Studies on complex and hypercomplex multidimensional analytic signals

• Autorzy: Snopek K. M.
• Języki publikacji: EN

Abstrakty

EN

Within the last twenty years, the theory of complex and hypercomplex multidimensional signals has been still developing and finding new interesting applications in various fields This monograph, devoted to analytic complex and hypercomplex signals, presents mutual relations between these two approaches in signal- and frequency domains. Complex and hypercomplex signals are n-dimerisional (n-D) generalizations of corresponding complex and hypercomplex numbers belonging to Cayley-Dickson and Clifford algebras. The brief look into the properties of the Cayley-Dickson algebras of quaternions, octonions and sedenions and also into the Clifford algebras of biquaternions and bioctonions is presented. The tables of multiplication of imaginary units in all considered algebras have been constructed. The main part of the book concerns complex and hypercomplex multidimensional signals with spectra defined using various Fourier transformations (FTs). The n-D Clifford FT is recalled and the new n-D Cayley-Dickson FT is introduced. For n = 2, both are equivalent and known as the Quaternion FT (QFT). For n = 3, the new Cayley-Dickson FT, named the Octonion FT (OFT), is introduced The relation between the 3-D complex FT and the OFT is derived basing on multiplication rules in the algebra of octonions. The n-D complex and hypercomplex analytic signals are defined in the signal domain using the n-D Convolution with the corresponding n-D delta distribution. Such a idea was proposed by Hahn in 1992 for multidimensional complex signals with single-orthant spectra. Here, it is applied for quaternion and octonion analytic signals. Their spectra are defined in analogy to the Hahn's approach, as the inverse QFT (resp. inverse OFT) of a single-quadrant (resp. single-octant) spectrum. The complete set of definitions of 2-D and 3-D analytic signals with spectra in all orthants is included. It has been noticed that there are some conjugate pairs between them and basing on this fact, the lower rank signals have been defined. Basing on definitions of complex, quaternion and octonion analytic signals, it is possible to define the local amplitudes and local phase functions of a signal. As the dimension of a signal is higher, the number of polar components increases. It has been shown that in 2-D, a signal is completely defined by two amplitudes and two phase functions in the complex case and in the quaternion case, by a single amplitude and three phases represented by Euler angles. In case of separable signals, the number of components is reduced. In 3-D, the complex analytic signal is represented by four amplitudes and four phase functions, while the octonion analytic signal has a single amplitude and seven phase functions. In this work, the hypothesis concerning the mutual relations between the complex and octonion polar representation is discussed.

Słowa kluczowe

EN

analytical signal; Cayley-Dickson algebras; Clifford algebras; complex signals; hypercomplex signals

PL

sygnał analityczny; sygnał złożony; algebra Cayley-Dickson'a; algebra Clifforda

• Wydawca: Oficyna Wydawnicza Politechniki Warszawskiej
• Czasopismo: Prace Naukowe Politechniki Warszawskiej. Elektronika
• Rocznik: 2013
• Tom: z. 190
• Strony: 1--169
• Opis fizyczny: Bibliogr. 229 poz., rys., tab., wykr.

Twórcy

autor

Snopek K. M.

• Instytut Radioelektroniki, Wydział Elektroniki i Technik Informacyjnych, Politechnika Warszawska

