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Envelope detection using generalized analytic signal in 2D QLCT domains

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Abstract

The hypercomplex 2D analytic signal has been proposed by several authors with applications in color image processing. The analytic signal enables to extract local features from images. It has the fundamental property of splitting the identity, meaning that it separates qualitative and quantitative information of an image in form of the local phase and the local amplitude. The extension of analytic signal of linear canonical transform domain from 1D to 2D, covering also intrinsic 2D structures, has been proposed. We use this improved concept on envelope detector. The quaternion Fourier transform plays a vital role in the representation of multidimensional signals. The quaternion linear canonical transform (QLCT) is a well-known generalization of the quaternion Fourier transform. Some valuable properties of the two-sided QLCT are studied. Different approaches to the 2D quaternion Hilbert transforms are proposed that allow the calculation of the associated analytic signals, which can suppress the negative frequency components in the QLCT domains. As an application, examples of envelope detection demonstrate the effectiveness of our approach.

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References

  • Alieva, T., & Bastiaans, M. J. (1999). Powers of transfer matrices determined by means of eigenfunctions. Journal of the Optical Society of America A, 16(10), 2413–2418.

    Article  MathSciNet  Google Scholar 

  • Bayro-Corrochano, E., Trujillo, N., & Naranjo, M. (2007). Quaternion Fourier descriptors for the preprocessing and recognition of spoken words using images of spatiotemporal representations. Journal of Mathematical Imaging and Vision, 28(2), 179–190.

    Article  MathSciNet  Google Scholar 

  • Bülow, T., & Sommer, G. (1997). Multi-dimensional signal processing using an algebraically extended signal representation. In G. Sommer (Ed.), AFPAC 1997 LNCS (Vol. 1315, pp. 148–163). Heidelberg: Springer.

    Google Scholar 

  • Bülow, T., & Sommer, G. (2001). Hypercomplex signals-a novel extension of the analytic signal to the multidimensional case. IEEE Transactions on Signal Processing, 49(1), 2844–2852.

    Article  MathSciNet  Google Scholar 

  • Chen, L., Kou, K. I., & Liu, M. (2015). Pitt’s inequality and the uncertainty principle associated with t he quaternion Fourier transform. Journal of Mathematical Analysis and Applications, 423(1), 681–700.

    Article  MathSciNet  MATH  Google Scholar 

  • Cohen, L. (1995). Time-frequency analysis. Englewood Cliffs, NJ: Prentice-Hall.

    Google Scholar 

  • Ell, T. A., & Sangwine, S. J. (2007). Hyperomplex Fourier Transforms of Color Images. IEEE Transactions on Image Processing, 16(1), 22–35.

    Article  MathSciNet  MATH  Google Scholar 

  • Erden, M. F., Kutay, M. A., & Ozaktas, H. M. (1999). Repeated filtering in consecutive fractional Fourier domains and its application to signal restoration. IEEE Transaction on Signal Processing, 47, 1458–1462.

    Article  Google Scholar 

  • Felsberg, M., & Sommer, G. (2001). The monogenic signal. IEEE Transactions on Signal Processing, 49(12), 3136–3144.

    Article  MathSciNet  Google Scholar 

  • Fu, Y. X., & Li, L. Q. (2008). Generalized analytic signal associated with linear canonical transform. Optics Communications, 281, 1468–1472.

    Article  Google Scholar 

  • Gabor, D. (1946). Theory of communications. Proceedings of the IEE, 93, 429–457.

    Google Scholar 

  • Georgiev, S., & Morais, J. (2013). Bochner’s theorems in the framework of quaternion analysis. In E. Hitzer, S. Sangwine (Eds.), Quaternion and Clifford–Fourier transforms and wavelets trends in mathematics (Vol. 27, pp. 85–104) Springer Basel AG.

