ş 2 Keywords Luca Lucchese Kamera (ABC) ABSTRACT both low and problem. In order to show Camera Calibration karmaşık Rastrigin gibi az probleminde

Transcription

1 340 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00) The effectiveness of intelligent otimization techniqes in camea calibation Coşkn Özkan, Eme Beneş Eciyes Univesity, Engineeing Faclty, Comte Engineeing, Kaysei Eciyes Univesity, Engineeing Faclty, Geomatics Engineeing, Kaysei ş Keywos Lca Lcchesee Camea Calibation Atificial Bee Colony (ABC) ABSTRACT In this ae, it is aime to examine the effectiveness of the intelligent otimization algoithms to otimize the camea aametes with esect to the calibation metho intoce by Lca Lcchese (LL). The motivation of the intelligent otimization algoithms is thatt they ae so effective, flexible an easy aatable fo the eal comlex oblems. The selecte otimization algoithms ae Atificial Bee Colony (ABC), Diffeential Evoltion (DE), Genetic Algoithm (GA) an Paticle Swam (PSO). These algoithms excet ABC have been se effectively fo many comlex oblems. ABC has ecently eveloe an its effectiveness has not been teste fo a tye of the camea calibation oblem. Bt it is highly caable of geneating goo soltions fo many benchmak fnctions sch as Rosenbock an Rastigin with both low an vey highh imensions. The othe atificial intelligent otimization algoithms ae also the fist time being se in this camea calibation oblem. In oe to show the effectiveness of thesee intelligent otimization algoithms, thei eslts have been comae with the conventional eivative-base Levenbeg-Maqat (LM). Kamea kalibasyonna zeki otimizasyon yöntemleinin etkinliği ÖZET B çalışmanın amacı, liteatüe Lca Lcchese ın öneiği kamea kalibasyon metona ait moel aameteleinin otimizasyonna zeki otimizasyon yöntemleinin etkinlikleinin incelenmesii. B algoitmala, geçek kamaşık ünya oblemleinin çözümüne olkça etkili, esnek ve kolay aate eilebili olklaı için tecih eilmişlei. Seçilen algoitmala, Yaay Aı Anahta Kelimele Kolonisi (ABC), Difeansiyel Gelişim (DE), Genetik Algoitma (GA) ve Lca Lcchese Kamea Paçacık Süü Otimizasyon (PSO) algoitmalaıı. B algoitmalaan ABC Kalibasyon haiç iğelei biçok geçek ünya obleminin çözümüne kllanılmıştı. Yaay Aı Kolonisi (ABC) ABC algoitması yeni geliştiilmiş olğ için liteatüe Rosenbock ve Rastigin gibi az ve çok boytl bi çok temel fonksiyon üzeine test eilmiş fakat henüz etkinliği b ti bi kamea kalibasyon oblemine aaştıılmamıştı. Diğe zeki otimizasyon yöntemlei e b ti bi kamea kalibasyon oblemi için ilk efa kllanılmaktaı. Algoitmalaın etkinliklei, sonçla Levenbeg-Magat yöntemiyle kaşılaştıılaak otaya konlmşt. * Soml yaza (Coesoning atho) e-osta: cozkan@eciyes.e.t

