The unified theory of n-dimensional complex and hypercomplex analytic signals

The paper is devoted to the theory of n-D complex and hypercomplex analytic signals with emphasis on the 3-dimensional (3-D) case. Their definitions are based on the proposed general n-D form of the Cauchy integral. The definitions are presented in signaland frequency domains. The new notion of lower rank signals is introduced. It is shown that starting with the 3-D analytic hypercomplex signals and decreasing their rank by extending the support in the frequency-space to a so called space quadrant, we get a signal having the quaternionic structure. The advantage of this procedure is demonstrated in the context of the polar representation of 3-D hypercomplex signals. Some new reconstruction formulas are presented. Their validation has been confirmed using two 3-D test signals: a Gaussian one and a spherical one.

[1]  Thomas Bülow,et al.  Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case , 2001, IEEE Trans. Signal Process..

[2]  Dimitrios S. Alexiadis,et al.  Estimation of Motions in Color Image Sequences Using Hypercomplex Fourier Transforms , 2009, IEEE Transactions on Image Processing.

[3]  Patrick R. Girard,et al.  Quaternions, Clifford Algebras and Relativistic Physics , 2007 .

[4]  Stephen J. Sangwine,et al.  Hypercomplex Fourier Transforms of Color Images , 2001, IEEE Transactions on Image Processing.

[5]  S. Hahn.  Hilbert Transforms in Signal Processing , 1996 .

[6]  S. Hahn,et al.  Multidimensional complex signals with single-orthant spectra , 1992, Proc. IEEE.

[7]  K. Alamdar,et al.  Reduction to the Pole of Magnetic Anomalies Using Analytic Signal , 2009 .

[8]  Stefan L. Hahn,et al.  Complex Signals with Single-orthant Spectra as Boundary Distributions of Multidimensional Analytic Functions , 2003 .

[9]  Vaclav Smil,et al.  Book of numbers , 1996, Nature.

[10]  Gerald Sommer,et al.  Geometric Computing with Clifford Algebras , 2001, Springer Berlin Heidelberg.

[11]  Vector product algebras , 2008, 0810.5464.

[12]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. II. A/lgorithms and applications , 1992, Proc. IEEE.

[13]  Stefan L. Hahn,et al.  The N-dimensional complex delta distribution , 1996, IEEE Trans. Signal Process..

[14]  Boualem Boashash,et al.  Estimating and interpreting the instantaneous frequency of a signal. I. Fundamentals , 1992, Proc. IEEE.

[15]  K. Snopek.  New Hypercomplex Analytic Signals and Fourier Transforms in Cayley-Dickson Algebras , 2009 .

[16]  V. Mironov,et al.  Octonic representation of electromagnetic field equations , 2009 .

[17]  T. Ell.  Hypercomplex spectral transformations , 1992 .

[18]  W. Massey.  Cross products of vectors in higher dimensional euclidean spaces , 1983 .

[19]  A. Kwasniewski.  Glimpses of the Octonions and Quaternions History and Today’s Applications in Quantum Physics , 2008, Advances in Applied Clifford Algebras.

[20]  Reformulation of electromagnetism with octonions( , 2006 .

[21]  S. Sangwine.  Fourier transforms of colour images using quaternion or hypercomplex, numbers , 1996 .

[22]  M. Beiki.  Analytic signals of gravity gradient tensor and their application to estimate source location , 2010 .

[23]  Boualem Boashash,et al.  The relationship between instantaneous frequency and time-frequency representations , 1993, IEEE Trans. Signal Process..

[24]  V. Pavlovic,et al.  Complex analytic signals applied on time delay estimation , 2008 .

[25]  Lawrence Charles Paulson,et al.  Quaternions , 1873, Nature.

[26]  STEFAN L. HAHN,et al.  Comparison of Properties of Analytic , Quaternionic and Monogenic 2-D Signals , 2003 .

[27]  Thomas Bülow,et al.  Hypercomplex spectral signal representations for the processing and analysis of images , 1999 .

[28]  Stephen J. Sangwine,et al.  Colour image filters based on hypercomplex convolution , 2000 .

[29]  Soo-Chang Pei,et al.  Commutative reduced biquaternions and their Fourier transform for signal and image processing applications , 2004, IEEE Transactions on Signal Processing.

[30]  Thomas Bülow,et al.  Non-commutative hypercomplex Fourier tranforms of multidimensional signals , 2001 .