Abstract
In the Clifford algebra setting of a Euclidean space on the boundary of a domain it is natural to define a monogenic (analytic) signal to be the boundary value of a monogenic (analytic) function inside the domain. The question is how to define a canonical phase and, correspondingly, a phase derivative. In this paper we give an answer to these questions in the unit ball and in the upper-half space. Among the possible candidates of phases and phase derivatives we decided that the right ones are those that give rise to, as in the one dimensional signal case, the equal relations between the mean of the Fourier frequency and the mean of the phase derivative, and the positivity of the phase derivative of the shifted Cauchy kernel.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlfors, L.V.: Clifford numbers and Möbius transformations in R n. In: Chisholm, J.S.R., Common, A.K. (eds.) Clifford Algebras and Their Applications in Mathematical Physics, Proceedings of the NATO and SERC Workshop Held in Canterbury, September 15–27, 1985, NATO Advanced Science Institutes Series C: Mathematical and Physical Sciences, vol. 183, pp. 167–175. D. Reidel Publ., Dordrecht (1986)
Axelsson A., Kou K.I., Qian T.: Hilbert transforms and the Cauchy integral in Euclidean space. Stud. Math. 193(2), 161–187 (2009)
Bell, S.: The Cauchy Transform, Potential Theory and Conformal Mapping, Studies in Advanced Mathematics. Steven R. Bell, Purdue University, August 14 (1992)
Brackx F., De Knock B., De Schepper H., Eelbode D.: On the interplay between the Hilbert transform and conjugate harmonic functions. Math. Methods Appl. (29)(12), 1435–1450 (2006)
Brackx F., De Schepper H., Eelbode D.: A new Hilbert transform on the unit sphere in R m. Complex Var. Elliptic Equ. 51(5–6), 453–462 (2006)
Brackx F., Delanghe R., Sommen F.: Clifford Analysis, vol. 76. Pitman, Boston (1982)
Bülow, T., Sommer, G.: Multi-dimensional sigal processing using an algebraically extended signal representation. In: Sommer, G., Koenderink, J.J. (eds.) Proceedings International Workshop on Algebraic Frames for the Perception-Action Cycle (AFPAC 97), Kiel. LNCS, vol. 1315, pp. 148–163. Springer, Berlin (1997)
Bülow, T., Sommer, G.: Algebraically extended representations of Multi-dimensional signals. In: Proceedings 10th Scandinavian Conference on Image Analysis, Lappeenranta, Finland, pp. 559–566 (1997)
Bülow, T.: Hypercomplex Spectral Signal Representations for the Processing and Analysis of Images. PhD thesis, University of Kiel, Germany (1999)
Brackx F., Van Acker N.: A conjugate Poisson kernel in Euclidean space. SIMON STEVIN Q. J. Pure Appl. Math. 67(1–2), 3–14 (1993)
Cohen L.: Time-Frequency Analysis: Theory and Applications. Prentice Hall, Inc., Upper Saddle River (1995)
Dang, P., Qian, T.: Hardy-Sobolev derivatives of phase and amplitude, and their applications. Preprint (2011)
Dang P., Qian T., You Z.: Hardy-Sobolev spaces decomposition in signal analysis. J. Fourier Anal. Appl. 17(1), 36–64 (2011)
Delanghe R., Sommen F., Soucek V.: Clifford Algebra and Spinor Valued Functions. Kluwer, Dordrecht (1992)
Felsberg, M.: Low-Level Image Processing with the Structure Multivector. Bericht Nr. 0203. Institut für Informatik und Praktische Mathematik der Christian-Albrechts-Universität zu Kiel, Kiel (2002)
Felsberg M., Sommer G.: The monogenic signal. IEEE Trans. Signal Process. 49(12), 3136–3144 (2001)
Garnett J.B.: Bounded Analytic Functions. Academic Press, London (1981)
Lounesto, P.: Clifford Algebras and Spinors, 2nd edn. London Mathematical Society Lecture Note Series, vol. 286. Cambridge University Press, London (2001)
Li C., Mcintosh A., Qian T.: Clifford algebras, Fourier transforms, and singular convolution operators on Lipschitz surfaces. Rev. Mat. Iberoamericana 10, 665–721 (1994)
Picinbono B.: On instantaneous amplitude and phase of signals. IEEE Trans. Signal Process. 45(3), 552–560 (1997)
Peetre J., Qian T.: Moebius covariance of iterated Dirac operators. J. Aust. Math. Soc. (Ser. A) 56, 403–414 (1994)
Qian T.: Analytic signals and harmonic measure. J. Math. Anal. Appl. 314(2), 526–536 (2006)
Qian T., Chen Q.-H., Li L.-Q.: Analytic unit quadrature signals with non-linear phase. Phys. D Nonlinear Phenom. 203(1–2), 80–87 (2005)
Qian T.: Phase derivative of Nevanlinna functions and applications. Math. Methods Appl. Sci. 32, 253–263 (2009)
Qian T.: Intrinsic mono-component decomposition of functions: an advance of Fourier theory. Math. Methods Appl. Sci. 33(7), 880–891 (2010)
Qian T., Wang Y.-B.: Adaptive Fourier series—a variation of greedy algorithm. Adv. Comput. Math. 34(3), 279–293 (2011)
Qian T., Yang Y.: Hilbert transforms on the sphere with the Clifford analysis setting. J. Fourier Anal. Appl. 15(6), 753–774 (2009)
Stein E., Weiss G.: Introduction to Fourier Analysis on Euclidean. Princeton University Press, NJ (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Daniele Struppa.
This project sponsored by the National Natural Science Funds for Young Scholars (No. 10901166), Sun Yat-Sen University Operating Costs of Basic Research Projects to Cultivate Young Teachers (No. 11lgpy99) and by Research Grant of University of Macau UL017/08-Y2/MAT/QT01/FST, and the FDCT project on Clifford and Harmonic Analysis (014/2008/A1).
Rights and permissions
About this article
Cite this article
Yang, Y., Qian, T. & Sommen, F. Phase Derivative of Monogenic Signals in Higher Dimensional Spaces. Complex Anal. Oper. Theory 6, 987–1010 (2012). https://doi.org/10.1007/s11785-011-0210-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11785-011-0210-x