Skip to main content
SpringerLink
Log in

Correspondence-free Structure from Motion

  • Published:
International Journal of Computer Vision Aims and scope Submit manuscript

Abstract

We present a novel approach for the estimation of 3D-motion directly from two images using the Radon transform. The feasibility of any camera motion is computed by integrating over all feature pairs that satisfy the epipolar constraint. This integration is equivalent to taking the inner product of a similarity function on feature pairs with a Dirac function embedding the epipolar constraint. The maxima in this five dimensional motion space will correspond to compatible rigid motions. The main novelty is in the realization that the Radon transform is a filtering operator: If we assume that the similarity and Dirac functions are defined on spheres and the epipolar constraint is a group action of rotations on spheres, then the Radon transform is a correlation integral. We propose a new algorithm to compute this integral from the spherical Fourier transform of the similarity and Dirac functions. Generating the similarity function now becomes a preprocessing step which reduces the complexity of the Radon computation by a factor equal to the number of feature pairs processed. The strength of the algorithm is in avoiding a commitment to correspondences, thus being robust to erroneous feature detection, outliers, and multiple motions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Antone, M. and Teller, S. 2002. Scalable, extrinsic calibration of omni-directional image networks. International Journal of Computer Vision, 49:143–174.

    Article  MATH  Google Scholar 

  • Arfken, G. and Weber, H. 1966. Mathematical Methods for Physicists. Academic Press.

  • Basri, R. and Jacobs, D.W. 2003. Lambertian reflectance and linear subspaces. IEEE Trans. Pattern Anal. Mach. Intell., 25(2):218–233.

    Article  Google Scholar 

  • Chirikjian, G. and Kyatkin, A. 2000. Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups. CRC Press.

  • Daniilidis, K. and Spetsakis, M. 1996. Understanding noise sensitivity in structure from motion. In Visual Navigation, Y. Aloimonos (Ed.). Lawrence Erlbaum Associates: Hillsdale, NJ, pp. 61–88.

    Google Scholar 

  • Deans, S. 1981. Hough transform from the radon transform. IEEE Trans. Pattern Analysis and Machine Intelligence, 3:185–188.

    Google Scholar 

  • Dellaert, F., Seitz, S., Thorpe, C., and Thrun, S. 2000. Structure from motion without correspondence. In CVPR. Hilton Head Island, SC.

  • Driscoll, J. and Healy, D. 1994. Computing fourier transforms and convolutions on the 2-sphere. Advances in Applied Mathematics, 15:202–250.

    Article  MATH  MathSciNet  Google Scholar 

  • Fermuller, C. and Aloimonos, J. 1995. Direct perception of three-dimensional motion from patterns of visual motion. Science, 270:1973–1976.

    Article  Google Scholar 

  • Gallier, J. 2005. Notes on group actions, manifolds, lie groups, and lie algebras. http://www.cis.upenn.edu/~cis610/lie1.pdf/.

  • Geyer, C. and Daniilidis, K. 2001. Catadioptric projective geometry. International Journal of Computer Vision, 43:223–243.

    Article  Google Scholar 

  • Geyer, C., Sastry, S., and Bajcsy, R. 2004. Euclid meets fourier. In Workshop on Omnidirectional Vision. Prague.

  • Helgason, S. 2000. Groups and Geometric Analysis: Integral Geometry, Invariant Differential Operators, and Spherical Functions. American Mathematical Society.

  • Horn, B. and Weldon, E. 1988. Direct methods for recovering motion. International Journal of Computer Vision, 2:51–76.

    Article  Google Scholar 

  • Jin, H., Favaro, P., and Soatto, S. 2003. A semi-direct approach to structure from motion. The Visual Computer, 19:1–18.

    Article  MATH  Google Scholar 

  • D.H. Jr., Rockmore, D., Kostelec, P., and Moore, S. 2003. FFTs for the 2-sphere—improvements and variations. The Journal of Fourier Analysis and Applications, 9(4):341–385.

  • Kostelec, P.J. and Rockmore, D.N. 2003. FFTs on the rotation group. In Working Paper Series. Santa Fe Institute.

