Curve Representaton for Outlnes of Planar Images usng Multlevel Coordnate Search MHAMMAD SARFRAZ and NAELAH AL-DABBOUS Department of Informaton Scence Kuwat Unversty Adalya Campus, P.O. Box 5969, Safat 1060 KUWAIT prof.m.sarfraz@gmal.com Abstract: - Ths paper proposes an optmzaton technque for the outlne capture of planar mages. Ths s nspred by a global optmzaton algorthm based on multlevel coordnate search (MCS). 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 multlevel coordnate search has been used to optmze the shape parameters n the descrpton of the generalzed cubc splne ntroduced. The splne method ultmately produces optmal results for the approxmate vectorzaton of the dgtal contour obtaned from the generc shapes. It provdes an optmal ft as far as curve fttng s concerned. The proposed algorthm s fully automatc and requres no human nterventon. Implementaton detals are suffcently dscussed. Some numercal and pctoral results are also demonstrated to support the proposed technque. Key-Words: - Optmzaton, multlevel coordnate search, Generc shapes, curve fttng, cubc splne 1 Introducton Capturng and vectorzng outlnes of mages 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 [-5, -5]. Computatonally economcal and optmally good soluton s an ultmate objectve to acheve the vectorzed outlnes of mages for planar objects. 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 [10-16, 1], 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. Curve segmentaton s advantageous n a way that t gves a rough geometry of the shape. Approaches used to acheve ths task, n the lterature, are polygonal approxmatons [8, 1], crcular arc approxmatons [10,15,17,18, ] and approxmatons usng cubcs or hgher order splne functons [,14, 4-5]. A non-parametrc domnant pont detecton algorthm was proposed n [8], t used the domnant ponts for polygonzaton of dgtal curves. The problem wth polygonal approxmaton s that these approaches are rarely used for shape analyss. A combnaton of lne segments and crcular arcs for object approxmaton s used n [17, 18]. A scheme to construct a curvature contnuous conc splne s proposed n [15]. Ths approach presented the conc splne curve fttng and farng algorthm usng conc arc scalng. The smoothng s done by removng unwanted curvature extrema. Smlar algorthms for data fttng by arc splne curves are presented n []. A method for segmentaton of curves nto lne segments and crcular arcs by usng types of breakponts s proposed n [10]. Advantage of ths technque s that t s threshold free and transformaton nvarant. Fve categores of breakponts have been defned. The lne and conc segmentaton and mergng s based on these breakponts. Least square fttng s mostly adopted n approxmatons, whch uses splnes and hgher order polynomals. Some approaches are based on actve contour models known as snakes. These technques are also based on parameterzaton. Enhancement to the scheme by adjustng both number and postons E-ISSN: 4-87 6 Issue, Volume 1, February 01
of control ponts of the actve splne curve s shown n [14]. Ths scheme s based on curve approxmaton usng teratve optmzaton wth B- splne curve by squared dstance mnmzaton. Another way, other than parametrc form, s to use mplct form of the polynomal. Curve reconstructon problem s solved by approxmatng the pont clouds usng mplct B-splne curve [1]. The authors have used trust regon algorthm n optmzaton theory as mnmzaton heurstcs. Technques descrbed for fttng mplctly defned algebrac splne curves and surfaces to scattered data by smultaneously approxmatng ponts and assocated normal vectors are proposed n [19, 0, 1]. The proposed work, n ths paper, s nspred by the fast growng area of soft computng. A good amount of lterature has been produced on varous heurstcs [0-] for optmzaton problems. The proposed work s motvated by an optmzaton algorthm based on multlevel coordnate search (MCS) by Huyer and Neumaer [0]. It motvates the author to an optmzaton technque proposed for the outlne capture of planar mages. It s an extenson of the work n [5]. In ths paper, the data pont set represents any generc shape whose outlne s requred to be