Bibliografia

• [Abl1996] Abłamowicz R., Lounesto P., Parra J. M. (Ed.), Clifford Algebras with Numeric and Symbolic Computations, Birkauser, 1996.
• [Abl2004a] Abłamowicz R. (Ed.), Clifford Algebras: Applications to Mathematics. Physics, and Engineering, Birkauser, 2004.
• [Abl2004b] Abłamowicz R., Sobczyk G., Lectures on Clifford (Geometric) Algebras and Applications, Birkauser, 2004.
• [Ale2009] Alexiadis D. S., Sergiadis G. D., “Estimation of Motions in Color Image Sequences Using Hypercomplex Fourier Transforms,” IEEE Trans. Image Processing, vol. 18, no. 1, pp. 168-187, January 2009.
• [Alex2012] Alexeyeva L. A., “Biquaternions algebra and its applications by solving of some theoretical physics equations,” Int. J. Clifford Analysis, Clifford Algebras and their Applications, vol. 7, no. 1, 19 pages, 2012.
• [Alf2005] Alfsmann D., Gockler H. G., “Design of Hypercomplex Allpass-Based Paraunitary Filter Banks applying Reduced Biquaternions,” The International Conference on Computer as a Tool, EUROCON'2005, Belgrade, Serbia&Montenegro, Novmber 21-24, 2005, pp. 92-95.
• [All2004] Allen R. L., Mills D.W., Signal analysis - time, frequency, scale and structure, IEEE Press, Wiley-Interscience, 2004.
• [Alt1986] Altmann S. L., Rotations, Quaternions, and Double Groups, New York: Oxford University Press, 1986.
• [And2004] Andreis D., Canuto E. S., “Orbit dynamics and kinematics with full quaternions," Proc. of the American Control Conference, Boston, Massachusetts, June 30-July 2, 2004, pp. 3660-3665.
• [Ass2010] Assefa D., Mansinha L., Tiampo K.. F., Rasmussen H., Abdella K.., “Local quaternion Fourier transform and color texture analysis,” Signal Processing, vol. 60, no. 6, pp. 1825-1835, June 2010.
• [Att2008] Attoh-Okine N., Bentil D., Barner K., Zhang R„ “The Empirical Mode Decomposition and the Hilbert-Huang Transform,” EURASIP Journal on Advances in Signal Processing, vol. 251518, November 6, 2008.
• [Bae2001] Baez J. C., “The octonions”, Bull. Amer. Math. Soc., vol. 39, no. 2, pp. 145-205, 2001. Available: http://math.ucr.edu/home/baez/octonions.
• [Bah2011] Bahri M., “Quaternion Algebra-Valued Wavelet Transform, “ Applied Math. Sciences, vol. 5, no. 71, pp. 3531-3540, 2011.
• [Bal2009] Bales J. W., “The Cayley-Dickson Calculator”, Department of Ivtathematics, Tuskegee University. Available: http://jwbales.us/.
• [Bal2009a] Bales J. W., “The Sedenion Product Calculator”, Available: http://jwbales.us/sedenion.html.
• [Bal2009b] Bales J. W., “A Tree for Computing the Cayley-Dickson Twist”, Missouri J. Math. Sciences, vol. 21, no. 2, pp. 83-93, 2009.
• [Bas2003] Bas P., Le Bihan N., Chassery J.-M., “Color Image Watermarking Using Quaternion Fourier Transform,” Proc. 1CASSP, Hong Kong, 2003, 4 pages.
• [BasA2012] Basit A., Aziz W., Zafar F., “Implementation of SSB Modulation/Demodulation using Hilbert Transform in MATLAB," Journal of Expert Systems, World Science Publisher, USA, vol. 1, no. 3, pp. 79-83, 2012.
• [Bat2010] Batrakov D. O., Golovin D. V., Simachev A. A., Batrakova A. G., “Hilbert Transform Application to the Impulse Signal Processing,” Ultrawideband and Ultrashort Impulse Signals, September 6-10, 2010, Sevastopol, Ukraine, pp. 113-115.
• [Bayl1996] Baylis W. E., Clifford (Geometric) Algebras: With Applications in Physics, Mathematics, and Engineering, Springer, 1996.
• [Bayr2001b] Bayro-Corrochano E. J., “Geometric Neural Computing”, IEEE Trans. Neural Networks, vol. 12, no. 5, pp. 968-986, Sep. 2001.
• [Bayr2001b] Bayro-Corrochano E. ”The Geometry and Algebra of Kinematics,” in Geometric Computing with Clifford Algebras, G. Sommer (Ed.), Springer, pp. 455-488, 2001.
• [Bayr2006] Bayro-Corrochano E., “The theory and use of the quaternion wavelet transform,” J. Math. Imaging and Vision, vol. 24, pp. 19-35, 2006.
• [Bayr2010] Bayro-Corrochano E. J., “Clifford Support Vector Machine,” IEEE Trans. Neural Networks, vol. 21, no. 11, pp. 1731-1746, Nov. 2010.
• [Bih2001] Le Bihan N., Mars J., “New 2D attributes based on complex and hypercomplex analytic signal,” 71st Meeting of Society of Exploration Geophysicists SEG01, San Antonio, September 2001, 4 pages.
• [Bih2003] Le Bihan N., Sangwine S. J., “Color Image Decomposition Using Quaternion Singular Value Decomposition,” Intern. Conf. on Visual Information Engineering, July 7-9, 2003, pp. 113-116.
• [Bor2011] Borges M. F., Roque M. D., Marao J. A., “Sedenions of Cayley-Dickson and the Cauchy-Riemann Like Relations”, Int. J. Pure and Applied Mathematics, vol. 68, no. 2, pp. 165-188, 2011.
• [Bou2000] Boudreaux-Bartels G. F. “Mixed Time-Frequency Signals Transformations,” in The Transforms and Applications Handbook, Second Ed., A. D. Pouiarikas (Ed.), CRC Press, IEEE Press, 2000.
• [Bra1986] Bracewell R. N., The Fourier transform and its applications, McGraw Hill, 1986.
• [Brac1982] Brackx F., Delanghe R., Sommen F., Clifford Analysis, Pitman, Boston, 1982.
• [Brac2006] Brackx F., N. de Schepper, F. Sommen, “The Two-dimensional Clifford-Fourier Transform," J. Math. Imaging, vol. 26, pp. 5-18, 2006.
• [Bre1965] Bremermann H., Distributions, Complex variables and Fourier Transforms, Reading, MA: Addison-Wesley, 1965.
• [Buc1977] Buccella F., Falcioni M., “Octonions and Unified Theories with Orthogonal Groups,” Lettere al Nuovo Cimento, vol. 18, no.14, pp. 441-446, April 1977.
• [Buch2005] Buchholz S., “A Theory of Neural Computation with Clifford Algebras,” PhD dissertation, Kiel, 2005. Available: http://www.informatik.uni-kiel.de/inf/Sommer/doc/Dissertationen/Sven_Buchholz/diss.pdf.
• [Buch2008] Buchholz S., Sommer G., On Clifford neurons and Clifford multi-layer perceptrons in Neural Networks, vol. 21, pp. 925-935, 2008.
