利用充分光滑的S形函数构造Meyer小波
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摘要
为了在信号或图像的重构中获得较好的平滑效果,必须尽量增大小波的正则性或者连续可微性。在Meyer小波构造中S形函数的选取影响着Meyer小波的可微性、光滑性和衰减速度等性质,所以S形函数的选取至关重要。给出一种构造充分光滑的S形函数的方法,并以一个充分光滑的非多项式S形函数为例,将其作为BP神经网络中的激励函数进行函数逼近得到好的逼近效果且训练次数少。然后通过充分光滑的S形函数得到Meyer小波的尺度函数,给出相应的具有充分光滑、高阶消失矩且无穷次可微性的频谱有限的Meyer小波。最后把充分光滑的Meyer小波与剪切波变换结合进行图像去噪,与传统的Meyer小波剪切波变换去噪相比较,峰值信噪比高于传统的Meyer剪切波变换且去噪后的图像纹理和边缘信息保留更加完整。
In order to obtain better smooth effect in signal or image reconstruction, the regularity or continuous differentiability of wavelet must be increased as much as possible. The selection of sigmoid function in Meyer wavelet construction affects the differentiability, smoothness and attenuation speed of Meyer wavelet, so it is exceedingly essential to select sigmoid function. Firstly, a method of constructing fully smooth sigmoid function is given, and a sufficiently smooth non-polynomial sigmoid function is taken as an example, which is used as the excitation function in BP neural network for function approximation to obtain good approximation results. Then, the scale functions of Meyer wavelets are obtained by fully smooth sigmoid functions. Meyer wavelets with limited spectrum are given with sufficient smooth, high-order vanishing moments and infinitesimal differentiability. Finally, the full smooth Meyer wavelet and the shearlet transform are combined to denoise the image. Compared with the traditional Meyer wavelet shearlet transform, the peak signal-to-noise ratio is higher and the denoised image texture and edge information are more complete.
In order to obtain better smooth effect in signal or image reconstruction, the regularity or continuous differentiability of wavelet must be increased as much as possible. The selection of sigmoid function in Meyer wavelet construction affects the differentiability, smoothness and attenuation speed of Meyer wavelet, so it is exceedingly essential to select sigmoid function. Firstly, a method of constructing fully smooth sigmoid function is given, and a sufficiently smooth non-polynomial sigmoid function is taken as an example, which is used as the excitation function in BP neural network for function approximation to obtain good approximation results. Then, the scale functions of Meyer wavelets are obtained by fully smooth sigmoid functions. Meyer wavelets with limited spectrum are given with sufficient smooth, high-order vanishing moments and infinitesimal differentiability. Finally, the full smooth Meyer wavelet and the shearlet transform are combined to denoise the image. Compared with the traditional Meyer wavelet shearlet transform, the peak signal-to-noise ratio is higher and the denoised image texture and edge information are more complete.
引文
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[2] MALLATS.A Wavelet Tour of Signal Processing [J].Wavelet Analysis & Its Applications,1999:74.
[3] 丁锋,秦峰伟.小波降噪及Hilbert变换在电机轴承故障诊断中的应用[J].电机与控制学报,2017,21(6):89.
[4] 孙鹏跃,唐小妹,黄仰博,等.利用实时小波降噪的抗电离层闪烁载波跟踪[J].国防科技大学学报,2017,39(5):38.
[5] 尤波,李忠杰,黄玲,等.手部抓取动作特征提取算法研究[J].电机与控制学报,2017,21(12):75.
[6] 刘玉琪.基于随机森林算法的人体运动模式识别研究[D].北京:北京邮电大学,2018.
[7] MALLAT S.Multiresolution Approximations and Wavelet Orthonormal Bases of L2(R) [J].Transactions of the American Mathematical Society,1989,315(1):69.
[8] 樊启斌.小波分析[M].武汉大学出版社,2008:92.
[9] MEYER Y.Uncertainty Principle,Hilbertian Bases and Operator Algebras [J].Paris Bourbaki Seminar,1986,145-146(4):209.
[10] 张占利.一类椭圆型方程不适定问题的Meyer小波正则化方法[J].数学进展,2016,45(3):390.
[11] JUN QIN,PENGFEI SUN.Applications and Comparison of Continuous Wavelet Transforms on Analysis of A-wave Impulse Noise [J].Archives of Acoustics,2016,40(4):84.
[12] SELVAN A A,RADHA R.An Optimal Result for Sampling Density in Shift-invariant Spaces Generated by Meyer Scaling Function [J].Journal of Mathematical Analysis & Applications,2017,451(1):197.
[13] BRANI VIDAKOVIC.Statistical Modeling by Wavelets [M].Beijing:Higher Education Press,2007:52.
[14] 彭玉华.小波变换与工程应用[M].北京:科学出版社,1999:21.
[15] QIAN L I,WANG Z.Image Compression Based on Wavelet Analysis [J].Communications Technology,2010,25(2):1778.
[16] 汪可,张书琦,李金忠,等.基于灰度图像分解的局部放电特征提取与优化[J].电机与控制学报,2018,22(5):25.
[17] EASLEY G,LABATE D,LIM W Q.Sparse Directional Image Representations Using the Discrete Shearlet Transform[J].Applied & Computational Harmonic Analysis,2008,25(1):25.
[18] 程义平.基于平移不变性的剪切波变换医学超声图像去噪算法[D].杭州:浙江工业大学,2016.
[19] ABAZARI R,LAKESTANI M.A Hybrid Denoising Algorithm Based on Shearlet Transform Method and Yaroslavsky’s Filter [J].Multimedia Tools & Applications,2018,77(14):17829.
[20] 潘庆先,董红斌,韩启龙,等.一种基于BP神经网络的属性重要性计算方法[J].中国科学技术大学学报,2017(1):18.