A NOVEL FAMILY OF MATHEMATICAL SELF-CALIBRATION ADDITIONAL PARAMETERS FOR AIRBORNE CAMERA SYSTEMS

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1 A NOVEL FAMILY OF MATHEMATICAL SELF-CALIBRATION ADDITIONAL PARAMETERS FOR AIRBORNE CAMERA SYSTEMS R. Tang*, D. Fritsch, M. Cramer Institute for Photogrammetr (ifp), Universit of Stuttgart, Geschwister-Scholl-Str. 4D, 774, Stuttgart, German (rongfu.tang, dieter.fritsch, EuroCOW KEY WORDS: Airborne cameras, camera self-calibration, additional parameters, Legendre APs, function approimation ABSTRACT: Self-calibration additional parameters (APs) have identified their significant role in photogrammetric calibration and orientation since 97s. However, the traditional APs might not be adequate for the digital airborne camera calibration. A novel famil of mathematical self-calibration APs is presented in this paper. We point out that photogrammetric self-calibration can, to a ver large etent, be considered as a function approimation or, more precisel, a curve fitting problem in mathematics. Based on the rigorous approimation theor, the whole famil of Legendre APs, which is derived from well-defined orthogonal Legendre Polnomials, is developed. Legendre APs are in general efficient to calibrate all the frame airborne cameras. The can also be considered as the superior generalization of the conventional APs developed b Ebner and Grün. A solution strateg b using Legendre APs is also suggested for in-situ camera sstem calibration. Etensive tests on various cameras including DMC, UltracamX, UltracamXp and DigiCAM illustrate the good performance of Legendre APs. The optimal theoretical accurac can be achieved b appling Legendre APs, if a dense pattern of ground control points (GCPs) is available. The comparisons with the traditional APs show the theoretical and practical advantages of Legendre APs.. INTRODUCTION Camera calibration is an essential subject in photogrammetr. Self-calibration b using additional parameters (APs) has been widel accepted and substantiall utilized for camera calibration in photogrammetric societ. Traditionall, two tpes of selfcalibration APs were developed for analogue camera calibration: phsical and mathematical. The development of phsical APs was mainl attributed to D. C. Brown (Brown, 97) for close-range camera calibration and these APs were later etended b attaching additional polnomials for aerial application (Brown, 976). Mathematical APs (or polnomials APs ) were proposed b Ebner (976) and Grün (978), who used two- and four- order orthogonal bivariate polnomials respectivel. The polnomials APs are often criticized as have no foundations based on observable phsical phenomena (Clarke and Frer, 998), even though the can continuousl reduce the image residuals. These APs, even though being widel used for man ears even though in digital era, might be inadequate to fit the distinctive features of digital airborne cameras, such as push-broom, multi-head, virtual images composition, multiple image formats, etc.. A considerable progress was made recentl for the digital camera calibration and some new APs have been developed. Fraser (997) analzed the digital close-range camera calibration, based on the classical work of Brown. Cramer (9) and Jacobsen et al. () reported comprehensive empirical tests, in which different APs were emploed to compensate the image distortion. However, lots of the APs do not own solid phsical or mathematical foundation and some are limited in calibration efficienc. For eample, the input parameter b of Grün APs, which is.4 l where l is the side length of the square analogue image, is set as. ( l l ) where l and l are the length and width of rectangular digital image format. Consequentl, Grün APs are orthogonal on the generated fictitious grid points, of which some ma la beond the image format, as illustrated in Fig.. The incorporation of navigation sensors into airborne camera sstems also demands calibrating the whole sstem rather than camera lens distortion onl (Honkavaara, 4, Cramer et al., ). Fig. the fictitious grid points for rectangular image format All the above motivate our present work on airborne camera calibration. We start with a mathematical viewpoint of selfcalibration and then develop a novel famil of polnomials APs, which has rigorous mathematical foundation. The new socalled Legendre APs are theoreticall capable to calibrate the distortion of all the frame-format airborne cameras. We also suggest an eas but effective strateg for camera sstem calibration. Etensive empirical tests will be performed to evaluate Legendre APs in the sstem calibration. * Corresponding author.

