分数阶小波变换及应用研究
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摘要
随着信息时代的到来,作为信息载体的信号及相关处理技术获得了迅猛的发展,信号处理技术已经渗透到生物医学、地球物理、射电天文、机械工程、通信系统等各个方面。时频分析作为一种重要的信号处理方法,近年来受到越来越广泛的关注,其中比较经典的方法有小波变换(WT)、分数阶Fourier变换(FRFT)等。随着时频分析技术的不断发展,分数阶小波变换(FRWT)应运而生,成为目前信号分析领域的一个学术前沿。
本论文研究分数阶小波变换及其应用,主要研究工作为:
(1)分析了分数阶小波变换的研究现状,阐述了分数阶小波变换的定义、性质及光学实现。给出了实现分数阶小波变换的方法,从分数阶小波域与时、频域间的关系可以看出FRWT实质上是一种统一的时频变换,能够同时反映信号在时域和频域的信息,适合于处理非平稳信号。
(2)提出了一种分数阶小波时频域的信号去噪新方法。用输出信号信噪比作为判据,采用遗传算法寻找分数阶小波变换的最优分数阶p值,通过分数阶小波变换将带噪信号映射到最优分数阶小波时频域内,对变换后的信号进行窄带通滤波,最后通过分数阶小波逆变换对信号进行重构,实现分数阶小波域内的信号去噪。以带噪Bumps信号和语音信号为例的去噪实验结果表明,采用新方法去噪后的信号信噪比明显提高,在抑制噪声的同时可以有效保留细节信息。
(3)针对数字图像信息加密问题,提出了一种基于分数阶小波变换的图像置乱新方法。新方法将分数阶小波变换与图像置乱加密相结合,首先选择三组密钥即分数阶数、小波分解尺度和置乱映射,对原始图像进行已知阶数和分解尺度的分数阶小波变换,然后在分数阶小波域内对变换后的图像进行映射置乱,得到加密后的密文图像。图像的加密实验结果表明,新方法的密钥空间大,保密性高。
With the coming of information age, signal which is the carrier of information, and its processing techniques have obtained a rapid development. Signal processing techniques has penetrated into many fields, such as biomedical, geophysics, radio astronomy, mechanical engineering, communication systems and so on. As an important signal processing method, time-frequency analysis has been more and more concerned in recent years. Wavelet transform(WT) and fractional Fourier transform (FRFT) are classic methods in time-frequency analysis. With the continuous development of WT and FRFT, fractional wavelet transform (FRWT) as a new time-frequency analysis method came into being, and it became a focus of frontier signal processing area.
In this paper, research on fractional wavelet transform and its applications is made. The main research works can be summarized as follows:
(1) The recent research situation of fractional wavelet transform is analyzed. The definition, nature and optical realization of FRWT are summarized. Besides, the method to realize fractional wavelet transform is given. According to the relationship of the fractional wavelet domain and time-frequency domain, FRWT is a time-frequency transform. It can reflect the information in time domain and frequency domain at the same time, and is suitable for non-stationary signal processing.
(2) A novel signal de-noising method based on FRWT is proposed. The optimal fractional order of FRWT is obtained by a genetic algorithm according to the SNR of output signals. The noisy signal is transformed into optimal factional wavelet time-frequency domain by an optimal FRWT. Then, the transformed signal is filtered by a narrow band-pass filter. Finally, the signal is reconstructed by an inverse FRWT. De-noising results of noisy Bumps signal and speech signal show that the SNR of output signals can be effectively improved. Results also show that the proposed method can preserve detail information effectively and reduce the noise at the same time.
(3) For ensuring the security of digital image information, a novel image scrambling encryption method based on FRWT is proposed. We can encrypt the image by FRWT and scrambling transform. Firstly, factional orders, a series of scaling factors and a scrambling algorithm are chosen. The original image is transformed into factional wavelet time-frequency domain by a FRWT. Then, the transformed image is scrambled by a scrambling algorithm, and the encrypted image is got. Encryption simulations prove its possibility.