captured. We present an teratve process to acheve our objectves. The algorthm comprses of varous phases to acheve the target. Frst of all, t fnds the contour of the gray scaled btmap mage. Secondly t detects corners. These phases are consdered as preprocessng steps. The next phase detects the corner ponts on the dgtal contour of the generc shape under consderaton. The dea of multlevel coordnate search (MCS) s then used to ft a generalzed cubc splne whch passes through the corner ponts. It globally optmzes the shape parameters n the descrpton of the generalzed cubc splne to provde a good approxmaton to the dgtal curve. In most of the cases, corner ponts are not enough to approxmate the dgtal object and hence some more ponts are also needed. These ponts are known as break ponts or knots as they are used to break a segment for better approxmaton. For onwards dscusson, the set of corner ponts together wth the break ponts wll be called as the set of sgnfcant ponts. In the fourth phase of the proposed algorthm, for each teraton, we wll nsert a pont as knot n every pece (f needed) n a manner that the dstance, d, of the computed pont on the splne curve and ts correspondng contour pont s greater than a threshold ε. Ths process ncreases the set of sgnfcant ponts and hence needs multlevel coordnate search to be employed agan for the updated set of sgnfcant ponts to ft an optmal splne curve. Ths process contnues untl t rectfes the soluton and helps towards the objectve optmzaton n a global fashon. We stop the teratve process when all d s are less than ε. The proposed splne method, usng multlevel coordnate search, ultmately produces optmal results for vectorzng the dgtal contour of the generc shapes. It provdes an optmal ft as far as curve fttng s concerned. The organzaton of the paper s as follows, Secton dscusses about pre-processng step whch ncludes fndng the boundary of planar object and corner detecton algorthm for fndng the sgnfcant ponts. Secton s about the nterpolant form of cubc splne curves and computaton of ts assocated tangents. The process of multlevel coordnate search s explaned n Secton 4. Overall methodology of curve fttng s explaned n Secton 5, t ncludes the dea of knot nserton as well as the algorthm desgn for the proposed vectrozaton scheme. Demonstraton of the proposed scheme s presented n Secton 6. Fnally, the paper s concluded n Secton 7. Preprocessng The proposed scheme starts wth frst fndng the boundary of the generc shape and then usng the output to fnd the corner ponts or the sgnfcant ponts. Forthcomng Sectons.1 and. wll explan these phases..1 Fndng Boundary of Generc Shapes The mage of the generc shape can be acqured ether by scannng or by some other mean. The qualty of scanned mages s dependent upon factors such as paper qualty and scannng resoluton. The better the resoluton and paper qualty, the better wll be the mage. The am of boundary detecton s to produce an object s shape n graphcal or nonscalar representaton. Chan codes [6, 7, 7-8] are the most wdely used representatons. Other wellknown representatons are syntactc technques, boundary approxmatons and scale-space technques. The beneft of usng chan code s that t gves the drecton of edges. The boundary ponts are selected as contour ponts based on ther corner strength and fluctuatons. To arrange the extracted boundary ponts n a sequence (clockwse drecton), a boundary tracng s performed, usng the algorthm n [6] for boundary tracng. Demonstraton of the E-ISSN: 4-87 6 Issue, Volume 1, February 01
