基于离散Gabor变换的信号稀疏时频表示
发布时间：2018-07-25 13:45
【摘要】：离散Gabor变换是一种重要的时频分析工具,已经在数字信号处理、数字图像处理、系统建模中得到广泛的应用。在过去的十年里,稀疏变换已经被证明是一种全新的、有效的数学工具并成功应用于语音处理、图像去噪、压缩感知等工程领域。本文将稀疏变换理论应用于离散Gabor变换,研究了基于离散Gabor变换的信号稀疏时频表示方法。传统的离散Gabor变换在过抽样情况下是冗余的变换,且包含大量的非零变换系数,信号的稀疏表示可利用尽可能少的非零Gabor变换系数来表示原始信号,所以基于稀疏理论的离散Gabor变换能够提高Gabor时频谱的分辨率和集中度从而使离散Gabor变换更加有效地运用于非平稳信号分析和处理。主要研究内容和创新成果如下：提出了一种基于离散Gabor变换的信号稀疏时频表示方法。离散Gabor变换中窗函数宽度直接决定了Gabor时频谱的聚集性和时频分辨率。首先我们利用Gabor时频谱熵度量确定了信号在离散Gabor变换中最优窗函数的宽度,然后将离散Gabor变换转换成带有lp稀疏约束的凸优化方程,最后根据稀疏模型求出近似解。由于基于l1范数的稀疏约束模型得到的解往往不够稳定,该模型容易导致解过于稀疏并且破坏解的内在结构,而使用基于l1-l2混合范数的稀疏约束模型得到的解具有较好的稀疏性和稳定性,因此该模型具有实际的工程应用价值。实验也表明基于此稀疏约束模型获得的Gabor时频谱具有较好的时频集中度并且在降噪方面具有更好的效果。提出了一种基于多窗离散Gabor变换的信号稀疏时频表示方法。多窗离散Gabor变换可以克服单窗离散Gabor变换具有固定时频分辨率的缺点,基于多窗离散Gabor变换的结构化稀疏时频表示方法可以对信号进行有效的分解和分析。首先将多窗离散Gabor变换转换成带有混合范数(lp,q)约束下的凸优化方程,然后根据不同的混合范数使用相应的软门限函数,最后使用块坐标下降法获得稀疏Gabor系数。实验表明所提出的方法能获得更高精度的时频谱。提出了一种基于矩阵分解和快速傅里叶变换的对偶窗快速求解算法。由于在稀疏分析中,需要分析窗对应的对偶窗来综合还原信号,因此研究对偶窗的快速求解算法十分必要。本文提出了一种基于矩阵分解和快速傅里叶变换的对偶窗的快速求解方法。该方法首先根据离散Gabor变换的完备性条件得到了变换窗的新双正交关系式,然后对新双正交关系式的线性方程组进行简化并分解成一定数量的独立线性子方程组,每一子方程组可利用快速傅里叶变换求解对偶窗,从而可节省大量的计算时间,实验验证了方法的有效性和快速性。提出了一种基于加权线性组合分析窗的离散Gabor变换及其权值求解算法。在传统的多窗离散Gabor变换中,Gabor组合时频谱的时频精度不仅取决于所选择的分析窗还取决于这些窗的线性组合权值。本文据此提出了一种基于加权线性组合分析窗的离散Gabor变换算法,利用变换系数稀疏性原则从而将加权线性组合分析窗的离散Gabor变换转换成带有l1-l2范数约束下的稀疏方程,进而根据稀疏变换理论求解出窗函数的权值。由于求解窗函数权值的迭代过程中需要计算组合分析窗对应的综合窗序列,所以使用前面提出的对偶窗序列的快速求解算法可以减少运算时间和加快运算速度。实验表明了所提出的离散Gabor变换的有效性。
[Abstract]:Discrete Gabor transform is an important time frequency analysis tool, which has been widely used in digital signal processing, digital image processing and system modeling. In the past ten years, sparse transformation has been proved to be a new and effective mathematical tool and successfully applied to the engineering fields of speech processing, image denoising, compression perception and so on. In this paper, the sparse transform theory is applied to discrete Gabor transform, and the time-frequency representation method of sparse signal based on discrete Gabor transform is studied. The traditional discrete Gabor transform is redundant in oversampling, and contains a large number of non zero transform coefficients. The sparse representation of the signal can make use of the least non zero Gabor transform coefficients. The discrete Gabor transform based on sparse theory can improve the resolution and concentration of the Gabor spectrum and make the discrete Gabor transform more effectively used for non-stationary signal analysis and processing. The main research content and innovation results are as follows: a sparse time-frequency representation based on discrete Gabor transform is proposed. Method. The width of the window function in the discrete Gabor transform directly determines the aggregation and time frequency resolution of the Gabor spectrum. Firstly, we use the spectrum entropy measure of Gabor to determine the width of the optimal window function in the discrete Gabor transform, and then convert the discrete Gabor transform into a convex optimization equation with LP sparse constraint. Finally, the thinning is based on sparsity. The approximate solution is obtained by the model. The solution obtained by the sparse constraint model based on the L1 norm is often not stable. The model can easily lead to the sparse solution and destroy the inner structure of the solution, and the solution obtained by the sparse constraint model based on the mixed norm of L1-L2 has better sparsity and stability. Therefore, the model has practical engineering. The experiment also shows that the Gabor time spectrum based on this sparse constraint model has better time frequency concentration and better effect on noise reduction. A method of signal sparse time frequency representation based on multi window discrete Gabor transform is proposed. Multi window discrete Gabor transform can be used to overcome single window discrete Gabor transformation. The shortcoming of time-frequency resolution is that the structured sparse time-frequency representation method based on multi window discrete Gabor transform can effectively decompose and analyze the signal. First, the multi window discrete Gabor transform is converted into a convex optimization equation with a mixed norm (LP, q) constraint, and then the corresponding soft threshold function is used according to the different mixed norm, finally, the corresponding soft threshold function is used. Finally, the corresponding soft threshold function is used. The block coordinate descent method is used to obtain the sparse Gabor coefficient. The experiment shows that the proposed method can obtain higher precision time spectrum. A fast algorithm for the dual window based on matrix decomposition and fast Fourier transform is proposed. In the sparse analysis, the pair window corresponding to the window is needed to synthesize the reduction signal, so the dual is studied. In this paper, a fast solution method of dual windows based on matrix decomposition and fast Fourier transform is proposed in this paper. This method first obtains a new biorthogonal formula of the transformation window according to the completeness condition of the discrete Gabor transform, and then simplifies and divides the linear equations of the new double normal cross relation. In order to solve a certain number of independent linear subsets, each subgroup can use fast Fourier transform to solve the dual window, thus saving a lot of calculation time. Experiments verify the effectiveness and speediness of the method. A discrete Gabor transform based on weighted linear combination analysis window and its weight calculation algorithm are proposed. In the multi window discrete Gabor transform, the time frequency accuracy of the frequency spectrum of the Gabor combination depends not only on the selected analysis window but also on the linear combination weights of these windows. In this paper, a discrete Gabor transform algorithm based on the weighted linear combination analysis window is proposed, and the weighted linear combination analysis window is used to make use of the thinning principle of the transform coefficients. The discrete Gabor transform is converted into a sparse equation with L1-L2 norm constraints, and then the weight of the window function is solved according to the sparse transformation theory. The experimental results show the effectiveness of the proposed discrete Gabor transform.
