Fgs. 3 and 3(d) are the results for the conc scheme, respectvely, wthout and wth nserton of ntermedate 305

6 Internatonal Journal of Machne Learnng and Computng, Vol., No. 3, June 0 ponts. Fgs. 4 and 5 show the mplementaton results of a Tua mage. Fgs. 4, 4, 4 are respectvely the orgnal mage of the Plane, ts outlne, outlne together wth the corner ponts detected. Fg. 5 shows the mplementaton results of the two algorthms for the Tua mage n Fg. 4. Fgs. 5 and 5 are the results for the lnear scheme, respectvely, wthout and wth nserton of ntermedate ponts. Smlarly, Fgs. 5 and 5(d) are the results for the conc scheme, respectvely, wthout and wth nserton of ntermedate ponts. Fgs. 6 and 7 show the mplementaton results of a Fork mage. Fgs. 6, 6, 6 are, respectvely, the orgnal mage of the Fork, ts outlne, outlne together wth the corner ponts detected. Fg. 7 shows the mplementaton results of the two algorthms for the mage n Fg. 6. Fgs. 7 and 7 are the results for the lnear scheme, respectvely, wthout and wth nserton of ntermedate ponts. Smlarly, Fgs. 7 and 7(d) are the results for the conc scheme, respectvely, wthout and wth nserton of ntermedate ponts. One can see that the approxmaton s not satsfactory when t s acheved over the corner ponts only. Ths s specfcally due to those segments whch are bgger n sze and hghly curvy n nature. Thus, some more treatment s requred for such outlnes. Ths s the reason that the dea to nsert some ntermedate ponts s demonstrated n the algorthms. It provdes excellent results. The dea of how to nsert ntermedate ponts s not explaned here due to lmtaton of space. It wll be explaned n a subsequent paper. TABLE I: NAMES AND CONTOUR DETAILS OF IMAGES Image Name # of Contours # of Contour # of Intal Corner Lllah [5+6] 4 Tua [045+85] nterpolaton, and total tme taken for conc nterpolaton. TABLE III: COMPARISON OF NUMBER OF INITIAL CORNER POINTS, INTERMEDIATE POINTS AND TOTAL TIME TAKEN (IN SECONDS) FOR CONIC INTERPOLATION APPROACHES Image # of Intermedate n Conc Interpolaton Total Tme Taken for Conc Interpolaton Wthout Wth Intermedate Intermedate Lllah.bmp Tua.bmp Fork.bmp VIII. CONCLUSION Two optmzaton technques are proposed for the outlne capture of planar mages. Frst technque uses smply a lnear nterpolant and a straght forward method based on dstrbuton of corner and ntermedate ponts. Second technque uses the smulated annealng to optmze a conc splne to the dgtal outlne of planar mages. By startng a search from certan good ponts (ntally detected corner ponts), an mproved convergence result s obtaned. The overall technque has varous phases ncludng extractng outlnes of mages, detectng corner ponts from the detected outlne, curve fttng, and addton of extra knot ponts f needed. The dea of smulated annealng has been used to optmze the shape parameters n the descrpton of a conc splne ntroduced. The two methods ultmately produce optmal results for the approxmate vectorzaton of the dgtal contours obtaned from the generc shapes. The schemes provde an optmal ft wth an effcent computaton cost as far as curve fttng s concerned. The proposed algorthms are fully automatc and requre no human nterventon. The author s also thnkng to apply the proposed methodology for another model curve namely