method can be seen n Fgure 1(b) whch s the contour of the btmap mage shown n Fgure 1(a).. Detectng Corner Ponts Corners n dgtal mages gve mportant clues for the shape representaton and analyss. Generally objects nformaton can be represented n terms of ts corners, whch play a very vtal role n object recognton, shape representaton and mage nterpretaton [1,8]. These are the ponts that partton the boundary nto varous segments. The strategy of gettng these ponts s based on the method proposed n [1]. The detals of ths procedure s left for the reader to see n [1]. The demonstraton of the algorthm s made on Fgure 1(b). The corner ponts of the mage are shown n Fgure 1(c). (a) (b) (c) Fgure 1. Pre-processng Steps: (a) Orgnal Image, (b) Outlne of the mage, (c) Corner ponts acheved. Curve Fttng wth Cubc Splne The motve of fndng the corner ponts, n Secton., was to dvde the contour nto peces. Each pece contans the data ponts n between two subsequent corners nclusve. Ths means that f there are m corner ponts cp 1, cp,, cp m then there wll be m peces p 1, p,, p m. We treat each pece separately and ft the splne [9, 4-5] to t. Frst pece ncludes all the contour ponts n between cp 1 and cp nclusve. Second pece contans all contour ponts n between cp and cp nclusve. Consequently, the m th pece contans all contour ponts between cp m and cp 1 nclusve. In general, the th pece contans all the data ponts between cp and cp +1 nclusve. After breakng the contour of the mage nto dfferent peces, we ft the splne curve to each pece. For ths purpose we have used pecewse parametrc cubc splne nterpolant. The splne formulaton globally s C 1 contnuous..1 Cubc Splne Interpolant The cubc splne s defned as follows: P ( t) (1 ) F (1 ) V (1 ) W where ( t t ) [ t t t, ( ) ), h 1 t 1 t h F 1 (1) and h D h D1 V F, W F 1 Equaton (4) can be rewrtten as P t, t ( t) R0, ( t) F R1, ( t) V R, ( t) 1 R, ( t) F 1 () where R 0, ( t) (1 t), R1, ( t) t(1 t), () R, ( t) t (1 t), R, ( t) t, The functons R j,, j = 0,1,, are Bernsten Bézer lke bass functons, such that j0 R j, ( t) 1 (4) From the Bernsten-Bézer theory, t follows that the curve segment P t, t 1 les n the convex hull of the control ponts {F, V, W, F +1 } and s varaton dmnshng wth respect to the control polygon jonng these ponts. To get the control ponts F, V, W, F 1, we make use of a Bernsten-Bézer representaton where we can mpose the Hermte nterpolaton condtons: P( t ) F and P ( t ) D, (5) where F and F +1 are corner ponts of th pece. D and D +1 are the correspondng tangents at corner ponts. To construct the parametrc C 1 cubc splne m nterpolant on the nterval [ t 0, t n ] we have F R, 0,1,...,n, as nterpolaton data, at knots t, 0,1,...,n. The dervatves D R can be found out by the mposton of C 1 constrants on the pecewse cubc form. The C 1 constrants can be wrtten as: P ( t ) P( t ). (6) The tangent vectors are calculated as follows: ( P P0 ) D0 ( P1 P0 ) D a ( P P 1 ) (1 a )( P 1 P ), (7) ( P n Pn ) D ( n Pn Pn 1 ) m E-ISSN: 4-87 64 Issue, Volume 1, February 01
where P 1 P a. P P P P 1 1 Snce, the objectve of the paper s to come up wth an optmal technque whch can provde a decent curve ft to the dgtal data. Therefore, the nterest would be to compute the curve n such a way that sum square error of the computed curve wth the actual curve (dgtzed contour) s mnmzed. Mathematcally, the sum squared dstance s gven by: S m j1 P u ) P, t t, t, 0,1,..., n 1 (, j, j, j 1 (8) where P,j = (x,j, y,j ), j = 1,,,m, (9) 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. ( t p( t)) can be consdered as the nterpolaton scheme appled n R to data ( tf) wth dervatves (1 D ) Z Ths s a consequence of the property that the nterpolant s able to reproduce lnear functons. In partcular, for v w 1 the scalar data F t and dervatves D 1 Z the nterpolant reproduces the functon t F F (a) +1. Generalzed Cubc Splne Interpolant The curve ftted n Secton.1 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 cubc splne. Ths secton deals wth the generalzed form of cubc splne. It ntroduces two parameters v and w n the descrpton of cubc splne defned as follows: P ( t) (1 ) F (1 ) V (1 ) W where V F 1 (10) F hv D, F 1 h w D1 W (11) It follows from Bézer theory that the curve segment P(t) les n the convex hull of the control ponts F, V, W, F 1 and s varaton dmnshng wth respect to the control polygon jonng these ponts. It should also be observed that the weghts are ndependent and that one could take v w a constant for all Z wthout loss of generalty. However, we fnd t useful to consder the two scalar weghts v and w on [ t t 1) n the development of the theory. Remark 1. If Pt () s the nterpolant for scalar data F R wth dervatves D R Z then F (b) F +1 (c) F +1 F Fgure. (a) Based behavor of the generalzed cubc for dfferent values of v wth w = 0, (b) Based behavor of the generalzed cubc for v = 0 and dfferent values of w, (c) nterval tenson behavor of the generalzed cubc for the same values of v and w. E-ISSN: 4-87 65 Issue, Volume 1, February 01
