Abstract
The fractional transforms are a class of powerful tool for the presentation of time–frequency domains in the field of signal processing. Based on the convolution algorithm of discrete fractional Fourier transform and gyrator transform, we propose a generalized framework defining a class of fractional transforms. By choosing various phase filters, the fractional transform can be employed for different computational tasks of information processing. The several properties of typical fractional transform are reserved in this definition scheme. Under the model of the convolution-based transform, fractional Fourier transform and gyrator transform are synthesized. Moreover, the transform can be implemented by an optical 4f system with phase-only filtering easily, which is a useful tool in the application of optical information processing. Numerical results are given for demonstrating the proposed transform and its application.
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References
Chen, H., Du, X., Liu, Z.: Optical spectrum encryption in associated fractional Fourier transform and gyrator transform domain. Opt. Quant. Electron. 48, 12 (2016a). doi:10.1007/s11082-015-0291-2.
Chen, H., Tanougast, C., Liu, Z., Hao, B.: Securing color image by using hyperchaotic system in gyrator transform domains. Opt. Quant. Electron. 48, 396 (2016b). doi:10.1007/s11082-016-0669-9.
Goodman, J.W.: Introduction to Fourier Optics, pp. 243–251. McGraw-Hill, New York (1968)
Lang, J., Tao, R., Wang, Y.: Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function. Opt. Commun. 283, 2092–2096 (2010)
Liu, Z., Liu, S.: Randomization of the Fourier transform. Opt. Lett. 32, 478–480 (2007a).
Liu, Z., Liu, S.: Random fractional Fourier transform. Opt. Lett. 32, 2088–2090 (2007b)
Liu, Z., Zhao, H., Liu, S.: A discrete fractional random transform. Opt. Commun. 255, 357–365 (2005)
Liu, Z., Ahmad, M.A., Liu, S.: A discrete fractional angular transform. Opt. Commun. 281, 1424–1429 (2008)
Liu, Z., Chen, D., Ma, J., Wei, S., Zhang, Y., Dai, J., Liu, S.: Fast algorithm of discrete gyrator transform based on convolution operation. Optik 122, 864–867 (2011a)
Liu, Z., Zhang, Y., Zhao, H., Ahmad, A.A., Liu, S.: Optical multi-image encryption based on frequency shift. Optik 122, 1010–1013 (2011b)
Liu, Z., Gong, M., Dou, Y., Liu, F., Lin, S., Ahmad, M.A., Dai, J., Liu, S.: Double image encryption by using Arnold transform and discrete fractional angular transform. Opt. Lasers Eng. 50, 248–255 (2012)
Liu, Z., Tan, J., Liu, W., Wu, J., Wu, Q., Liu, S.: A diffraction model of direction multiplexing method for hiding multiple images. J. Mod. Opt. 61, 1127–1132 (2014)
Liu, Z., Guo, C., Tan, J., Wu, Q., Pan, L., Liu, S.: Iterative phase-amplitude retrieval with multiple intensity images at output plane of gyrator transforms. J. Opt. 17, 025701 (2015). doi:10.1088/2040-8978/17/2/025701.
Lohmann, A.W.: Image rotation, Wigner rotation, and the fractional Fourier transform. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 10, 2181–2186 (1993)
Ozaktas, H.M., Zalevsky, Z., Kutay, M.A.: The Fractional Fourier Transform with Applications in Optics and Signal Processing, 1–100. Wiley, New York (2000)
Pei, S.C., Hsue, W.L.: The multiple-parameter discrete fractional Fourier transform. IEEE Signal Process. Lett. 13, 329–332 (2006)
Pei, S.C., Hsue, W.L.: Random discrete fractional Fourier transform. IEEE Signal Process. Lett. 16, 1015–1018 (2009)
Pei, S.C., Yeh, M.H.: Improved discrete fractional Fourier transform. Opt. Lett. 22, 1047–1049 (1997)
Rodrigo, J.A., Alieva, T., Calvo, M.L.: Experimental implementation of the gyrator transform. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 24, 3135–3139 (2007)
Yang, X., Tan, Q., Wei, X., Xiang, Y., Yan, Y., Jin, G.: Improved fast fractional-Fourier transform algorithm. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 21, 1677–1681 (2004)
Zhang, Z.-C.: New convolution structure for the linear canonical transform and its application in filter design. Optik 127, 5259–5263 (2016)
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 61571160, 61377016, 61575055 and 61575053), by Program for New Century Excellent Talents in University (No. NCET-12-0148), by the Fundamental Research Funds for the Central Universities (No. HIT.BRETIII.201406), the China Postdoctoral Science Foundation (Nos. 2013M540278 and 2015T80340), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry, China.
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Dou, J., He, Q., Peng, Y. et al. A convolution-based fractional transform. Opt Quant Electron 48, 407 (2016). https://doi.org/10.1007/s11082-016-0685-9
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DOI: https://doi.org/10.1007/s11082-016-0685-9