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The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties

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Abstract

The uncertainty principle is a fundamental concept in the context of signal and image processing, just as much as it has been in the framework of physics and more recently in harmonic analysis. Uncertainty principles can be derived by using a group theoretic approach. This approach yields also a formalism for finding functions which are the minimizers of the uncertainty principles. A general theorem which associates an uncertainty principle with a pair of self-adjoint operators is used in finding the minimizers of the uncertainty related to various groups.

This study is concerned with the uncertainty principle in the context of the Weyl-Heisenberg, the SIM(2), the Affine and the Affine-Weyl-Heisenberg groups. We explore the relationship between the two-dimensional affine group and the SIM (2) group in terms of the uncertainty minimizers. The uncertainty principle is also extended to the Affine-Weyl-Heisenberg group in one dimension. Possible minimizers related to these groups are also presented and the scale-space properties of some of the minimizers are explored.

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References

  1. S.T. Ali, J.P. Antoine, and J.P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, 2000.

  2. A.C. Bovik, M. Clark, and W.S. Geisler, “Multichannel texture analysis using localized spatial filters,” IEEE Transactions on PAMI, Vol. 12, No. 1, pp. 55–73, 1990.

    Google Scholar 

  3. J.G. Daugman, “Uncertainty relation for resolution in space, spatial frequency, and orientation optimized by two-dimensinal visual cortical filters,” J. Opt. Soc. Amer., Vol. 2, No. 7, pp. 1160–1169, 1985.

    Google Scholar 

  4. S. Dahlke and P. Maass, “The affine uncertainty principle in one and two dimensions,” Comput. Math. Appl., Vol. 30, Nos. 3–6, pp. 293–305, 1995.

  5. R. Duits, M. Felsberg, L. Florack, and B. Platel, “α-scale-spaces on a bounded domain,” in: Proceedings of Scale space Methods in Computer Vision, LNCS 2695, Springer, 2003, pp. 424–510.

  6. R. Duits, L. Florack, J. de Graaf, and B. Ter Haar Romeny, “On the axioms of scale space theory,” Journal of Mathematical Imaging and Vision, Vol. 20, pp. 267–298 2004.

    Article  MathSciNet  Google Scholar 

  7. M. Felsberg and G. Sommer, “Scale-adaptive filtering derived from the Laplace equation,” in: Pattern Recognition, Volume 2032 of LNCS, B. Radig and S. Florczyk (Eds.), Springer: Berlin, 2001 pp. 95–106.

  8. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever, “Linear scale-space,” Journal of Mathematical Imaging and Vision, 1994, Vol. 4, pp. 325–351

  9. G. Folland, “Harmonic analysis in phase space,” Princeton University Press, Princeton, NJ, USA, 1989.

  10. D. Gabor, “Theory of communication,” J. IEEE, Vol. 93, pp. 429–459 1946.

    Google Scholar 

  11. G. Gilboa, N. Sochen, and Y. Y. Zeevi, “Image Enhancement and Denoising by Complex Diffusion Processes,” IEEE Transactions on Pattern Analysis and Machine Intelligence (PAMI), Vol. 25, No. 8, pp. 1020–1036 2004.

  12. A. Jain and F. Farrokhnia, “Unsupervised texture segmentation using Gabor filters,” Pattern Recognition, Vol. 24, pp. 1167–1186 1991.

    Article  Google Scholar 

  13. C. Kalisa and B. Torresani, “N-dimensional affine Weyl-Heisenberg wavelets,” Annales de L’Instut Henri Poincare, Physique Theorique, Vol. 59, No. 2, pp. 201–236 1993.

  14. J.R. Klauder, “Path Integrals for affine variables,” in: Functional Integration, Theory and Applications, J.P. Antoine and E. Tirapegui (Eds.), Plenum Press, New York and London, 1980, pp. 101–119.

  15. J. R. Klauder and B.S. Skagerstam, “Coherent States Applications in Physics and Mathematical Physics,” World Scientific, Singapore, 1985.

  16. T.S. Lee, “Image representation using 2D gabor-wavelets,” IEEE Transactions on PAMI, Vol. 18, No. 10, pp. 959–971, 1996.