Remark. If Pt () s the nterpolant for planar data m F R, m, we need to have some specfc parametrzaton over t. Although, there are number of parametrzaton schemes n the lterature, we wll prefer to use the chord lenght parametrzaton as follows: t t 0 0; t 1 F F, 1,,,... 1 n, (1) where F F 1 denotes the dstance of the th chord segment. Some more detals about the parametrzaton s gven n the next secton. (a) (b) (c) Fgure. (a) Global based behavor of the generalzed cubc for dfferent values of v wth w = 0, (b) Global based behavor of the generalzed cubc for v = 0 and dfferent values of w, (c) Global nterval tenson behavor of the generalzed cubc for the same values of v and w. The followng tenson propertes of the Hermte lke form are now mmedately apparent from (10) and (11). Based Tenson: The based tenson behavor s possble for the curve desgnng. For the based tenson behavor to the left, one can observe that for any Z one can have the followng: lmv F (1) v 0 Ths behavor follows from equaton (11). For the demonstraton, see Fgure (a) for local behavor n one nterval of the desgn curve, and see Fgure (a) for global behavor n the whole curve. Smlarly, for the based tenson behavor to the rght, one can observe that for any Z one can have the followng: lm W F. (14) w 0 1 Ths behavor follows from equaton (11), see Fgures (b) and (b) for local and global behavor respectvely. Interval Tenson: The nterval tenson behavor s possble for the curve desgnng too. For any Z one can have the followng: lmv F and lm W F 1 (15) v 0 w 0 and the nterpolant (10) reduces to: P( t) (1 ) F F. (16) lm v, w 0 1 For the demonstraton, see Fgure (c) for local behavor n one nterval of the desgn curve, and see Fgure (c) for global behavor n the whole curve. (a) (b) Fgure 4. Curve fttng on a data of a fork (a) Default cubc splne, (b) Varety of shape control used. A varety of shape control has been demonstrated n the Fgure 4, where dfferent shape parameter values have been used to get a desred shape n Fgure 4(e). The Fgure 4(a) the default cubc splne ftted on the data ponts of a fork mage. Remark. The case v w r s that of the nterval tenson method. Obvously, the parameter values v w 1/ provde the specal case of cubc splne of Secton.1. Otherwse, these parameters can be used to loose or tght the curve. Ths paper proposes an evolutonary technque, namely multlevel coordnate search (MCS), to E-ISSN: 4-87 66 Issue, Volume 1, February 01
optmze these parameters so that the curve ftted s optmal. 4 Multlevel Coordnate Search Mult-level coordnate search (MCS) s a global optmzaton technque [0]. It guarantees the converge of the optmal soluton f the functon s contnuous n the neghborhood of a global mnmzer. It works by combnng two types of searchng: global searchng and local searchng. The advantage s that f the optmal value s somewhere near the current poston, local search makes sure that the algorthm does not dvert to dstant locatons n the soluton space. It also reduces the tme to reach the exact optmal value after reachng near t. MCS makes use of other complex mplemented technques such as global lne search and bound constraned quadratc program solver [0]. Dervaton of the MCS algorthm and underlyng theory can be found n [0]. A detaled descrpton of the mappng of the MCS technque on our problem s gven n the Secton 5. 5 Proposed Approach The proposed approach to the curve problem s descrbed here n detal. It ncludes the phases of problem matchng wth MCS usng cubc splne, descrpton of parameters used for MCS, curve fttng, and the overall algorthm desgn. 