A Study on Geometric Model of CCD Camera Based on Projective Geometry

Transcription

1 Amercan Journal of Informaton Scence and Computer Engneerng Vol. 1, No. 2, 2015, pp A Study on Geometrc Model of CCD Camera Based on Projectve Geometry Sun Yu * Dept. of Computer Scence & Technology, X an Unversty of Scence and Technology, X an, Chna Abstract Amng at enhancng the spatal coordnates precson solved from 2D mages, a ne method for calculaton of 3D model coordnate transformaton beteen mage sde and object sde space s proposed based on projectve geometry. CCD camera magng geometrc prncple s analyzed based on perspectve transformaton and cross-rato nvarant propertes. Imagng geometrc model based on lnear feature s establshed usng the relatonshp of parallel, perpendcular and ntersectng beteen straght lnes. Wth acqured photographs and camera parameters, the shape and sze of the correspondng scene space s deduced and proved on the bass of the cross rato nvarablty of collnear ponts th perspectve projecton. The coordnate transformaton beteen mage sde and object sde space of the 3D model s calculated. And perspectve correspondence from mage sde to object sde space s bult. Expermental results prove that calculaton accuracy of the spatal ponts coordnates can be enhanced usng mage geometrc nformaton. The method can be appled to enhance the precson of mage feature extracton and locaton of geometrc features such as crcle, and to ncrease the measurement precsong of the photogrammetry system and 3-D coordnate reconstructon. Keyords CCD Camera, Projectve Geometry, Perspectve Transformaton, Cross Rato, Imagng Model Receved: May 2, 2015 / Accepted: May 6, 2015 / Publshed onlne: June 5, 2015 The Authors. Publshed by Amercan Insttute of Scence. Ths Open Access artcle s under the CC BY-NC lcense Introducton Accordng to the to-dmensonal mages acqured at dfferent veponts to reconstruct the orgnal three-dmensonal model space objects s a hot research n photogrammetry and computer vson feld. Transformaton relatonshp beteen them from to-dmensonal mages to three-dmensonal reconstructon s determned by the camera magng geometry [1-3].Camera magng geometry s the bass of three-dmensonal reconstructon of data processng, manly descrbed camera mage space and object space coordnate transformaton relatons [4-6]. Lterature [7-8] ere studed to explore space and precse postonng and testng of the crcle. About camera lens dstorton problems, lterature[9] calculated camera parameters based cross-rato nvarance prncple advantage of fttng functons; [10] usng a nonlnear optmzaton method acheved a calbrated lens dstorton parameters; [11] usng the propertes of the secton plane conjugate establshed a geometrc model of mage dstorton. In ths paper, the establshment of CCD camera magng geometry model are derved to calculate the mage sde perspectve space, and the perspectve correspondence of object space boundary contour and mage boundary contour are analyszed, dependng on the camera parameters. 2. Camera Imagng Prncple The typcal camera model s based on the prncple of collnear, each pont n the object space through a straght lne through the center of the projecton mage projected onto the plane * Correspondng author E-mal address: sunyu@xust.edu.cn

2 34 Sun Yu: A Study on Geometrc Model of CCD Camera Based on Projectve Geometry [12-13]. Usng camera to shoot the object, gettng a pcture of an object, the correspondng relatons beteen space objects and the objects n the mage s projectve geometry of the central projecton, also knon as perspectve projecton [14]. As shon n Fgure 1, the perspectve center s the center of photography S, namely the optcal center of the camera. Optcal axs for SS, perspectve axs s the ntersecton l of a mage plane (A'B'C'D') and objects plane (ABCD). Wthout loss of generalty, assumng the angle beteen the mage plane and the plane of the object s α. The follong further studes the correspondence of perspectve plane and object plane. Fg. 1. Geometry model of camera magng. mage s, from S S pedal forx. perpendcular to do the perspectve axsl, On the mage plane s knon A' B' C' D' E' F' 2 a, AD ' ' BC ' ' G' H' 2b, S as the center of rectangle A' B' C' D ', therefore, G' S SH' b, E' S SF' a Imagng Geometry Deduced Establshed the plane geometry relatons over the optcal axs, perpendcular to perspectve axs are shon n Fgure 1. The camera optcal center of S,camera focal length of f,then ss f, the dstance from optcal center to object plane n optcal axs s d, d ss,the angle beteen mage plane and object plane of α, SXS SSI SSI ',camera ptchng angle θ, e have SSH' SSG' θ ; tan θ b/ f. sn θ b/ b + f, cos θ f / b + f As shon n Fgure 1, makng perpendcular from the center of photography to the object space plane, dong SI GX, ntersecton G' X to pont I '.In the kno: Rt SSI',easy to S f tan α, SI' f /cosα (1) In the Rt SIS, can be derved: SI dcos α, S I dsnα (2) By the formula (1), the formula (2) can be derved: I' I dcos α f /cosα (3) In the Rt XSS, can be derved: S X ( d f)/sn α, SX ( d f)cotα (4) Rt SIG,because of tan θ b/ f, θ arctan( b/ f),can be derved: Fg. 2. The mage plane. 3. Correspondence beteen the Image Space and Object Space 3.1. Informaton of Image Plane Shoot a mage, t s rectangular n shape, and ts length and dth s knon. Settng mage plane s a rectangle A'B'C'D', a length of 2a, dth 2b. as the mage center, from to S perspectve axs dong vertcal lne, pedal of. Center of the SH' SG f b ' + (5) In the Rt SIG, because of GSI α + θ,by the formula (2),can be derved: SG SI/ cos( α + θ) dcosα/ cos( α + θ) (6) GI SGsn( α + θ) dcosαtan( α + θ) (7) GS GI S I d(cosαtan( α + θ) sn α) d(( f snα + bcos α)/( f btan α) sn α) (8)