本论文研究分数阶小波变换及其应用,主要研究工作为:
(1)分析了分数阶小波变换的研究现状,阐述了分数阶小波变换的定义、性质及光学实现。给出了实现分数阶小波变换的方法,从分数阶小波域与时、频域间的关系可以看出FRWT实质上是一种统一的时频变换,能够同时反映信号在时域和频域的信息,适合于处理非平稳信号。
(2)提出了一种分数阶小波时频域的信号去噪新方法。用输出信号信噪比作为判据,采用遗传算法寻找分数阶小波变换的最优分数阶p值,通过分数阶小波变换将带噪信号映射到最优分数阶小波时频域内,对变换后的信号进行窄带通滤波,最后通过分数阶小波逆变换对信号进行重构,实现分数阶小波域内的信号去噪。以带噪Bumps信号和语音信号为例的去噪实验结果表明,采用新方法去噪后的信号信噪比明显提高,在抑制噪声的同时可以有效保留细节信息。
(3)针对数字图像信息加密问题,提出了一种基于分数阶小波变换的图像置乱新方法。新方法将分数阶小波变换与图像置乱加密相结合,首先选择三组密钥即分数阶数、小波分解尺度和置乱映射,对原始图像进行已知阶数和分解尺度的分数阶小波变换,然后在分数阶小波域内对变换后的图像进行映射置乱,得到加密后的密文图像。图像的加密实验结果表明,新方法的密钥空间大,保密性高。
With the coming of information age, signal which is the carrier of information, and its processing techniques have obtained a rapid development. Signal processing techniques has penetrated into many fields, such as biomedical, geophysics, radio astronomy, mechanical engineering, communication systems and so on. As an important signal processing method, time-frequency analysis has been more and more concerned in recent years. Wavelet transform(WT) and fractional Fourier transform (FRFT) are classic methods in time-frequency analysis. With the continuous development of WT and FRFT, fractional wavelet transform (FRWT) as a new time-frequency analysis method came into being, and it became a focus of frontier signal processing area.
In this paper, research on fractional wavelet transform and its applications is made. The main research works can be summarized as follows:
(1) The recent research situation of fractional wavelet transform is analyzed. The definition, nature and optical realization of FRWT are summarized. Besides, the method to realize fractional wavelet transform is given. According to the relationship of the fractional wavelet domain and time-frequency domain, FRWT is a time-frequency transform. It can reflect the information in time domain and frequency domain at the same time, and is suitable for non-stationary signal processing.
(2) A novel signal de-noising method based on FRWT is proposed. The optimal fractional order of FRWT is obtained by a genetic algorithm according to the SNR of output signals. The noisy signal is transformed into optimal factional wavelet time-frequency domain by an optimal FRWT. Then, the transformed signal is filtered by a narrow band-pass filter. Finally, the signal is reconstructed by an inverse FRWT. De-noising results of noisy Bumps signal and speech signal show that the SNR of output signals can be effectively improved. Results also show that the proposed method can preserve detail information effectively and reduce the noise at the same time.
(3) For ensuring the security of digital image information, a novel image scrambling encryption method based on FRWT is proposed. We can encrypt the image by FRWT and scrambling transform. Firstly, factional orders, a series of scaling factors and a scrambling algorithm are chosen. The original image is transformed into factional wavelet time-frequency domain by a FRWT. Then, the transformed image is scrambled by a scrambling algorithm, and the encrypted image is got. Encryption simulations prove its possibility.
引文
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[2] S. Soltani, P. Simard, D. Boichu. Estimation of the self-similarity parameter using the wavelet transform[J]. Signal Processing, 2004, 84(1): 117-123.
[3] Felix C. A. Fernandes, Ivan W. Selesnick, Rutger L. C. van Spaendonck, et al. Complex wavelet transforms with allpass filters[J]. Signal Processing, 2003, 83(8): 1689-1706.
