Accuracy of Measuring Camera Position by Marker Observation

Transcription

1 J. Software Engneerng & Applcatons, 2010, 3, do: /jsea Publshed Onlne October 2010 ( Accuracy of Measurng Camera Poston by Marker Observaton Vladmr A. Grshn Space Research Insttute (IKI), Russan Academy of Scences, Moscow, Russa. Emal: Receved July 14 th, 2010; revsed August 10 th, 2010; accepted August 14 th, ABSTRACT A lower bound to errors of measurng object poston s constructed as a functon of parameters of a monocular computer vson system (CVS) as well as of observaton condtons and a shape of an observed marker. Ths bound justfes the specfcaton of the CVS parameters and allows us to formulate constrants for an object trajectory based on requred measurement accuracy. For makng the measurement, the boundares of marker mage are used. Keywords: Computer Vson System, Camera Poston Measurement, Marker Observaton, Lower Bound to Errors 1. Introducton CVSs are wdely appled for a soluton of moton control problems. Ths fact s assocated by the followng condtons. Frst, the computatonal capablty of avalable processors allows for the real-tme processng of large volumes of nformaton formed by TV cameras. The nformaton processng tme proves to be acceptable to a number of practcal problems [1-6]. Second, the ncreasng applcaton of computer-aded control systems of unmanned aeral vehcles requres the enhancement of the vector of measured parameters to solve the automatc landng problem [5]. Another task s a dockng problem (ncludng the spacecraft dockng), whch requres precse measurng a relatve poston for solvng the termnal control task [6]. As an example we can refer to the dockng the frst European Automated Transfer Vehcle (ATV) Jules Verne to the Internatonal Space Staton (ISS) on 3 Aprl In the above experment, a specal computer vson system was used for measurng the relatve spatal and angular poston. All of these facts stmulate nterest n estmaton of the potental accuracy (lower bounds to errors) of measurng the poston parameters as a functon of the marker shape, ts observaton condton and techncal parameters of the CVS. Ths allows us to evaluate an applcablty of CVSs to solvng control problems under specfc condtons as well as to optmze the CVS parameters from the vewpont of ensurng the requred accuracy of measurements. There are a small number of publcatons devoted to the problems of determnng the current coordnates measurement precson estmaton. Most publcatons are based on expermental approach (full-scale experments or stochastc smulaton) to the measurement precson estmaton. For obtanng relable estmaton, such approach requres too much tme and addtonally the full-scale experments are very expensve. In [7], the Cramér Rao bound s constructed to camera poston estmaton by dockng marker observaton. For poston estmaton, a set of the marker features (ponts of nterest) are used, namely corners, contrast spots and others. Ths approach s sutable for the case of small or medum marker observaton dstance. In such dstances the vsble sze of marker s of order tens or hundreds of pxels n any drecton. In the present paper, we consder the approach, whch s sutable for large dstances by usng the boundares between marker mage and background. Ths approach allows obtanng lower bound to errors of measurng object poston wth small computatonal expenses. It allows n one s turn to optmze CVS parameters and marker shape for a specfed set of the observaton condtons. In Secton 2, we formulate the assumptons for constructng the bound to errors. In Secton 3, we construct a Cramér Rao bound to the measurement errors and, n Secton 4, we present expermental results. 2. Assumptons Made When Constructng a Bound We make the followng smplfyng assumptons to est- Copyrght 2010 ScRes.