3 Amercan Journal of Informaton Scence and Computer Engneerng Vol. 1, No. 2, 2015, pp Object Space Boundary Shape Analyss and Proof The follong dscusson s on geometry features of the rectangular mage obtaned by the camera. Combnaton of the above analyss, and further proof s derved as follos: object along the optcal axs of the plane dstanced, then SS f, SS d. Perspectve transformaton perspectve has the follong propertes: the axs parallel to the perspectve of a set of parallel chords, stll parallel to the axs of a rear perspectve. Fg. 3. Geometrc relaton of object space Object sde boundary contour analyss shon n Fgure 4, n ESFand E' SF' can be derved by the follong formula: ES S F ad/ f (9) (c) boundary 3 Fg. 4. Object space boundary contour analyss (a) boundary 1 As shon n Fgure 4(a), th (9),you can kno E' F'/ EF f / d,therefore: EF 2 ad/ f (10) In the same ay n Fgure 3, because of edge AB ' '// l, AB// l,so AB// lcd, // l,therefore AB// CD. As shon n Fgure 4(b),n SAB:BG GA. Because of A' B' SG' f + b, then: AB SG dcosαcos( α + θ) (b) boundary 2 AB 2ad f btanα (11) As shon n Fgure 4(c), n SCD:DH HC.

4 36 Sun Yu: A Study on Geometrc Model of CCD Camera Based on Projectve Geometry C' D' SH' f + b CD SH GH + SI (12) ( SG', SS; SH', SX) ( G' S, H' X) ( GS, HX) (13) Wth (4), deprved (13): Integrated the above conclusons, e analyzed the object space contour shape. As shon n Fgure 5, l SS X, GH lgh, AB,and GH to splt up and don, e can derve the quadrlateral s sosceles trapezod. Gettng to see, therefore, the rectangle A' B' C' D ' of mage plane s correspondng th the sosceles trapezod ABCD of object plane. G' H' SX ( G' S, H' X) SH ' G' X, 2 b ( d f)cotα 2( d f)cotα b (( d f)cot α + b) ( d f)cotα + b GH SX ( GS, HX) S H GX ( GS + SH) SX GS/ SH + 1 S H ( GS + S X) GS / S X + 1 Accordng to above ( GS, HX ) calculatng formula, th formula (8), can obtans H,havng: b/ f cosαcos2 α( d f + btan α) SH 1 f / d + b / df tan α (14) And accordng to HX SX SH, calculated HX. Fg. 5. Object space boundary contour. From the above analyss, the rectangular mage plane s perspectve correspondng th sosceles trapezod perspectve object space, t also llustrates a phenomenon, to other long-term segment are parallel to the perspectve axs, far aay from the eyes of the pece looks shorter than the dstance near ths length. 5. The Determnaton of the Sze of Perspectve Correspondng Contour In projectve geometry, the center projecton has the follong property: Property projectve transformaton keep collnear ponts and the nature of the cross rato unchanged. That s to say, f the four ponts are collnear, after projectve transformaton, the correspondng four ponts are stll collnear, and keepng the nature of the cross rato unchanged, namely, cross( ABC, ;, D) cross( Ac, BcCc ;, Dc). Accordng to above property, dscussed the coplanar harness SG, SS, SH, SX cross rato. We can be converted to calculate the four ponts cross rato on G' X and GX. Comng here, you can use formula (2) and (4) then smplfyng formula, obtanng: At ths pont, usng equaton (2), formula (14), the formula (12) can be smplfed to gve 2 2 a ( dsn a SH) + d cos α CD f + b (15) Fnally, as shon n Fgure5, accordng to the sze of the rectangular mage, calculatng a perspectve ve correspondng deduced trapezodal space sze, ncludng the upper and loer end of the trapezod, and the hgh and the poston of a projecton center n the trapezod. Based on the above analyss and dervaton, you can get the follong conclusons: (1)Wth a rectangular mage correspondng sosceles trapezod perspectve, perspectve axs s parallel to the bottom of an sosceles trapezod, small perspectve axs near bottom, bg bottom far. (2) Gven the sze of the rectangular mage, the camera focal length, angle beteen the plane and the dstance beteen the camera and the object can be calculated sosceles trapezod poston correspondng to the sze perspectve. (3) The lne segment s bsected magng center and s vertcal to the perspectve axs on the rectangle magng, n space plane, ts perspectve lne segment s vertcal to the perspectve axs, but magng central perspectve pont s more close than the lne segment center from the perspectve axs.