2 The rest is organized as follows. In Section, the mathematical principle of self-calibration is reviewed and Legendre APs are constructed. The practical tests are demonstrated in Section, followed b the comparisons between Legendre APs and traditional ones would be discussed in Section 4. This work is concluded in the final section.. LEGENDRE SELF-CALIBRATION APS The collinearit equation which is the mathematical fundamental of photogrammetr reads as follows. r X X r Y Y r Z Z c, r X X r Y Y r Z Z r X X r Y Y r Z Z c, r X X r Y Y r Z Z Where and () denote image distortion, random error. The other notations can be seen in tetbooks such like Kraus (7). The image distortion terms, (, ) and (, ), are two-variable functions whose form is unknown. The have to be approimated b some models, i.e., self-calibration APs. In general, two modeling approaches are possible. On the one hand, if the phsical properties of the distortion are readil understood, then the distortion can be represented b some specific functions, like the phsical APs developed b Brown. On the other hand, if the precise knowledge on the distortion is not available, we need to approimate the distortion in some more abstract means. There are advantages and disadvantages for both modeling approaches. The first model can be quite accurate but usuall case-dependent, while the second one are ver generall effective but ma face the risk of overparameterization. We consider the second approach. We would like to find a mathematical approach to model and compensate the distortion. It shall be orthogonal, mathematicall rigorous, and generall effective for all airborne cameras. As image measurements are available to fi the distortion, the distortion of unknown form can be approimated b the linear combination of mathematical basis functions. The coefficients can be estimated during the adjustment process. Therefore, photogrammetric self-calibration can to a ver large etent be considered as a function approimation or, more precisel, a curve fitting problem in mathematics. Therefore, we will start with the general principle of the mathematical approimation as follows.. Orthogonal polnomials approimation The algebraic polnomial approimation is founded on the Weierstrass Theorem (Mason & Handscomb, ). It indicates that an univariate function can be approimated with arbitrar accurac b a polnomial of sufficientl high degree. Among all the possible forms, the orthogonal polnomials (OPs) are often favored in both theoretical and practical applications due to man elegant properties. The OPs can be categorized into two tpes: discrete and continuous. The former is orthogonal on specific discrete measurements while the later is orthogonal over the whole domain of definition. For the curve fitting problem, the analtical form of the function is unknown while some sample measurements are available. The unknown function can be approimated b the combination of OPs. If the number of the measurements is close to the degree of the used polnomials, the discrete OPs are usuall emploed and can be obtained b orthogonalization process. It is noteworth that the discrete OPs are orthogonal on the measured locations onl, but not necessaril on the others. Else, if the measurements are ver dense and the number is much larger than the polnomials degree, the continuous OPs is preferred. More theoretical materials can be seen in such as Berztiss (964) and Mason & Handscomb (). Legendre Polnomials, denoted b { Lm( )} m,,... where m indicates the order, are a series of continuous OPs over [,], i.e., L ( ), () m, L ( ) L ( ) d m n (), m n m n Legendre polnomials grant the optimal approimation in the least-square sense (Mason & Handscomb, ) and are widel used in man applications. The first few normalized Legendre Polnomials are listed in appendi A. The bivariate OPs can be generalized from the univariate cases. The could be much more complicated, depending on the twodimensional definition domain. Particularl, the twodimensional generalization on the rectangular domain turns out to be rather