# Infinitesimal Plane-Based Pose Estimation

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## Abstract

Estimating the pose of a plane given a set of point correspondences is a core problem in computer vision with many applications including Augmented Reality (AR), camera calibration and 3D scene reconstruction and interpretation. Despite much progress over recent years there is still the need for a more efficient and more accurate solution, particularly in mobile applications where the run-time budget is critical. We present a new analytic solution to the problem which is far faster than current methods based on solving Pose from \(n\) Points (PnP) and is in most cases more accurate. Our approach involves a new way to exploit redundancy in the homography coefficients. This uses the fact that when the homography is noisy it will estimate the true transform between the model plane and the image better at some regions on the plane than at others. Our method is based on locating a point where the transform is best estimated, and using only the local transformation at that point to constrain pose. This involves solving pose with a local non-redundant 1st-order PDE. We call this framework Infinitesimal Plane-based Pose Estimation (IPPE), because one can think of it as solving pose using the transform about an infinitesimally small region on the surface. We show experimentally that IPPE leads to very accurate pose estimates. Because IPPE is analytic it is both extremely fast and allows us to fully characterise the method in terms of degeneracies, number of returned solutions, and the geometric relationship of these solutions. This characterisation is not possible with state-of-the-art PnP methods.

## Keywords

Plane Pose SfM PnP Homography## Notes

### Acknowledgments

This research has received funding from the EU FP7 ERC research Grant 307483 FLEXABLE. Code is available at http://www.tobycollins.net/research/IPPE.

