Quaternionic Local Phase for Low-level Image Processing Using Atomic Functions

Part of the Trends in Mathematics book series (TM)


In this work we address the topic of image processing using an atomic function (AF) in a representation of quaternionic algebra. Our approach is based on the most important AF, the up(??) function. The main reason to use the atomic function up(??) is that this function can express analytically multiple operations commonly used in image processing such as low-pass filtering, derivatives, local phase, and multiscale and steering filters. Therefore, the modelling process in low level-processing becomes easy using this function. The quaternionic algebra can be used in image analysis because lines (even), edges (odd) and the symmetry of some geometric objects in R2 are enhanced. The applications show an example of how up(??) can be applied in some basic operations in image processing and for quaternionic phase computation.


Quaternionic phase 


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  1. [1]
    E. Bayro-Corrochano. The theory and use of the quaternion wavelet transform. Journal of Mathematical Imaging and Vision, 24:19–35, 2006.MathSciNetCrossRefGoogle Scholar
  2. [2]
    J. Bernd. Digital Image Processing. Springer-Verlag, New York, 1993.Google Scholar
  3. [3]
    J. Bigun. Vision with Direction. Springer, 2006.Google Scholar
  4. [4]
    F. Brackx, R. Delanghe, and F. Sommen. Clifford Analysis, volume 76. Pitman, Boston, 1982.Google Scholar
  5. [5]
    T. Bülow. Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD thesis, University of Kiel, Germany, Institut für Informatik und Praktische Mathematik, Aug. 1999.Google Scholar
  6. [6]
    M. Felsberg. Low-Level Image Processing with the Structure Multivector. PhD thesis, Christian-Albrechts-Universität, Institut für Informatik und Praktische Mathematik, Kiel, 2002.Google Scholar
  7. [7]
    M. Felsberg and G. Sommer. The monogenic signal. IEEE Transactions on Signal Processing, 49(12):3136–3144, Dec. 2001.MathSciNetCrossRefGoogle Scholar
  8. [8]
    A.S. Gorshkov, V.F. Kravchenko, and V.A. Rvachev. Estimation of the discrete derivative of a signal on the basis of atomic functions. Izmeritelnaya Tekhnika, 1(8):10, 1992.Google Scholar
  9. [9]
    G.H. Granlund and H. Knutsson. Signal Processing for Computer Vision. Kluwer, Dordrecht, 1995.Google Scholar
  10. [10]
    V.M. Kolodyazhnya and V.A. Rvachov. Atomic functions: Generalization to the multivariable case and promising applications. Cybernetics and Systems Analysis, 46(6), 2007.Google Scholar
  11. [11]
    P. Kovesi. Invariant Measures of Image Features from Phase Information. PhD thesis, University of Western Australia, Australia, 1996.Google Scholar
  12. [12]
    V. Kravchenko, V. Ponomaryov, and H. Perez-Meana. Adaptive digital processing of multidimensional signals with applications. Moscow Fizmatlit, Moscow, 2010.Google Scholar
  13. [13]
    E. Moya-Sánchez and E. Bayro-Corrochano. Quaternion atomic function wavelet for applications in image processing. In I. Bloch and R. Cesar, editors, Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, volume 6419 of Lecture Notes in Computer Science, pages 346–353. Springer, 2010.Google Scholar
  14. [14]
    E. Moya-Sánchez and E. Vázquez-Santacruz. A geometric bio-inspired model forrecognition of low-level structures. In T. Honkela, W. Duch, M. Girolami, and S. Kaski, editors, Artificial Neural Networks and Machine Learning – ICANN 2011, volume 6792 of Lecture Notes in Computer Science, pages 429–436. Springer, 2011.Google Scholar
  15. [15]
    M.N. Nabighian. Toward a three-dimensional automatic interpretation of potential field data via generalized Hilbert transforms: Fundamental relations. Geophysics, 49(6):780–786, June 1982.CrossRefGoogle Scholar
  16. [16]
    S. Petermichl. Dyadic shifts and logarithmic estimate for Hankel operators with matrix symbol. Comptes Rendus de l’Académie des Sciences, 330(6):455–460, 2000.MathSciNetMATHGoogle Scholar
  17. [17]
    S. Petermichl, S. Treil, and A. Volberg. Why are the Riesz transforms averages of the dyadic shifts? Publicacions matematiques, 46(Extra 1):209–228, 2002. Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002.Google Scholar
  18. [18]
    I.W. Selesnick, R.G. Baraniuk, and N.G. Kingsbury. The dual-tree complex wavelet transform. IEEE Signal Processing Magazine, 22(6):123–151, Nov. 2005.CrossRefGoogle Scholar
  19. [19]
    B. Svensson. A Multidimensional Filtering Framework with Applications to Local Structure Analysis and Image Enhancement. PhD thesis, Linköping University, Linkoöping, Sweden, 2008.Google Scholar

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© Springer Basel 2013

Authors and Affiliations

  1. 1.CINVESTAVZapopanMexico

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