straightforward. Namel, if { p ( )} are a m m,,... series of univariate OPs over [,], then p (, ) p ( ) p ( ) (4) m, n m n m,,... n,,... are complete bivariate OPs over the rectangular domain [,] [,], satisfing p (, ) p (, ) dd () m, n i, j, if m i and n j, else Complete indicates that an two-variable function can be approimated well b the p (Koornwinder, 97).. Self-calibration APs mn, Let b and b denote the width and length of the image format, respectivel. B scaling we obtain, l (, b ) L / b (6) m m l (, b ) L / b (7) n n where and are the metric image coordinates, L and m are univariate Legendre Polnomials ( mn,,,,...). The first few l (, b ) are, l (, b ) m m l (, b ) / b L n

3 4 6 l (, b ) / b / l (, b ) / b / b / b 4 l (, b ) / b / b / 8 l (, b ) 6 / b 7 / b / b / l (, b ) / b / b / /6 Similar formulae of l (, b ) could be derived. Denote n n f f (, ; b, b ) l (, b ) l (, b ) (8) m, n m, n m n then f are the bivariate OPs over the rectangular frame mn, mn, b, b b, b and f. Considering the image, mn distortion is tpicall in the order of m, we obtain p b mn, 6 multipling f with for numerical stabilit. mn, mn, mn, p 6 f m, n m, n, 6 p mn (9), p can be ordered leicographicall as Eq. (), following Koornwinder (97). Obviousl, b b p p,, p,, p, p, p,,, p, p, p, p,,,, p, p, p, p, p 4,,,,,4 () p p dd i, j m, n if i m or j n () b b It indicates that if the image measurements are densel distributed, then p (, ) p (, ) i, j k k m, n k k if i m or j n () k Eq. () implies that p is (almost) orthogonal over the mn, mn, all image measurements. Therefore, the bivariate distortion ( ) ( ( ) ) in,, Eq. () could be approimated b a series of continuous OPs m M, n p N mn, m, n ( m M, n p N mn, m, n ), where M and N ( M and N ) are the chosen maimum degrees which are not necessaril equal. Further, si of them should be eliminated, as done b Ebner (976) and Grün (978). Speciall, the constant terms p in (, ) and (, ) are nothing but the, principle point offset; p, p, p and p in (, ),,,, are highl correlated with p, p, p and p in,,,, (, ), respectivel. Thus, the number of the unknown parameters is ( M )( N ) ( M )( N ) 6. As eamples, the APs with M M and N N are obtained in Eq. () with 66 unknown parameters ( a, i,,..., 66 ). The APs with 4 unknowns, M M 4 and N N, are given in Eq. (4). So far the whole famil of APs has been completel constructed. The input of APs includes the image length and width ( b and b ), and the chosen degrees ( and M, N ). Usuall, it can further adopt N, M M M i M and N N N in practice. This class of APs is based on Legendre Polnomials and thus called Legendre APs. a p a p a p a p a p a p a p,,, 4,, 6, 7, a p a p a p a p a p a p 8, 9, 4,,,, a p a p a p a p a p a p 4,4, 6 4, 7, 8, 9,4 a p a p a p a p a p a p,, 4,, 4,4, a p a p a p a p a p 6, 7 4, 8,4 9,, a p a p a p a p a p 4,4,,4 4 4,, a p a p a p a p a p a p a p,, 6,, 4, 7, 8, a p a p a p a p a p a p 9, 4, 4 4, 4, 4, 44, a p a p a p a 4,4 46, 47 4, 48 p, a 49 p, a p,4 a p a p a p a p a p a p,, 4, 4,,4 6, a p a p a p a p a p 7, 8 4, 9,4 6, 6, a p a p a p a p a p 6 4,4 6, 64,4 6 4, 66, a p a p a p a p a p a p a p (),,, 4,, 6, 7, a p a p a p a p a p a p 8, 9, 4,,,, a p a p a p a p a p a p 4 4,, 6, 7 4, 8, 9 4, a p a p a p a p a p a p a p,,,, 4,,, a p a p a p a p a p a p, 4, 4, 6, 7, 8, a p a p a p a p a p a p 9 4,,, 4,, 4 4, (4)

4 . Overall sstem calibration Nowadas the integrated navigation sstems are incorporated as a part of the digital airborne camera sstems. The GPS/IMU incorporation, on the one hand, accelerates the photogrammetric mapping and remarkabl reduces the number of ground control points (GCPs). On the other hand, it brings etra sstematic effects, e.g., the misalignment between the camera and the navigation instruments and the drift/shift effect in the direct georeferencing measurements. Therefore, the overall camera sstem calibration becomes a must in current photogrammetr. For the sstem calibration, one most challenging work could be to minimize the coupling effect of the different sstematic errors. The decoupling is of vital importance in