2 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00). INTRODUCTION Camea calibation is an imotant ste in many fiels sch as comte vision an image ocessing. The main iea in the camea calibation is to etemine the camea tansfomation aametes fom D image oints to the coesoning 3D sace oints. Ths, camea calibation is esecially se to extact metic infomation fom D images []. The nmbe of camea aametes can change eening to the tye of camea calibation aoach. Basically thee ae two tyes of camea aametes: (i) intenal aametes that ceate mathematically the inne geomety of a camea when the image exose, they ae incial oint cooinates, effective focal length an istotions, (ii) extenal aametes that efine the angla attites in tems of oll, itch an yaw angles an the ositional islacements with esect to an object cooinate system. Althogh these two tyes of aametes can be comte thogh the enant measements ing the calibation, it is the main ose to etemine the intenal aametes fo a calibation ocess. In the close ange an the instial hotogammetic comte vision alications, eening on the wiely sage of the igital cameas, intenal an extenal aametes ae geneally etemine togethe in a mlti-view geomety in a kin of self calibation. The calibation mathematical moels can be assemble as linea o nonlinea moels []. Althogh the linea calibation algoithms sch as Diect Linea Tansfomation [3] ae easy to aly to the oblem, the mathematical moel oes not eesent the eal hysical moel becase of ignoing some aametes like lens istotions. Ths, the soltion becomes lack of stability an accacy. So, these moels can be only aoximate soltions. The aametes of the linea moels can be easily estimate by a least sqaes metho. In aition to the aametes of linea moels, nonlinea calibation moels take into accont the lens istotions as aial an tangential, asect atio as well. Althogh the nonlinea soltions ae moe ealistic an obst hysical moels with aitional aametes, they eqie iteative otimization algoithms fo aamete estimation. So, the comtational comlexity of nonlinea moels is highe than the linea systems. The most wiely se metho fo soltion of these systems is the Levenbeg-Maqat (LM) algoithm. In oe to convege a soltion, vey goo initial vales of aametes ae eqie in LM. Many calibation methos have been eveloe so fa [, 3-6]. They iffe fom each othe base on which aametes ae taken into accont. Althogh thee ae iffeent aoaches to the calibation oblem, the nelying mathematical moel geneally se is the colineaity incile, i.e. the colineaity eqation is the basic eqation on which the most of the calibation methos een. It can be efine the object oint, ojection cente an the coesoning image oint mst be collinea. The colineaity eqation sets the hysical moel of a light ay fom the object sace cooinates to the coesoning image cooinates in tems of the camea lens system. Anothe calibation metho (LL) consiee in this sty was intoce by Lca Lcchese [5]. In this metho, the way of homogahy is fom the image fame to efeence image fame wheeas it is sally fom sace to image cooinates at many methos. Anothe iffeence comes with sing calibation boa. 3D object cooinates ae not se in LL. The cooinates of the vital efeence image ae eive fom 3D cooinates of calibation boa with the constant istance. In this sty, Atificial Bee Colony (ABC), Diffeential Evoltion (DE), Genetic Algoithm (GA) an Paticle Swam (PSO) Algoithms ae se to calibate LL moel in comaison with Levenbeg- Maqat metho. Besies ABC algoithm is the fist time being se in a camea calibation oblem DE, GA an PSO that have been se fo othe calibation moels sch as Tsai [7-9] ae the fist time being se fo LL moel. ABC an PSO ae the membe of olation base swam intelligence algoithms [0, ] while DE an GA ae olation base evoltionay algoithms [, 3]. All of them ae heistic an iteative algoithms [, 4-6]. Atificial bee colony algoithm ABC algoithm is insie by the behavio of the bee colonies to fin ot foo soces. All of the bees ae eesente with thei ositions an by changing aametes of ositions they ty to fin otimal soltion. ABC woks iteatively an iteations contine ntil minimm objective vale eqal o smalle then the goal. Thee ae thee tyes of bee in ABC: (i) emloye bee that is going to the foo soce visite by itself eviosly, (ii) onlooke bee that waits on the ance aea fo making ecision to choose a foo soce an (iii) scot bee that caies

