A NEW METHOD FOR STEREO- CAMERAS SELF-CALIBRATION IN SCHEIMPFLUG CONDITION

Transcription

1 A NEW METHOD FOR STEREO- CAMERAS SELF-CALIBRATION IN SCHEIMPFLUG CONDITION S. Hamroun 1, H. Louhch 1, H. Ben Assa 1, M. Elhajem 2 1 Unté de Métrologe en Mécanque des Fludes et Thermque, Ecole Natonale d Ingéneurs, Monastr, 5000, Tunse. 2 Laboratore de Mecanque des Fludes et Acoustque (LMFA), INSA de Lyon, Unversté de Lyon, France H : Tel.: ; Fax: ; Emal: hamroun_salha@yahoo.fr KEYWORDS: Stereo PIV, Self-calbraton, Schempflug condton, macro-partcle ABSTRACT: The calbraton of a Schempflug SPIV devce s actually acheved n two steps from a bundle adjustment technque. The SPIV recordng system s frst calbrated by magng the planar target at dfferent locaton and then the laser plane equaton s obtaned from a geometrcal optmzaton. The msalgnments between the laser sheet plane and the calbraton plane always remans sgnfcant. In ths paper we demonstrated How to calbrate a camera model drectly nto the laser sheet. A new schempflug model s appled to calbrate a cameras system. Ths new approach s based on the use of a target formed by sphercal macro-partcle. Expermental results on real data show the relevance of the proposed method. 1 Introducton The Schempflug condton enables to focus a vdeo-camera when oblque vewng at low aperture number. It s used n Stereoscopc Partcle Image Velocmetry (SPIV) for measurng the three-dmensonal velocty feld n a secton of a flow [1][2]. Ths condton s performed by tltng the sensor (or the lens) such that the object plane, the lens plane and the mage plane ntersect n a common lne (fgure 1). The three velocty components are reconstructed from mappng functons. One mappng functon per camera s used to relate any 3D locaton to ts mage. The parameters of each mappng functon are recovered at the calbraton stage. A precse target s usually placed n a reference plane for the calbraton of SPIV cameras. If the target s a planar one, t has to be accurately moved to dfferent locatons n depth. Other mult-level targets, on whch dfferent z-postons are present, are avalable for the specal case of SPIV calbraton. In [3], accurate axal dsplacements are requred by the calbraton protocol. All these calbraton methods may be unstable f the postonng of the target s not accurate enough. Besdes a method usng a sngle mage of a dot grd target was proposed n [4] but gven the focal length and sensor pxel ptches of the PIV camera. Self-calbraton does not use such addtonal accurate nformaton on the recordng system. A non lnear least mean square procedure allows the (bundle) adjustment of the vector of the calbraton parameters after ther rough ntalzaton. The coordnates of the calbraton ponts are ncluded n ths vector. Such an approach can present two practcal advantages [5]. Frstly t does not requre metrologcal calbraton targets so that calbraton targets can be home-made and easly adjusted to be studed feld of vew. Secondly t relaxes the constrant of an accurate postonng of the calbraton plate by a controlled translaton stage devce. The SPIV recordng system s frst calbrated by magng the planar target at dfferent locaton [6] and then the laser plane equaton s obtaned from a geometrcal optmzaton.

2 Despte these advantages, the msalgnments between the laser sheet plane and the calbraton plane always remans sgnfcant [7]. The am of ths paper s to combne ths two mentoned steps n one level usng a new camera model n schempflug condton. It conssts of makng the calbraton drectly nto the laser sheet. The new camera model n schempflug condton s descrbed n secton 2. In secton 3, we present the technque of Feature ponts detecton. The homography between mage plane when magng the laser plane s determned n secton 4. Expermental results on real data show how the relevance of the proposed method. 2 Schempflug camera model Self-calbraton from multple mages. The Schempflug camera model s based on the pn-hole assumpton. In the latter, the mage plane s parallel to the lens plane but n the Schempflug camera model the mage plane s tlted respect to the lens plane as can be seen n the Fg. 1. The condtons of the α angle (twstng of the x0 x lens plane around the X cam axs see (3)) and the laser plane should be the followng : so that z0 z 1 1 1,(see Fg. 1) d d f 0 Fg. 1. Coordnate systems n Schempflug condton. In the pnhole camera model, the extrnsc parameters are the rotaton matrx R and the translaton vector T to change from the world plane to the camera plane, and the ntrnsc parameters correspond to the focal length multpled by the pxel to mllmeter factors (F x,f y ), ntersecton of the optcal axs wth the mage plane (C x,c y ) and the dstorton parameters (equaton (1)), where s s any real number dfferent from zero. See later on the radal dstorton defnton.