the sense that each sstematic error must be independentl and appropriatel calibrated and the calibration results are block-invariant. For this purpose, we suggest the joint application of the Legendre APs (for calibrating the image distortion) with other correction parameters, i.e., the three interior orientation (IO) parameters used for correcting the principle point offset and the focal length deformation, and GPS/IMU drift/shift and misalignment correction parameters. The low correlation must be warranted among these calibration parameters and between them and eterior orientation (EO). As will be seen in Section 4, the correlations between Legendre APs and EO, and between Legendre APs and other correlation parameters, are fairl small. The low correlation is one advantage of Legendre APs over the traditional APs.. PRACTICAL TESTS The Legendre APs are empiricall tested b using the data from the recent DGPF project (German Societ for Photogrammetr, Remote Sensing and Geoinformation), which was performed under the umbrella of DGPF and carried out in the test field Vaihingen/Enz nearb Stuttgart, German. This project aims at an independent and comprehensive evaluation on the performance of digital airborne cameras, as well as offering a standard empirical dataset for the net ears. Four flights data of the frame cameras are adopted: DMC (GSD cm, ground sample distance), DMC (GSD 8cm), UltracamX (GSD cm) and UltracamX (GSD 8cm). For each flight, we are interested in two most often contets: the in-situ calibration one and the operational project one. The former contet is with high side overlapping ( 6%) and dense GCPs and the later with low side overlapping ( %) and few GCPs. The block configuration of four flights is not detailed here and the readers are referred to Cramer () and DGPF website () for the project details as well.. In-situ calibration contet The sstem calibration strateg in Section. is adopted for all the blocks. Particularl, IMU misalignment, horizontal GPS shift, IO parameters and Legendre APs with M N M N are emploed. The order of Legendre APs is empiricall selected b the compromise between achieving the optimal accurac and reducing overparameterization. The derived eternal accurac, indicated b self calibrating, would be compared to the theoretical accurac and the without APs one, for which the same correction parameters ecept Legendre APs are used. The derived eternal accurac is demonstrated in Fig.. B comparing Self calibrating with Without APs, the refinement of Legendre APs is significant in all tests, up to cm in the DMC (GSD cm) block. Moreover, all the self calibrating accurac reaches ver close to the theoretical one and it means that the optimal accurac has been achieved. The closeness keeps well when the presumptions of the std. dev. of the GPS observations are varied from cm to cm (though not illustrated here). All the self calibrating accurac reaches to / GSD in the horizontal directions and / GSD in the vertical directions in four blocks. It is also interesting to notice that although the DMC and UltracamX cameras are differentl manufactured, ver similar eternal accurac can be obtained b using Legendre APs in the blocks of similar configuration, i.e., similar GSD, similar forward and side overlapping levels and similar GCPs distribution. This fact, independent of the used cameras, coincides well with our photogrammetric accurac epectation. Now look at the estimation of the precision of the image measurements. The posterior std. dev. estimation is.6,.4,.89 and.78 (unit: m ) for DMC (GSD cm, GSD 8cm) and UltracamX (GSD cm, GSD 8cm) blocks, respectivel. These values are around. piel size, which are m and 7. m for DMC and UltracamX cameras, respectivel. The well match the epected precision of the automatic tie point transfer techniques, which are.