3 34 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00 ot anom seach [4]. Half of the colony consists of emloye bees an the othe at of the colony consists of onlooke bees. In each foo soce thee is only one emloye bee. Ding the iteations ositions of foo soces ae change by bees. At the beginning of the algoithm emloye bees go to thei stating ositions anomly an mease fitness of ositions. Afte that onlooke bee chooses its stating oint by fitness of foo soce of emloye bee. The foo soce of the most convenient fitness vale has a highe obability of choosing by onlooke. Bees stat seach fom this initial oint fo iteation. Althogh in a foo soce thee is only one emloye bee nmbe of onlooke bees can change becase onlooke bees on t have owneshi of foo soce. In othe wos, the nmbe of emloye bees is eqal to the nmbe of foo soces aon the hive. In seach ocess, anomly chosen aametes of the osition of anomly chosen foo soce is sbtacting fom own aametes. Reslts ae mltilie by a nmbe oce in [-,] inteval an the octs ae ae to the aametes. If fitness vale of new osition oce with this way is moe convenient than evios foo soce is chance. At each iteation seaching new osition of a foo soce is execte by bees at this foo soce. The emloye bee whose foo soce is exhaste by the emloye an onlooke bees becomes a scot. Limit an colony size vales ae the aametes of ABC to be tne. The main stes of the algoithm can be given like this: (i) Initialize. (ii) REPEAT. (a) Place the emloye bees on the foo soces in the memoy; (b) Place the onlooke bees on the foo soces in the memoy; (c) Sen the scots to the seach aea fo iscoveing new foo soces. (iii) UNTIL (eqiements ae met). In stana ABC fo a foo soce jst one osition aamete is change. Bt it is seen that this aoach is not sfficient fo the calibation oblem. Theefoe aitionally a etbation ate aamete is emloye [7]. This aamete etemines the change obability fo each aamete.. METHODOLOGY a. Images Boget s images available on the intenet wee emloye [4]. Images contain black-an-white checkeboa of high contast with a size of 3cm. Thanks to this feate contol oints can be etemine on the images by sing Hais Cone Detecto [8-0]. b. Camea Paametes Intenal aametes efine whee a light ay that came into the camea falls onto the image lane. The contact oint of the otical axis of lens to the image lane ae calle incial oint an qantize with x an y image cooinate ais. Becase of the imefection of the otical system of a eal igital camea, incial oint vey selom coincies with actal hysical cente of its image lane [5, ]. Accoing to the esective ojection, a 3D oint is mae into D image oint as Eq. X x f x Z Y y f y Z () In esective ojection f x an f y ae also intenal aametes. Effective focal length f is the istance between image lane an the otical cente of lens an escibe as ff x w an ff y h whee w an h is with an height of a ixel esectively. Also f x /f y atio efines asect atio[5, -3]. The lens istotions ae seaate into two tyes as aial an tangential istotions. The fome one aises becase of the eficient cvate of the lens sface an the latte one is case by misalignment of lens cente. So, tangential istotion has two comonents along the iections of x an y. Howeve the aial istotion is mainly affecte by the aial istance fom the incial oint. Becase of the istotions, ixel cooinates of the contol oints o not coincie with the coect laces. Theefoe a staight line can be seen as a bening line. Raial an tangential istotions ae exesse Eq. an Eq. 3, esectively.

4 343 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00 Δx Δy s s q q δ k 6 k k 3 +K () [ ( + (x x ) ) + (x x )(y y )] [ ( + (y y ) ) + (x x )(y y (3) In the above eqations, k, k an k 3 ae the aial istotion coefficients,, an 3 ae the tangential istotion aametes an x an y efine image cooinate have istotions. an q aametes ae (x q + x 3 ) + (y y ) (4) Extenal aametes etemine the osition an the oientation of a camea accoing to a secific cooinate system. Roll, itch an yaw angles ae thee of extenal aametes with that otation matix R R cos θcosϕ sin θcosω + cos θsin ϕsin ω sin θsin ω + cos θsin ϕcosω sin θcosϕ cosθcos ω + sin θsin ϕsin ω cos θsin ω + sin θsin ϕcosω (5) sin ϕ cosϕsin ω cos ϕcosω Tanslation elements along x, y, an z iections ae othe extenal aametes showe as a vecto T [T,T,T ] (6) x y z c. Lca Lcchese Metho Calibation metho intoce by Lcchese looks like Tsai o Zhang methos. The main iffeence comes fom the calibation allet se. This moel ses an imagine efeence image as a calibation allet athe than a eal allet. It is assme that the efeence image is geneate by an ieal inhole camea (C ) having aallel CCD axes to allet an having othogonal otical axis to allet intesecte at cente [5]. This ieal camea locate at a istance L+f fom the allet. The L istance is chosen abitay with a conition of comising whole imaginay allet. This ieal camea oes not have any istotions an its focal length is the same as the eal camea. As seen in Fige, image I acqie by ieal camea C lace with L istance fom allet P an in O`X`Y`Z` object cooinates. On the othe han image I is geneate by C camea at iffeent lace fom C. in OXYZ cooinate system. In LL metho, images ae geneate fom iffeent ositions an oientations. In Fige, o x y is the cooinate system of efeence image an ~ o ~ x ~ y is the cooinate system of eal images. Becase of imefection of lens, eal image cente of oxy cooinate system is not coincie with cente of ~ o ~ x ~ y cooinate system. Camea aametes ae obtaine via esective ojection of all images to efeence image. Tansfomation between O`X`Y`Z` an OXYZ is consiee as a two stages ocess (Fige ). Image I obtaine by C has wa an istotions. At the fist stage, lens istotions ae coecte an I image is geneate by aial an tangential istotion eqations.