## References

- Ansar, A., & Daniilidis, K. (2003). Linear pose estimation from points or lines.
*Pattern Analysis and Machine Intelligence (PAMI)*,*25*, 282–296.Google Scholar - Barreto, J., Roquette, J., Sturm, P., & Fonseca, F. (2009). Automatic camera calibration applied to medical endoscopy. In:
*British machine vision conference (BMVC)*.Google Scholar - Bouguet, J. Y. A camera calibration toolbox for matlab. http://www.vision.caltech.edu/bouguetj/calib_doc/. Accessed May 2013.
- Brown, M., Majumder, A., & Yang, R. (2005). Camera-based calibration techniques for seamless multiprojector displays.
*Visualization and Computer Graphics*,*11*, 193–206.CrossRefGoogle Scholar - Chen, P., & Suter, D. (2009). Error analysis in homography estimation by first order approximation tools: A general technique.
*Journal of Mathematical Imaging and Vision*,*33*, 281–295.CrossRefMathSciNetGoogle Scholar - Collins, T., Durou, J. D., Gurdjos, P., & Bartoli, A. (2010). Single-view perspective shape-from-texture with focal length estimation: A piecewise affine approach. In
*3D data processing visualization and transmission (3DPVT10)*.Google Scholar - Dhome, M., Richetin, M., & Lapreste, J. T. (1989). Determination of the attitude of 3D objects from a single perspective view.
*Pattern Analysis and Machine Intelligence (PAMI)*,*11*, 1265–1278.CrossRefGoogle Scholar - Faugeras, O., Luong, Q. T., & Papadopoulou, T. (2001).
*The geometry of multiple images: The laws that govern the formation of images of a scene and some of their applications*. Cambridge, MA: MIT Press.Google Scholar - Fiore, P. D. (2001). Efficient linear solution of exterior orientation.
*Pattern Analysis and Machine Intelligence (PAMI)*,*23*, 140–148.CrossRefGoogle Scholar - Fischler, M. A., & Bolles, R. C. (1981). Random sample consensus: A paradigm for model fitting with applications to image analysis and automated cartography.
*Communications of the ACM*,*24*, 381–395.CrossRefMathSciNetGoogle Scholar - Gao, X. S., Hou, X., Tang, J., & Cheng, H. F. (2003). Complete solution classification for the perspective-three-point problem.
*Pattern Analysis and Machine Intelligence (PAMI)*,*25*, 930–943.CrossRefGoogle Scholar - Gao, X. S., Hou, X. R., Tang, J., & Cheng, H. F. (2003). Complete solution classification for the perspective-three-point problem.
*Pattern Analysis and Machine Intelligence (PAMI)*,*25*(8), 930–943.CrossRefGoogle Scholar - Geiger, A., Moosmann, F., Car, M., & Schuster, B. (2012). A toolbox for automatic calibration of range and camera sensors using a single shot. In
*International conference on robotics and automation (ICRA)*.Google Scholar - Haralick, R. M., Lee, C. N., Ottenberg, K., & Nölle, M. (1994). Review and analysis of solutions of the three point perspective pose estimation problem.
*International Journal of Computer Vision (IJCV)*,*13*, 331–356.CrossRefGoogle Scholar - Haralick, R. M., Lee, D., Ottenburg, K., & Nölle, M. (1991). Analysis and solutions of the three point perspective pose estimation problem. In
*Computer vision and pattern recognition (CVPR)*.Google Scholar - Harker, M., & O’Leary, P. (2005). Computation of homographies. In
*British computer vision conference (BMVC)*.Google Scholar - Hartley, R., & Zisserman, A. (2004).
*Multiple view geometry in computer vision*. Cambridge: Cambridge University Press.CrossRefzbMATHGoogle Scholar - Hesch, J. A., & Roumeliotis, S. I. (2011). A direct least-squares (DLS) method for PnP. In
*International conference on computer vision (ICCV)*.Google Scholar - Hilsmann, A., Schneider, D., & Eisert, P. (2011). Template-free shape from texture with perspective cameras. In
*British machine vision conference (BMVC)*.Google Scholar - Horaud, R., Dornaika, F., Lamiroy, B., & Christy, S. (1997). Object pose: The link between weak perspective, paraperspective and full perspective.
*International Journal of Computer Vision (IJCV)*,*22*, 173–189.CrossRefGoogle Scholar - Hung, Y., Harwood, D., & Yeh, P.-S., (1984).
*Passive ranging to known planar point sets*. Technical Report. College Park, MD: University of Maryland.Google Scholar - Kato, H., & Billinghurst, M. (1999). Marker tracking and HMD calibration for a video-based augmented reality conferencing system. In
*International workshop on augmented reality (IWAR)*.Google Scholar - Lepetit, V., Moreno-Noguer, F., & Fua, P. (2009). EPnP: An accurate O(n) solution to the pnp problem.
*International Journal of Computer Vision (IJCV)*,*81*, 155–166.CrossRefGoogle Scholar - Li, S., Xu, C., & Xie, M. (2012). A robust O(n) solution to the perspective-n-point problem.
*Pattern Analysis and Machine Intelligence (PAMI)*.Google Scholar - Lobay, A., & Forsyth, D. A. (2004). Recovering shape and irradiance maps from rich dense texton fields. In
*Computer vision and pattern recognition (CVPR)*.Google Scholar - Lobay, A., & Forsyth, D. A. (2006). Shape from texture without boundaries.
*International Journal of Computer Vision (IJCV)*,*67*, 71–91.CrossRefGoogle Scholar - Lowe, D. G. (2004). Distinctive image features from scale-invariant keypoints.
*International Journal of Computer Vision (IJCV)*,*60*, 91–110.CrossRefGoogle Scholar - Lu, C. P., Hager, G. D., & Mjolsness, E. (2000). Fast and globally convergent pose estimation from video images.
*Pattern Analysis and Machine Intelligence (PAMI)*,*22*, 610–622.CrossRefGoogle Scholar - Munoz-Salinas, R. ArUco: Augmented reality library from the university of cordoba. http://www.uco.es/investiga/grupos/ava/node/26. Accessed May 2013.
- Oberkampf, D., DeMenthon, D., & Davis, L. S. (1996). Iterative pose estimation using coplanar feature points.
*Computer Vision and Image Understanding (CVIU)*,*63*, 495–511.CrossRefGoogle Scholar - Ohta, Y., Maenobu, K., & Sakai, T. (1981). Obtaining surface orientation from texels under perspective projection. In
*International joint conferences on artificial intelligence (IJCAI)*.Google Scholar - Poelman, C., & Kanade, T. (1993). A paraperspective factorization method for shape and motion recovery. Technical Report.Google Scholar
- Quan, L., & Lan, Z. (1999). Linear n-point camera pose determination. In
*Pattern analysis and machine intelligence (PAMI)*. Google Scholar - Schweighofer, G., & Pinz, A. (2006). Robust pose estimation from a planar target.
*Pattern Analysis and Machine Intelligence (PAMI)*,*28*, 2024–2030.CrossRefGoogle Scholar - Sturm, P. (2000). Algorithms for plane-based pose estimation. In
*Computer vision and pattern recognition (CVPR)*.Google Scholar - Taubin, G. (1991). Estimation of planar curves, surfaces, and nonplanar space curves defined by implicit equations with applications to edge and range image segmentation.
*Pattern Analysis and Machine Intelligence (PAMI)*,*13*, 1115–1138.CrossRefGoogle Scholar - Triggs, B. (1999). Camera pose and calibration from 4 or 5 known 3D points. In
*International conference on computer vision (ICCV)*.Google Scholar - Vedaldi, A., & Fulkerson, B. VLFeat: An open and portable library of computer vision algorithms. http://www.vlfeat.org/. Accessed May 2013.
- Zhang, C. X., & Hu, Z. Y. (2005). A general sufficient condition of four positive solutions of the p3p problem.
*Journal of Computer Science and Technology*,*20*, 836–842.CrossRefMathSciNetGoogle Scholar - Zhang, Z. (2000). A flexible new technique for camera calibration.
*Pattern Analysis and Machine Intelligence (PAMI)*,*22*, 1330–1334.CrossRefGoogle Scholar