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Detection of Outer Rotations on 3D-Vector Fields with Iterative Geometric Correlation and its Efficiency

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Abstract

Correlation is a common technique for the detection of shifts. Its generalization to the multidimensional geometric correlation in Clifford algebras has been proven a useful tool for color image processing, because it additionally contains information about a rotational misalignment. But so far the exact correction of a three-dimensional outer rotation could only be achieved in certain special cases.

In this paper we prove that applying the geometric correlation iteratively has the potential to detect the outer rotational misalignment for arbitrary three-dimensional vector fields.

We further present the explicit iterative algorithm, analyze its efficiency detecting the rotational misalignment in the color space of a color image. The experiments suggest a method for the acceleration of the algorithm, which is practically tested with great success.

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References

  1. Lisa Gottesfeld Brown: A survey of image registration techniques. ACM Computing Surveys 24(4), 325–376 (1992)

    Article  Google Scholar 

  2. Roxana Bujack, H. De Bie, N. De Schepper and G. Scheuermann, Convolution products for hypercomplex Fourier transforms. Accepted for publication in J. Math. Imaging Vision, 2013.

  3. Roxana Bujack, Gerik Scheuermann, Eckhard Hitzer: A General Geometric Fourier Transform Convolution Theorem. Advances in Applied Clifford Algebras 23(1), 15–38 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Roxana Bujack, Gerik Scheuermann, Eckhard Hitzer: Detection of Total Rotations on linear 2D-Vector Fields with Iterative Geometric Correlation. AIP Conference Proceedings 1493, 190–199 (2012)

    Google Scholar 

  5. William Kingdon Clifford: Applications of Grassmann’s Extensive Algebra. American Journal of Mathematics 1(4), 350–358 (1878)

    Article  MathSciNet  Google Scholar 

  6. J. Ebling and G Scheuermann, Clifford convolution and pattern matching on vector fields. In: Visualization, VIS 2003. IEEE (2003), 193-200.

  7. Julia Ebling, Visualization and Analysis of Flow Fields using Clifford Convolution. PhD thesis, University of Leipzig, Germany, 2006.

  8. Andreas Görlitz, Helmut Seibert and Dietmar Hildenbrand, Registration of Multichannel Images using Geometric Algebra. In: V. Skala and E. Hitzer, Editors, Proceedings Workshop on Computer Graphics, Computer Vision and Mathematics, Brno University of Technology, Czech Republic, (2010).

  9. David Hestenes, New Foundations for Classical Mechanics. Kluwer Academic Publishers, Dordrecht, Netherlands, 1986.

  10. David Hestenes and Garret Sobczyk, Clifford Algebra to Geometric Calculus, D. Reidel Publishing Group, Dordrecht, Netherlands, 1984.

  11. C.D. Kuglin and D.C. Hines, The Phase Correlation Image Alignment Method, Proceedings of the IEEE, International Conference on Cybernetics and Society, New York, USA, 1975, 163-165.

  12. Jose Mennesson, Christophe Saint-Jean and Laurent Mascarilla, Color Object Recognition Based on a Clifford Fourier Transform. In: Leo Dorst and Joan Lasenby, Editors, Guide to Geometric Algebra in Practice, Springer London 2011, 175-191.

  13. C. Eddie Moxey, Todd A. Ell and Steven J. Sangwine, Vector correlation of color images. 1st European Conference on Color in Graphics, Imaging and Vision, Society for Imaging Science and Technology, 2002.

  14. C. Eddie Moxey, Steven J. Sangwine and Todd A. Ell, Hypercomplex Correlation Techniques for Vector Images, Signal Processing, IEEE Transactions, 51 (7) (2003), 1941-1953.

  15. Christian Perwass: Geometric Algebra with Applications in Engineering. Springer, Berlin Heidelberg (2009)

    MATH  Google Scholar 

  16. Azriel Rosenfeld and Avinash C. Kak, Digital Picture Processing, Academic Press, Inc., Orlando, FL, USA, 2nd Edition, 1982.

  17. Steven J. Sangwine, Todd A. Ell and C. Eddie Moxey, Vector phase correlation. 1st European Conference on Color in Graphics, Imaging and Vision, Society for Imaging Science and Technology, address=Poitiers, France, 37 (25), (2001), 1513-1515.

  18. Gerik Scheuermann, Topological Vector Field Visualization with Clifford Algebra, PhD thesis, University of Kaiserslautern, Germany, 1999.

  19. Michael Schlemmer, Pattern Recognition for Feature Based and Comparative Visualization, PhD thesis, Universität Kaiserslautern, Germany, 2011.

  20. Barbara Zitová, Jan Flusser: Image registration methods: a survey. Image and Vision Computing 21(11), 977–1000 (2003)

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. Universität Leipzig, Institut für Informatik, Augustusplatz 10, 04109, Leipzig, Germany

    Roxana Bujack & Gerik Scheuermann

  2. College of Liberal Arts, Department of Material Science, International Christian University, 181-8585, Tokyo, Japan

    Eckhard Hitzer

Authors
  1. Roxana Bujack
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  2. Gerik Scheuermann
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  3. Eckhard Hitzer
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Correspondence to Roxana Bujack.

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Bujack, R., Scheuermann, G. & Hitzer, E. Detection of Outer Rotations on 3D-Vector Fields with Iterative Geometric Correlation and its Efficiency. Adv. Appl. Clifford Algebras 24, 403–421 (2014). https://doi.org/10.1007/s00006-013-0411-7

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  • DOI: https://doi.org/10.1007/s00006-013-0411-7

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