5 344 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00) Fige. Camea calibation geomety of LL [5]. x y x y + (x + (y x ) δ y ) δ + Δx x s + Δy s (7) Aftewas, I is elate to the efeence image I thogh esective ojection as Eq.. Fige. Geometic Tansfomation [5]. In Eq. 8, coefficients of (a,a,a,a,b,b,c,c ) ae the fowa homogahy coefficients, whichh eens on the camea aametes. a(x x ) + a(y x c (x x ) + c (y a (x x ) + a (y y c (x x ) + c (y y ) + b y ) + y ) + b y ) + (8) In matix fom fowa homogahy coefficients ae invese of the backwa homogahy coefficients. They ae obtaine thogh a a c a a c b α b α γ α α γ β β (9) f f x y α, α, α, α D f yd fxd D 3 + τ x 3 + τy 3 β f x, β f y, γ, γ D D fx D T n D τz, τn, n {X,Y,Z} L + f 3 f D y (0)

6 345 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00 The calibation with LL metho has 0 intenal aametes that ae the same fo alll images an have 6 extenal aametes that ae iffeent fo all images. Theefoee nmbes of aametes to calclate can be obtaine with m *6 + 0 wheee m is the nmbe of image. 3. APPLICATION () Objectivee fnction that is necessay fo otimization algoithms contains calibation metho. This fnction gets aametes oce by otimization algoithm an comte esective ojection of image cooinates. As a eslt of calibation metho caniate allet cooinates ae flowchat of the alication in illstate in Fige(3). oce. this ae The was Since the ltimate sage of the calibation aametes is to comte the object sace cooinates of image oints, we have two tyes of comte ata of x an y cooinates. The efomance mease of the oblem is the oot mean sqaes eo (RMSE) of x an y iections Since the calibation oblem hanling in this ae is a nonlinea fnctional otimization oblem, Levenbeg-Maqat otimization metho is emloye. LM is a eivative base fnctional otimization metho between Gass-Newton of the lineaization of the obsevation eqations, soltion is obtaine in a an Steeest Decent [4]. Becase manne of iteation. So, the initial vales of the nknown aametes ae eqie to be able to statt the iteation. The close the initial vales to the eal vales ae, the less iteation is ocesse an the moe stable the soltion is. The initial vales eqie fo LM ae etemine by the same way given in [5]. The extenal aametes ajste by LM ae not given in a tabla fomat as in othe otimization methos becase of the hge table size. The intenal aametes ajste by LM ae given in Table in ose of comaison. The LM eslts fo homogahy ae visalize in Fige4 fo image. x makes eesent the cone oints on the eal image, i.e. fowa homogahy fom eal image to efeence image an the + makes eesent the cone oints of the efeence image. The qality of the homogahy can also be easily seen fom the gahics RMSE n n (x i x ) i i + n n ( y i i i ) y () Whee n is the nmbe of contol oints, x an y ae the image cooinates of the contol oints, x` an y` ae the comte vital efeence image cooinates. Whole system eo is comte with RMSE vale. Paametes with the minimm vale of RMSE mease oblem. ae consiee as the soltion of the In this sty, all images ae incle to calibation oceee in a mlti-vieimages wee sense by the same camea, the geomety. In that case, since all intenal aametes mst be the same fo all images. So, the otimization algoithms ty to estimatee 0 intenal aametes fo whole images an 6 extenal aametes fo each image. Ths, fo 5 images thee ae totally 60 nknowns. This inteeency among images thogh intenal aametes makes oblem qite comlex to hanle. Fige 3. Stcte of the se calibation system

7 346 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00. Fige 4. LM homogahies of image.

