Vectorization of Image Outlines Using Rational Spline and Genetic Algorithm

Transcription

1 01 Internatonal Conference on Image, Vson and Computng (ICIVC 01) IPCSIT vol. 50 (01) (01) IACSIT Press, Sngapore DOI: /IPCSIT.01.V50.4 Vectorzaton of Image Outlnes Usng Ratonal Splne and Genetc Algorthm Muhammad Sarfraz 1, Msbah Irshad and Malk Zawwar Hussan 1 Department of Informaton Scence, Kuwat Unversty, Adalya Campus, P.O. Box 5969, Safat 1060, Kuwat Department of Mathematcs, Unversty of the Punjab, New Campus, Lahore, Pakstan Abstract. A ratonal splne approach has been ntroduced for the outlne capture of the planar mages. The dea of genetc algorthm(ga) has been ncorporated to optmze the shape parameters n the descrpton of the ratonal splne. The proposed approach has varous phases ncludng detectng dgtal outlnes (contours), fndng corner ponts on the dgtal outlnes, and fttng the splne usng GA. The proposed method ultmately produces optmal results for the approxmate vectorzaton of the dgtal contours obtaned from the generc shapes. Demonstratons and llustraton of the results also make the essental part of the paper. Keywords: vectorzaton, corner ponts, generc shapes, curve fttng, 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. Computatonally economcal and optmally good soluton s an ultmate objectve to acheve the vectorzed outlnes of mages for planar objects. Curve modellng [1] 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 [-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. Ths work s a presentatonof an approach usng ratonal cubc splne nterpolaton. It s nspred by an optmzaton algorthm based on genetc algorthm (GA). 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 [5] of the gray scaled btmap mages. Secondly, t uses the dea of corner ponts [] 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. Ratonal cubc splnenterpolant s then used to vectorze the outlne. The dea of genetc algorthm(ga) s used to ft a ratonal cubc splne whch passes through the corner ponts. It globally optmzes the shape parameters n the descrpton of the ratonal cubc splne to provde a good approxmaton to the dgtal curve. 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 s about the nterpolant form Correspondng author. Tel.: ; fax: E-mal address:muhammad.sarfraz@ku.edu.kw.

2 ofratonal cubc splne curves. Overall methodology of curve fttng, usng genetc algorthm, s explanedn Secton 4, t ncludes the dea of knot nserton as well as the algorthm desgn for the proposed vectorzaton scheme. Demonstraton of the scheme s presented n Secton 5. Fnally, the paper s concluded n Secton 6.. Preprocessng 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 [7], n ths paper, have been used for ths purpose. Demonstraton of the method can be seen n Fgure 1(b) whch s the contour of the btmap mage shown n Fgure 1(a). 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 [1]. The demonstraton of the algorthm s made on Fgure 1(b). The corner ponts of the mage are shown n Fgure 1(c). Fg. 1: Pre-processng Steps: (a) Orgnal Image, (b) Outlne of the mage, (c) Corner ponts acheved, (d) Ftted Outlne of the mage.. Curve Fttng and Splne 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 1, cp,,cp m then there wll be m peces p 1, p,, p m. We treat each pece separately and ft the splne to t. 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. To construct the m parametrc splne 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 curve ftted by a ratonal cubc 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 ratonal cubc splne. Ths secton deals wth a form of ratonal cubc splne. It ntroduces shape parameters v s n the descrpton of ratonal cubc splne defned as follows: where P (1 ) vv (1 ) vw (1 ) P 1 P ( t), (1) (1 ) v (1 ) v (1 ) V P, v h D W h D 1 P 1. v D and D +1 are the correspondng tangents at corner ponts P and P +1 of the th pece. For open curves, the tangent vectors are calculated as follows:

3 ( P P0 ) D0 ( P1 P0 ) P 1 P D a ( P P 1 ) (1 a )( P 1 P ),where a. ( P P 1 P P P 1 n Pn ) D ( n Pn Pn 1 ) For closed curves, the condtons are as follows: F1 Fn1, Fn1 F1 D a ( P P 1 ) (1 a )( P 1 P ), 0,1,..., n Obvously, the parameters v 's, when equal to, provde the specal case of cubc Hermte nterpolaton. If v 's are too large, then the ratonal cubc functon (1) converges to the lnear nterpolant. Ths paper proposes an evolutonary technque, namely genetc algorthm (GA), to optmze these parameters so that the curve ftted s optmal. 4. Proposed Approach In ths Secton, the proposed scheme to the curve fttng problem s descrbed. It ncludes the phases of problem matchng wth Genetc Algorthm usng ratonal cubc splne functon, descrpton of parameters used for GA and curve fttng Problem mappng Snce, the objectve of the paper s to come up wth optmal technque 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 m j1 P t ) P, t t, t, 0,1,..., n 1 (, j, j, j 1 wherep,j = (x,j, y,j ), j = 1,,,m, 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. The Genetc Algorthm formulaton of the problem dscussed n ths paper s descrbed n detal. For the best fttng of the curve to gven data, the valuesof parameters v s are requred so that the sums S ' s are mnmal. Genetc Algorthm s used to optmze theses values for the ftted curve. We start wth ntal populaton of values of v s chosen randomly. Successve applcatons of search operatons to ths populaton leads to optmal values of v s. 4.. Intalzaton Once we have the btmap mage shown n Fgure 1(a) and Fgure (a), the boundary of the mage can be extracted (see Fgure 1(b) and Fgure (b)) usng the method descrbed n Secton. After the boundary ponts of the mage are found, the next step s to detect corner ponts as mentoned n Secton. The 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 parametersv s s randomly selected.fgure 1(c) and Fgure (c) show boundary of the btmap mages together wth detected corner ponts. Table 1 gves number of contour ponts and ntal corner ponts of the mages. 4.. Algorthm for ratonal cubc splne usng GA The overall scheme can be explaned n the form of an algorthm. The summary of the algorthm, desgned for optmal curve desgn usng ratonal cubcnterpolant, s as follows: Step AG.1: Step AG.: Input the mage. Extract the contours from the mage n Step AG.1.,