3 Fg. 2. Intersecton of the pont n the real world n the tlted plane and n the perpendcular plane taken from[14]. s. xpx Fx 0 cx xw s. y px 0 Fy cy R yw T s z w To ths pnhole model equaton (1), two extra parameters (α,) are added. α s the between Ycam axs and v vector (whch can be arbtrarly chosen as the vertcal drecton of the tlted CCD plane), and β s the angle between the u vector (whch can be arbtrarly chosen as the horzontal drecton of the CCD tlted plane) and the vector Xcam, see Fg. 2. The orgns of the tlted and the perpendcular plane are the same, as the perpendcular plane can be arbtrarly placed. Cameras under Schempflug condton have been used n Steroscopc Partcle Image Velocmetry (SPIV), but n ths works [8], [9], they consder a rotaton matrx for small angles between the perpendcular plane and the tlted plane, resultng n three degrees of freedom when only two degrees are needed. In [10] the rotaton angles are small because they corrected the tlted lens of the camera due to the weght on the lens. Another nterestng work, [11], consders the nclnaton of the sensor as an addtonal dstorton, the radal dstorton beng the same as n Brown model [12]. They change the tangental dstorton addng a rotaton matrx. Ths s only useful for small angles. Fnally [13] develops a system of eleven lnear equatons n whch only one Schempflug angle s ncluded. In order to transform ponts from the tlted plane to the perpendcular plane ( Fg. 2), we have, the followng scenaro. The pont pw ( xw, yw, zw) s a pont n the world and r s a straght lne whch goes from the pont p to the optcal center o. Ths lne crosses the tlted mage plane and the perpendcular mage plane. The vector u s defned as u (cos, 0,sn ). That vector u les n the plane Xcam, Zcam. The other vector v s perpendcular to the u and has an α angle wth the Ycam vector. Hence the vector v ( sn *sn, cos,sn * cos ). It s verfed that u * v 0, (where u s the transpose to the vector u n matrx notaton). The followng all transposed vectors are noted the same way. The Equaton of the lne s: r: * pw *( xw, yw, zw) (2) The projected pont p per n the perpendcular plane at a dstance f s: (1)

4 p ( p, p, f ) (3) per perx pery In [14], Legarda has defned the projected pont p tlt n the tlted plane at a dstance f as followng: p (0,0, f ) p * u p * v (4) tlt tltx tlty From (3), (4) the straght lne passng though the pont p tlt ntersects n the perpendcular plane: *((0,0, f ) p * u p * v) ( p, p, f ) (5) Resultng: tltx tlty perx pery f ( f p *sn p *sn *cos ) (6) tltx Fnally we obtan the pont n the perpendcular plane p per the tlted one p tlt : ( p, p ) *( p *(cos,0) p *(sn *cos,cos ) ) (7) tlty perx pery tltx tlty To do just the opposte transformaton,.e, to calculate the pont n the tlted plane usng the nformaton n the perpendcular plane, the result can be easly derved: pperx ptltx 1 *( p pery *tan *tan ) cos (10) ppery ptlty 1 * cos Where λ 1 s defned as: tan 1 f f pperx *tan ppery * (11) cos In order to consder the lens dstorton, we transform frst the observed ponts (n the tlted plane) to the perpendcular one. For these transformed ponts, n the perpendcular plane, only radal dstorton[12] have to be appled due to the radal symmetry of the lenses respect to ther optcal axs. p x p x ( p x, p y) per dst per und pper xund per per p y p y ( p x, p y) per dst per und pper yund per per (12) ( p x, p y) ( k r k r k r ) p x per per p x per und per ( p x, p y) ( k r k r k r ) p y per per p y per und per (13) r s the dstance from the pxel to the prncpal pont of the mage, x x, y y. The radal dstorton coeffcents are: k 1, k 2, k 3, where p x p 0 p y p 0 per r x y. 3 Feature ponts detecton per We propose to use ponts tangent to dots and ther projectons n mages to perform calbraton [15]. The tangency ponts wll be determned by selectng b-tangents.e. tangents commons to a par of dots. In the case of two separated ellpses, there exst four b-tangents, two nternal (LL,