【学位授予单位】：安徽大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TN911.7
，
本文编号：2144005
[Abstract]:Discrete Gabor transform is an important time frequency analysis tool, which has been widely used in digital signal processing, digital image processing and system modeling. In the past ten years, sparse transformation has been proved to be a new and effective mathematical tool and successfully applied to the engineering fields of speech processing, image denoising, compression perception and so on. In this paper, the sparse transform theory is applied to discrete Gabor transform, and the time-frequency representation method of sparse signal based on discrete Gabor transform is studied. The traditional discrete Gabor transform is redundant in oversampling, and contains a large number of non zero transform coefficients. The sparse representation of the signal can make use of the least non zero Gabor transform coefficients. The discrete Gabor transform based on sparse theory can improve the resolution and concentration of the Gabor spectrum and make the discrete Gabor transform more effectively used for non-stationary signal analysis and processing. The main research content and innovation results are as follows: a sparse time-frequency representation based on discrete Gabor transform is proposed. Method. The width of the window function in the discrete Gabor transform directly determines the aggregation and time frequency resolution of the Gabor spectrum. Firstly, we use the spectrum entropy measure of Gabor to determine the width of the optimal window function in the discrete Gabor transform, and then convert the discrete Gabor transform into a convex optimization equation with LP sparse constraint. Finally, the thinning is based on sparsity. The approximate solution is obtained by the model. The solution obtained by the sparse constraint model based on the L1 norm is often not stable. The model can easily lead to the sparse solution and destroy the inner structure of the solution, and the solution obtained by the sparse constraint model based on the mixed norm of L1-L2 has better sparsity and stability. Therefore, the model has practical engineering. The experiment also shows that the Gabor time spectrum based on this sparse constraint model has better time frequency concentration and better effect on noise reduction. A method of signal sparse time frequency representation based on multi window discrete Gabor transform is proposed. Multi window discrete Gabor transform can be used to overcome single window discrete Gabor transformation. The shortcoming of time-frequency resolution is that the structured sparse time-frequency representation method based on multi window discrete Gabor transform can effectively decompose and analyze the signal. First, the multi window discrete Gabor transform is converted into a convex optimization equation with a mixed norm (LP, q) constraint, and then the corresponding soft threshold function is used according to the different mixed norm, finally, the corresponding soft threshold function is used. Finally, the corresponding soft threshold function is used. The block coordinate descent method is used to obtain the sparse Gabor coefficient. The experiment shows that the proposed method can obtain higher precision time spectrum. A fast algorithm for the dual window based on matrix decomposition and fast Fourier transform is proposed. In the sparse analysis, the pair window corresponding to the window is needed to synthesize the reduction signal, so the dual is studied. In this paper, a fast solution method of dual windows based on matrix decomposition and fast Fourier transform is proposed in this paper. This method first obtains a new biorthogonal formula of the transformation window according to the completeness condition of the discrete Gabor transform, and then simplifies and divides the linear equations of the new double normal cross relation. In order to solve a certain number of independent linear subsets, each subgroup can use fast Fourier transform to solve the dual window, thus saving a lot of calculation time. Experiments verify the effectiveness and speediness of the method. A discrete Gabor transform based on weighted linear combination analysis window and its weight calculation algorithm are proposed. In the multi window discrete Gabor transform, the time frequency accuracy of the frequency spectrum of the Gabor combination depends not only on the selected analysis window but also on the linear combination weights of these windows. In this paper, a discrete Gabor transform algorithm based on the weighted linear combination analysis window is proposed, and the weighted linear combination analysis window is used to make use of the thinning principle of the transform coefficients. The discrete Gabor transform is converted into a sparse equation with L1-L2 norm constraints, and then the weight of the window function is solved according to the sparse transformation theory. The experimental results show the effectiveness of the proposed discrete Gabor transform.
【学位授予单位】：安徽大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TN911.7
，
本文编号：2144005
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