  • Georgiev, S., Morais, J., Kou, K.I., & Sprößig, W. (2013) Bochner–Minlos theorem and quaternion Fourier transform. In E. Hitzer, S. Sangwine (Eds.), Quaternion and Clifford–Fourier transforms and wavelets trends in mathematics (Vol. 27, pp. 105–120) Springer Basel AG.

  • Gröchenig, K. (2001). Foundations of time-frequency analysis. Boston, Mass: Birkhäuser.

    Book  MATH  Google Scholar 

  • Gürlebeck, K., & Sprössig, W. (1989). Quaternionic analysis and elliptic boundary value problems. Berlin: Akademie Verlag.

    MATH  Google Scholar 

  • Gürlebeck, K., & Sprössig, W. (1997). Quaternionic calculus for engineers and physicists. Chichester: Wiley.

    MATH  Google Scholar 

  • Hahn, S. L. (1992). Multidimensional complex signals with single-orthant spectra. Proceedings of the IEEE, 80(8), 1287–1300.

    Article  Google Scholar 

  • Havlicek, J.P., Havlicek, J.W., & Bovik, A.C. (1997). The analytic image. In: ICIP 97. Proceedings of international conference on image processing (Vol. 2, pp. 446–449).

  • Havlicek, J.P., Havlicek, J.W., Mamuya, N.D., & Bovik, A.C. (1998). Skewed 2D Hilbert transforms and computed AM-FM models. In ICIP 98. Proceedings of international conference on image processing (Vol. 1, pp. 602–606).

  • Hitzer, E. M. S. (2007). Quaternion Fourier transform on quaternion fields and generalizations. Advances in Applied Clifford Algebras, 17, 497–517.

    Article  MathSciNet  MATH  Google Scholar 

  • Kou, K., Morais, J., & Zhang, Y. (2013). Generalized prolate spheroidal wave functions for offset linear canonical transform in Clifford analysis. Mathematical Methods in the Applied Sciences, 36, 1028–1041.

    Article  MathSciNet  MATH  Google Scholar 

  • Kou, K.I., Ou, J., & Morais, J. (2013). On uncertainty principle for quaternionic linear canonical transform. In Abstract and applied analysis (Vol. 2013, Article ID 725952, 14 p). doi:10.1155/2013/725952.

  • Kou, K. I., Ou, J., & Morais, J. (2016). Uncertainty principles associated with quaternionic linear canonical transforms. Mathematical Methods and Applied Sciences,. doi:10.1002/mma.3724.

    MathSciNet  MATH  Google Scholar 

  • Kou, K. I., & Morais, J. (2014). Asymptotic behaviour of the quaternion linear canonical transform and the Bochner–Minlos theorem. Applied Mathematics and Computation, 247, 675–688.

    Article  MathSciNet  MATH  Google Scholar 

  • Kohlmann, K. (1996). Corner detection in natural images based on the 2D Hilbert Transform. Signal Processing, 48, 225–234.

    Article  MATH  Google Scholar 

  • Kovesi, P. (1999). Image features from phase congruency. Videre Journal of Computer Vision Research, 1(3), 1–26.

    Google Scholar 

  • Kravchenko, V., & Shapiro, M. (1996). Integral representations for spatial models of mathematical physics., Research notes in mathematics London: Pitman Advanced Publishing Program.

    MATH  Google Scholar 

  • Kravchenko, V. (2003). Applied quaternionic analysis. Research and exposition in mathematics (Vol. 28). Lemgo: Heldermann Verlag.

    Google Scholar 

  • Larkin, K. G., Bone, D. J., & Oldfield, M. A. (2001). Natural demodulation of two-dimensional fringe patterns. I. General background of the spiral phase quadrature transform. Journal of the Optical Society of America, 18(8), 1862–1870.

    Article  Google Scholar 

  • Liu, Y. L., Kou, K. I., & Ho, I. T. (2010). New sampling formulae for non-bandlimited signals associated with linear canonical transform and nonlinear Fourier atoms. Signal Processing, 90(3), 933–945.