5 RR). In ths way, 6 ntersectons between b-tangents can be added as nvarant ponts to the 8 tangency ponts as shown n Fg.3. In order to get more feature ponts. For these ponts whch does not requre to mark dots, there s no offset due to the perspectve projecton. Fg. 3. Btangents and feature ponts extracton (a) Orgnal mage (b) Common tangents( sold lnes: external tangents, dashed lnes : nternal tangents) and assocated feature ponts (flled crcles : tangent ponts, rngs : btangent ntersecton ponts. To obtan ths tangency ponts, the block dagram of the fgure below summarzes the dfferent step: 4 Laser plane equaton Fg. 4. Blok dagram of the tangency ponts approach After calbraton of the stereo system, the rgd transformaton (R s,t s ) from the second camera coordnate system to the frst one for the th vew s gven by:

6 (1) (1) (2) (2) R t Rh th R t (14) Let us assume that optcal dstortons are corrected. As the projectons are perspectve ones, there exsts a pure homography transformaton H between ponts x (1) (of pxel homogeneous coordnates u (1) ) and x (2) (of pxel homogeneous coordnates u (2) ) : s u H u (15) (2) (1) Where H s (3 3) matrx defned wth only 3 unknown parameters and s s a scalar ntroduced n order to normalze the thrd component of u (2). In fact, the mage u (2) s the mage of the back projecton of u (1) on the laser plane. The laser plane beng defned by the dstance d from the frst projecton center and by the normal vector n=(n x,n y,n z ) T, the homography H can be expressed as: (1) (1) (1) (2) (2) p 0 u0 p x x px 0 u0 0 T T (1) (1) (1) (2) 2 Rh t 0 p h y vo py H 0 py v n n (1) x d y d nz d 1 p (16) T The purpose s to fnd the unknowns 1 a, b, c The soluton ( a, b, c ) T can be estmated by mnmzng the sum vector between measured poston current value 1 H ( a, b, c) u s 5 Expermental results. (1). (2) d n whch leads to the laser plane equaton. 2mn T where 1 u (correspondng to the th measured poston s the devaton (1) u ) and the The method mentoned n secton 2 for SPIV self-calbraton has been appled on a par of /2" CCD cameras (Sony XC 700) mounted on the Schempflug devces. The CCD cameras were equpped wth f=50mm lenses focused on the measurement plane wth a dstance of 500 mm and vewng t at 45. Ths new method s appled usng prnted target and macro-partcle nto laser sheet Usng prnted target. Fve photographs of ellpses of prnted target are recorded by each camera at dfferent postons as shown n Fg. 5(a). The tangency ponts obtaned usng the method descrbed n secton 3 are llustrates n Fg. 5(b). The results of calbraton are shown n table 1. As can be seen the use of btangents allows accurate estmates: for example, the estmate of Schempflug tlt (theoretcal values: 2.8, 0 ) s , Once can notce a relatve

7 devaton (nferor to 1%) of mage dstance p x wth respect to focal length f x =8000 as the lens was separated from the CCD devce for Schempflug tlt. (a) (b) Fg. 5. (a) Photographs of ellpses target at dfferent postons (b) Btangents and tangency ponts for ellpses at dfferent postons Intrnsc parameters Standard devaton p x (pxel) p y (pxel) u 0 (pxel) v 0 (pxel) a 1 a 2 a 3 α(degree) (degree) , ,0 02 Table 1. Results of self-calbraton wth prnted target Usng macro partcle nto laser sheet We have keep the same cameras system. We have just replace the prnted target by macropartcles n the laser plane. An Argon contnue laser sheet (λb =488nm, λv = 514.5nm) wth a maxmum power of 7W. The laser plane was located n the depth of feld of the cameras. In these experments, the target s a cross-sectons of two mm sphercal soap bubbles flled wth smoke were llumnated by laser sheet(about 1 mm thck). These Cross-sectons marked the laser plane as two crcular dots whch can be maged nto large ellpses n mage planes as shown n Fg. 6.

8 Fg. 6. test of two soap bubbles In order to calbrate the system of cameras n Schempflug model, 5 vews of the target were recorded by each camera. The ponts belongng to the btangents of the two ellpses (Fg. 7) are computed accordng the algorthm gven n secton 3. The cameras were calbrated usng the tangency ponts. For each camera, relable results are obtaned as shown n table 2. Fg. 7. Feature ponts belongng to the btangents of the two ellpses Optmal parameters of the rght cameras Optmal parameters of the left cameras Intrnsc parameters σ Intrnsc parameters σ u 0 (pxel) u 0 (pxel) v 0 (pxel) v 0 (pxel) f (pxel) f (pxel) α(degree) α(degree) (degree) (degree) Table 2. Results of Schempflug self-calbraton usng macro-partcles.