• [Bül1999] Bülow T., “Hypercomplex spectral signal representation for the processing and analysis of images,” in Bericht Nr. 99-3, Institut fur lnformatik und Praktische Mathematik, Christian-Albrechts-Universitat Kiel, August 1999.
• [Bül2001a] Bülow T., Felsbetg M., Sommer G., “Non-Commutative Hypercomplex Fourier Transforms of Multidimensional Signals” in Geometric Computing with Clifford Algebra, G. Sommer, ed., Berlin: Springer-Verlag, pp. 187-207, 2001.
• [Bül2001b] Bülow T., Sommer G., “Local Hypercomplex Signal Representations and Applications" in Geometric Computing with Clifford Algebra, G. Sommer, ed., Berlin: Springer-Verlag p 255-289, 2001.
• [Bül2001c] Bülow T., Sommer G., “The Hypercomplex Signal - A Novel Extension of the Analytic Signal to the Multidimensional Case,” IEEE Trans. Signal Processing, vol. 49, no. 11, pp. 2844-2852, November 2001.
• [Cak2001] Cakrak F., Loughlin P. J., “Multiwindow Time-Varying Spectrum with lnstantateous Bandwidth and Frequency Constraints,” IEEE Trans. Signal processing, vol. 49, no. 8, pp. 1656-1666, August 2001.
• [Car1988] Carmody K., “Circular and hyperbolic quaternions, octonions, and sedenions,” Appl. Math. Comput., vol. 28, pp. 47-72, 1988.
• [Car1997] Carmody K., “Circular and hyperbolic quaternions, octonions, and sedenions - further results,” Appl. Math. Comput., vol. 84, no. 1, pp. 27-47, 1997.
• [Cha2010] Chanyal B.C., Bisht P. S., Negi O. P. S. “Generalized Octonion Electrodynamics,” Int. J., Theor. Physics, vol. 49, pp. 1333-1343, 2010.
• [Chan2008] Chan W., Choi H., Baraniuk R., “Coherent multiscale image processing using dual-tree quaternion wavelets,” IEEE Trans. Image process., vol. 17, no. 7, pp. 1069-1082, July 2008.
• [Chou1992] Chou J. C. K., “Quaternion Kinematic and Dynamic Differential Equations,” IEEE Trans. On Robotics and Automation, vol. 8, no. 1, pp. 53-64, February 1992.
• [Chr2010] Chrisitianto V., Smarandache F., “A Derivation of Maxwell Equations in Quaternion Space,” Progress in Physics, vol. 2, pp. 23-27, April 2010.
• [Cla1980a] Claasen T. C. A. M., Mecklenbrauker W. F. G., “The Wigner distribution - a Tool for Time-Frequency Signal Analysis, Part I - Continuous-Time Signals,” Philips J. Research, vol. 35, pp. 217-250, 1980.
• [Cla1980b] Claasen T. C. A. M., Mecklenbrauker W. F. G., “The Wigner Distribution - a Tool for Time-Frequency Signal Analysis, Part II - Discrete-Time Signals,” Philips J. Res., vol. 35, pp. 276-300, 1980.
• [Cla1980c] Claasen T. C. A. M., Mecklenbrauker W. F. G., “The Wigner Distribution - a Tool for Time-Frequency Signal Analysis, Part III - Relations with other Time-Frequency Signal Transformations,” Philips J. Res., vol. 35, pp. 372-389, 1980.
• [Cla1983] Claasen T. C. A. M., Mecklenbrauker W. F. G., “The Aliasing Problem in Discrete-Time Wigner Distributions,” IEEE Trans. Acoust. Speech and Signal Processing, vol. ASSP-31, no.5, pp. 1067-1072, October 1983.
• [Cli1873] Clifford W. K., “Preliminary sketch of biquaternions,” Proc. London Math. Soc., pp. 381-396, 1873.
• [Coh1989] Cohen L., “Time-Frequency Distributions - A Review,” Proc. IEEE, vol. 77, no. 7, pp. 941-981, 1989.
• [Cont1997] Conte E. “On the Generalization of the Physical Laws By Biquaternions: An Application to the Generalization of Minkowski Space-Time,” Physics Essays, vol. 10, no. 3, pp. 437-441, September 1997.
• [Conw1996] Conway J. H., Guy R. K., Cayley Numbers. In: The Book of Numbers, New York, Springer-Verlag, pp. 234-235, 1996.
• [Conw2003] Conway J., Smith D., On Quaternions and Octonions: Their Geometry, Arithmetic and Symmetry A. K. Peters Ltd., 2003.
• [Dan2001] Daniilidis K., “Using the Algebra of Dual Quaternions for Motion Alignement,” in Geometric Computing with Clifford Algebras, G. Sommer (Ed.), Springer, pp. 489-500, 2001.
• [Den2007] Denis P., Carre P., Fernandez-Maloigne C., “Spatial and spectral quaternionic approaches for colour images,” Elsevier, Computer Vision and Image Understanding, vol. 107, pp. 74-87, 2007.
• [DenL2006] Deng L., “A new approach of data hiding within speech based on Hash and Hilbert Transform,” ICSNC '06 Proceedings of the International Conference on Systems and Networks Communication, October 29-November 03, 2006, Tahiti, French Polynesia, page 13.
• [Dim1992] Dimitrov V. S., Cooklev T. V., Donevsky B. D., “On the Multiplication of Reduced Biquaternions and Applications, 44 Inf. Process. Letters, vol. 43, no. 3, pp. 161-164, Sept. 1992.
• [Din2012] Ding G., “Bi-quaternion form of electromagnetism theory,” Scientia Sinica Physica, Mechanica & Astronomica, vol. 42, no. 10, pp. 1029-1039, 2012.
• [Dir1930] Dirac P. A. M., The Principles of Quantum Mechanics, Oxford, U. K..: Plenum, 1930.
• [Dor2002] Dorst L., Mann S., “Geometric Algebra: A Computational Framework for Geometrical Applications,” IEEE Computer Graphics and Applications, vol. 22, no. 3, pp. 24-31, May-June 2002.
• [Dzh2007] Dzhunushaliev V., “Toy Models of a Nonassociative Quantum Mechanics,” Advances in High Energy’ Physics, Hindawi Publishing Corp., vol. 2007, Article ID 12387, 10 pages.
• [Ebl2005] Ebling J., Scheuermann G., “Clifford Fourier Transform on Vector Fields,” IEEE Trans. Visualization and Computer Graphics, vol. 11, no. 4, pp. 469-479, July/August 2005.
• [Ell1992] Ell T. A., “Hypercomplex Spectral Transformations,” Ph. D. dissertation, Univ. of Minnesota, Minneapolis, 1992.
• [Ell1993] Ell T. A., “Quaternion-Fourier transforms for analysis of 2-dimensional linear time-invariant partial-differential systems,” Proc. 32nd IEEE Conf. on Decision and Control, December 15-17, 1993, San Antonio, TX, USA, vol. 1-4, pp. 1830-1841.
• [Ell2007] Ell T. A., Sangwine S. J. , „Hypercomplex Fourier Transforms of Color Images," IEEE Trans. Image Processing, vol. 16, no. 1, pp. 22-35, January 2007.