-. piel size for aerial images.. Operational project contet There are 4 GCPs and % side overlapping level in each block, which is much weaker than the in-situ calirbation contet. The IMU misalignment, IO parameters and the Legendre APs with M M 4, N N are emploed in the adjustment. Using Legendre APs of lower degree tries to avoid the potential overparameterization. This derived eternal accurac is analogousl denoted as self calibrating one. Due to 4 GCPs available onl, the GPS/IMU observations have to be weighted carefull to achieve best accurac. We also evaluate the qualit of the in-situ calibration in last sub-section. The calibration results of IO and image distortion in Section. are utilized as known and fied values in the adjustment of the corresponding reduced operational block, i.e., the cameras are assumed being calibrated and need no further self-calibration. The derived eternal accurac is named as after calibration. We compare after calibration with self calibrating, without APs and theoretical ones. The adjust accurac in four blocks is illustrated in Fig.. From those results, the self-calibrating Legendre APs help improve the eternal accurac and the after calibration ields further refinement, more than / GSD in DMC (GSD 8cm) block. The after calibration accurac is ver close to the optimal theoretical one in ever block. Therefore, these tests not onl recognize the sufficient accurac obtained b Legendre APs in the operational projects, but also confirm again their great efficienc in the in-situ calibration. More discussions would be appeared in Tang et. al (). It is worth mentioning that the Legendre APs have also been assessed b the flight data of other airborne cameras in other test fields, like medium-format DigiCAM and large-format UltracamXp. The similar good results are confirmed while the details are not published here.

5 unit: cm Fig. Eternal accurac in four in-situ calibration blocks, dense GCPs and p6%-q6% ( without APs indicates without using Legendre APs onl) Fig. Eternal accurac in four operational project blocks, 4 GCPs and p6%-q% ( without APs indicates without using Legendre APs onl) 4. DISCUSSIONS In this section, we make comparisons between Legendre APs and the conventional APs from the theoretical and practical viewpoints. Based on the standard 6% forward overlapping level and a small few photographic measurements in analogue time, Ebner and Grün built polnomials APs of two and four orders, respectivel. These APs are restricted in the assumed regular and grid points configurations, respectivel. Both APs can be obtained b orthogonalizaiton and elimination of si highl correlated parameters, and finall get the models of and 44 unknown parameters respectivel. It is known, somewhat confusingl, that both APs can improve the accurac even when the regular grid patterns are not satisfied. This is a source of criticism posed on these polnomials APs. All these bewilderments can be clarified easil b using the theor of function approimation. First, Ebner and Grün APs are merel special orthogonal arrangements of polnomials. In fact, the belong to discrete OPs in the mathematical jargon; see the mathematical materials in Section.. Second, the mathematical principle behind Ebner and Grün APs is still Weierstrass theorem, eactl the same with Legendre APs. Therefore, the irregular patterns onl affect the correlations among APs rather than their effect in compensating lens distortion. That is wh the still work even though tie points do not satisf the grid pattern. Third, it is also eas to understand from the approimation view wh Ebner APs sometimes achieve quite poor performance. That is, the distortion is too comple for two-order polnomials to well approimate. Higher degree s polnomials are required and that is the reason wh Grün APs perform better in general. Two eamples are illustrated for the comparison on the eternal accurac in DMC (GSD cm) calibration and operational blocks, in Fig. 4 and Fig. respectivel. It is clear that Ebner APs obtain quite poor accurac, particularl in Fig. 4. However, Legendre APs must be preferred to these two traditional polnomials APs. First, from the mathematical viewpoint, continuous polnomials are more reasonable than discrete polnomials for calibration purpose, i.e., Legendre APs are more theoreticall