9 A 6 th vew s recorded by each camera and used to fnd the laser equaton. Table 3 shows the results of the laser plane equaton usng the homographc method. θ 1 s the angle under (ox), θ 2 s the angle under (oy), θ 3 s the angle under (oz). Poston of the laser plane θ 1 θ 2 θ 3 d Table 3. Results of the laser plane parameters 6. Concluson Self-calbraton of Schempflug cameras allows easy calbraton by vewng a target at dfferent locatons n the depth of feld. The cameras parameters are recovered from the bundle adjustment technque usng the new schempflug model proposed by Legarda[14]. The laser equaton s obtaned from a geometrcal optmzaton. The msalgnments between the laser sheet plane and the calbraton plane always remans sgnfcant. We have proposed a method to combne the two steps n one level. However, the cameras parameters and the parameters of the laser plane can be obtaned smultaneously. The target can be postoned drectly n the laser sheet. Ths technque allows to gettng rd of the problem of correctng msalgnment between the calbraton plane and the laser plane. Our approach s based on the use of a target formed by a macro-partcle whch can be entrely seen n the laser plane. The feature ponts are determned usng the tangency ponts technque. Such a procedure leads to detect a ffteen feature ponts at a subpxel precson. The obtaned expermental results confrm the relevance of the proposed method. References [1] Hnsch K D, Hnrchs H, Roshop A, Dreesen F 1993 Holograpc and stereoscopc advance n 3DPIV. Holographc Partcle Image Velocmetry. Proc. of Fluds Engneerng Dvson, AmercanSocety of mechncal Engneers. E P Rood (Washngton, DC:ASME), (148), [2] Prasad A K, Jens K 1995 Schempflug stereocamera for partcle mage velocmetry n lqud flow. Appled Optcal (34), [3] Soloff S, Adran R, Lu Z C 1997 Dstorton Compensaton for generalsed stereoscopc partcle mage velocmetry Measurement Scence and Technology, (8), [4] Qunot G, Rambert A, Lusseyran F, Gougat P 2001 Smple and accurate PIV camera calbraton usng a sngle target mage and camera focal length 4th Int. Symp. Partcle Image Velocmetry (Gottngen: Germany), [5] Fournel T, Lavest J M, Collange F 2003 Self-calbraton of PIV vdeo-cameras n Schempflug condton Espagne: Sprnger.

10 [6] Louhch H, Fournel T, Lavest J M, Ben Assa H 2007 Self-Calbraton of Schempflug cameras: an easy protocol Meas. Sc. Technol [7] Lavest J M, Vala M, Dhome M 1998 Do we really need an accurate calbraton pattern to acheve a relable camera calbraton In Proc. of ECCV98, (Freburg: Germany), [8] Louhch H, Fournel T, Lavest J, BenAssa H 2006 Camera selfcalbraton n schempflug condton for ar flow nvestgaton In proc of Advances n Vsual Computng. Second Internatonal Symposum, ISVC, Part II, pp , [9] Louhch H, Fournel T, Lavest J, BenAssa H 2007 self-calbraton of schempflug cameras: An easy protocol Measurement Scence and Technology, vol. 18, no. 8, pp [10] Hag C, Hepke C, Wggenhagen M 2006 Lens nclnaton due to nstable fxngs detected and verfed wth vd/vde 2634 part 1. [11] Wang J, Sh F, Zhang J, Lu Y 2008 A new calbraton model of camera lens dstorton Pattern Recogn., vol. 41, no. 2, pp [12] Brown D 1966 Decenterng dstorton of lenses vol. 32, no. 3, pp [13] L J, Guo Y, Zhu J, Ln X, Xn Y, Duan K, Tang Q 2007 Large depth-of-vew portable three-dmensonal laser scanner and ts segmental calbraton for robot vson Optcs and Lasers n Engneerng, vol. 45, pp [14] Legarda A, Izagurre A, Arana N, Iturrospe A 2011 Internatonal Workshop on Electroncs, Control, Measurement and Sgnals - ECMS. [15] Fournel, T., Louhch, H., Barat, C., Menudet, J.F.(2006). Schemplug self-calbraton based on tangency ponts. In proc. of 12TH Internatonal Symposum of Flow Vsualsaton, (Göttngen, Germany).