8 347 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00) The secific algoithm aametes of ABC, DE, PSO an GA algoithms wee etemine afte many tial an eo ocesses. The etaile exlanations abot these algoithm aametes can be fon in many efeences. The vales of some of them ae given in Table. Polation o colony size (N) fo all algoithms was set 50 an all the algoithms wee n fo 000 iteations. These moel vales shol be tne as ecisely as ossible fo gaantying them to n effectively. Unlike LM which nees the initial vales fo the nknowns, ABC, DE, GA an PSO nee only secific woking intevals fo the nknowns. These intevals mst cove fo all nknown vales in oe that seach sace wol be too lage to fin a soltion. Bt if they ae too lage, it makes the algoithm convege an accetable soltion vey ifficlt. In oe to obtain the otimal inteval vales, fistly each image is seaately otimize. Fo the intenal aametes, otimal intevals ae obtaine taking the minimm an maximm vales coming fom single fame soltions. Fo the extenal aametes, the otimal intevals ae comte by aing an sbtacting an aeqate little vale to the extenal aametes fom single fame soltions. Conseqently, the seach sace incling the eal vales is mae naowe. Ths, it is sose to obtain moe stable soltions. Afte moe ealistic inteval vales ae comte fom these inivial oeations, the main otimization ocess is one with these intevals fo 60 nknowns fo 5 aallel ns an the eslts with smallest RMSE vales wee chosen. Table shows the otimize intenal aametes comaing to LM with RMSE vales. Accoing to the RMSE vales, none of the intelligent algoithms gave an comaable soltion to LM. Among them, ABC an DE have the highe efomances esectively. Homogahies of image fo ABC wee comte an visalize in Fige 5 Table. Paametes of ABC, DE, PSO an GA algoithms. ABC DE PSO GA Petbation Rate 0.9 Cossove Rate 0.9 Ineteia Weights ( ) CosoveFnction Scattee Scale Facto 0.6 Scale Facto 0.6 Acceleato Weights (.-.) Selection Fnction StochasticUnifom Limit (Nx)/ Mtation Fnction Gassian Cossove Rate 0.9 Table. The otimize intenal aametes fom ABC, DE, GA, PSO. ABC DE GA PSO LM x Y k 9.76x x0-06.5x x x0-07 k 3.45x0 -.55x x x0-6.05x0-3 k 3-5.x x0-6.3x x x x x x x x x x x x x x x x x x0-06 f x f y RMSE

9 348 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00 Fige 5. ABC homogahies of image. 4. CONCLUSIONS In this ae the sage ossibilities of intelligent otimization algoithms, ABC, DE, GA an PSO have been examine with esect to the ecently oose camea calibation aoach of LL. Esecially as a novel metho ABC has not been se in any calibation oblem yet. Both the eo vales an the gahical homogahies show that intelligent otimization methos o not give satisfactoy eslts. Fo esecially high imensional oblems, this sitation is not sising fo DE, GA an PSO. Bt in ecent liteates ABC has showe vey goo efomances fo iffeent benchmak oblems. Althogh thee ae some algoithm aamete combinations to be tne fo these algoithms it can be accately concle these algoithms ae not as stable an obst as Levenbeg-Maqat. In site of