    Article  MATH  Google Scholar 

  • Mawardi, B., Hitzer, E., Hayashi, A., & Ashino, R. (2008). An uncertainty principle for quaternion Fourier transform. Computers & Mathematics with Applications, 56(9), 2411–2417.

    Article  MathSciNet  MATH  Google Scholar 

  • Mawardi, B., Hitzer, E., Ashino, R., & Vaillancourt, R. (2010). Windowed Fourier transform for two-dimensional quaternionic signals. Applied Mathematics and Computation, 216, 2366–2379.

    Article  MathSciNet  MATH  Google Scholar 

  • Pei, S.-C., Ding, J.-J., & Chang, J.-H. (2001). Efficient implementation of quaternion Fourier transform, convolution, and correlation by 2-D complex FFT. IEEE Transations on Signal Processing, 49(11), 2783–2797.

    Article  MathSciNet  Google Scholar 

  • Pei, S. C., & Ding, J. J. (2003). The generalized radial Hilbert transform and its applications to 2-D edge detection (any direction or specified directions), In: ICASSP’03. Proceedings of the IEEE international conference on acoustics, speech, and signal processing (Vol. 3, pp. 357–360).

  • Sangwine, S. J., & Ell, T. A. (2007). Hypercomplex Fourier transforms of color images. IEEE Transactions on Image Processing, 16(1), 22–35.

    Article  MathSciNet  MATH  Google Scholar 

  • Wolf, K. B. (1979). Integral transforms in science and engineering, Vol. 11, chapter 9: canonical transforms. New York, NY: Plenum Press.

  • Xu, G., Wang, X., & Xu, X. (2009). Generalized Hilbert transform and its properties in 2D LCT domain. Signal Processing, 89, 1395–1402.

    Article  MATH  Google Scholar 

  • Yang, Y., & Kou, K. I. (2014). Uncertainty principles for hypercomplex signals in the linear canonical transform domains. Signal Processing, 95, 67–75.

    Article  Google Scholar 

  • Yin, M., Liu, W., Shui, J., & Wu, J. (2012). Quaternion wavelet analysis and application in image denoising. Mathematical Problems in Engineering, 2012, 493976. doi:10.1155/2012/493976.

  • Zalevsky, Z., Mendlovic, D., Kutay, M. A., Ozaktas, H. M., & Solomon, J. (2001). Improved acoustic signals discrimination using fractional Fourier transformbased phase-space representations. Optics Communications, 190(1–6), 95–101.

    Article  Google Scholar 

Download references

Acknowledgments

The authors acknowledge financial support from the National Natural Science Funds for Young Scholars (Nos. 11401606, 11501015) and University of Macau Nos. MYRG2015-00058-FST, MYRG099(Y1-L2)-FST13-KKI and the Macao Science and Technology Development Fund FDCT/094/2011A, FDCT/099/2012/A3. The second author also acknowledges financial support from the Guangdong Natural Science Foundation (Grant Nos. 2014A030307016, 2014A030313422). The third author’s work is supported by the Asociación Mexicana de Cultura, A. C.

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Authors and Affiliations

  1. School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, Guangdong, China

    Kit Ian Kou & Ming-Sheng Liu

  2. Department of Mathematics, University of Macau, Taipa, Macau

    Kit Ian Kou & Cuiming Zou

  3. Departamento de Matemáticas, Instituto Tecnológico Autónomo de México, Río Hondo #1, Col. Progreso Tizapan, 01080, Mexico, DF, Mexico

    João Pedro Morais

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  1. Kit Ian Kou
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  2. Ming-Sheng Liu
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  4. Cuiming Zou
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Correspondence to Kit Ian Kou.

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Kou, K.I., Liu, MS., Morais, J.P. et al. Envelope detection using generalized analytic signal in 2D QLCT domains. Multidim Syst Sign Process 28, 1343–1366 (2017). https://doi.org/10.1007/s11045-016-0410-7

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  • DOI: https://doi.org/10.1007/s11045-016-0410-7

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