Title

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

Journal title

Bulletin of the Polish Academy of Sciences Technical Sciences

Yearbook

2011

Volume

59

Issue

No 2

Authors

Hahn, S. ; Snopek, K.

Divisions of PAS

Nauki Techniczne

Coverage

167-181

Date

2011

Identifier

DOI: 10.2478/v10175-011-0021-2 ; ISSN 2300-1917

Source

Bulletin of the Polish Academy of Sciences: Technical Sciences; 2011; 59; No 2; 167-181

References

Sangwine S. (1996), Fourier transforms of colour images using quaternion or hypercomplex numbers, Electron. Lett, 32, 21, 1979, doi.org/10.1049/el:19961331 ; Bülow T. (1999), Hypercomplex spectral signal representation for the processing and analysis of images. ; Ell T. (2007), Hypercomplex Fourier transforms of color images, IEEE Trans. Image Processing, 16, 1, 22, doi.org/10.1109/TIP.2006.884955 ; Alexiadis D. (2009), Estimation of motions in color image sequences using hypercomplex Fourier transforms, IEEE Trans. Image Processing, 18, 1, 168, doi.org/10.1109/TIP.2008.2007603 ; Sangwine S. (2000), Color image filters based on hypercomplex convolution, IEEE Proc. Vis. Image Signal Process, 147, 2, 89, doi.org/10.1049/ip-vis:20000211 ; Pei S.-C. (2004), Commutative reduced biquaternions and their Fourier transform for signal and image processing applications, IEEE Trans. Signal Process, 52, 7, 2012, doi.org/10.1109/TSP.2004.828901 ; Bülow T. (2001), Non-Commutative Hypercomplex Fourier Transforms of Multidimensional Signals, Geometric Computing with Clifford Algebra. ; A.K. Kwaśniewski, "Glimpses of the octonions and quaternions history and today's applications in quantum physics", <i>ArXiv e-prints</i> <a target="_blank" href='http://aps.arxiv.org/PScache/arxiv/pdf/0803/0803.0119v1.pdf'>http://aps.arxiv.org/PScache/arxiv/pdf/0803/0803.0119v1.pdf</a> ; Hahn S. (1992), Multidimensional complex signals with singleorthant spectra, Proc. IEEE, 80, 8, 1287, doi.org/10.1109/5.158601 ; Hahn S. (1996), Hilbert Transforms in Signal Processing. ; Hahn S. (2003), Complex signals with single-orthant spectra as boundary distributions of multidimensional analytic functions, Bull. Pol. Ac.: Tech, 2, 2, 155. ; Bülow T. (2001), The hypercomplex signal - a novel extension of the analytic signal to the multidimensional case, IEEE Trans. Signal Processing, 49, 11, 2844, doi.org/10.1109/78.960432 ; Hahn S. (2004), Comparison of properties of analytic, quaternionic and monogenic 2-D signals, WSEAS Trans. Computers, 3, 3, 602. ; Hahn S. (1996), The n-dimensional complex delta distribution, IEEE Trans. Signal Proc, 44, 1833, doi.org/10.1109/78.510632 ; Snopek K. (2009), New hypercomplex analytic signals and Fourier transforms in Cayley-Dickson algebras, Electronics and Telecommunications Q, 55, 3, 403. ; Conway J. (1996), Cayley Numbers. The Book of Numbers. ; Girard P. (2007), Quaternions, Clifford Algebras and Relativistic Physics. ; T.A. Ell, "Hypercomplex spectral transforms", <i>Ph.D. Dissertation</i>, Univ. Minnesota, Minneapolis, 1992. ; Sommer G. (2001), Geometric Computing with Clifford Algebras. ; Massey W. (1983), Cross products of vectors in higher dimensional Euclidean spaces, Amer. Math. Monthly, 90, 10, 697, doi.org/10.2307/2323537 ; Darpo E. (2009), Vector product algebras, Bull. London Math. Soc, 41, 898, doi.org/10.1112/blms/bdp066 ; Veličkovic Z. (2008), Complex analytic signals applied on time delay estimation, Facta Universitatis, Series: Physics, Chemistry and Technology, 6, 1, 11, doi.org/10.2298/FUPCT0801011V ; Ansari A. (2009), Reduction of the pole of magnetic anomalies using analytic signal, World Appl. Sc. J, 7, 4, 405. ; Beiki M. (2010), Analytic signals of gravity gradient tensor and their application to estimate source location, Geophysics, 75, 6, 159. ; Boashash B. (1992), Estimating and interpreting the instantaneous frequency of a signal. Part I: Fundamentals, Proc. IEEE, 80, 4, 520, doi.org/10.1109/5.135376 ; Boashash B. (1992), Estimating and interpreting the instantaneous frequency of a signal. Part II: Algorithms and applications, Proc. IEEE, 80, 4, 539. ; Lovell B. (1993), The relationship between instantaneous frequency and time-frequency representations, IEEE Trans. Signal Process, 41, 3, 1458, doi.org/10.1109/78.205756 ; Tolan T. (2006), Reformulation of electromagnetism with octonions, Il Nuovo Cimento, 121B, 1, 43. ; Mironov V. (2009), Octonic representation of electromagnetic filed equations, J. Math. Physics, 50, 012901, 1. ; The derivation of (F2) is delivered to us by Prof. K. Howell.