一类四元数线性正则变换的不确定性原理
Uncertainty Principle for a Kind of Quaternionic Linear Canonical Transform
DOI: 10.12677/AAM.2014.33020,
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作者:
付应雄, 熊 珍：湖北大学数学与统计学学院，武汉
关键词:
四元数；左边四元数线性正则变换；不确定性原理；Quaternion； Left-Sided Quaternionic Linear Canonical Transform； Uncertainty Principle
摘要:
基于左边四元数线性正则变换的相关性态，本文建立左边四元数线性正则变换的不确定性原理。其表明四元数值的信号在时域和频域中方差的乘积具有下界，仅有二维的高斯信号能满足不确定性原理的等式。
Abstract: In this paper, based on the properties of the left-sided quaternionic linear canonical transform (QLCT), an uncertainty principle is established for the left-sided QLCT. It states that the product of the variances of quaternion-valued signals in the spatial and frequency domains has a lower bound and only a 2D Gaussian signal minimizes the uncertainty principle.
文章引用:
付应雄, 熊珍. 一类四元数线性正则变换的不确定性原理[J]. 应用数学进展, 2014, 3(3): 134-139. http://dx.doi.org/10.12677/AAM.2014.33020
参考文献
|
[1]
|
Hamilton, W.R. (1866) Elements of Quaternions. Longmans, Green and Co., London.
|
|
[2]
|
Kantor, I.L. and Solodov-nikov, A.S. (1989) Hypercomplex number: An elementary introduction to algebras. Springer-Verlag, New York.
|
|
[3]
|
Pei, S.C. and Cheng, C.M. (1997) A novel block truncation coding of color image using a quater-nion-moment-pre- serving principle. IEEE Transactions on Communications, 45, 583-595.
|
|
[4]
|
Sangwine, S.J. (1998) Color image edge detector based on quaternion convolution. Electronics Letters, 34, 969-971.
|
|
[5]
|
Sangwine, S.J. and Ell, T.A. (2007) Hypercomplex Fourier transforms of color images. IEEE Transactions on Image Processing, 16, 22-35.
|
|
[6]
|
Evans, C.J., Sangwine, S.J. and Ell, T.A. (2000) Hypercomplex color-sensitive smoothing filters. IEEE International Conference on Image Processing, 1, 541-544.
|
|
[7]
|
Mawardi, B., Hitzer, E., Hayashi, A. and Ashino, R. (2008) An uncertainty principle for quaternion Fourier transform. Computers and Mathematics with Applications, 56, 2411-2417.
|
|
[8]
|
Moshinsky, M. and Quesne, C. (1971) Linear canonical transformations and their unitary representa-tions. Journal of Mathematical Physics, 12, 1772-1783.
|
|
[9]
|
Collins, S.A. (1970) Lens-system diffraction integral written in terms of matrix optics. Journal of the Optical Society of America, 60, 1168-1177.
|
|
[10]
|
Healy, J.J. and She-ridan, J.T. (2010) Fast linear canonical transforms. Journal of the Optical Society of America A, 27, 21-30.
|
|
[11]
|
Ozaktas, H.M., Kutay, M.A. and Zalevsky, Z. (2001) The fractional Fourier transform with applications in optics and signal processing. Wiley, New York, 93-95.
|
|
[12]
|
Tao, R., Qi, L. and Wang, Y. (2004) Theory and applications of the fractional Fourier transform. Tsinghua University Press, Beijing, 73-76.
|
|
[13]
|
Barshan, B., Kutay, M.A. and Ozaktas, H.M. (1997) Optimal filters with linear canonical transformations. Optics Communications, 135, 32-36.
|
|
[14]
|
Pei, S.C. and Ding, J.J. (2003) Eigenfunctions of the offset Fourier, fractional Fourier, and linear canonical transforms. Journal of the Optical Society of America A, 20, 522-532.
|
|
[15]
|
Sharma, K.K. and Joshi, S.D. (2006) Signal separation using linear canonical and fractional Fourier transforms. Optics Communications, 256, 454-460.
|
|
[16]
|
Xu, G.L., Wang, X.T. and Xu, X.G. (2009) Generalized Hilbert transform and its properties in 2D LCT domain. Signal Process, 89, 1395-1402.
|
|
[17]
|
Tao, R., Li, B.Z., Wang, Y. and Aggrey, G.K. (2008) On sampling of band-limited signals associated with the linear canonical transform. IEEE Transactions on Signal Processing, 56, 5454-5464.
|
|
[18]
|
Stern, A. (2008) Uncertainty principles in linear canonical transform domains and some their implications in optics. Journal of the Optical Society of America A, 25, 647-652.
|
|
[19]
|
Kou, K.I., Ou, J.Y. and Morais, J. (2013) On uncertainty principle for quaternionic linear canonical transform. Abstract and Applied Analysis, Article ID: 725952.
|