Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography (한국측량학회지)

Korean Society of Surveying, Geodesy, Photogrammetry and Cartography (한국측량학회)
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Single Photo Resection Using Cosine Law and Three-dimensional Coordinate Transformation

코사인 법칙과 3차원 좌표 변환을 이용한 단사진의 후방교회법

  • Hong, Song Pyo (Department of GIS Engineering, Namseoul University) ;
  • Choi, Han Seung (GIS Research Center, Geospatial Information Technology Co., Ltd) ;
  • Kim, Eui Myoung (Department of Spatial Information Engineering, Namseoul University)
  • Received : 2019.06.03
  • Accepted : 2019.06.26
  • Published : 2019.06.30
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Abstract

In photogrammetry, single photo resection is a method of determining exterior orientation parameters corresponding to a position and an attitude of a camera at the time of taking a photograph using known interior orientation parameters, ground coordinates, and image coordinates. In this study, we proposed a single photo resection algorithm that determines the exterior orientation parameters of the camera using cosine law and linear equation-based three-dimensional coordinate transformation. The proposed algorithm first calculated the scale between the ground coordinates and the corresponding normalized coordinates using the cosine law. Then, the exterior orientation parameters were determined by applying linear equation-based three-dimensional coordinate transformation using normalized coordinates and ground coordinates considering the calculated scale. The proposed algorithm was not sensitive to the initial values by using the method of dividing the longest distance among the combinations of the ground coordinates and dividing each ground coordinates, although the partial derivative was required for the nonlinear equation. In addition, since the exterior orientation parameters can be determined by using three points, there was a stable advantage in the geometrical arrangement of the control points.

사진측량에서 단사진의 후방교회법은 이미 알고 있는 카메라의 내부표정요소, 지상좌표, 사진좌표를 이용하여 촬영당시 카메라의 위치와 자세에 해당하는 외부표정요소를 결정하는 방법이다. 본 연구에서는 코사인 법칙과 선형식기반의 3차원 좌표변환식을 이용하여 카메라의 외부표정요소를 결정할 수 있는 단사진의 공간후방교회법 알고리즘을 제안하였다. 제안한 알고리즘은 먼저 렌즈왜곡이 보정된 정규좌표를 코사인 법칙을 이용하여 지상좌표와 이에 대응되는 정규좌표간의 축척을 계산하였다. 그리고 나서 축척을 고려한 정규좌표와 지상좌표를 이용하는 선형방정식 기반의 3차원 좌표변환식을 적용하여 외부표정요소를 결정하였다. 제안한 알고리즘은 비선형방정식으로 편미분이 필요하나 지상좌표의 조합 중 가장 긴 거리를 구하여 각 지상좌표에 나누는 방법을 이용하여 초기값에 민감하지 않은 장점이 있었다. 또한, 세 점을 이용하여도 외부표정요소를 결정할 수 있기 때문에 기준점의 기하학적 배치에 안정적인 장점이 있었다.

Keywords

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Fig. 1. Geometric constraints at two corresponding points.

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Fig. 2. Geometric constraints at three corresponding points.

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Fig. 3. Implementation procedure of the proposed algorithm

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Fig. 4. Calibration target

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Fig. 5. Mean reprojection error per image

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Fig. 6. Used image

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Fig. 7. Layout of point pairs

Table 1. Camera calibration result

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Table 2. Point ID used

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Table 3. Experimental results

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Table 4. Experiments using normalized coordinate

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References

  1. Bouguet, J.Y. (2015), Camera calibration toolbox for Matlab, Caltech Vision, URL: http://www.vision.caltech.edu/bouguetj/calib_doc (last date accessed: 22 May 2019).
  2. Fischler, M.A. and Bolles, R.C. (1981), Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Communications of the ACM. Vol. 24, No. 6, pp. 381-395. https://doi.org/10.1145/358669.358692
  3. Gao, X.S., Hou, X.R., Tang, J., and Cheng, H.F. (2003), Complete solution classification for the perspective-threepoint problem. IEEE transactions on pattern analysis and machine intelligence, Vol. 25, No. 8, pp. 930-943. https://doi.org/10.1109/TPAMI.2003.1217599
  4. Grafarend, E.W. and Shan, J. (1997), Closed-form solution of P4P or the three-dimensional resection problem in terms of Mobius barycentric coordinates, Journal of Geodesy, Vol. 71, No. 4, pp. 217-231. https://doi.org/10.1007/s001900050089
  5. Guan, Y., Cheng, X., Zhan, X., and Zhou, S. (2008), Closed-form solution of space resection using unit quaternion. In Artigo apresentado no XXI ISPRS Congress, 3-11 July 2008, Beijing, China, Vol. XXXVII, Part B3b, pp. 3-11.
  6. Jiang, G.W., Jiang, T., Wang, Y., and Gong, H. (2007). Space resection independent of initial value based on unit quaternions, Acta Geodaetica et Cartographica Sinica, Vol. 36, No. 2, pp. 169-175. https://doi.org/10.3321/j.issn:1001-1595.2007.02.010
  7. Hong, S.P. and Kim, E.M. (2019), Comparison of point-based 3d transformation methods, Proceedings of Journal of Korean Society for Geospatial Information System, Korean Society for Geospatial Information Science, 31-1 May, Busan, Korea, pp. 19-22. (in Korean)
  8. Kabsch, W. (1976), A solution for the best rotation to relate two sets of vectors, Acta Crystallographica Section A: Crystal Physics, Diffraction, Theoretical and General Crystallography, Vol. 32, No. 5, pp. 922-923. https://doi.org/10.1107/S0567739476001873
  9. Kim, E.M. and Choi, H.S. (2018), Analysis of the accuracy of quaternion-based spatial resection based on the layout of control points, Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography, Vol. 36, No. 4, pp. 255-262. (in Korean with English abstract) https://doi.org/10.7848/KSGPC.2018.36.4.255
  10. Lepetit, V., Moreno-Noguer, F., and Fua, P. (2009). EPnP: an accurate o(n) solution to the PnP problem. International Journal of Computer Vision, Vol. 81, No. 2, pp. 155-166. https://doi.org/10.1007/s11263-008-0152-6
  11. Lim, J.H. (2018), Optimization Theory, Jang-Hwan Publishing, Goyang.
  12. Luhmann, T., Robson, S., Kyle, S., and Harley, I. (2011), Close Range Photogrammetry Principles, techniques and applications, Whittles Publishing, Caithness.
  13. Mazaheri, M. and Habib, A. (2015), Quaternion-based solutions for the single photo resection problem, Photogrammetric Engineering & Remote Sensing, Vol. 81, No. 3, pp. 209-217. https://doi.org/10.14358/PERS.81.3.209-217
  14. Mikhail, E.M., Bethel, J.S., and McGlone, J.C. (2001), Introduction to Modern Photogrammetry, John Wiley & Sons Inc., New York, N.Y.
  15. Nielsen, A.A. (2013). Least squares adjustment: linear and nonlinear weighted regression analysis. Applied Mathematics and Computer Science/National Space Institute, Technical University of Denmark, URL: http://www2.imm.dtu.dk/pubdb/views/edoc_download.php/2804/pdf/imm2804.pdf (last date accessed: 22 May 2019).
  16. Strang, G. (2016), Introduction to Linear Algebra, Wellesley-Cambridge Press, Massachusetts, M.A.
  17. Torr, P.H.S. and Zisserman, A. (2000), Mlesac: a new robust estimator with application to estimating image geometry, Computer Vision and Image Understanding, Vol. 78, No. 1, pp. 138-156. https://doi.org/10.1006/cviu.1999.0832