[4] Elias Masry. Covariance and spectral properties of the wavelet transform and discrete wavelet coefficients of second-order random fields[J]. Signal Processing, 1998, 64(2): 131-143.
[5] Erdal Din?, Dumitru Baleanu. Mathematical Methods in Engineering: A review on the wavelet transform applications in analytical chemistry[B]. Netherlands: Springer Netherlands, 2007. 265-284.
[6] Erdal Dinc, Dumitru Baleanu, Giuseppina Ioele, et al. Multivariate analysis of paracetamol, propiphenazone, caffeine and thiamine in quaternary mixtures by PCR, PLS and ANN calibrations applied on wavelet transform data[J]. Journal of Pharmaceutical and Biomedical Analysis, 2008, 48(5): 1471-1475.
[7] A. Arneodo, Y. D'Aubenton-Carafa, B. Audit. What can we learn with wavelets about DNA sequences[J]. Physica A: Statistical and Theoretical Physics, 1998, 249(1-4): 439-448.
[8] A. Arneodo, Y. d'Aubenton-Carafa, E. Bacry. Wavelet based fractal analysis of DNA sequences[J]. Physica D: Nonlinear Phenomena, 1996, 96(1-4): 291-320.
[9]陶然,邓兵,王越.分数阶Fourier变换在信号处理领域的研究进展[J].中国科学E辑:信息科学,2006,36(2):113-136.
[10]罗慧,王友仁,崔江.基于最优分数阶傅里叶变换的模拟电路故障特征提取新方法[J].仪器仪表学报,2009,30(5): 997-1001.
[11] Yangquan Chen, Rongtao Sun, Anhong Zhou. An improved Hurst parameter estimator based on fractional Fourier transform[J]. Telecommunication Systems, 2010, 43(3-4): 179-206.
[12] Mendlovic D, Zalevsky Z, Mas D,et al. Fractional wavelet transform[J]. Applied Optics, 1997, 36(20): 4801-4806.
[13] Ying Huang. The fractional wave packet transform[J]. Multidimensional Systems and Signal Processing, 1998, 4(9): 399-402.
[14] Unser M, Blu T. Construction of fractional spline wavelet bases[M]. Bellingham: SPIE-INT SocOptical Engineering, 1999: 422-431.
[15] Unser M, Blu T. Fractional splines and wavelets[J]. SIAM Review, 2000, 42(1): 43-67.
[16] Blu T, Unser M. The fractional spline wavelet transform: definition end implementation[C]. IEEE International Conference on Acoustics, Speech, and Signal Processing, Istanbul, Turkey, 2000, 6: 512-515.
[17] Alpana Bhagatji, Naveen K. Nishcahl, Arun K. Gupta, et al. Extended fractional wavelet joint transform correlator[J]. Optics Communications, 2008, 281(1): 44-48.
[18] Rutuparna Panda, Madhumita Dash. Fractional generalized splines and signal processing [J]. Signal Processing, 2006, 9(86): 2340-2350.
[19] F. Abdelliche, A. Charef, M.L. Talbi, et al. A fractional wavelet for QRS detection[C]. The 2nd International Conference on Information and Communication Technologies, Damascus, 2006, 4: 1186-1189.
[20]黄思齐.分数阶小波变换[J].测控技术.2009,28(02):12-16.
[21] Yuan Lin. Wavelet-fractional Fourier transforms[J]. Institute of Physics Publishing. 2008, 17(1): 170-179.
[22] F Abdelliche, A Charef, M.L Talbi, et al. A fractional wavelet for QRS detection[J]. Information and Communication Technologies, 2006, 1: 1186-1189.
[23] F Abdelliche1, A Charef. R-peak detection using a complex fractional wavelet[C]. International Conference on Electrical and Electronics Engineering, Bursa, Turkey, 2009, 11: 267-270.
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