2 Accuracy of Measurng Camera Poston by Marker Observaton 907 mate the methodc errors: The resoluton of the optcal system s the same over the frame area. There are no geometrcal dstortons of the optcal system (or they are compensated for durng the preprocessng of mages). The optcal system s calbrated durng ts manufacturng and the calbraton error s neglgble. The exposure tme tends to zero, so smearng of the pcture due to the moton of the object durng shootng can be neglected. The precson of marker localzaton s lmted by sgnal to nose rato. The parameters of ths nose law are the same over the area of a frame. The pxel sze of CCD matrx tends to zero. All of these assumptons, except for the last one, are qute easly realzable at moderate cost. In regard to the last assumpton, t s ntroduced for smplfcaton of analyss. Wthout ths smplfcaton, an analytcal soluton s very dffcult. Apparently, t s possble to obtan some asymptotc estmatons of addtonal object poston measurement errors, whch s condtoned by lmted sze of CCD matrx pxels. In any case, ths problem should be a subject of separate analyss. Thus, the used model has no error sources except for the mage nose. 3. Cramér Rao Bound to Measurng Errors The constructon and applcaton of a lkelhood functon and Cramér Rao bound for measurement errors are extensvely descrbed n the lterature [8-10] and others. A lkelhood functon s used for constructng the Cramér Rao lower bound to the varance of estmated parameters. The schematc vew of the marker shootng s shown n Fgure 1. The marker s placed n the coordnate s orgn. The optcal system forms the mage of observed marker n the plane of a CCD matrx. The space poston of the TV camera and ts orentaton gves a vector of parameters A that should be estmated. Camera coordnate system s shown n the Fgure 2. Projecton center C of the camera s placed on the end of vector R (Fgure 1), whch s turned wth respect to a normal of the surface of marker on angle n the plane p whch pass through axs OZ and s prelmnarly rotated on the azmuth on angle relatvely plane XOZ. In the ntal camera poston vectors e1, e2 and e3 are gven by the coordnates as follows: 3 e1 0,0, 1 e2 1, 0, 0 e 0,1,0. The above three vectors are rotated by an angle together wth the projecton center of camera C n the plane p. So, the obtaned coordnates of the vectors are the followng: e e, e, e e2 e21, e22, e23 e e, e, e. Let, and be three rotaton angles around the vectors e 1, e and 2 e 3 respectvely. The frst rotaton s the rotaton by the angle. Snce the TV camera s space stablzed so that, the mage of observed marker s n the center of the vson area, t s possble to suppose the angles and small enough ( 0, 0 ). Hence, the rotaton operators by the angles and are approxmately commutatve. The coordnates of any -th marker pont X, Y, Z taken n camera coordnate system are the followng: Fgure 1. TV camera poston. Fgure 2. Camera coordnate system. Copyrght 2010 ScRes.

3 908 Accuracy of Measurng Camera Poston by Marker Observaton X X X YY Y Z Z Z, where X, Y, Z s the coordnates of camera projecton center C. The coordnates of the -th pont of the marker n CCD matrx are calculated by: f a X e Ye Ze a X e31ye 32 Z e a Xe Ye Ze, where f -s a focal dstance of the camera. For the specfed camera s spatal and angular postons, the -th pont X, Y, Z taken n the coordnates of CCD matrx depends on the parameters: 1 1 XYZ,,,,, 2 2 XYZ,,,,,. Snce we consder an observaton of marker from medum and long dstances, the measurement angular errors of and as well as the translaton errors n the drecton of the vectors e 3 and e 2 are heavly correlated. So, we estmate the precson only for four parameters, that are gven by a vector A rvu,,,. Axs r s parallel to e 1, v s parallel to e 2 and u s parallel to e 3. The constructon and applcaton of the lkelhood functon are well known from [8-10] and others. Ths lkelhood functon s used for constructng the Cramér Rao lower bound to the varance of estmated parameters. The lkelhood functon depends on parameters beng under estmaton. The estmatons of the parameters are defned by the values that provde the extremum of the lkelhood functon: P A extr, where P A s the lkelhood functon. The necessary condton of extremum s gven by: P A A 0, 1,...,4. Accordngly, we can use a logarthm of the lkelhood functon for fndng of extremum of PA. Analogous condton of extremum can be: ln P A A 0, 1,...,4. Covarance matrx of estmated parameters s: R J 1, where J s the Fsher nformaton matrx, whch s calculated from the lkelhood functon. Accordng to the Cramér Rao nequaltes, the lower bounds to the varances of unbased estmaton errors are gven by: 2 2 r R11 A, v R2 2 A, 2 2 u R33 A, R44 Α. We estmate the covarance for the estmaton of vector A. For ths goal, we frst determne the Fsher nformaton matrx, whch s expressed va the second dervatves of the lkelhood functon as follows: A A A 2 ln P ln P ln P Jj E E, AAj A Aj where E[...] s a mathematcal expectaton. Let s consder an observed mage of marker: where, ω s ω, A ω, s ω A s the marker mage and ω s 2 N0 nose wth ntensty 2. Wthout loss of commonness, we can suppose that a brghtness value of marker mage s ω, A s equal to one, and a brghtness of remanng part of the cadre s zero. In reference [11], an expresson of Fsher Informaton Matrx was derved for the case of one-dmensonal sgnal. For the two-dmensonal case, ths expresson can be easly obtaned by the same way:, s, 2 s ω A ω A Jj E d N ω, 0 A Aj where s a marker mage area and dω s an elementary square n. In general case, the calculaton of the Fsher nformaton matrx requres to determne the above mathematcal expectaton E[...]. In our case, the expresson n square brackets s determnstc, and therefore we obtan the followng elements of the Fsher nformaton matrx: J j 0, s, 2 s ω A ω A d N ω A A Let s consder dervatves. The A s the vector of parameters that gves the camera poston. The fnte dfference approxmaton of the dervatve s defned as follows: j (1) Copyrght 2010 ScRes.

