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Abstract

Mathematical formulation of certain natural phenomena exhibits group structure on topological spaces that resemble the Euclidean space only on a small enough scale, which prevents incorporation of conventional inference methods that require global vector norms. More specifically in computer vision, such underlying notions emerge in differentiable parameter spaces. Here, two Riemannian manifolds including the set of affine transformations and covariance matrices are elaborated and their favorable applications in distance computation, motion estimation, object detection and recognition problems are demonstrated after reviewing some of the fundamental preliminaries.

Keywords

Region Covariance Riemannian Geometry Detection Tracking Regression Classification 

References

  1. 1.
    Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd edn. Academic Press, London (1986)zbMATHGoogle Scholar
  2. 2.
    Rossmann, W.: Lie Groups: An Introduction Through Linear Groups. Oxford Press (2002)Google Scholar
  3. 3.
    Forstner, W., Moonen, B.: A metric for covariance matrices. Technical report, Dept. of Geodesy and Geoinformatics, Stuttgart University (1999)Google Scholar
  4. 4.
    Pennec, X., Fillard, P., Ayache, N.: A Riemannian framework for tensor computing. International Journal of Computer Vision 66(1), 41–66 (2006)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Tuzel, O., Porikli, F., Meer, P.: Region covariance: A fast descriptor for detection and classification. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3952, pp. 589–600. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Porikli, F.: Integral histogram: A fast way to extract histograms in Cartesian spaces. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition, San Diego, CA, vol. 1, pp. 829–836 (2005)Google Scholar
  7. 7.
    Georgescu, B., Shimshoni, I., Meer, P.: Mean shift based clustering in high dimensions: A texture classification example. In: Proc. 9th International Conference on Computer Vision, Nice, France, pp. 456–463 (2003)Google Scholar
  8. 8.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Commun. Pure Appl. Math. 30, 509–541 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Tuzel, O., Subbarao, R., Meer, P.: Simultaneous multiple 3D motion estimation via mode finding on Lie groups. In: Proc. 10th International Conference on Computer Vision, Beijing, China, vol. 1, pp. 18–25 (2005)Google Scholar
  10. 10.
    Hastie, T., Tibshirani, R., Freidman, J.: The Elements of Statistical Learning. Springer, Heidelberg (2001)zbMATHGoogle Scholar
  11. 11.
    Miller, E., Chefd’hotel, C.: Practical non-parametric density estimation on a transformation group for vision. In: Proc. IEEE Conference on Computer Vision and Pattern Recognition (2003)Google Scholar
  12. 12.
    Porikli, F., Pan, P.: Regressed importance sampling on manifolds for efficient object tracking. In: Proc. 6th IEEE Advanced Video and Signal based Surveillance Conference (2009)Google Scholar
  13. 13.
    Viola, P., Jones, M.: Robust real-time object detection. International Journal of Computer Vision (2001)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Fatih Porikli
    • 1
  1. 1.Mitsubishi Electric Research LabsCambridgeUSA

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