cubc. It mght mprove the approxmaton process. Ths work s n progress to be publshed as a subsequent work. Fork [693] 0 ACKNOWLEDGMENT M. Sarfraz thanks Kuwat Unversty for supportng ths work through Project# [WI 05/0]. TABLE II: COMPARISON OF NUMBER OF INITIAL CORNER POINTS, INTERMEDIATE POINTS AND TOTAL TIME TAKEN (IN SECONDS) FOR CONIC INTERPOLATION APPROACHES Image # of Intermedate n Lnear Interpolaton Total Tme Taken For Lnear Interpolaton Wthout Wth Intermedate Intermedate Lllah.bmp Tua.bmp Fork.bmp Tables I, II and III summarze the expermental results for dfferent btmap mages. These results hghlght varous nformaton ncludng contour detals of mages, corner ponts, ntermedate ponts, total tme taken for lnear REFERENCES [] X. Yang, Curve Fttng and Farng usng Conc Spnes, Computer Aded Desgn, vol. 6, no. 5, pp , 004. [] X. N. Yang and G. Z. Wang, Planar Pont Set Farng and Fttng by Arc Splnes, Computer Aded Desgn, vol. 33(), pp , 00. [3] M. Sarfraz, Desgnng Objects wth a Splne, Computer Mathematcs, vol. 87, no. 4, pp , 00. [4] M. Sarfraz, Representng Shapes by Fttng Data usng an Evolutonary Approach, Computer-Aded Desgn & Applcatons, vol., no. -4, pp 79-86, 004. [5] M. Sarfraz and M. A. Khan, An Automatc Algorthm for Approxmatng Boundary of Btmap Characters, Future Generaton Computer Systems, vol. 0, no. 8, pp , 004. [6] M. Sarfraz, Some Algorthms for Curve Desgn and Automatc Outlne Capturng of Images, Image and Graphcs, vol. 4, no., pp , 004. [7] Z. J. Hou and G. W. We, A New Approach to Edge Detecton, Pattern Recognton, vol. 35, no. 7, pp ,

7 Internatonal Journal of Machne Learnng and Computng, Vol., No. 3, June 0 [8] M. Sarfraz, Computer-Aded Reverse Engneerng usng Smulated Evoluton on NURBS, Vrtual & Physcal Prototypng, vol., no. 4, pp , 006. [9] M. Sarfraz, M. Ryazuddn, and M. H. Bag, Capturng Planar Shapes by Approxmatng ther Outlnes, Computatonal and Appled Mathematcs, vol. 89, no. -, pp , 006. [0] M. Sarfraz, A. Rasheed, A Randomzed Knot Inserton Algorthm for Outlne Capture of Planar Images usng Cubc Splne, n Proc. nd ACM Symposum on Appled Computng (ACM SAC-07), Seoul, Korea, pp. 7 75, 007. [] H. Kano, H. Nakata, C. F. Martn, Optmal Curve Fttng and Smoothng usng Normalzed Unform B-Splnes: A Tool for Studyng Complex Systems, Appled Mathematcs and Computaton, vol.69, no., pp. 96-8, 005. [] Z. Yang, J. Deng, F. Chen, Fttng Unorganzed Pont Clouds wth Actve Implct B-Splne Curves, Vsual Computer, vol., no. 8-0, pp (005). [3] G. Lavoue, F. Dupont, A. Baskurt, A New Subdvson Based Approach for Pecewse Smooth Approxmaton of 3D Polygonal Curves, Pattern Recognton, vol. 38, pp. 39-5, 005. [4] H. Yang, W. Wang, and J. Sun, Control Pont Adjustment for B-Splne Curve Approxmaton, Computer Aded Desgn, vol. 36, pp , 004. [5] J. H. Horng, An Adaptve Smoothng Approach for Fttng Dgtal Planar Curves wth Lne Segments and Crcular Arcs, Pattern Recognton Letters, vol. 4, no. -3, pp , 003. [6] B. Sarkar, L. K. Sngh, and D. Sarkar, Approxmaton of Dgtal Curves wth Lne Segments and Crcular Arcs usng Genetc Algorthms, Pattern Recognton Letters, vol. 4, pp , 003. [7] M. Sarfraz, Vectorzng Image Outlnes usng Splne Computng Approach, n Proc. 4th Internatonal Conf. Machne Learnng and Computng (ICMLC 0), Hong