reasonable than these two APs. This is basicall due to the wide-accepted assumption that the geometric distortion of each image is homogeneous in a single photogrammetric block. It implies equivalentl that all measurements in all images are put into one single image dimension for the calibration purpose. Consequentl, ver dense image measurements encompass the image format for almost all the cases. As a simple eample, considering a block including images, each image contains a ver small amount of measurements, sa around. It turns out to be roughl measurements usable for self-calibration, much more than the number of unknown APs (usuall quite smaller than ). Therefore the continuous polnomials (Legendre polnomials) should be favored. Second, Ebner and Grün APs are single order polnomials while Legendre APs are a whole famil of polnomials. Thus, Legendre APs offer much more fleibilit for applications. Third, Legendre APs are advantageous in low correlations. An eample in DMC (GSD cm) calibration block is illustrated in Table., where <. indicates the percentage of correlations smaller than. and ma denotes the maimum correlations. It is demonstrated that Legendre APs have much lower correlations with IO and IMU than Grün APs. The intra-correlations among APs (denoted b intra-corr ) show that Legendre APs are more orthogonal. Fig. 4 and Fig. also illustrate that Legendre APs deliver slightl better accurac. In fact, Legendre APs can also be seen as the superior generalization of the traditional polnomials APs from the mathematical viewpoint. DMC (GSD cm) Fraser () Brown () Ebner() Grün(44) Legendre (66) X Y 47GCPs/8ChPs, p6%-q6% Fig. 4 Comparison on the eternal accurac in DMC (GSD cm) calibration block (47GCPs/8ChPs, p6%-q6%)

6 unit: cm unit: cm 4 DMC (GSD cm) Fraser () Brown () Ebner() Grün(44) Legendre (66) X Y 4GCPs/8ChPs, p%-q6% Fig. Comparison on the eternal accurac in DMC (GSD cm) operational block (4GCPs/8ChPs, p%-q6%) DMC (GSD cm) Legendre APs(4) Legendre APs(66) X Y 4GCPs/8ChPs, p6%-q% Fig. 6 Insignificant impact of moderate overparameterization on the eternal accurac in DMC (GSD cm) operational block (4GCPs/8ChPs, p%-q6%) Table. Correlation analsis in DMC (GSD cm) calibration block (47GCPs/8ChPs, p6%-q6%) APs corr. EO IO IMU Intra-corr Grün <. % 8% 8% 88% APs (44) ma Brown <. 98% 78% 86% 78% APs () ma Legendre APs (66) <. % 97% % 96% ma We also compare Legendre APs with the phsical APs proposed in Brown (97) (Fraser (997) as well) and Brown (976). Although Brown model achieves the comparable accurac as Legendre APs do in Fig. 4 and Fig., the later possesses much better performance in low correlations, as demonstrated in Table. In fact, the image distortion of the multi-head airborne cameras is not dominated b the radial-smmetric distortion anmore; and this is the main reason wh Fraser model delivers rather poor accurac (Fig. ). The mathematical APs are sometimes criticized for overparameterization. It is true that mathematical APs achieve general effectiveness at the price of more parameters than phsical counterparts. However, we urge here that moderate overparameterization is tolerable and would not degrade remarkabl the accurac. Fig. 6 depicts an eample of DMC (GSD cm) operational block ( flight lines, 4 GCPs, % side overlap and 8 images). The Legendre APs of 66 and 4 unknowns are applied. It is seen that no significant difference is observed. The influence of redundant APs can be further reduced in the blocks of higher overlapping and more GCPs. From the view of curve fitting, this tolerance of moderate redundant APs is mainl due to the number of image measurements being much larger than number of APs.. CONCLUSIONS We proposed a new class of self-calibration Legendre APs for calibrating digital frame-format airborne cameras. The prime theoretical foundations of Legendre APs are mathematical polnomial approimation and the renowned Weierstrass Theorem. This is one significant theoretical development in this work. It is thus guaranteed that Legendre APs of proper degree can, at least theoreticall, calibrate