Two-dimensional quaternion Fourier transform of type II and quaternion wavelet transform

A two-dimensional quaternion Fourier transform (QFT) defined with the kernel e - i+j+k/√3 ω · x is proposed. Some fundamental properties, such as convolution theorem and Plancherel theorem are established. The wavelet transform is extended to quaternion algebra using the kernel of the QFT.

[1]  Soo-Chang Pei,et al.  Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT , 2001, IEEE Trans. Signal Process..

[2]  Mawardi Bahri,et al.  Clifford Algebra Cl3, 0-Valued Wavelet Transformation, Clifford Wavelet Uncertainty inequality and Clifford Gabor Wavelets , 2007, Int. J. Wavelets Multiresolution Inf. Process..

[3]  Bo Yu,et al.  Continuous wavelet transforms on the space L2(R, H;dx) , 2004, Appl. Math. Lett..

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

[5]  Mawardi Bahri,et al.  An uncertainty principle for quaternion Fourier transform , 2008, Comput. Math. Appl..

[6]  T. Ell,et al.  Quaternion-Fourier transforms for analysis of two-dimensional linear time-invariant partial differential systems , 1993, Proceedings of 32nd IEEE Conference on Decision and Control.

[7]  Nicolas Le Bihan,et al.  Fast Complexified Quaternion Fourier Transform , 2006, IEEE Transactions on Signal Processing.

[8]  Sofia C. Olhede,et al.  The Monogenic Wavelet Transform , 2009, IEEE Transactions on Signal Processing.

[9]  Mawardi Bahri.  Quaternion Algebra-Valued Wavelet Transform , 2011 .

[10]  Rémi Vaillancourt,et al.  Two-dimensional quaternion wavelet transform , 2011, Appl. Math. Comput..

[11]  Ming Zhu,et al.  Quaternion Fourier-Mellin moments for color images , 2011, Pattern Recognit..

[12]  Mawardi Bahri,et al.  Clifford Algebra-Valued Wavelet Transform on Multivector Fields , 2011 .

[13]  Eckhard Hitzer,et al.  Directional Uncertainty Principle for Quaternion Fourier Transform , 2010, 1306.1276.

[14]  Eckhard Hitzer,et al.  Clifford Fourier Transform on Multivector Fields and Uncertainty Principles for Dimensions n = 2 (mod 4) and n = 3 (mod 4) , 2008 .

[15]  Jun Zhou,et al.  Quaternion wavelet phase based stereo matching for uncalibrated images , 2007, Pattern Recognit. Lett..

[16]  Eduardo Bayro-Corrochano,et al.  The Theory and Use of the Quaternion Wavelet Transform , 2005, Journal of Mathematical Imaging and Vision.

[17]  Stephen J. Sangwine,et al.  Hypercomplex Fourier Transforms of Color Images , 2007, IEEE Trans. Image Process..

[18]  Henning Rasmussen,et al.  Local quaternion Fourier transform and color image texture analysis , 2010, Signal Process..

[19]  Leonardo Traversoni.  Image Analysis Using Quaternion Wavelets , 2001 .

[20]  Eckhard Hitzer,et al.  Quaternion Fourier Transform on Quaternion Fields and Generalizations , 2007, ArXiv.