• [Far2004] Farrell J., Mathematics for Game Developers, Premier Press, Inc., 2004.
• [Fel2001a] Felsberg M., Biilow T., Sommer G, “Commutative Hypercomplex Fourier Transforms of Multidimensional Signals” in Geometric Computing with Clifford Algebras G. Sommer ed., Springer-Verlag Berlin Heidelberg, pp. 209-229, 2001.
• [Fel2001b] Felsberg M., Brilow T., Sommer G, Chernov V.M, “Fast Algorithms of Hypercomplex Fourier Transforms” in Geometric Computing with Clifford Algebras G. Sommer ed., Springer-Verlag Berlin Heidelberg, pp. 231-254, 2001.
• [Fel2001c] Felsberg M., Sommer G., “The Monogenic Signal”, IEEE Trans. Signal Processing, vol. 49, pp. 3136-3144, 2001.
• [Feld2008] Feldman M.. Theoretical analysis and comparison of the Hilberl transform decomposition methods”, Mechanical Systems and Signal Processing, vol. 22/3, pp. 509-519, 2008.
• [Feld2009] Feldman M., Bucher I., Rolberg J., “Experimental Identification of Nonlinearities under Free and Forced Vibration using the Hilbert Transform,” Journal of Vibration and Control, vol. 15, no. 10, pp. 1563-1579, 2009.
• [Feld2011] Feldman M., Hilbert Transform Applications in Mechanical Vibration, John Wiley&Sons Ltd 2011.
• [Feld2012] Feldman M., “Nonparametric identification of asymmetric nonlinear vibration systems with the Hilbert transform,” Journal of Sound and Vibration, vol. 331, pp. 3386-3389, 2012.
• [Fer2013] Fernandez-Maloigne C. (Ed.), Advanced Color Image Processing and Analysis, Springer Science+Bussiness Media, New York, 2013.
• [Fla1999] Flandrin P., Time-frequency/time~scale analysis. Academic Press, 1999.
• [Fra2010] Franchini S., Vassallo G., Sorbell F., ”A brief introduction to Clifford algebra”, Universita degli studi di Palermo, Dipartimento di Ingegneria Informatica, Technical Report No. 2, February 2010.
• [Fun1988] Funda J., Paul R.P., “A Comparison of Transforms and Quaternions in Robotics,” Proceedings, IEEE Int. Conf. Robotics and Automation, Philadelphia, September 24-29, 1988, vol. 2, pp. 886-891.
• [Gab1946] Gabor D., “Theory of communications”, J. Inst. E.E., Part III, vol. 93, pp. 429-457, November 1946.
• [Gai2013] Gai S., Yang G., Zhang S., “Multiscale texture classification using reduced quaternion wavelet transform”, Int. J. Electronics and Communication, vol. 67, no. 3, pp. 233-241, March 2013.
• [Gao2012] Gao C. Zhou J., Lang F., Pu Q, Liu C., “Novel Approach to Edge Detection of Color Image Based on Quaternion Fractional Directional Differentation,” Advances in Automation and Robotics, vol. 1, pp. 163-170, 2012.
• [Gau1996] Gaunaurd G. C., Strifors H. C., “Signal Analysis by Means of Time-Frequency (Wigner-Type) Distributions - Applications to Sonar and Radar Echoes,” Proc. IEEE, vol. 84, no. 9, pp. 1231-1248, September 1996.
• [Gir2008] Girard P. R., Quaternions, Clifford Algebras and Relativistic Physics, Birkhauser, Basel Boston Berlin, 2008.
• [Gra1997] Grassin S., Garello R., “Spectral Analysis of Waves on the Ocean Surface Using High Resolution Time-Frequency Representations,” 7th Inter. Conf. Electronic Engineering in Oceanography, 23-25 June 1997, no. 439, pp. 153-159, 1997.
• [Grav2006] Gravelle M., “Quaternions and their Applications to Rotation in 3D Space,” May 1, 2006, 18 pages. Available: http://www.morris.umn.edu/academic/math/Ma4901/gravelle.pdf.
• [Gürl1997] Gürlebeck K., Sproösig W., Quaternionic and Clifford Calculus for Physicists and Engineers, John Wiley&Sons, New York, 1997.
• [Hah1987] Hahn S., Teoria modulacji i detekcji, Wydawnictwa Politechniki Warszawskiej, Warszawa 1987.
• [Hah1992a] Hahn S. L., “Multidimensional Complex Signals with Single-Orthant Spectra,” Proc. IEEE, vol. 80, no. 8, pp. 1287-1300, August 1992.
• [Hah1992b] Hahn S. L., “Amplitudes, Phases and Complex Frequencies of 2-D Gaussian Signals,” Bull. Polish Ac. Sciences, Tech. Sciences, vol. 40, no. 3, pp. 289-311, 1992.
• [Hah1996a] Hahn S. L., Hilbert Transforms in Signal Processing, Artech House Inc., 1996.
• [Hah1996b] Hahn S. L., “The N-Dimensional Complex Delta Distribution,” IEEE Trans. Signal Processing, vol. 44, no. 7, pp. 1833-1837, July 1996.
• [Hah2000] Hahn S. L., “The theory of time-frequency distributions with extension for two-dimensional signals,” Metrology and Measurements Systems, vol. VIII, no. 2, pp. 113-143, 2000.
• [Hah2001] Hahn S. L., “A review of methods of time-frequency analysis with extension for signal plane frequency plane analysis/4 Kleinheubacher Berichte, Band 44, pp. 163-182, 2001.
• [Hah2002] Hahn S. L., “The Relationships of Analytic and Quaternionic Representation of 2-D Signals,” X National Symposium of Radio Science, Poznań, March 14-15, 2002, pp. 328-332.
• [Hah2002a] Hahn S. L., ”Hilbert Transforms,” in The Transforms and Applications Handbook, Second Edition, Poularikas A.D. (Ed.), CRC Press, IEEE Press, 2000.
• [Hah2003] Hahn S. L., “Complex Signals with Single-Orthant Spectra as Boundary Distributions of Multidimensional Analytic Functions,” Bull. Polish Academy of Sciences, Technical Sciences, vol. 51, no. 2, pp. 155-161,2003.
• [Hah2003a] Hahn S. L., “On the uniqueness of the definition of (he amplitude and phase of the analytic signal,” Signal Processing, vol. 83, pp. 1815-1820, 2003.
• [Hah2007] Hahn S. L, “The History of Applications of Analytic Signals in Electrical and Radio Engineering,” Intern. Conf. on “Computer as a Tool” EUROCON, Warsaw, September 9-12, 2007, pp. 2627-2631.
• [HahS1994] Hahn S. L., Snopek K. M., “The decomposition of two-dimensional images into amplitude and phase patterns,” Kleinheubacher Berichte, Band 37, pp. 91-99, 1994.
