We propose a non-iterative solution to the PnP problem—the estimation of the pose of a calibrated camera from n 3D-to-2D point correspondences—whose computational complexity grows linearly with n. This is in contrast to state-of-the-art methods that are O(n 5) or even O(n 8), without being more accurate. Our method is applicable for all n≥4 and handles properly both planar and non-planar configurations. Our central idea is to express the n 3D points as a weighted sum of four virtual control points. The problem then reduces to estimating the coordinates of these control points in the camera referential, which can be done in O(n) time by expressing these coordinates as weighted sum of the eigenvectors of a 12×12 matrix and solving a small constant number of quadratic equations to pick the right weights. Furthermore, if maximal precision is required, the output of the closed-form solution can be used to initialize a Gauss-Newton scheme, which improves accuracy with negligible amount of additional time. The advantages of our method are demonstrated by thorough testing on both synthetic and real-data.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Abdel-Aziz, Y. I., & Karara, H. M. (1971). Direct linear transformation from comparator coordinates into object space coordinates in close-range photogrammetry. In Proc. ASP/UI symp. close-range photogrammetry (pp. 1–18).
Ansar, A., & Daniilidis, K. (2003). Linear pose estimation from points or lines. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(5), 578–589.
Arun, K. S., Huang, T. S., & Blostein, S. D. (1987). Least-squares fitting of two 3-D points sets. IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(5), 698–700.
Chang, Y., & Rockwood, A. P. (1994). A generalized de Casteljau approach to 3D free-form deformation. In ACM SIGGRAPH (pp. 257–260).
DeMenthon, D., & Davis, L. S. (1995). Model-based object pose in 25 lines of code. International Journal of Computer Vision, 15, 123–141.
Dhome, M., Richetin, M., & Lapreste, J.-T. (1989). Determination of the attitude of 3d objects from a single perspective view. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(12), 1265–1278.
Fiore, P. D. (2001). Efficient linear solution of exterior orientation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23(2), 140–148.
Fischler, M. A., & Bolles, R. C. (1981). Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography. Communications ACM, 24(6), 381–395.
Gao, X. S., Hou, X. R., Tang, J., & Cheng, H. F. (2003). Complete solution classification for the perspective-three-point problem. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(8), 930–943.
Golub, G. H., & Van Loan, C. F. (1996). Matrix computations. Johns Hopkins studies in mathematical sciences. Baltimore: Johns Hopkins Press.
Haralick, R. M., Lee, D., Ottenburg, K., & Nolle, M. (1991). Analysis and solutions of the three point perspective pose estimation problem. In Conference on computer vision and pattern recognition (pp. 592–598).
Hartley, R., & Zisserman, A. (2000). Multiple view geometry in computer vision. Cambridge: Cambridge University Press.
Horaud, R., Conio, B., Leboulleux, O., & Lacolle, B. (1989). An analytic solution for the perspective 4-point problem. Computer Vision, Graphics, and Image Processing, 47(1), 33–44.
Horaud, R., Dornaika, F., & Lamiroy, B. (1997). Object pose: The link between weak perspective, paraperspective, and full perspective. International Journal of Computer Vision, 22(2), 173–189.
Horn, B. K. P., Hilden, H. M., & Negahdaripour, S. (1988). Closed-form solution of absolute orientation using orthonormal matrices. Journal of the Optical Society of America, 5(7), 1127–1135.
Kipnis, A., & Shamir, A. (1999). Cryptanalysis of the HFE public key cryptosystem by relinearization. In Advances in cryptology—CRYPTO’99 (Vol. 1666/1999, pp. 19–30). Berlin: Springer.
Kumar, R., & Hanson, A. R. (1994). Robust methods for estimating pose and a sensitivity analysis. Computer Vision and Image Understanding, 60(3), 313–342.
Lepetit, V., & Fua, P. (2006). Keypoint recognition using randomized trees. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(9), 1465–1479.
Lowe, D. G. (1991). Fitting parameterized three-dimensional models to images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(5), 441–450.
Lu, C.-P., Hager, G. D., & Mjolsness, E. (2000). Fast and globally convergent pose estimation from video images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(6), 610–622.
McGlove, C., Mikhail, E., & Bethel, J. (Eds.) (2004). Manual of photogrametry. American society for photogrammetry and remote sensing (5th edn.).
Moreno-Noguer, F., Lepetit, V., & Fua, P. (2007). Accurate non-iterative o(n) solution to the pnp problem. In IEEE international conference on computer vision. Rio de Janeiro, Brazil.
Oberkampf, D., DeMenthon, D., & Davis, L. S. (1996). Iterative pose estimation using coplanar feature points. Computer Vision and Image Understanding, 63, 495–511.
Quan, L., & Lan, Z. (1999). Linear N-point camera pose determination. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(7), 774–780.
Schweighofer, G., & Pinz, A. (2006). Robust pose estimation from a planar target. IEEE Transactions on Pattern Analysis and Machine Intelligence, 28(12), 2024–2030.
Sederberg, T. W., & Parry, S. R. (1986). Free-form deformation of solid geometric models. ACM SIGGRAPH, 20(4).
Skrypnyk, I., & Lowe, D. G. (2004). Scene modelling, recognition and tracking with invariant image features. In International symposium on mixed and augmented reality (pp. 110–119). Arlington, VA.
Stewènius, H., Engels, C., & Nister, D. (2006). Recent developments on direct relative orientation. International Society for Photogrammetry and Remote Sensing, 60, 284–294.
Triggs, B. (1999). Camera pose and calibration from 4 or 5 known 3D points. In International conference on computer vision (pp. 278–284).
Umeyama, S. (1991). Least-squares estimation of transformation parameters between two point patterns. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(4).
About this article
Cite this article
Lepetit, V., Moreno-Noguer, F. & Fua, P. EPnP: An Accurate O(n) Solution to the PnP Problem. Int J Comput Vis 81, 155 (2009). https://doi.org/10.1007/s11263-008-0152-6
- Pose estimation
- Absolute orientation