A Riemannian Fourier Transform via Spin Representations

  • T. Batard
  • M. Berthier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8085)

Abstract

We introduce a new Riemannian Fourier transform for color image processing. The construction involves spin characters and spin representations of complex Clifford algebras. Examples of applications to low-pass filtering are presented.

Keywords

Spin Representation Spinor Bundle Color Image Processing Monogenic Signal Spin Character 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Batard, T., Berthier, M.: Clifford Fourier transform and spinor representation of images. In: Hitzer, E., Sangwine, S.J. (eds.) Quaternion and Clifford Fourier Transforms and Wavelets. Trends in Mathematics (TIM). Birkhauser, Basel (to appear)Google Scholar
  2. 2.
    Batard, T., Berthier, M.: Spinor Fourier Transform for Image Processing (preprint submitted)Google Scholar
  3. 3.
    Batard, T., Saint Jean, C., Berthier, M.: A Metric Approach to nD Images Edge Detection with Clifford Algebras. J. Math. Imaging Vis. 33, 296–312 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Batard, T., Berthier, M., Saint Jean, C.: Clifford Fourier Transform for Color Image Processing. In: Bayro Corrochano, E., Scheueurmann, G. (eds.) Geometric Algebra Computing for Engineering and Computer Science, pp. 135–162. Springer (2010)Google Scholar
  5. 5.
    Brackx, F., De Schepper, N., Sommen, F.: The Two Dimensional Clifford Fourier Transform. J. Math. Imaging Vis. 26, 5–18 (2006)MATHCrossRefGoogle Scholar
  6. 6.
    Brackx, F., De Schepper, N., Sommen, F.: The Fourier transform in Clifford analysis. Adv. Imag. Elect. Phys. 156, 55–203 (2008)CrossRefGoogle Scholar
  7. 7.
    Bujack, R., Scheueurmann, G., Hitzer, E.: A general geometric Fourier transform. In: Gürlebeck, K. (ed.) 9th Int. Conf. on Clifford Algebras and their Applications in Mathematical Physics, Weimar, Germany, July 15-20, 19 pages (2011)Google Scholar
  8. 8.
    Bujack, R., Scheueurmann, G., Hitzer, E.: A general geometric Fourier transform convolution theorem. To appear in Adv. Appl. Clifford Alg.Google Scholar
  9. 9.
    Chevalley, C.: The Algebraic Theory of Spinors and Clifford Algebras, New Edn. Springer (1995)Google Scholar
  10. 10.
    Dahlke, S., Kutyniok, G., Steidl, G., Teschke, G.: Shearlet Coorbit Spaces and Associated Banach Frames. Appl. Comput. Harmon. Anal. 27, 195–214 (2009)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    De Bie, H.: Clifford algebras, Fourier transforms and quantum mechanics. To appear in Math. Methods Appl. Sci.Google Scholar
  12. 12.
    De Bie, H., Xu, Y.: On the Clifford Fourier transform. Int. Math. Res. Not. IMRN (22), 5123–5163 (2011)Google Scholar
  13. 13.
    Demarcq, G., Mascarilla, L., Berthier, M., Courtellemont, P.: The Color Monogenic Signal. Application to Color Edge Detection and Color Optical Flow. J. Math. Imaging Vis. 40, 269–284 (2011)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Ebling, J., Scheueurmann, G.: Clifford Fourier Transform on Vector Fields. IEEE Transactions on Visualization and Computer Graphics 49(11), 2844–2852 (2001)Google Scholar
  15. 15.
    Felsberg, M., Sommer, G.: The Monogenic Signal. IEEE Transactions on Signal Processing 49(12), 3136–3144 (2001)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Frankel, T.: The Geometry of Physics - An Introduction, Revised edn. Cambridge University PressGoogle Scholar
  17. 17.
    Helgason, S.: Differential Geometry, Lie Groups and Symmetric Spaces. Academic Press, London (1978)MATHGoogle Scholar
  18. 18.
    Hestenes, D., Sobczyk, G.: Clifford Algebra to Geometric Calculus. Reidel, Dordrecht (1984)MATHCrossRefGoogle Scholar
  19. 19.
    Lawson, H.B., Michelson, M.-L.: Spin Geometry. Princeton University Press, Princeton (1989)MATHGoogle Scholar
  20. 20.
    Lounesto, P.: Clifford Algebras and Spinors, 2nd edn. London Mathematical Society Lecture Note Series, vol. 286. Cambridge University Press (2001)Google Scholar
  21. 21.
    Nakahara, M.: Geometry, Topology and Physics, 2nd edn. Graduate Student Series in Physics. Taylor and Francis (2003)Google Scholar
  22. 22.
    Postnikov, M.: Leçons de géométrie: groupes et algèbres de Lie. MIR (1985)Google Scholar
  23. 23.
    Smach, F., Lemaire, C., Gauthier, J.-P., Miteran, J., Atri, M.: Generalized Fourier Descriptors with Applications to Objects Recognition in SVM Context. J. Math. Imaging Vis. 30(1), 43–71 (2008)CrossRefGoogle Scholar
  24. 24.
    Vilenkin, N.J.: Special Functions and the Theory of Group Representations. Translations of Mathematical Monographs, vol. 22. American Mathematical Society, Providence (1968)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • T. Batard
    • 1
  • M. Berthier
    • 2
  1. 1.XLIM-SIC Lab.Poitiers UniversityFrance
  2. 2.MIA Lab.La Rochelle UniversityFrance

Personalised recommendations