5.1 Problem Mappng Ths secton descrbes about the MCS formulaton of the soluton to the problem n detal. Our nterest s to optmze the values of shape parameters v and w, n the descrpton of the splne n Secton., such that the defned curve fts as close to the orgnal contour segment as possble. We use MCS for the optmzaton of these two varables for the ftted curve. Hence the dmensonalty of the soluton space s, and each pont n MCS represents a par of values for v and w. We start wth an ntal set of ponts that are taken to be the corner ponts of the -dmensonal soluton space and the mdponts along the two drectons. Snce the soluton space s bounded, wth boundary values as - 1 and 1 for both the dmensons, the ntal ponts are chosen at these corners. Then we make boxes of the soluton spaces usng these ponts. For each pont, we also compute and store the objectve functon value and assocate each wth one of the boxes. Now each box corresponds to a range of values of v and w. From all these boxes (ranges of v and w values), we frst select the one havng an assocated pont wth the lowest functon value. In ths box, we apply local search and try to fnd the optmum n the determned drecton of mnmzaton wthn the box. If the v and w par found n ths box s not the optmal soluton, then ths box s splt. That s, the range of v and w values wthn ths box s further splt nto smaller mutually exclusve ranges. Each new range s assocated wth a new representatve pont n the soluton space and ts ftness value. The shoppng basket s hence kept updated wth these ranges and ftness values. Note that we apply MCS ndependently for each segment of a contour between two consecutve corner ponts that we have dentfed usng corner pont algorthm. MCS s appled sequentally on each of the segments, generatng an optmzed ftted curve for each segment. The algorthm s run untl the maxmum level of allowed splttng s reached, or an optmal value s reached. Once, all the contour segments are exhausted and stll the desred global optmum soluton s not acheved, MCS s appled agan. MCS s appled sequentally on each of the segments. 5.1.1 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.1. 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 Secton.. The ntal soluton of splne parameters (v and w) are randomly selected wthn the range [-1, 1]. 5.1. Intalzaton After an ntal approxmaton for the segment s obtaned, better approxmatons are obtaned through MCS to reach the optmal soluton. We experment wth our system by approxmatng each segment of the boundary usng the generalzed cubc splne of Secton.. Ths splne method s a varaton of the well-known Hermte cubc splne. Ths modfed Hermte cubc splne 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 least square method. Each boundary segment s approxmated by the splne. The shape parameters v and w, n the cubc E-ISSN: 4-87 67 Issue, Volume 1, February 01
splne, provde greater flexblty over the shape of the curve. These parameters are adjusted usng MCS to get the optmal ft. Once an ntal ft for a partcular segment s obtaned, the parameters of the ftted curve (v's and w's) are adjusted to get better ft. Here, we try to mnmze the sum squared error. Usng MCS, 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. Smlarly, another parameter that defnes how much local search to do s also specfed. Fgure 5. Calculaton of Intermedate Pont (a hollow bullet). 5.1. Segmentaton usng Intermedate Ponts 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 and parametrc curve exceeds some predefned threshold, see Fgure 5. Such ponts are termed as ntermedate ponts. A new parametrc curve s ftted for each new segment. 5. The Algorthm We can summarze all the phases from dgtzaton to optmzaton dscussed n the prevous sectons. The algorthm of the proposed scheme s contaned on varous steps as shown n the Pseudo code n Fgure 6. A detaled descrpton, descrbng the whole system wth step by step flow, s shown n the flowchart demonstrated n Fgure 7. 