5 Amercan Journal of Informaton Scence and Computer Engneerng Vol. 1, No. 2, 2015, pp Concluson In ths paper, the dscusson about the camera perspectve magng model and the correspondence beteen the above outlne for the correspondng characterstcs of pnhole camera's perspectve, the nature of projectve geometry based on perspectve transformaton, the mage sze and camera parameters are knon, the shape of the object space s derved, locaton and sze, the exact expresson for the to-dmensonal mage and perspectve correspondence beteen space objects provde a theoretcal bass that can be used lke a square space and object space coordnate converson three-dmensonal reconstructon, has some theoretcal sgnfcance and applcaton value. At the same tme e see the need for future research n the camera dstorton practcal problems do further research and dscusson. References [1] FENG Wenhao, LI Jansong, YAN L. Creaton of Dstorton Model for Dgtal Camera Based on 2D DLT [J]. Geomatcs and Informaton Scence of Wuhan Unversty, 2004, 29(3): (n Chnese). [2] ZHAO Shuangmng, RAN Xaoya, GUO Xnhong. Geometrc model of CE-1 stereo camera [J]. Scence of Surveyng and Mappng, 2011, 36(6): (n Chnese). [3] LIU Songln, HA Changlang, HAO Xangyang. Lne Scanor Cameras Imagng Model Based on Machne Vson [J]. Journal of Zhengzhou Insttute of Surveyng and Mappng, 2006, 23(5): (n Chnese). [4] Janne H, Oll S. A Four-step Camera Calbraton Procedure th Implct Image Correcton [EB/OL]. kla97.pdf. ( ). [5] Atknsn K B. Close Range Photogrammetry and Machne Vson. Department of Photogrammetry and Surveyng, Unversty of London, [6] Fraser C S. The Metrc Impact of Reducton Optcs n Dgtal Cameras. Photogrammetrc Record, 1996, 15(87):437~446. [7] LI Zhanl, LIU Me, SUN Yu. Research on calculaton method for the projecton of crcular target center n photogrammetry [J]. Chnese Journal of Scentfc Instrument, 2011, 32(10): (n Chnese). [8] ZHOU F Q, ZHANG G J, JIANG J. Hgh accurate non-contact method for measurng geometrc parameters of spatal crcle [J]. Chnese Journal of Scentfc Instrument,2004, 25( 5) : (n Chnese). [9] LIN G Y, CHEN X, ZHANG W G. Hgh precson measurement method of parameters of plane crcle based on stereo vson [J]. Applcaton Research of Computers, 2010, 27( 3) : (n Chnese). [10] WANG B F, FAN SH H. Matchng Homologous ponts based on photographs n computer vson [J]. Journal of Zhengzhou Insttute of Surveyng and Mappng, 2006, 23( 6) : (n Chnese). [11] LOURAKIS M I A, ARGYROS A A. The desgn and mplementaton of a generc sparse bundle adjustment softare package based on the Levenberg-Marquardt algorthm [R]. Techncal Report FORTH-ICS/TR-340, 2004.