图像处理的超小波分析与变分方法研究
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摘要
超小波分析和变分方法是当前数学图像处理和计算机视觉等领域最具代表性的两种研究方法。超小波分析是应用现代调和分析的概念和方法以及群表示理论，在小波分析的基础上发展起来的，是小波理论的新进展。其目的旨在检测、表示、处理某些高维空间中的数据，而这些数据的某些重要特征又集中于低维子空间中。对于含线奇异、面奇异的二维或高维函数，超小波分析显示出了比小波分析更好的“稀疏”表示能力。与计算调和分析方法不同，变分方法是图像处理中的另一有效工具，在图像去噪、图像增强、边缘检测等方面已经取得很多成功的应用。本文以图像处理为应用背景，围绕超小波分析和变分方法进行了一些有益的探索和研究，并取得了初步的研究成果。主要研究结果如下：
1、讨论了曲线波变换和反应扩散方程，并结合曲线波变换和反应扩散方程提出两种图像去噪算法。
详细讨论Nordstr m能量泛函极小化问题，从Nordstr m能量泛函的欧拉方程出发，通过重新定义合适的控制函数提出一种新型反应扩散滤波器模型和图像去噪算法。该滤波器模型不是直接求解偏微分方程或者是泛函极值，而是求解数字形式的非线性代数方程组，求解过程简单。讨论了模型的优缺点，通过引入平滑算子和新的扩散函数对该模型进行了改进，有效克服了反应扩散数字滤波过程中的斑点噪声和扩散系数的病态。分析了该滤波器模型的迭代过程，从理论上证明了该滤波器模型的性质和滤波迭代算法的收敛性。数值实验表明所给滤波器模型对不同类型、不同程度噪声污染的图像都有较好的处理效果。
为了抑制曲线波变换去噪中出现的“虚假”效应和去除反应扩散数字滤波过程的斑点噪声，提出一种结合曲线波变换和新型反应扩散滤波器模型的图像去噪算法。实验结果表明，所提算法在有效去噪和保持边缘的同时，一定程度上也克服了曲波变换本身的伪Gibbs效应和类曲波伪曲线现象，视觉效果较好。
2、讨论了波原子变换的基本特征、构造及数值实现，研究了波原子在图像处理中的应用。提出两种基于波原子变换的图像去噪算法。
一种是基于波原子系数全变差最小的图像去噪算法。由于硬阈值算子的不连续性和波原子变换的FFT周期化过程，在去噪图像的不连续点附近产生了新方向性纹理失真和伪Gibbs震荡，而全变差正则化可以抑制这些震荡。将超小波分析和变分方法有机地结合起来，提出了基于波原子系数全变差最小化的图像去噪算法。该算法首先对降质图像利用波原子变换和非线性阈值，然后根据保留的变换系数确定可行域建立模型，最后利用投影梯度算法对其进行求解。实验结果表明，所提算法在有效抑噪和保持边缘的同时，能够有效地抑制伪吉布斯震荡，取得较为理想的视觉效果。
另一种是基于Cycle Spinning思想的波原子变换图像去噪算法。由于波原子变换不具有平移不变性，对系数阈值后会产生伪Gibbs现象，而Cycle Spinning可以很好地避免这些失真。将波原子变换和Cycle Spinning有效结合，提出了基于CycleSpinning的波原子变换图像去噪算法。实验结果表明，所提算法可以很好地减少图像在波原子阈值去噪过程中出现的伪Gibbs现象，且能更多地保留图像的纹理细节，视觉效果明显地较曲线波、小波方法要好。
3、基于对偶树复小波变换，提出一种图像质量评价的结构相似性指标。对图像采用小波变换进行多分辨率分析，可以将图像分解为不同尺度下的子带图像，这样图像的边缘结构就表达成不同尺度下的小波系数。对偶树复小波变换具有平移不变性和更好的方向选择性，可以更好地表达图像的细节与边缘信息。针对SSIM指标对平移、尺度变化和旋转非常敏感的缺点，提出了基于对偶树复小波变换的结构相似度指标。通过几个数值实验验证了所提指标对平移、尺度和旋转变化的鲁棒性，在图像去噪的应用说明了所提指标的有效性。
4、通过考虑不同的图像空间，详细分析了图像调和修补、全变差修补算法的误差。重点讨论了基于小波的图像修补问题的三个模型：图像的全变差小波修补、基于曲率驱动的小波域图像修补及小波域图像修补的空域实现算法，并对这三种算法进行仿真实验。数值仿真实验表明，TV模型选择全变差的最小化来推进图像修补的进程，也能系统性地抑制图像中的噪声；曲率驱动的小波域图像修补利用曲率正则化标准惩罚了边缘线的长度和曲率沿边缘线的积分，保证了边缘线曲率的连续性；小波域图像修补的空域实现算法是利用空域的全变差修补技术完成小波域丢失系数的图像修补。
In the field of mathematical imaging and computer visual, beyond wavelet analysisand variation method are the most representative methods at current．Beyond waveletanalysis is a new development based on wavelet theory, which come from the conceptand method in the application of modern harmonic analysis and theory of grouprepresentation. Beyond wavelet analysis aims to test, represent, and process the data insome high dimension space, but some of the important characteristics of these datafocus on low-dimensional subspace. For two or higher dimensional functions includingline or surface singular, beyond wavelet analysis shows the more sparseness of therepresentation ability than the wavelet. Difference from the calculation harmonicanalysis method, variation method is another effective tool in image processing. Therehave found a lot of successful applications in the image denoising and imageenhancement, edge detection and so on. For the background of image processingapplication, this dissertation has made some useful exploration and research around thebeyond wavelet analysis and variation methods and got some preliminary researchachevements.The main research results are as follows:
1. Curvelet transform and reaction diffusion equation are discussed in detail.Twoimage denoising algorithms based on curvelet transform and reaction diffusion equationare presented.