5. MCS Parameters Although MCS sets default values (see Table 1) of the algorthm varables, t gves the opton of manpulatng some parameters that defne varous factors affectng ts performance. One of the factors s that how much weght MCS should gve to global searchng as opposed to local searchng. The hgher ths value, the more global level search wll be done. Fgure 6. Pseudo-code Algorthm wth MCS. An ntal set of startng soluton ponts have to be specfed for the system to start wth. MCS requres an ntal guess for the soluton. It s ths startng state parameters that affect the performance of the algorthm. If the startng soluton s very near the optmal soluton, t s more lkely to fnd the optmal soluton readly than f the startng soluton s dstant from the optmal soluton. An acceptable error value has to be defned, so that f the system comes wthn ths error range from the optmal value, t termnates wth the found soluton. An overall constranng factor s the maxmum number of epochs that the algorthm may run, so that t does not run ndefntely f t s not reachng a stable soluton after that number of epochs. The drecton of optmzaton of the ftness functon has to be specfed.e. specfc value that has to be attaned. The default value s negatve nfnty and t can be used for our problem snce the lowest value for our objectve functon s zero. The dmenson of the problem has to be defned as the number of nputs that wll be passed to MCS, and the allowable range of these varables. Table 1 shows the MCS parameter settngs that have been used for the proposed curve fttng optmzaton problem n a global way. E-ISSN: 4-87 68 Issue, Volume 1, February 01
Fgure 7. Flowchart of the system wth MCS. E-ISSN: 4-87 69 Issue, Volume 1, February 01
Table 1. Parameter Settngs for MCS. MCS parameters Values Range of nput parameters v and w [-1,1] Ftness Functon Optmzaton Target Dmenson of problem (number of nputs to MCS) Weght gven to global search versus local search Maxmum number of teratons (epochs) Stoppng relatve error (f dstance from optma s less than ths, the algorthm termnates) Intal Set of Soluton Ponts 0 (functon mnmzaton) 0 00 1e-4 Corner-ponts and Mdponts of soluton space Number of Steps n Local Search 50 Fgure 8 shows the mplementaton results of the algorthm wth MCS. Fgures 8(a), 8(b), 8(c) and 8(d) are respectvely the orgnal mage of an Arabc language word "Ilm", ts outlne, outlne together wth corner ponts detected, and the ftted outlne together wth corner ponts and ntermedate ponts. (a) (c) (b) (d) Fgure 9. Demonstraton of splne fttng at dfferent teratons (average per segment) usng MCS wthout Intermedate Pont: (a) 1 st teraton, (b) 5 th teraton, (c) 10 th teraton, (b) 6th (fnal) teraton. (a) (b) (c) (d) Fgure 8. Pre-processng Steps: (a) Orgnal Image, (b) Outlne of the mage, (c) Corner ponts acheved, (d) Ftted Outlne of the mage. 6 Demonstraton The algorthm n Secton 5 has been mplemented practcally and the proposed curve scheme has been mplemented wth and wthout ntermedate pont ncorporaton. Use of MCS has provded pleasng and effcent results. We evaluate the performance of our system by fttng parametrc curves to dfferent bnary mages. Fgures 9(a), 9(b), 9(c) and 9(d) demonstrate, respectvely, 1 st, 5 th, 10 th and last (6 th ) teraton of the algorthm usng MCS wthout nsertng any ntermedate pont. Snce the number of teratons may not be same for each segment, therefore the number of teratons for the whole curve mentoned here actually represents the average number of teratons per segment. Ths experment s done wthout nsertng any ntermedate pont n any of the curve segment. One can notce that after some teratons, although an approxmaton curve has been acheved, stll t s requred to have some further mprovements. Fgures 8(d) demonstrates the mprovement n the output. Ths s done by nsertng some approprate ntermedate ponts n the desred curve segments. The process of such an nserton has been explaned n Secton 5.1.. One can notce that after some nsertons, a pleasng approxmate curve has been acheved. However, some cost of havng some ntermedate ponts has been pad. Ths cost, the author beleves, s bearable as the computaton tme consumed s not very sgnfcant as compared to the tme pad to acheve not a good approxmaton n Fgure 9. Moreover, accuracy E-ISSN: 4-87 70 Issue, Volume 1, February 01