• [HahS2000] Hahn S. L., Snopek K. M., “The feasibility of the extension of the exponential kernel (Choi-Williams) for 4-dimensional signal-domain/frequency-domaindistributions,” Kleinheubacher Berichte, Band 43, pp. 90-97, 2000.
• [HahS2001] Hahn S. L., Snopek K. M., “Wigner Distributions and Ambiguity Functions in Image Analysis,” Computer Analysis of Images and Patterns, 9th International Conference CAIP 2001, Warsaw, Poland, pp. 537-546, September 2001.
• [HahS2002a] Hahn S. L., Snopek K. M., “Comparison of selected Cohen's class double-dimensional distributions, ” Kleinheubacher Berichte, Band 45, pp. 111-115, 2002.
• [HahS2002b] Hahn S. L., Snopek K. M., “Double-dimensional Distributions: Another Approach to “Quartic” Distributions,” IEEE Transactions on Signal Processing, vol. 50, no. 12, pp. 2987-2997, December 2002.
• [HahS2004a] Hahn S. L., Snopek K. M., “On the Frequency-domain Definition of the Monogenic Signal”, Report No. 1, Institute of Radioelectronics WUT, 2004.
• [HahS2004b] Hahn S. L., Snopek K. M., “Comparison of Properties of Analytic, Quaternionic and Monogenic 2-D Signals,” WSEAS Transactions on Computers, Issue 3, vol. 3, pp. 602-611, July 2004.
• [HahS2004c] Hahn S.L., Snopek K.M., “The derivation of the Wigner distribution of the quaternionic signal (...)” Report No. 2, Institute of Radioelectronics WUT, 2004. Available: http://www.ire.pw.edu.pl/~ksnopek.
• [HahS2005a] Hahn S. L., Snopek K. M. “Wigner Distributions and Ambiguity Functions of 2-D Quaternionic and Monogenic Signals,” IEEE Transactions on Signal Processing, vol. 53, no. 8, pp. 3111-3128, August 2005.
• [HahS2005b] Hahn S. L., Snopek K. M., “Pseudo-Wigner Distribution with Extension for 4-D Distributions Including Double-dimensional Distributions”, Report No. 1, Institute of Radioelectronics WUT, 2005.
• [HahS2007] Hahn S. L., Snopek K. M., “Audio watermarking using two anti-slope chirps and windowed double-dimensional Wigner distributions”, Report No. 1, Institute of Radioelectronics WUT, 2007.
• [HahS2010a] Hahn S. L., Snopek K. M., “Various Approaches to the Theory of Complex and Hypercomplex Analytic Signals”, Report No. 2, Institute of Radioelectronics WUT, 2010.
• [HahS2010b] Hahn S. L, Snopek K. M., “The Unified Theory of Complex and Hypercomplex Analytic Signals”, Report No. 1, Institute of Radioelectronics WUT, 2010.
• [HahS2011a] Hahn S. L., Snopek K. M., “The Unified Theory of n-Dimensional Complex and Hypercomplex Analytic Signals, ” Bull. Polish Ac. Sci., Tech. Sci., vol. 59, no. 2, pp. 167-181, 2011.
• [HahS2011b] Hahn S. L., Snopek K. M., “The Quasi-analytic Multidimensional Signals”, Report No. 1, Institute of Radioelectronics WUT, 2011.
• [HahS2013] Hahn S. L., Snopek K. M., “Quasi-analytic Multidimensional Signals”, Bull. Polish Ac. Sci.,ech. Sci., 2013 (revised).
• [Ham1847] Hamilton W. R. , “On quaternions,” Proc. Royal Irish Academy, vol. 3, pp. 1-16, 1847.
• [Ham1866] Hamilton W. R., Elements of quaternions, Longmans, Green and Co., London, 1866.
• [Hay1994] Haykin S., Communication systems, John Wiley&Sons, Inc., 1994.
• [Hit2007] Hitzer E. M. S ., “Quaternion Fourier Transform on Quaternion Fields and Generalizations,” Adv. Appl. Clifford Alg., vol. 17, no. 3, pp. 497-517, 2007.
• [Hit2013] Hitzer E. M. S. , “Applications of Clifford’s Geometric Algebra,” Adv. Appl. Clifford Algebras, vol. 23, no. 2, pp. 377-404, 2013.
• [Hoh2013] Hohne C., Boehm R., Prager J., “Application of 2-dimensional Analytic Signals with Single-Quadrant Spectra for Processing of SAFT-Reconstructed Images”, Multidimensional Systems and Signal Processing, Springer Science+Business Media, New York, published on line 9 February 2013.
• [Hua1998] Huang N. E., Shen Z., Long S. R., Wu M. L., Shih H. H., Zheng Q., Yen N. C., Tung C. C., Liu H.H., “The empirical mode decomposition and Hilbert spectrum for nonlinear and non-stationary time series analysis,” Proc. Roy. Soc. London A., vol. 454, pp. 903-995, 1998.
• [Hua2000] Huang N. E., “A new Method for Nonlinear and Nonstationary Time Series Analysis: Empirical Mode Decomposition and Hilbert Spectral Analysis,” Proc. SPIE, Wavelet Applications VII, vol. 4056, pp. 197-209, 2000.
• [Hua2005] Huang N. E.,Shen S. S. P. (Eds.), Hilbert-Huang Transform and Its Applications, World Scientific Publishing Co. Pte. Ltd., 2005.
• [Ima2000] Imaeda K. , Imaeda M., “Sedenions: algebra and analysis,” Applied Mathematics and Computation, vol. 115, pp. 77-88, 2000.
• [JinL2010] Jin L., Liu H., Xu X., Song E., “Quaternion-based color image filtering for impulsive noise suppression,” J. Electronic Imaging, vol. 19, no. 4, 12 pages, October-December 2010.
• [JinS2010] Jing S., Jing-yu Y., “Quaternion Frequency Watermarking Algorithm for Color Images,” Int. Conf. Multimedia Technology, October 29-31,2010, Ningbo, pp. 1-4.
• [Kan1989] Kantor I. L., Solodovnikov A. S., Hypercomplex numbers, Springer Varlag, New York, 1989.
• [Kap2008] Kaplan A., “Quaternions and Octonions in Mechanics,” Revisla de la Union Matematica Argentina, vol. 49, no. 2, pp. 45-53, 2008.
• [Kha2012] Khalil M. I., “Applying Quaternion Fourier Transforms for Enhancing Color Images,” Int. J. of Image, Graphics and Signal Processing, MECS, vol. 4, no. 2, pp. 9-15, 2012.
• [Koh2012] Kohli S. S., Makwana N., Mishra N., Sagar B., “Hilbert Transform Based Adaptive ECG R-Peak Detection Technique,” Int. J. Electrical and Computer Engineering (IJECE), vol. 2, no. 5, pp. 639-643, October 2012.
• [Kön1996] König D., Böhme J. F., “Wigner-Ville Spectral Analysis of Automotive Signals Captured at Knock,” Applied Signal Processing, vol. 3, pp. 54-64, 1996.
• [Köp2006] Köplinger J., “Dirac equation on hyperbolic octonions,” Appl. Math. Comp., vol. 182, no. 1, pp. 443-446, November 2006.