    Google Scholar 

  17. S. Marcelja, “Mathematical description of the response of simple cortical cells,” J. Opt. Soc. Amer., Vol. 70, pp. 1297–1300, 1980.

    Article  MathSciNet  Google Scholar 

  18. Th. Paul and K. Seip, “Wavelets in quantum mechanics,” in: Wavelets and their Applications, M.B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer and L. Raphael (Eds.), Jones and Bartlett: Boston, 1992 pp. 303–322.

  19. M. Porat and Y.Y. Zeevi, “The generalized Gabor scheme of image representation in biological and machine vision,” IEEE Transactions on PAMI, Vol. 10, No. 4, pp. 452–468, 1988.

    MATH  Google Scholar 

  20. M. Porat and Y.Y. Zeevi, “Localized texture processing in vision: Analysis and synthesis in the gaborian space,” IEEE Transactions on Biomedical Engineering, Vol. 36, No. 1, pp. 115–129, 1989.

    Article  Google Scholar 

  21. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, and M.A. Viergever, “Linear scale-space,” Journal of Mathematical Imaging and Vision, Vol. 4, No. 4, pp. 325–351, 1994.

    Article  MathSciNet  Google Scholar 

  22. G. Teschke, “Construction of generalized uncertainty principles and wavelets in bessel potential spaces,” International Journal of Wavelets, Multiresolution and Information Processing, Vol. 3, No. 2, 2005.

  23. B. Torresani, “Wavelets associated with representationsn of the affine Weyl-Heisenberg group,” Journal of Mathematical Physics, Vol. 32, No. 5, pp. 1273–1279, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  24. C. Sagiv, N. Sochen, and Y.Y. Zeevi, “Gabor space geodesic active contours,” Algebraic Frames for the Perception-Action Cycle, Lecture Notes in Computer Science, G. Sommer and Y.Y. Zeevi (Eds.), Springer: Berlin, Vol. 1888, 2000.

  25. C. Sagiv, N.A. Sochen, and Y.Y. Zeevi, “Geodesic active contours applied to texture feature space,” Proceedings of Scale-Space and Morphology in Computer Vision, Lecture Notes in Computer Science, M. Kerckhove (Ed.), Springer: Berlin, 2001, Vol. 2106 pp. 344–352.

  26. C. Sagiv, N.A. Sochen, and Y.Y. Zeevi, “Gabor feature space diffusion via the minimal weighted area method,” in Proceedings of Energy Minimization Methods in Computer Vision and Pattern Recognition, Lecture Notes in Computer Science, M. Figueiredo, J. Zerubia and A.K. Jain (Eds.), Springer, Berlin, 2001, Vol. 2134, pp. 621–635.

  27. C. Sagiv, N.A. Sochen, and Y.Y. Zeevi, “Texture segmentation via a diffusion-segmentation scheme in the Gabor feature space,” in Proceedings of Texture 2002, The 2nd International Workshop on Texture Analysis and Synthesis, M. Chantler (Ed.), 2002, pp. 123–128, Copenhagen.

  28. C. Sagiv, N.A. Sochen, and Y.Y. Zeevi, “Integrated active contours for texture segmentation,” IEEE Transactions on Image Processing, Vol. 15, No. 6, 2006.

  29. C. Sagiv, N.A. Sochen, and Y.Y. Zeevi, “Scale-space generation via uncertainty principles,” in Proceedings of Scale Space and PDE Methods in Computer Vision, R. Kimmel, N. Sochen, J. Weickert (Eds.), Hofgeismar, Germany, 2005, pp. 351–362.

  30. J. Segman and W. Schempp, “Two ways to incorporate scale in the Heisenberg group with an intertwining operator,” Journal of Mathematical Imaging and Vision, Vol. 3, No. 1, pp. 79–94, 1993.

    Article  MATH  MathSciNet  Google Scholar 

  31. J. Segman and Y.Y. Zeevi, “Image analysis by wavelet-type transform: Group theoretic approach,” J. Mathematical Imaging and Vision, Vol. 3, pp. 51–75 1993.

