Detecting Maximum Inscribed Rectangle Based On Election Campaign Algorithm Qing-Hua XIE1,a,*, Xiang-Wei ZHANG1,b, Wen-Ge LV1,c and Si-Yuan CHENG1,d

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1 6th Internatonal onference on Advanced Desgn and Manufacturng Engneerng (IADME 2016) Detectng Maxmum Inscrbed Rectangle Based On Electon ampagn Algorthm Qng-Hua XIE1,a,*, Xang-We ZHAG1,b, Wen-Ge LV1,c and S-Yuan HEG1,d 1 School of Electromechancal Engneerng, Guangdong Unversty of Technology, Guangzhou chna a xqhcool@163.com, bxwzhang@gdut.edu.cn, clvwenge@163.com, dmdesgn@gdut.edu.cn *orrespondng author Keywords: Maxmum Inscrbed Rectangle, onstraned Grd Graph, Electon ampagn Algorthm, Optmzaton. Abstract. Detectng maxmum nscrbed rectangle on rregularly shaped raw materal s a key problem of cuttng and nestng process n natural stone producton plant. The present rulng method s the artfcal realzaton, and has the dsadvantages of low effcency, poor utlzaton rate and so on. Based on the constraned grd graph and Electon ampagn Algorthm, optmzaton mathematcal model s used to detect the maxmum nscrbed rectangle. A new detecton method s proposed and get the better utlzaton rate method for producton. The expermental results show that the optmzaton method has a good applcablty to the rregularly shape, and has a hgh operatonal characterstcs to acheve the object of maxmum nscrbed rectangle detecton wth hgh precson. Introducton Stone slab s drectly cut from the natural rock blocks. It has the rregular profle and varous surface defects, such as the gall stone, holes, cracks and other defects [1]. It s dffcult to get the best utlzaton rate of cuttng and nestng stone slab, artfcal realzaton s usually the soluton of ths problem [2, 3]. Dgtal mage processng technology provdes a techncal support to determne such as the mnmum external rectangle and maxmum nscrbed rectangle parallel to the coordnate axes [4]. Detectng maxmum nscrbed rectangle on rregularly shaped s not a tradton mage processng method. Also, the optmal layout and cuttng of rectangular stone s feasble method to ncrease utlzaton rate of a rectangular materal [5]. Get the maxmum nscrbed rectangle on rregularly shaped and then process the optmal layout and cuttng wll be the better utlzaton rate soluton for natural stone producton plant. In ths paper, we use constraned grd graph to detect the overlappng area of the object shape and test rectangle. The author uses the electon campagn algorthm to fnd out the center and drecton of the test rectangle whch has the maxmum area and no overlappng area wth the object shape. After calbraton, the dgtal mages of objects wth rregular shapes are collected by the camera, and the mage s processed by the mage correcton mothed. The corrected mages wll be used to get the object shape and constraned grd graph. The next step, the optmzaton processng of mathematcal model compute the closest soluton, the defect of maxmum nscrbed rectangle complete. onstraned Grd Graph Accordng to the actual use of stone cuttng precson, we use vrtual grd graphcs wtch has the spacng of requred precson n the mage. Each pxel of mage s dvde to equal vrtual grd just lke the sub pxel method [6]. A sample profle retreval each vrtual grd n the regon, the defnton of each grd s 0, grd out of the regon s not 0. For example, the actual graphc dscretzaton s shown n Fgure 1. For the nterpretaton of clarty, we abstract some exceptons for sample enlarged graphcs. If you can retreve the pcture element n the vrtual grd area and small area, s defned as 0, sad the need to dscharge at the correspondng poston; the remanng non 0, as shown n Fgure 1 (b). Fg. 1 (c) sad there s a hole n the mddle of the pcture. All dscharge can be consdered as the target number s non-zero The authors - Publshed by Atlants Press 815

2 Fg. 1 Dscrete grds and correspondng pxel values As s shown n Fgure 2, the mages of stone slabs have has the rregular profle and varous surface features. Fgure 3 represent the constraned grd graphs. After dscrete transformaton processng, the mages of stone slabs converted nto mnmum vrtual rectangular grd graphs ncludes a set of grd nformaton value. Fg. 2 Sample of Stone Slab Optmzaton Processng Mathematcal Model Fg. 3 onstraned Grd Graph of Stone Slab If the nscrbed rectangle has the maxmum area and has not overlappng area wth the outsde area of the object shape, the soluton should be the area we detect for. The Mathematcal Model descrbes as below. A max( w* l) S( r, c) R (1) B( r, c, ) R B( r, c, ) s the grd at the border of the non-zero, S( r, c) s the grd n the nscrbed rectangle area, A s the area measure of nscrbed rectangle, the optmzaton program fnd the poston on the border of the shape and detect an angle of the nscrbed rectangle, then fnd the wdth and heght whch get the maxmum area of the rectangle and test whether the overlappng area exsts. 816