• [Köp2007a] Köplinger J., “Signature of gravity in conic sedenions,” Appl. Math. Comp., vol. 188, pp. 942-947, 2007.
• [Köp2007b] Köplinger J., “Gravity and electromagnetism on conic sedenions,” Appl. Math. Comp., vol. 188, pp. 948-953, 2007.
• [Kui1999] Kuipers J. B., Quaternions and Rotation Sequences, Princeton: Princeton University Press, 1999.
• [Kwa2012] Kwaśniewski A. K., “Glimpses of the Octonions and Quaternions history and Today’s Applications in Quantum Physics,” Adv. Appl. Clifford Algebra, vol. 22, pp. 87-105, 2012.
• [Laz2011] Lazarov B. S., Matzen R., Elesin Y., “Topology optimization of pulse shaping filters using the Hilbert transform envelope extraction,” Journal of Structural and Multidisciplinaiy Optimization, vol. 44, no. 3, pp. 409-419, September 2011.
• [Len2001] Lengyel E., Mathematics fo r 3D Programming and Computer Graphics, Charles River Media, 2001.
• [Li2012] Li C., Li B., Xiao L., Hu Y., Tian L., “A Watermarking Method Based on Hypercomplex Fourier Transform and Visual Attention,” J. Information & Computational Science, vol. 9, no. 15, pp. 4485-4492, 2012.
• [Ma2008] Ma X., Xu Y., Yang X., “Color image watermarking using local quaternion Fourier spectra analysis,” IEEE Int. Conf. Multimedia and Expo, June 23-26, 2008, Hannover, pp. 233-236.
• [Mar2011] Martin C. S., Kim S.-W. (Eds.), Progress in Pattern Recognition, Image Analysis, Computer Vision and Applications, Proceedings of the 16th Iberoamerican Congress CIARP’2011, Pucon, Chile, November 2011, Springer-Verlag Berlin Heidelberg 2011.
• [Mei1998] Meister L., “Quaternions and their application in photogrammetry and navigation,” Habilitationsschrift, TU Bergakademie, Freiberg, 1998.
• [Mei2000] Meister L., “Mathematical Modelling in Geodesy Based on Quaternion Algebra,” Phys.Chem. Earth (A), vol. 25, no. 9-11, pp. 661-665, 2000.
• [Mel2007] Melges D. B., Infantosi A. F. C , Ferreira F. R., Rosas D. B., “Using the Discrete Hilbert Transform for the Comparison Between Trace Alternant and High Voltage Slow Patterns Extracted from Full-Term Neonatal EEG,” World Congress on Medical Physics and Biomedical Engineering, IFMBE Proceesings, vol. 14, pp. 1111-1114, 2007.
• [Mer1999] Mertins A., Signal Analysis, John Wiley&Sons Ltd, 1999.
• [Mox2003] Moxey C. E, Sangwine S. J., Ell T. A., “Hypercomplex Correlation Techniques for Vector Images,” IEEE Trans. Sig. Proc., vol. 3 1, no. 7, pp. 1941-1953, July 2003.
• [Muk2002] Mukundan R., “Quaternions: From Classical Mechanics to Computer Graphics, and Beyond,” Proc. 7th ATCM Conf, pp. 97-106, 2002.
• [Nei1999] O’Neill J. C., Williams W. J., “A function of time, frequency, lag and Doppler,” IEEE Trans. Signal Processing, vol. 47, pp. 789-799, March 1999.
• [Nei2000] O’Neill J. C., Williams W. J., “Virtues and vices of quartic time-frequency distributions,” IEEE Trans. Signal Processing,, vol. 48, pp. 2641-2650, September 2000.
• [Nie2006] Nie K., Barco A., Zeng F. G., “Spectral and temporal cues in cochlear implant speech perception,” Ear Hear, vol. 27, no. 2, pp. 208-217, April 2007.
• [Pan2011] Panicaud B., “Clifford Algebra (C) for Applications to Field Theories,” Int. J. Theoretical Physics, vol. 50, no. 10, pp. 3186-3204, 2011.
• [Pei2001] Pei S.-C. , Ding J.-J. , Chang J.-H., “Efficient Implementation of Quaternion Fourier Transform, Convolution, and Correlation by 2-D Complex FFT,” IEEE Trans. Sig. Proc., vol. 49, no. 11, pp. 2783-2797, November 2001.
• [Pei2001a] Pei S.-C., Ding J.-J., Chang J. “Color pattern recognition by quaternion correlation,” IEEE Int. Conf. Image Process., Thessaloniki, Greece, October 7-10, 2010, pp. 894-897.
• [Pei2004] Pei S.-C. , Chang J.-H., Ding J.-J., “Commutative Reduced Biquaternions and Their Fourier Transform for Signal and Image Processing Applications,” IEEE Trans. Sig. Proc., vol. 52, no. 7, pp. 2012-2031, July 2004.
• [Pei2010] Pei S.-C., Hsiao Y.-Z., “Colour image edge detection Rusing quaternion quantized localized chase,” 18th European Signal processing conference (EUSCIPCO’2010), Aalborg, Denmark, August 23-27, 2010, pp. 1766-1770.
• [Pou2000] The Transforms and Applications Handbook, Second Edition, Editor-in-Chief A.D. Poularikas, CRC Press, IEEE Press, 2000.
• [Pri1965] Price R., Hofstetter E. m., “Bounds on the Volume and Height Distributions of the Ambiguity Function,” IEEE Trans. Inform. Theory, vol. 11, pp. 207-214, April 1965.
• [Qi2008] Qifeng Qi, Yueming Flu, Chaoyang Zhang, “A New Color Pattern Recognition Algorithm Based on Quaternion Correlation,” IEEE Intern. Conf. on Networking, Sensing and Control, Sanyam, April 6-8, 2008, pp. 462-466.
• [Rap1985] Rapaport D. C., “Molecular Dynamics Simulation Using Quaternions,” J. Computational Physics, vol. 60, pp. 306-314, 1985.
• [Rec2011] Recio-Spinoso A., Fan Y.-FI., Ruggero M. A., “Basilar-Membrane Responses to Broadband Noise Modeled Using Linear Filters with Rational Transfer Functions,” IEEE Trans. Biomedical Engineering, vol. 58, no. 5, pp. 1456-1465, May 2011.
• [Red2002] Redfield S. A., Huynh Q. Q., “Hypercomplex Fourier transforms applied to detection for side-scan sonar,” OCEANS’02 MTS/IEEE, October 29-31,2002, vol. 4, pp. 2219-2224.
• [Roo2007] Rooney J., “William Kingdon Clifford (1845-1879),” in Distinguished Figures in Mechanism and Machine Science, M. Ceccarelli (Ed.) Springer, pp. 79-116, 2007.
• [Run2001] Rundblad-Labunets E., Labunets V., “Spatial-Color Clifford Algebras for Invariant Image Recognition,” in Geometric Computing with Clifford Algebras G. Sommer ed., Springer-Verlag Berlin Heidelberg, pp. 155-185, 2001.
• [Rya2000] Ryan J., Sprößig W. (Ed.), Clifford Algebras and Their Applications in Mathematical Physics: Volume 2: Clifford analysis, Birkhauser 2000.