    Article  MATH  MathSciNet  Google Scholar 

  32. M. Zibulski and Y.Y. Zeevi, “Analysis of multiwindow Gabor-type schemes by frame methods,” Applied and Computational Harmonic Analysis, Vol. 4, pp. 188–221 1997.

    Article  MATH  MathSciNet  Google Scholar 

  33. P. Antoine, R. Murenzi and P. Vandergheynst, “Directional wavelets revisited: Cauchy wavelets and symmetry detection in patterns,” Appl. Comput. Harmon. Anal., Vol. 5, pp. 314-345, 1999.

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Authors and Affiliations

  1. Department of Applied Mathematics, University of Tel Aviv, Ramat-Aviv, Tel-Aviv, 69978, Israel

    Chen Sagiv & Nir A. Sochen

  2. Department of Electrical engineering, Technion—Israel Institute of Technology, Technion City, Haifa, 32000, Israel

    Yehoshua Y. Zeevi

Authors
  1. Chen Sagiv
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  2. Nir A. Sochen
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  3. Yehoshua Y. Zeevi
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Correspondence to Chen Sagiv.

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Chen Sagiv, finished her B.Sc studies in Physics and Mathematics in 1990 and her M.Sc. studies in Physics in 1995 both in the Tel-Aviv University. After spending a few years in the high-tech industry, she went back to pursue her PhD in Applied Mathematics at the Tel Aviv university. Her main research interests are Gabor analysis and the applications of differential geometry in image processing, especially for texture segmentation.

Nir A. Sochen completed his B.Sc. studies in Physics, 1986, and his M.Sc. in theoretical physics, 1988, at the University of Tel-Aviv. He received his Ph.D. in Theoretical physics, 1992, from the Université de Paris-Sud while conducting his research in the Service de Physique Théorique at the Centre d’Etude Nucleaire at Saclay, France. He was the recipient of the Haute Etude Scientifique Fellowship, and pursued his research for one year at the Ecole Normal Superieure, Paris. He was subsequently an NSF Fellow in Physics at the University of California, Berkeley, where focus of research and interest shifted from quantum field theories and integrable models, related to high-energy physics and string theory, to computer vision and image processing. Upon returning to Israel he spent one year with the Physics Dept., Tel-Aviv University and two years with the Department of Electrical Engineering of the Technion. He is currently a Senior Lecturer in the Department of Applied mathematics, Tel-Aviv University, and a member of the Ollendorff Minerva Center, Technion. His main research interests are the applications of differential geometry and statistical physics in image processing and computational vision.

Yehoshua Y. (Josh) Zeevi is the Barbara and Norman Seiden Professor of Computer Sciences, Department of Electrical Engineering, Technion, where he served as the Dean 1994–1999. He is the Head of the Ollendorff Minerva Center for Vision and Image Sciences, and of the Zisapel Center for Nano-Electronics. He received his Ph.D. from U.C. Berkeley, was a Visiting Scientist at Lawrence Berkeley Lab; a Vinton Hayes Fellow at Harvard University, a Fellow-at-large at the MIT-NRP, a Visiting Senior Scientist at NTT, an SCEEE Fellow USAF on a joint appointment with MIT, a Visiting Professor at MIT, Harvard, Rutgers and Columbia Universities. His work on automatic gain control in vision led to the development of the Adaptive Sensitivity algorithms and Camera that mimics the eye, and he was a co-founder of i Sight Inc. He was also involved in development of Gabor representations and texture generators for helmet-mounted flight simulators. He is an Editor-in-Chief of J. Visual Communication and Image Representation, Elsevier, and the editor of three books. He has served on boards and international committees, including the Technion Board of Governors and Council, and the IEEE technical committee of Image and Multidimensional Signal Processing.

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Sagiv, C., Sochen, N.A. & Zeevi, Y.Y. The Uncertainty Principle: Group Theoretic Approach, Possible Minimizers and Scale-Space Properties. J Math Imaging Vis 26, 149–166 (2006). https://doi.org/10.1007/s10851-006-8301-4

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