the distortion of all the frame cameras. The ecellent performance of Legendre APs is demonstrated in the etensive tests on various airborne cameras, including DMC, DigiCam, UltracamX and UltracamXp. Legendre APs are generall effective and fleible for calibrating all digital frame airborne camera architectures, no matter which sstem design have been chosen b the camera manufacturer. In principle, the can be used for calibrating frame cameras of large-, medium- and small-format CCDs, mounted in singleand multi-head sstems. Moreover, the ver low correlation between Legendre APs and other parameters, such as those for eterior orientation (EO) and GPS/IMU offsets or misalignments, guarantees reliable calibration results. We also compare Legendre APs with other traditional APs. Both the theoretical investigations and practical eperiments show Legendre APs are superior to the conventional Ebner and Grün APs. Compared with the phsical APs, Legendre APs show its advantages in general efficienc and ver low correlations. Although the polnomials APs are sometimes dubbed empirical (McGlone et al., 4), Legendre APs are in fact more objective in man senses. It is allowed to conclude that Legendre APs are orthogonal, rigorous, generic and effective for the digital frame airborne cameras calibration. REFERENCES Berztiss, A., 964, Least squares fitting of polnomials to irregularl spaced data, SIAM Review, 6(), pp. -7. Brown, D., 97, Close-range camera calibration, Photogrammetric Engineering, vol. 7, pp Brown, D., 976, The bundle method --- progress and prospects, International Archives of Photogrammetr, (), ISP congress, Helsinki, pp.-. Clarke. T. and Frer J., 998, The development of camera calibration methods and models. Photogrammetric Record, 6(9), pp Cramer, M., 9, Digital camera calibration, EuroSDR official publication No., 7p. Cramer, M.,, The DGPF-test on digital airborne camera evaluation overview and test design, Photogrammetrie- Fernerkundung-Geoinformation (PFG), No., pp Cramer, M., Grenzdörffer, G. and Honkavaara, E.,, In situ digital airborne camera validation and certification the future standard? IAPRS, Vol. 8, 6 pages on CD-ROM. DGPF project,, Evaluation on digital airborne camera sstems, URL: Allg.html (last accessed: Jul ). Ebner, H., 976, Self-calibrating block adjustment, Bildmessung und Luftbildwesen, V.44, p Fraser, C., 997, Digital camera self-calibration, ISPRS Journal of Photogrammetr and Remote Sensing, (4), pp Grün, A., 978, Progress in photogrammetric point determination b compensation of sstematic errors and

7 detection of gross errors, Smposium of Comm. III of the ISP, Moscow, pp. -4. Honkavaara, E. et al. 6, Geometric test field calibration of digital photogrammetric sensors, ISPRS Journal of Photogrammetr and Remote Sensing, Vol. 6, Pages Jacobbsen, K., et al., DGPF-project: evaluation of digital photogrammetric camera sstems --- geometric performance. Photogrammetrie - Fernerkundung - Geoinformation (PFG), No., pp Koornwinder, T., 97, Two-variable analogues of the Classical orthogonal polnomials, Theor and Application of Special Functions (R. Aske ed.), Academic Press, New York, pp Kraus, K., 7, Photogrammetr: Geometr from Images and Laser Scans ( nd ed.), de Gruter, 49p. Mason, J. and Handscomb, D.,, Chebshev Polnomials, Chapman and Hall/CRC, 6p. McGlone C. et al. (ed.), 4, Manual of Photogrammetr ( th ed.), ASPRS, 4 Tang R., Fritsch, D. and Cramer M.,, New mathematical self-calibration models in aerial photogrammetr,. Wissenschaftlich-Technische Jahrestagung der DGPF, March (to be appeared). ACKNOWLEDGEMENTS Dipl. Ing Werner Schneider is greatl appreciated for his assistance. This work benefited much from his rich practical eperiences. We also thank Dipl. Ing Dirk Stallmann for his programming help. The first author is grateful to Chinese Education Ministr for the financial support during this stud in Stuttgart. APPENDIX The first few normalized orthogonal Legendre Polnomials over the interval, are following. L ( ) L ( ) L ( ) ( ) L ( ) ( ) L 8 4 ( ) ( ) 4 L 8 ( ) (6 7 ) L ( ) ( ) 6