• [Saa2006] Saad Z., “Hilbert transform” in Wiley Encyclopedia of Biomedical Engineering, 2006.
• [Sah2008] Sahu S., Biswal B. B., Subudhi B., “A Novel Method for Representing Robot Kinematics using Quaternion Theory,” IEEE Sponsored Conf. Computational Intelligence, Control and Computer Vision in Robotics&Automation (CICCRA’2008), pp. 76-82.
• [Sai2008] Said S., Le Bihan N., Sangwine S. J., “Fast Complexified Quaternion Fourier Transform,” IEEE Trans. Signal Proc., vol. 56, no. 4, pp. 1522-1531, April 2008.
• [San1996] Sangwine S. J., “Fourier transforms of colour images using quaternion or hypercomplex numbers,” Electron. Lett., vol. 32, no. 21, pp. 1979-1980, Oct. 1996.
• [San1998] Sangwine S. J., “Color image edge detector based on quaternion convolution,” Electronics Letters, vol. 34, no. 10, pp. 969-971, 1998.
• [San2000] Sangwine S. J., Ell T, A., “Colour image filters based on hypercomplex convolution,” IEEE Proc. Vision, Image and Signal Processing, vol. 49, pp. 89-93, 2000.
• [San2000a] Sangwine S. J., Evans C. J., Ell T, A., “Colour-sensitive edge detection using hypercomplex filters,” Proc. 10th European Signal Processing Conf. EUSIPCO, Tampere, Finland, 2000, vol. 1, pp. 107-110.
• [San2000b] Sangwine S. J., “Colour in image processing,” Electronic and Communication Engineering Journal, vol. 12, no. 5, pp. 211-219, 2000.
• [San2001] Sangwine S. J., Ell T. A., “Hypercomplex Fourier transforms of color images,” IEEE Int. Conf. Image Process., Thessaloniki, Greece, October 7-10, 2001, vol. 1, pp. 137-140.
• [San2007] Sangwine S. J., Le Bihan N., “Quaternion Polar Representation with a Complex Modulus and Complex Argument Inspired by the Cayley-Dickson Form,” Adv. Appl. Clifford Alg., 2008.
• [San2011] Sangwine S. J., Ell T. A., Le Bihan N., “Fundamental Representations and Algebraic Properties of Biquaternions or Complexified Quaternions,” Adv. Appl. Clifford Algebras, vol. 21, pp. 607-636, 2011.
• [Sch1965] Schwartz L., Methodes Mathematiqies pour les Sciences Physiques, Paris, France: Hermann, 1965.
• [SchP2010] Schreier P. J, Scharf L. L., Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, 2010.
• [Sha2012] Sharma N., Trikha M., Rajpoot V., “A Novel Approach for Single-Sideband Modulation using Hilbert Transform,” MIT Ini. Journal of Electrical and Instrumentation Engineering, vol. 2, no. 2, pp. 102-105, August 2012.
• [Sie1958] Siebert W.M., “Studies of Woodward's uncertainty function,” Quart. Progress Rep., Electron. Res. Lab., Cambridge, MA: Mass. Inst. Technol., Vol. 15, pp. 90-94, April 1958.
• [Smi2001] Smith Z. A., Delgutte B., Oxenham A. J., “Chimaeric sounds reveal dichotomies in auditory perception,” Nature, vol. 416, pp. 87-90, March 7, 2002.
• [Sno1999a] Snopek K. M., “Czterowymiarowa funkcja niejednoznaczności dwuwymiarowych sygnałów analitycznych,” Referaty IX Krajowego Sympozjum Nauk Radiowych URSI'99, Poznań, March 16-17, 1999, pp. 125-130.
• [Sno1999b] Snopek K.M., “The Comparison of the 4D Wigner Distributions and the 4D Woodward Ambiguity Functions,” Kleinheubacher Berichte, Band 42, pp. 237-246, 1999.
• [Sno2000a] Snopek K. M., “A review of the properties of the Cohen's class time-frequency distributions,” Proc. Applied Electronics 2000, Západočeská Univerzita v Plzni, 2000, Pilsen, September 6-7, 2000, pp. 131-136.
• [Sno2000b] Snopek K. M., “A Cohen's class distributions with separable kernels,” Proc. 1st International Conference on Signals and Electronic Systems ICSES 2000, Ustroń, October 17-20, 2000, 99-104.
• [Sno2001a] Snopek K. M., “The application of the concept of the dual-window pseudo-Wigner distribution in 4-D distributions,” Kleinheubacher Berichte, Band 44, pp. 191-197, 2001.
• [Sno2001b] Snopek K. M., “Rozkłady klasy Cohena sygnałów wielowymiarowych i ich zastosowania,” PhD dissertation, Oficyna Wydawnicza PW, 2001.
• [Sno2005a] Snopek K. M., “The study of properties of double-dimensional pseudo-Wigner distributions,” XI National Symposium of Radio Science, Poznań, April 7-8, 2005, 339-342.
• [Sno2005b] Snopek K. M., “Pseudo-Wigner and double-dimensional pseudo-Wigner distributions with extension for 2-D signals,” Electronics and Telecommun. Quarterly, vol. 51, pp. 9-21, Warsaw 2005.
• [Sno2006a] Snopek K. M., “New Insights into Wigner Distributions of Deterministic and Random Analytic Signals,” Proc. VI International Symposium on Signal Processing and Information Technology ISSPIT’06, Vancouver, August 27-30, 2006, pp. 74-379.
• [Sno2006b] Snopek K. M., “Wigner dostributions and ambiguity functions of radio-frequency telecommunication signals,” Proc. 6th International Conference on Signals and Electronic Systems ICSES’06, Łodź, September 17-20, 2006, pp. 175-178.
• [Sno2007] Snopek K. M., “Wigner distributions of noise and telecommunication stochastic processes,” Proc. 15th European Signal Processing Conference, September 3-7, 2007, Poznań, Poland, pp. 2239-2243.
• [Sno2008] Snopek K. M., “Badanie możliwości wykorzystania sygnałów “chirp” i rozkładów podwójnie wymiarowych w znakowaniu wodnym sygnałów audio,” Przegląd Telekomunikacyjny i Wiadomości Telekomunikacyjne, Zeszyt 4, pp. 205-208, 2008.
• [Sno2009] Snopek K. M., “New Hypercomplex Analytic Signals and Fourier Transforms in Cayley-Dickson Algebras,” Electr. Tel. Quarterly, vol. 55, no. 3, pp. 403-415, 2009.
• [Sno2011a] Snopek K. M., “The New Insight into the Theory of 2-D Complex and Quaternion Analytic Signals," Inti. Journal Electr. Telecomm., vol. 57, no. 3, pp. 285-291, 2011.
• [Sno2011b] Snopek K. M., “The n-D Analytic Signals and Fourier Spectra in Complex and Hypercomplex Domains,” 3th Int. Conf. on Telecommunications and Signal Processing, Budapest, August 18-20, 2011, pp. 423-427, 2011.