acheved n Fgure 8 s much hgher and vsually acceptable. (a) (b) (c) (d) (e) Fgure 10. Pre-processng steps for curve fttng (a) Image of a plane, (b) Extracted outlne (c) Intal corner ponts. Some more experments are done on dfferent mages. An mage of a plane s shown n Fgure 10(a), ts outlne s detected n Fgure 10(b), and the corner ponts are shown n 10(c). Fgures 10(d) and 10(e) demonstrate the ftted curves to the outlne of Fgure 10(b) correspondng to the proposed scheme wthout and wth nserton ponts respectvely. It can be notced that the ftted curve n Fgure 10(d) has a good approxmaton wthout nsertng extra ponts. However, nsertng extra ponts, has hghly refned the approxmaton n Fgure 10(e). to the scheme wthout and wth nserton ponts respectvely. It can be notced that the ftted curve n Fgure 1(a) has a good approxmaton, wthout nsertng extra ponts, except at two segments. However, nsertng extra ponts, has hghly refned the approxmaton everywhere n Fgure 1(b). Table. Names and contour detals of mages. Image Name # of Contours # of Contour Ponts Ilm.bmp 1 [1641] Plane.bmp [1106+61+8] (a) (b) (c) Fgure 11. Pre-processng steps for curve fttng (a) Image of a fork, (b) Extracted outlne (c) Intal corner ponts. (a) (b) Fgure 1. Cubc curve fttng (a) wthout ntermedate ponts (b) wth ntermedate ponts. Another experment s made on an mage of Fork n Fgure 11(a). Its outlne s detected n Fgure 11(b), and the corner ponts are shown n 11(c). Fgures 1(a) and 1(b) demonstrate the ftted curves to the outlne of Fgure 11(b) correspondng Fork.bmp 1 [69] Table. Comparson of number of ntal corner ponts, ntermedate ponts and total tme taken (n seconds) for cubc nterpolaton approaches. Image # of Intal Corner Ponts # of Intermedate Ponts n Cubc Interpolaton Total Tme Taken For Cubc Interpolaton Wthout Intermedate Ponts Wth Intermedate Ponts Ilm.bmp 18 4 46.1 164.17 Plane.bmp 1 1 56.766 100.58 Fork.bmp 10 18.48 70.97 Tables to 4 summarze the expermental results for dfferent btmap mages. These results hghlght varous nformaton ncludng contour detals of E-ISSN: 4-87 71 Issue, Volume 1, February 01
mages (Table ), ntermedate ponts (Table ), and number of teratons (Table 4). Table 4. Comparson of number of epochs taken by MCS cubc nterpolaton approach wth and wthout ntermedate ponts. Image # of Epochs taken by MCS Cubc Interpolaton Wthout Intermedate Ponts Cubc Interpolaton Wth Intermedate Ponts Ilm.bmp 459 8915 Plane.bmp 476 761 Fork.bmp 105 4690 4 Concluson A global optmzaton technque, based on multlevel coordnate search, s proposed for the outlne capture of planar mages. The proposed technque uses the multlevel coordnate search to optmze a cubc 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 multlevel coordnate search has been used to optmze the shape parameters n the descrpton of the generalzed cubc splne ntroduced. The splne method ultmately produces optmal results for the approxmate vectorzaton of the dgtal contour obtaned from the generc shapes. It provdes an optmal ft wth an effcent computaton cost as far as curve fttng s concerned. The proposed algorthm s fully automatc and requres no human nterventon. Implementaton detals are suffcently dscussed usng both wth and wthout nserton of ntermedate ponts. The proposed technque has been supported wth numercal and pctoral results demonstrated. Acknowledgement Ths work was supported by Kuwat Unversty, Research Grant No. [WI 01/09]. References: [1] D. Chetrkov, S. Zsabo, A Smple and Effcent Algorthm for Detecton of Hgh Curvature Ponts n Planar Curves, The Proceedngs of the rd Workshop of the Australan Pattern Recognton Group, 1999, pp. 1751-184. [] M. Sarfraz, Representng Shapes by Fttng Data usng an Evolutonary Approach, Internatonal Journal of Computer-Aded Desgn & Applcatons, Vol. 1(1-4), 004, pp 179-186. [] A. Goshtasby, Groupng and Parameterzng Irregularly Spaced Ponts for Curve Fttng, ACM Transactons on Graphcs, 000, pp. 185-0. [4] M. Sarfraz, M. A. Khan, An Automatc Algorthm for Approxmatng Boundary of Btmap Characters, Future Generaton Computer Systems, 004, pp. 17-16. [5] M. Sarfraz, Some Algorthms for Curve Desgn and Automatc Outlne Capturng of Images, Internatonal Journal of Image and Graphcs, 004, pp. 01-4. [6] Z. J. Hou, G.W.We, A New Approach to Edge Detecton, Pattern Recognton, 00, pp. 1559-1570. [7] P. Reche, C. Urdales, A. Bandera, C. Trazegnes, F. Sandoval, Corner Detecton by