  • Lowe, D. 2004. Sift (scale invariant feature transform): Distinctive image features from scale-invariant keypoints. International Journal of Computer Vision, 60:91–110.

    Article  Google Scholar 

  • Mahajan, D., Ramamoorthi, R., and Curless, B. 2006. A theory of spherical harmonic identities for BRDF/lighting transfer and image consistency. In European Conference on Computer Vision, vol. IV, pp. 41–55.

  • Makadia, A., Geyer, C., Sastry, S., and Daniilidis, K. 2005. Radon-based structure from motion without correspondences. In IEEE Conf. Computer Vision and Pattern Recognition, San Diego.

  • Makadia, A., Sorgi, L., and Daniilidis, K. 2004. Rotation estimation from spherical images. In Proc. Int. Conf. on Pattern Recognition. Cambridge, UK.

  • Mariottini, G. and Prattichizzo, D. 2005. EGT: A toolbox for multiple view geometry and visual servoing. IEEE Robotics and Automation Magazine, 3(12).

  • Maslen, D. and Rockmore, D. 1995. Generalized FFTs—a survey of some recent results. In Proceedings of the DIMACS Workshop on Groups and Computation.

  • Nayar, S. 1997. Catadioptric omnidirectional camera. In IEEE Conf. Computer Vision and Pattern Recognition. Puerto Rico, pp. 482–488.

  • Negahdaripour, S. and Horn, B. 1987. Direct passive navigation. IEEE Trans. Pattern Analysis and Machine Intelligence, 9:168–176.

    Article  MATH  Google Scholar 

  • Oliensis, J. 2000. A critique of structure from motion algorithms. Computer Vision and Image Understanding, 80:172–214.

    Article  MATH  Google Scholar 

  • Roy, S. and Cox, I. 1996. Motion without structure. In Proc. Int. Conf. on Pattern Recognition. Vienna, Austria.

  • Schröder, P. and Sweldens, W. 1995. Spherical wavelets: Efficiently representing functions on the sphere. In SIGGRAPH ’95: Proceedings of the 22nd annual conference on Computer graphics and interactive techniques. ACM Press: New York. pp. 161–172.

  • Shi, J. and Malik, J. 1998. Motion segmentation and tracking using normalized cuts. In Proc. Int. Conf. on Computer Vision.

  • Sugiura, M. 1990. Unitary Representations and Harmonic Analysis: An Introduction. second edition. North Holland, Amsterdam.

    MATH  Google Scholar 

  • Szeliski, R. and Kang, S.B. 1995. Direct methods for visual scene reconstruction. In IEEE Workshop on Representations of Visual Scenes, pp. 26–33.

  • Vidal, R. and Ma, Y. 2004. A unified algebraic approach to 2-D and 3-D motion segmentation. In European Conference on Computer Vision, pp. 1–15.

  • Wexler, Y., Fitzgibbon, A., and Zisserman, A. 2003. Learning epipolar geometry from image sequences. In IEEE Conf. Computer Vision and Pattern Recognition. Wisconsin.

Download references

Author information

Authors and Affiliations

  1. University of Pennsylvania, Philadelphia, PA, 19104, USA

    Ameesh Makadia & Kostas Daniilidis

  2. Carnegie Mellon University, Pittsburgh, PA, 15213, USA

    Christopher Geyer

Authors
  1. Ameesh Makadia
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christopher Geyer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Kostas Daniilidis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ameesh Makadia.

Additional information

The authors are grateful for support through the following grants: NSF-IIS-0083209, NSF-IIS-0121293, NSF-EIA-0324977, NSF-CNS-0423891, NSF-IIS-0431070, and ARO/MURI DAAD19-02-1-0383.

The author is grateful for the generous support of the ARO MURI program (DAAD-19-02-1-0383) while at U. C. Berkeley.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Makadia, A., Geyer, C. & Daniilidis, K. Correspondence-free Structure from Motion. Int J Comput Vis 75, 311–327 (2007). https://doi.org/10.1007/s11263-007-0035-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11263-007-0035-2

Keywords

Search

Navigation