4 Step AG.: Compute the corner ponts from the contour ponts n Step AG. usng the method n Secton. Step AG.4: Compute the dervatve values at the corner / ntermedate ponts. Step AG.5: Compute the best optmal values of the shape parameters v s usng GA. Step AG.6: Ft the splne curve method, of Secton, 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.10, ELSE locate the approprate ntermedate ponts (ponts wth hghest devaton) n the undesred curve peces. Step AG.8: Enhance and order the lst of the corner / ntermedate ponts acheved n Step AG. and AG.7. Step AG.9: GO TO Step AG.4. Step AG.10: STOP. 5. Demonstratons 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. : Results of the curve fttng: (a) Cubc Hermte ftted to cornersof the outlne of the mage, (b) Ftted ratonal cubc for the 1st teraton of GA, (c) Ftted ratonal cubc for the 5 th teraton of GAwth ntermedate ponts. Fg. : Pre-processng Steps: (a) Orgnal Image, (b) Outlne of the mage, (c) Corner ponts acheved. Fg. 4: Results of the curve fttng: (a) Cubc Hermte ftted to cornersof the outlne of the mage, (b) Ftted ratonal cubc for the 1st teraton of GA, (c) Ftted ratonal cubc for the 5 th teraton of GAwth ntermedate ponts. Fgure shows the mplementaton results of the algorthm for the mage Fork n Fgure 1(a). Fgures (a), (b) and (c) are the results for the scheme, respectvely, for the cubc Hermte splne (default curve),

5 1 st teraton of the GA, and the 5 th teraton of the GA respectvely. One can see the nserton of ntermedate ponts n Fgure (c). Table. 1: Names and contour detals of mages Image Name # of Contours # of Contour Ponts # of Intal Corner Ponts Fork Plane Fgures and 4 show the mplementaton results of a Plane mage. Fgures (a), (b), (c) are respectvely the orgnal mage of the Plane, ts outlne, outlne together wth the corner ponts detected. Fgure 4 shows the mplementaton results of the algorthm for the Plane mage n Fgures4(a-c). Fgures 4(a), 4(b) and 4(c) are the results for the scheme, respectvely, for the cubc Hermte splne (default curve), 1 st teraton of the GA, and the 5 th teraton of the GA respectvely. One can see the nserton of ntermedate ponts n Fgure 4(c). 6. Concludng Remarks An optmzaton technque s proposedfor the outlne capture of planar mages. It uses the GA to optmze a ratonal cubc splne to the dgtal outlne of planar mages. The dea of GA has been used to optmze the shape parameters n the descrpton of a ratonal cubc splne ntroduced. The expermental study shows that the method ultmately produces optmal results for the approxmate vectorzaton of the dgtal contours obtaned from the generc shapes. The scheme 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. 7. References [1] M. Sarfraz. Desgnng Objects wth a Splne, Internatonal Journal of Computer Mathematcs, Taylor & Francs, vol. 85(7) (008). [] D. Chetrkov, S. Zsabo. A Smple and Effcent Algorthm for Detecton of Hgh Curvature Ponts n Planar Curves, In Proceedngs of the rd Workshop of the Australan Pattern Recognton Group, pp (1999). [] 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, pp , ACM Press (007). [4] B. Sarkar, L.K.Sngh, D. Sarkar. Approxmaton of Dgtal Curves wth Lne Segments and Crcular Arcs usng Genetc Algorthms, Pattern Recognton Letters, vol. 4, pp (00). [5] M. Sarfraz, M.I. Sarfraz.. Capturng Image Outlnes usng Splne Computng Approach, The Proceedngs of The 5th Internatonal Conference on Sgnal-Image Technology & Internet Based Systems (SITIS-009), November 0 December 0, 009, Marrakech, Morocco, pp. 16-1, IEEE Computer Socety Press. [6] M. Sarfraz. Capturng Image Outlnes usng Smulated Annealng Approach wth Conc Splnes, The Proceedngs of the Internatonal Conference on Informaton and Intellgent Computng (ICIIC 011), Hong Kong, Chna, November 5-7, 011, pp , IPCSIT vol. 18, IACSIT Press, Sngapore. [7] M.S. Nxon,A.S. Aguado. Feature extracton and mage processng, Elsever, 008. [8] D.E. Goldberg. Genetc algorthms n search optmzatonand machne learnng,addson Wesley, Readng, MA.,1989. [9] M. Sarfraz.Vectorzng Outlnes of Generc Shapes by Cubc Splne usng Smulated Annealng, Internatonal Journal of Computer Mathematcs, Taylor & Francs, 010, Vol. 87(8), pp