Means of Contour Local Vectors, Electronc Letters, Vol. 8, No. 14, 00. [8] M. Marj, P. Sv, A New Algorthm for Domnant Ponts Detecton and Polygonzaton of Dgtal Curves, Pattern Recognton, 00, pp. 9-51. [9] M. Sarfraz, Desgnng Objects wth a Splne, Internatonal Journal of Computer Mathematcs, Taylor & Francs, Vol. 85, No. 7, 008. [10] Wu-Chh Hu, Multprmtve Segmentaton Based on Meanngful Breakponts for Fttng Dgtal Planar Curves wth Lne Segments and Conc Arcs, Image and Vson Computng, 005, pp. 78-789. [11] 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, 005, pp. 96-18. [1] Z. Yang, J. Deng, F. Chen, Fttng Unorganzed Pont Clouds wth Actve Implct B-Splne Curves, Vsual Computer, 005, pp. 81-89. [1] G. Lavoue, F. Dupont, A. Baskurt, A New Subdvson Based Approach for Pecewse Smooth Approxmaton of D Polygonal Curves, Pattern Recognton, 005, pp. 119-1151. [14] H. Yang, W. Wang, J. Sun, Control Pont Adjustment for B-Splne Curve E-ISSN: 4-87 7 Issue, Volume 1, February 01
Approxmaton, Computer Aded Desgn, 004, pp. 69-65. [15] X. Yang, Curve Fttng and Farng usng Conc Spnes, Computer Aded Desgn, 004, pp. 461-47. [16] M. Sarfraz, Computer-Aded Reverse Engneerng usng Smulated Evoluton on NURBS, Internatonal Journal of Vrtual & Physcal Prototypng, Vol. 1, No. 4, 006, pp. 4-57. [17] J. H. Horng, An Adaptve Smootng Approach for Fttng Dgtal Planar Curves wth Lne Segments and Crcular Arcs, Pattern Recognton Letters, 00, pp. 565-577. [18] B. Sarkar, L. K. Sngh,, D. Sarkar, Approxmaton of Dgtal Curves wth Lne Segments and Crcular Arcs usng Genetc Algorthms, Pattern Recognton Letters, 00, pp. 585-595. [19] J. C. Carr, R. K. Beatson, J. B. Cherre, T. J. Mtchell, W. R. Frght, B. C. McCallum, T. R. Evans, Reconstructon and Representaton of D Objects wth Radal Bass Functons, Proceedngs of SIGGRAPH 01, 6776, 001. [0] B. Juttler, A. Fels, A Least Square Fttng of Algebrac Splne Surfaces, Advance Computer Mathematcs, 00, pp. 15-15. [1] B. S. Morse, T. S. Yoo, D. T. Chen, P. Rhengans, K. R. Subramanan, Interpolatng Implct Surfaces from Scattered Surface Data usng Compactly Supported Radal Bass Functons, SMI 01 Proceedngs of the Internatonal Conference on Shape Modelng and Applcatons, 8998, IEEE Computer Socety, Washngton DC, 001. [] X. N. Yang, G. Z. Wang, Planar Pont Set Farng and Fttng by Arc Splnes, Computer Aded Desgn, 001, pp. 5-4. [] M. Sarfraz, M. Ryazuddn, M. H. Bag, Capturng Planar Shapes by Approxmatng ther Outlnes, Internatonal Journal of Computatonal and Appled Mathematcs, Vol. 189, No. 1-, 006, pp. 494 51. [4] M. Sarfraz, A. Rasheed, A Randomzed Knot Inserton Algorthm for Outlne Capture of Planar Images usng Cubc Splne, The Proceedngs of The th ACM Symposum on Appled Computng (ACM SAC-07), Seoul, Korea, 007, pp. 71 75, ACM Press. [5] M. Sarfraz, Outlne Capture of Images by Multlevel Coordnate Search on Cubc Splnes, Lecture Notes n Artfcal Intellgence: Advances n Artfcal Intellgence, A. Ncholson, X. L (Eds.): Vol. 5866, Sprnger- Verlag Berln Hedelberg, 009, pp. 66 645. [6] S. Krkpatrck, C. D. Gelatt Jr., M. P. Vecch, Optmzaton by Smulated Annealng, Scence, Vol. 0(4598), 198, pp. 671-680. [7] H. Freeman, L.S. Davs, A corner fndng algorthm for chan coded curves, IEEE Trans. Computers, Vol. 6, 1977, pp. 97-0. [8] M. Sonka, V. Hlavac, R. Boyle, Image processng, analyss, and machne vson. Brooks/Cole publcaton, 001, pp. 14-14. [9] N. Rchard, T. Glbert, Extracton of Domnant Ponts by estmaton of the contour fluctuatons, Pattern Recognton, Vol. (5), 00, pp. 1447-146. [0] W. Huyer, A. Neumaer, Global Optmzaton by Multlevel Coordnate Search, Journal of Global Optmzaton, Vol. 14, 1999, pp. 1-55. [1] Y. Kumar, S. K. Srvastava, A. K. Bajpa, N. Kumar, Development of CAD Algorthms for Bezer Curves/Surfaces Independent of Operatng System, WSEAS Transactons on Computers, Vol. 11, No. 6, 01, pp. 159-169. [] K. Thanushkod, K. Deeba, On Performance Analyss of Hybrd Intellgent Algorthms (Improved PSO wth SA and Improved PSO wth AIS) wth GA, PSO for Multprocessor Job Schedulng, WSEAS Transactons on Computers, Vol. 11, No. 5, 01, pp. 159-169. E-ISSN: 4-87 7 Issue, Volume 1, February 01