4 Accuracy of Measurng Camera Poston by Marker Observaton 909 ω, A ω, A ω, A A ω, A s s s s, A A A A 0,... A,...0. Fgure 3 shows the marker mage n the ntal poston s ω, A. In Fgure 4, the marker mages are shown for both the shfted poston s ω, A A and the ntal poston s ω, A. The gray colours of dfferent ntensty are used for markng dfference between both these mages. The dfference can be calculated by ntegratng an optcal flow on the contour of marker as follows: s ω, A nq,, ωc A 0, ω C, where n s the external normal ( n 1 ) wth respect to the marker mage boundary (contour), Q s the optcal flow, whch s caused by A, nq, s a scalar product of the vectors n and Q and C s the marker boundary. In such a way, we show that the surface ntegral (1) s reduced to the followng contour ntegral: J 2,, nq nq dl (2) j j N0 C Thus we have obtaned the expresson for any element of the Fsher nformaton matrx. For the one segment, the ntegral (2) can be numercally calculated, for nstance, by the trapezum method: N n n n1 n1 pj nq, nq, j nq, nq, j l N0 2 n1 (3) Fgure 5 explans the calculaton of scalar product nq,. The calculaton of nq, j s made smlarly. The q s the dfference between scalar product nq, n the ntegral (2) for ths segment, and n the expresson (3) for ths segment. Notce that q s proportonal to A 2 and tends to zero n condton of A 0. So we can neglect ths term. Calculaton of the expresson (3) should be performed for all sectons of the maker boundary. 4. Expermental Results To llustrate the applcaton of the obtaned relatons, we estmated the errors of calculatng poston parameters for the marker shown n Fgure 6. The marker s gven by the sosceles trangle. The base of the trangle equals to two meters and ts heght equals to three meters. The trangle has the round spot n hs centre. Contour (boundary) C of ths marker ncludes both external boundary of ths trangle and nternal boundary of the spot n the trangle centre. Let s specfy the followng camera parameters. The focal dstance of the optcal system s 18 mm. The feld of camera vew s Errors of poston are calculated for a set of values of angle : 5, 15, 25, 35, 45, 55, 65 (7 values) and set of values of angle : 0, 10, 20, (36 values). The dstance of the marker observaton s r 50 м. We put a nose ntensty to be equal to 0.2 ( 0.2 ). Fgure 7 shows the calculaton results for the mean square errors of coordnates and normalzed correlaton. The coordnates are measured n meters and the values of angles are measured n degrees. The errors are gven by the approprate surfaces over the matrx of sze 7 36 samples, where the matrx szes are determned by the sets of and values respectvely. Fgure 3. Trangle marker. Fgure 4. Dfference sω, A A sω, A. Fgure 5. Calculaton of the scalar product on the one segment of marker boundary. Copyrght 2010 ScRes.

5 910 Accuracy of Measurng Camera Poston by Marker Observaton Fgure 6. Marker shape. Accordngly to Fgure 7, for the dstance of 50 m and the nose ntensty 0.2, the range r can be measured wth error r m, as well as the dsplacement n a CCD matrx plane can be measured wth errors V, U m. Rotaton around the vector r can be measured wth the error As followed from Fgure 7, the functonal dependences of measurement errors and normalzed correlaton of lnear and angular coordnates are very complcated functons. We have consdered the maker of unform brghtness. In ths case, only the contrast boundary operates n the marker mage. The calculated precson values are much hgher than the smlar values n [7] that are based on usng a small set of features (ponts of nterest) of the marker. Usng the boundares of marker mage for measurement provdes an ncrease of the measurement precson. Menton should be made that optcal system dstortons and low resoluton of CCD camera can serously deterorate the precson of measurement. Jont analyss of nose and camera resoluton nfluence on the precson of measurement s complcated enough. The above values of the mean square error and the normalzed correlaton should be taken n an account n creatng the computer vson system. The sgnfcant values of the normalzed correlaton show the consderable dependences between control loops of object poston coordnates. Ths fact should be taken nto account n the control system. The development of a computer vson system should be carred out together wth the development of the marker shape. σ r σ v σ u σ γ Copyrght 2010 ScRes.