• [Sno2012] Snopek K. M. “The Study of Properties of n-D Analytic Signals in Complex and Hypercomplex Domains,” Radioengineering, vol. 21, no. 2, pp. 29-36, April 2012.
• [Sny1997] Snygg J., Clifford Algebra - a Computational Tool for Physicists, Oxford Univ. Press, 1997.
• [Som2001] Sommer G. (Ed.), Geometric Computing with Clifford Algebras: Theoretical Foundations and Applications in Computer Vision and Robotics, Springer-Verlag, 2001.
• [Stu1964] Stutt C.A., “Some Results on Real-Part/Imaginary-Part and Magnitude-Phase Relations in Ambiguity Functions,” IEEE Trans. Info. Theory, vol. 10, no. 4, pp. 321-327, October 1964.
• [Sun2011a] Sun J., Yang J., “A Secure Color Image Watermarking Algorithm Based on Holistic Quaternion Operation,” Adv. Information Science and Service Sciences (AISS), vol. 3, no. 10, pp. 363-374, Nov. 2011.
• [Sun2011b] Sun J., Yang J., Fu D., “Color Images Watermarking Algorithm Based on Quaternion Frequency Singular Value Decomposition,“ Information and Control, Shenyang, vol. 40, no. 6, pp. 813-818, 2011.
• [Sza1990] Szabatin J. Podstawy teorii sygnałów, Wydawnictwa Komunikacji i Łączności, Warszawa 1990.
• [Tia2000] Tian Y., “Matrix Representations of Octonions and Their Applications,” Adv. Appl. Clifford Algebra, vol. 10, no. 1, pp. 61-90, 2000.
• [Tol2006] Tolan T., Özdaş K., Tanişli M., “Reformulation of electromagnetism with octonions,” II Nuovo Cimento, vol. 121, no. 1, pp. 43-55, 2006.
• [Too2009] Took C. C., Mandic D. P., “The Quaternion LMS Algorithm for Adaptive Filtering of Hypercomplex Processes,” IEEE Trans. Signal Processing, vol. 57, no. 4, pp. 1316-1327, April 2009.
• [Tsu2006] Tsui T. K., Zhang X. P., Androutsos D., “Quaternion Image Watermarking using the Spatio-Chromatic Fourier Coefficients Analysis,” 14th annual ACM Int. Conf. on Multimedia, MM’2006, October 23-27, 2006, Santa Barbara, California, USA, pp. 149-152.
• [Ven2013] Venkatramana R. B. D. Yayachandra P. T., Color Image Processing Techniques using Quaternion Fourier Transforms, LAP LAMBERT Academic Publishing, 2013.
• [Vil1948] Ville J., “Theorie et Applications de la Notion de Signal Analytique,” Cables et Transmission, vol. 2A, pp. 61-74, 1948.
• [Wac2012] Wachinger C., Klein T., Navab N., “The 2D analytic signal for envelope detection and feature extraction on ultrasound images”, Medical Image Analysis, vol. 16, pp. 1073-1084, 2012.
• [Wan2010] Wang J., Liu L., “Specific Color-pair Edge Detection using Quaternion Convolution ,” 3rd International Congress on Image and Signal Processing (CISP’2010), pp. 1138-1141.
• [Wan2013] Wang X., Wang C., Yang H., Niu P., “A robust blind color image watermarking in quaternion Fourier transform domain,” J. Systems and Software, vol. 86, no. 2, pp. 255-277, February 2013.
• [Wen2009] Weng Z.-H., Weng Y., “Variation of Gravitational Mass in Electromagnetic Field,” Progress in Electromagnetics Research Symposium, March 23-27, 2009, Beijing, China, pp. 105-107.
• [Wen2010] Weng Z.-H., “Magneto-optic and electro-optic effects in electromagnetic and gravitational fields,” PIERS Proceedings, March 20-23, 2011, Marrakesh, Morocco, pp. 1785-1789.
• [Wen2010a] Weng Z.-H., “Wave Equations in Electromagnetic and Gravitational Fields," Progress in Electromagnetics Research Symposium, July 5-8, 2010, Cambridge, USA, pp. 971-975.
• [Wet2008] Wetula A., “A Hilbert Transform Based Algorithm for Detection of a Complex Envelope of a Power Grid Signals - an Implementation,” Journal of Electrical Power Quality and utilization, vol. XIV, no. 2, pp. 13-18, 2008.
• [Wie1985] Wie B., Barba P. M., “Quaternion Feedback for Spacecraft Large Angle Maneuvres,” J. Guidance Control and Dynamics, vol. 8, no. 3, 1985, pp. 360-365.
• [Wig1932] Wigner E. P., “On the quantum correction for thermodynamic equilibrium,” Phys. Rev., vol. 40, pp. 749-752, 1932.
• [Wil1995] Williams W. L., “Time-Frequency Distributions as Exploratory Tools in the Study of Biological Signals,” Proceedings of 1994 28th Asilomar Conference on Signals, Systems and Computers, pp. 373-377, 1995.
• [Wit2006] Witten B., Shragge J., “Quaternion-based Signal Processing,” Stanford Exploration Project, New Orleans Annual Meeting, pp. 2862-2866, 2006.
• [Woo1953] Woodward P. M., Probability and Information Theoiy with Applications to Radar, New York, Pergamon, 1953.
• [Wood1996] Wood J. C., Barry D. T., "Time-Frequency Analysis of Skeletal Muscle and Cardiac Vibrations,” Proc. IEEE, vol. 84, no. 9, pp. 1281-1294, September 1996.
• [Xu2010] Xu J., Ye L., Luo W., “Color Edge Detection Using Multiscale Quaternion Convolution, “ Ini. J. Imaging Systems and Technology\ vol. 20, no. 4, pp. 354-358, 2010.
• [Yan 2010] Yanshan L., “A new color image blind watermarking algorithm based on quaternion,” 10th Int. Conf. Signal Processing, October 24-28, 2010, Beijing, pp. 1698-1701.
• [YanG2013] Yan G., De Stefano A., Matta E., Feng R., "A novel approach to detecting breathing-fatigue cracks based on dynamic characteristics,” Journal of Sound and Vibration, vol. 332, no. 2, pp. 407-422, January 21, 2013.
• [Ye2007] Ye Y., Yu H., Wei Y., Wang G., “A General Local Reconstruction Approach Based on a Truncated Hilbert Transform,” Int. J. Biomedical Engineering, Hindawi Publishing Corp., vol. 2007, Article ID 63634, 8 pages, 2007.
• [Yeh2008] Yeh M.-H., “Relationships Among Various 2-D Quaternion Fourier Transforms,” IEEE Signal Processing Letters, vol. 15, pp. 669-672, 2008.
• [Yua1988] Yuan J. S. C., “Closed loop Manipulator Control with Quaternion Feedback,” IEEE Trans. Robotics and Automation, vol. 4, no. 4, pp. 434-439, 1988.
• [Zho2007] Zhou J., Xu J., Yang X., “Quaternion wavelet phase based stereo matching for uncalibrated images,” Pattern Recognition Letters, vol. 28, pp. 1509-1522, 2007.