10 349 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00 the fact that it is vey ha to exlain these eficiencies becase of the heistic nates of these algoithms, the enancies an scale iffeences among the intenal aamete may case the faile of these algoithms. The istotion aametes have elatively too small vales an conseqently vey sensitive to little changes, as well. So, this sitation may case instabilities an inceasing seach time, i.e. fining global minimm is eithe imossible o vey ifficlt. Anothe eason may be becase of the fact that the hysical eality of the igital imagey can be moele mathematically well enogh by nonlinea colineaity eqations with istotions. Desite the nmeical effectiveness of LM, since the camea calibation oblem is nonlinea, LM nees vey goo initial vales fo convegent stable soltions. So, it can be concle that ABC can be emloye as an initializing tool fo LM in the camea calibation oblem in tems of LL moel. Acknowlegments We gateflly acknowlege the eseach oject gant (EUBAP FBY ) fom Eciyes Univesity Fonation of Scientific Reseach Pojects. REFERENCES. Zhang, Z., A Flexible New Techniqe fo Camea Calibation, IEEE Tansactions on Patten Analysis an Machine Intelligence, (), , Zhang, Y. an Ji, Q., Camea Calibation with Genetic Algoithms, IEEE Tansactions on Systems, Man an Cybenetics, 3(), 0-30, Abel-Aziz, Y.I. an Kaaa, H.M., Diect Linea Tansfomation into Object Sace Cooinates in Close-Range Photogammety, Poceeings of Symosim on Close-Range Photogammety, Ubana-Chamaign, IL, 8, Boget, J.Y., Camea Calibation Toolbox fo Matlab, 00. htt:// /inex.html. 5. Lcchese, L., Geometic calibation of igital cameas thogh mlti-view ectification, Image an Vision Comting, 3, , Tsai, R.Y.A., Vesatile Camea Calibation Techniqe fo High Accacy 3D Machine Vision Metology sing off The Shelf TV Cameas an Lenses, IEEE Tansactions on Robotics an Atomation, 3(4), , Abella, A., Bochoicha, M., Khelifa, M.M.B., A Genetic Algoithm Alication to Steeo Calibation, Poceeings of IEEE Intenational Symosim on Comtational Intelligence in Robotics an Atomation, 85-90, Faga, L.G., Silva, I.V., Diect 3D Metic Reconstction fom Mltile Views sing Diffeential Evoltion, LNCS Alications of Evoltionay Comting, Singe-Velag, 4974, , Wang, D., T, Y., Zhang, T., Reseach on The Alication of PSO Algoithm in Non-Linea Camea Calibation, Poceeings of Intelligent Contol an Atomation WCICA, , Kaaboga, D. Bastk, B., On The Pefomance of Atificial Bee Colony (ABC) Algoithm, Alie Soft Comting, 8(), , Kenney, J., Ebehat, R. C., Paticle Swam Otimization, Poceeings of IEEE Confeence on Neal Netwoks, Piscataway, NJ, , Hollan, J.H., Aatation in Natal an Atificial System, Univesity of Michigan Pess, MI, Ston, R., Pice, K. Diffeential Evoltion: A Simle an Efficient Aative Scheme fo Global Otimization ove Continos Saces, Intenational Comte Science Institte, Bekeley, 995 (ft://ft.icsi.bekeley.e/b/techeots/995/t f) 4. Kaaboga, D., Bastk, B., A Powefl an Efficient Algoithm fo Nmeical Fnction Otimization: Atificial Bee Colony (ABC) algoithm, Jonal of Global Otimization, 39(3), 459-7, Pham, D.T., Kaaboga, D., Intelligent Otimization Techniqes, New Yok: Singe- Velag, R. Ston, "Diffeential Evoltion", 00 (htt:// 7. Kaaboğa, D., Bastk, B., Atificial Bee Colony (ABC) Otimization Algoithm fo Solving Constaine Otimization Poblems, LNCS Avances in Soft Comting: Fonations of Fzzy Logic an Soft Comting, 459, , Deanis, K.G., The Hais Cone Detecto, 004. (htt:// is_etecto.f) 9. Nixon, M., Agao, A., Comte Vision Demonstation, 005. (htt://ses.ecs.soton.ac.k/msn/book/new_emo/co nes/)

11 350 Eciyes Ünivesitesi Fen Bilimlei Enstitüsü Degisi 6 (4) (00 0. Schmi, C., Mon, R., Backhage, C., Evalation of Inteest Point Detectos, Intenational Jonal of Comte Vision, 37(), 5-7, Lcchese, L., Close-Fom Pose Estimation fom Metic Rectification of Colana Points, Poceeings of IEE Vision, Image an Signal Pocessing, 53(3), , Lcchese, L., Estimating The Pose an Focal Length of a Camea Fom The Pesective Pojection of a Plana Calibation Plate, Poceeings of Signal an Image Pocessing, Honoll, USA, Lcchese, L., Geometic Calibation of Digital Cameas. Pat I: Camea Moel, Sbixel Feate Extaction an Algoithm Initialization, Poceeings of Visalization, Imaging, an Image, Benalmáena, Sain, Noceal, J., Wight, S.J., Nmeical Otimization, Singe-Velag New Yok, 6-66, 999.