6 Accuracy of Measurng Camera Poston by Marker Observaton 911 ρ rv ρ ru ρ rγ ρ uv ρ vγ ρ uγ Fgure 7. Errors of estmated parameters and correlaton bonds between them (normalzed correlaton). For comparson, we estmated the errors of calculatng poston parameters for the T-shaped marker shown on Fgure 8. Ths marker has the same area as the marker on Fgure 6. Fgure 9 shows the calculaton results for the measurement error of coordnates. Accordngly to Fgure 9, the T-shaped marker provdes a slghtly hgher precson of poston parameters measurement. Fgure 8. T-shaped marker. Copyrght 2010 ScRes.

7 912 Accuracy of Measurng Camera Poston by Marker Observaton σ r σ v σ u σ γ Fgure 9. Errors of estmated parameters for T-shaped marker. 5. Conclusons The new method has been proposed for estmatng the errors of determnng the TV camera poston. Ths method s based on usng the marker mage of a gven shape. The method allows us to estmate the measurement errors dependng on shootng condtons and CVS parameters. The obtaned error's estmatons are useful for development of CVSs and partcularly for optmzaton of ther parameters. 6. Acknowledgement Ths work was supported by the Russan Foundaton for Basc Research, project no а. REFERENCES [1] C. De Wagter and J. A. Mulder, Towards Vson-Based UAV Stuaton Awareness, Proceedngs of the AIAA Gudance, Navgaton, and Control Conference, San Francsco, Calforna, August 2005, pp. AIAA [2] S. Sarpall, J. F. Montgomery and G. S. Sukhatme, Vsually-Guded Landng of an Unmanned Aeral Vehcle, IEEE Transactons on Robotcs and Automaton, Vol. 19, No. 3, June 2003, pp [3] S. Sarpall and G. S. Sukhatme, Landng a Helcopter on a Movng Target, Proceedngs of IEEE Internatonal Conference on Robotcs and Automaton (ICRA 2007), Roma, Italy, Aprl 2007, pp [4] T. Kubota, S. Sawa, T. Msu, T. Hashmoto, J. Kawaguch and A. Fujwara, Autonomous Landng System for MUSES-C Sample Return Msson, Proceedngs of the Ffth Internatonal Symposum on Artfcal Intellgence, Robotcs and Automaton n Space (ISAIRAS 99), ESA SP-440, Noordwjk, The Netherlands, June 1999, pp [5] C. S. Sharp, O. Shakerna and S. S. Sastry, A Vson System for Landng an Unmanned Aeral Vehcle, Proceedngs of IEEE Internatonal Conference on Robotcs and Automaton (ICRA 2001), Seoul, Korea, May 2001, Copyrght 2010 ScRes.

8 Accuracy of Measurng Camera Poston by Marker Observaton 913 pp [6] C. Aknl, Sem-Autonomous Termnal Phase Spacecraft Dockng Atttude Determnaton and Control, [7] V. A. Grshn, Precson Estmaton of Camera Poston Measurement Based on Dockng Marker Observaton, Pattern Recognton and Image Analyss, Vol. 20, No. 3, 2010, pp [8] H. Cramer, Mathematcal Methods of Statstcs, Prnceton Unversty Press, USA, [9] I. A. Ibragmov and R. Z. Khasmnsk, Asymptotc Estmaton Theory, Nauka, Moscow, [10] A. B. Kryanev and G. V. Lukn Mathematcal Methods for Processng Indetermnate Data, Fzmatlt, Moscow, [11] H. L. Van Trees, Detecton, Estmaton, and Modulaton Theory, Part 1: Detecton, Estmaton, and Lnear Modulaton Theory, John Wley & Sons Inc., New York, Copyrght 2010 ScRes.