Kong, Chna, March 0, 0, ASME. [8] S. Krkpatrck, C. D. Gelatt Jr., M. P. Vecch, "Optmzaton by Smulated Annealng", Scence, Vol. 0, no. 4598, pp , 983. [9] M. Sarfraz, and M. I. Sarfraz, Capturng Image Outlnes usng Splne Computng Approach, n Proc. 5th Internatonal Conf. Sgnal-Image Technology & Internet Based Systems (SITIS-009), November 30 December 03, 009, Marrakech, Morocco, pp. 6-3, 009, IEEE Computer Socety Press. [0] R. C. Gonzalez, R. E. Woods, and S. L. Eddns, Dgtal Image Processng Usng MATLAB, nd Ed., Gatesmark Publshng, 009. [] M. S. Nxon, A. S. Aguado, Feature extracton and mage processng, Elsever, 008. [] M. Sarfraz, Outlne Capture of Images by Multlevel Coordnate Search on Cubc Splnes, Lecture Notes n Artfcal Intellgence, A. Ncholson and X. L (Eds.): LNAI 5866, Sprnger-Verlag Berln Hedelberg, pp , 009. [3] D. Chetrkov and S. Zsabo, A Smple and Effcent Algorthm for Detecton of Hgh Curvature n Planar Curves, n Proc. 3 rd Workshop of the Australan Pattern Recognton Group, pp , 999. [4] A. A. Goshtasby Groupng and Parameterzng Irregularly Spaced for Curve Fttng, ACM Transactons on Graphcs, vol. 9, no. 3, pp , 000. [5] P. Reche, C. Urdales, A. Bandera, C. Trazegnes, and F. Sandoval, Corner Detecton by Means of Contour Local Vectors, Electronc Letters, vol. 38, no. 4, 00. [6] M. Marj and P. Sv, A New Algorthm for Domnant Detecton and Polygonzaton of Dgtal Curves, Pattern Recognton, vol. 36, no. 0, pp. 39-5, 003. [7] Wu-Chh Hu, Multprmtve Segmentaton Based on Meanngful Breakponts for Fttng Dgtal Planar Curves wth Lne Segments and Conc Arcs, Image and Vson Computng, vol. 3, no. 9, pp , 005. [8] H. Freeman, L. S. Davs, A corner fndng algorthm for chan-coded curves, IEEE Trans. Computers, Vol. 6, pp , 977. [9] N. Rchardand T. Glbert, Extracton of Domnant by estmaton of the contour fluctuatons, Pattern Recognton, vol. 35, pp , 00. [30] M. Sarfraz, Vectorzng Outlnes of Generc Shapes by Cubc Splne usng Smulated Annealng, Computer Mathematcs, Vol. 87, no. 8, pp , 00. Muhammad Sarfraz s a Professor n Kuwat Unversty, Kuwat. He receved hs Ph. D. from Brunel Unversty, UK, n 990. Hs research nterests, n general, are Computer Graphcs, Pattern Recognton, Computer Vson, Image Processng, and Soft Computng. He s currently workng on varous projects related to academa and ndustry. Prof. Sarfraz has advsed/supervsed more than 50 students for ther MSc and PhD theses. He s the Head of Research Commttee n Informaton Scence Department, Kuwat Unversty. He has publshed more than 00 publcatons n the form of varous Books, Book Chapters, journal papers and conference papers. Prof. Sarfraz s member of varous professonal socetes ncludng IEEE, ACM, IVS, IACSIT, Pattern Recognton Socety, and ISOSS. He s a Char, member of the Internatonal Advsory Commttees and Organzng Commttees of varous nternatonal conferences, Symposums and Workshops. He s the revewer, for many nternatonal Journals, Conferences, meetngs, and workshops around the world. He s the Edtor-n-Chef of three nternatonal journals. He s also Edtor/Guest Edtor of varous Internatonal Conference Proceedngs, Books, and Journals. He has acheved varous awards n educaton, research, and admnstratve servces. 307

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