3 1. Get Border Grd 2. Get Angle. 3. Get Wdth and Heght. 4. Test Overlappng. The grd n the overlappng area has non-zero feature and change number n the border area. 5. alculate the area measure. Select Wdth and Heght, test overlappng then change to bgger number and test agan. Electon ampagn Algorthm Electon campagn algorthm (EA) s a new optmzaton algorthm smulatng electon process. Electon canddates always pursue the maxmum support from voters by means of varous electon behavors. They get the overvew of ther support from the voters accordng to the sample survey, and then decde what to do on the next move. Soluton space s magned as voters and current solutons are magned as canddates n EA. The functon value of a feasble soluton s named as the prestge of a voter and the functon value of a current soluton s named as the prestge of a canddate. In EA, the canddates nfluence the voters round them, the voter s support s n proporton to the dstance between the canddates and the voters. The voter wll vote to the canddate whch he pefer the most. The sum of locaton coordnates of every voters supported the canddate powered by ts contrbuton s a new locaton coordnates, whch s named support focus, t s the next poston of the canddate. Such computatonal cycle s done contnually untl a canddate fnds the poston of the hghest support, whch s the global soluton of the optmzaton problems. Framework of the electon campagn algorthm s descrbed as the follows: Set the Algorthm Parameters. Man parameters of Electon campagn algorthm are: The number of canddates. The number of voters. The number of floatng voters. Target accuracy. Generate the anddates and alculate the Prestge of Them. Generated the defne number of canddates n feasble soluton feld on the unform dstrbuton. Use the objectve functon to calculate the prestge of each canddate. Generate the Voters. The unform dstrbuton s employed to generate the voters n feasble soluton feld. ompute the Investgate Mean Square of anddates. Hgher prestge of a canddate, smaller the mean square devaton of local voters, so that EA s able to converge to local optmzaton soluton rapdly and steadly. The followng formula s used to descrbe the relaton of the prestge and the mean square devaton of a canddate. ( PMax P ) (2) ( P P ) Max Mn Where represents the mean square devaton of canddate ; PMax and PMn are the maxmum and mnmum prestge of canddates. ext step, voters are generated around each canddate on the normal dstrbuton. ( P ) Max P ( ) V FV (3) ( P P ) Max Mn s the voter number of canddate, V s the number of all voters, FV s the number of floatng voters. Floatng voters are generated on the unform dstrbuton. Here, t s supposed that canddate s effect on voters declned lnearly. 817

4 alculate the Support of the Voters. anddates can nfluence the voters wthn ther effect range, the effect on voter V j from canddate s D Max D V j F V P j (4) D Max F V s the effect on voter V j j form canddate, V j s the global or local voter. Here, t s supposed that canddate s effect on voters declned lnearly. Then ompute the prestge of voters by means of objectve functon. The support of a voter s proportonal to hs prestge, and then the proportonal constant wll be reduced, so the prestge of a voter can used to denote the support of a voter drectly. A voter may be nfluenced by several canddates; the voter should dstrbute hs support to canddates proportonally on the magntude of effect from canddate to voter. The support from voterv j to canddate s FV j S V P j m Vj (5) max( F, F,, F, F ) 1 1V j 2V j V j mv j SV represents the support from the voter V j j to canddate. Fnd the Support Focus of anddates. A new poston coordnate wll acheve by means of summng the products of the support from the voters to the canddate and the poston coordnate of the voters. It s named the support focus. x * n j1 n j1 S V j Vj max( S, S,, S, S ) x V1 V2 V j Vn * x s the support focus of the canddate (6). The support focus of a canddate s obtaned by nvestgatng, whch depends on those voters whose dstances to the canddate are nearer and the prestge s hgher relatvely. The next post of the canddate should be the support focus, where the canddate wll have the hgher support. alculate the Prestge of the anddates and ompare the Prestge of the Voters wth the anddates. In order to jump out of local optmzaton soluton and ncrease search rate, the prestge of canddates are compared to that of voters, f the prestge of a voter s hgher than that of a canddate, the voter wth hgher prestge wll substtute for the canddate and the that canddate of lower prestge wll be elmnated n electon. heck Whether the ondton s Reach. heck whether the condton s reach, otherwse return to step B to execute the perod. Here, the condton could be the number of select canddates n the target accuracy or max cycle. Do that crcularly untl the hghest support s found. t t1 t P P (7) P represents the prestge of canddate n cycle t. When the absolute value of the dfference between the prestge of the canddates n the present cycle and the prevous cycle less than or equal to the target accuracy, the computaton stop. Do that crcularly untl the hghest support s found. Flow scheme of the electon campagn algorthm s shown n Fg

5 Generate anddates Generate Voters anddates Prestge Voters Prestge anddates Influence Voters Support Support Focus ompare Prestge Reach the ondton? The Optmzaton Pestge Fg. 4 Flow scheme of electon campagn algorthm Sample of Detectng Maxmum Inscrbed Rectangle Start on the grd of shape border, then get angle, wdth and heght, wth no overlappng. In all these soluton, fnd the one has the maxmum area [7, 8]. Usng Electon campagn algorthm, the number of canddates defne as 5, the number of voters s 10, the number of floatng voters s 5, and the target accuracy s set as 1%. The experment get the maxmum nscrbed rectangle of the stone slab, as s show n Fg. 3, t prove the feasblty of the algorthm. onclusons Fg. 5 Maxmum Inscrbed Rectangle of Stone Slab Wth the method of the transformaton from stone slab mage to constraned grd graph, the overlappng of object shape and test rectangle can be detected, all the rectangle meet the condtons can be found and the one has maxmum area wll be detect by the processng of Electon campagn algorthm optmzaton program, and reach the optmal soluton. The dsparty map obtaned from the results can show that method s feasble. Electon campagn algorthm has good performance to avod the soluton trapped n local optma. Ths s partcularly mportant when dealng wth hghly multmodal functons. For future work, another feature and dfferent energy functon wll be used n the detectng of maxmum nscrbed rectangle. 819

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