Nordstr m energy functional minimization problem is discussed in detail. From itsEuler equation, a new reaction diffusion filter model and related algorithm are given forimage denoising by new definition of appropriate control function. Our digitizedformulation leads to nonlinear algebraic equations instead of PDE’s and the analysisand application of the digital method needs no knowledge on numerical approximationof PDE’s. The advantages and disadvantages of the model are discussed. A smoothoperator and a new diffusion function are introduced to improve the model andeffectively avoid ill-posed diffusion coefficients and some spots in the process of usingthe reaction diffusion digital filter. The iterative process and the properties of the filtermodel are analyzed. The convergence of the iterative algorithm is proposed theoretically.The experiments demonstrate that the proposed model have preferable application forimage denoising with different types and degree noise．
In order to avoid the "false" effect in the curvelet transform and reduce the spots inthe process of using the reaction diffusion digital filter, the denoising algorithm based on the proposed reaction diffusion filter model and the curvelet transform is presented.Experimental results show that the proposed algorithms reduce noise effectively andkeep edges well. At the same time, more important is that the algorithm can overcomethe Pseudo-Gibbs effect and “class curvelet” Pseudo curve phenomenon in the curvelettransform with the ideal visual effect.
2. Basic characteristics, construction and digital realization of wave atoms arediscussed in detail, and its applications in image processing are introduced. Twoalgorithms for image denoising are proposed based on wave atoms transform.
One algorithm is based on total variation minimization of wave atoms coefficients.Due to discontinuity of hard threshold operators and FFT cycle process of wave atoms,Pseudo Gibbs and new orientation texture in the neighborhood of discontinuous pointwas found in the denoised images. These distortions can be seen as a total oscillatoryand TV regularization can better suppress the oscillatory. Combined the minimizationof total variation, the algorithm based on wave atoms is presented for image denoising.Firstly, the nonlinear threshold strategy associated with wave atoms is applied to thetransform coefficients of noisy image．And then，the feasible domain of the proposedmodel is determined by the coefficients remained. Finally, the projected gradientalgorithm is used to solve the proposed model. Experiments results show that theproposed algorithm can reduce noise efficiently and preserve edges well while thePseudo-Gibbs aritifacts was suppressed with better visual effects.
Another algorithm for image denoising is presented， which combine theadvantages of wave atoms transform and Cycle Spinning．Due to lack of translationinvariance of the wave atoms transform, image denoising method by coefficientthreshold would lead to Pseudo-Gibbs phenomena．Cycle Spinning is employed toavoid these artifacts．Experimental results show that the proposed algorithm can removenoise and remain edges，while Pseudo-Gibbs phenomena are suppressed efficiently withbetter visual effect and PSNR gains, compared with the methods like simple waveatoms or wavelet denoising using Cycle Spinning．
4. Based on the dual tree complex wavelet transform, the structural similarity index(CW-SSIM) is proposed. Using multi-resolution analysis, the image can be decomposedinto sub-band images in different wavelet scales and image edges and structure isexpressed into wavelet coefficients. The dual tree complex wavelet transform has theadvantages of translation invariance and good direction selectivity, and it can respondmore detail information and edges of images. The index SSIM is very sensitive totranslation, scale change, rotating, so the structural similarity index based on dual tree complex wavelet transform is presented. The index is robust for small range oftranslation and rotation. When evaluating the image similarity, the image preprocessingis not needed. The effectiveness of the proposed index is amply illustrated on a varietyof examples.
5. Some detailed discussions are made for image harmonic inpainting and totalvariation inpainting by considering some different image space. Three models based onwavelet transform and algorithms are discussed in image inpainting, which are totalvariation wavelet inpainting and image inpainting model based on curvature driven, aswell as the image domain realization for wavelet domain images inpainting. The TVwavelet inpainting model promotes the process of image repairing by total variationminimization and the minimization can also inhibit the noise of the image. Usingcurvature regularization standard, image inpainting model based on curvature driven haspunished the length of the edge line and the integral of curvature along the edge line,thus ensured the continuity of the curvature edge. The image domain realization forwavelet domain image inpainting is to use all inpainting technology in image domain inorder to finish the repairing of wavelet domain. Their effectiveness is illustrated on avariety of examples.
1、讨论了曲线波变换和反应扩散方程，并结合曲线波变换和反应扩散方程提出两种图像去噪算法。
详细讨论Nordstr m能量泛函极小化问题，从Nordstr m能量泛函的欧拉方程出发，通过重新定义合适的控制函数提出一种新型反应扩散滤波器模型和图像去噪算法。该滤波器模型不是直接求解偏微分方程或者是泛函极值，而是求解数字形式的非线性代数方程组，求解过程简单。讨论了模型的优缺点，通过引入平滑算子和新的扩散函数对该模型进行了改进，有效克服了反应扩散数字滤波过程中的斑点噪声和扩散系数的病态。分析了该滤波器模型的迭代过程，从理论上证明了该滤波器模型的性质和滤波迭代算法的收敛性。数值实验表明所给滤波器模型对不同类型、不同程度噪声污染的图像都有较好的处理效果。
为了抑制曲线波变换去噪中出现的“虚假”效应和去除反应扩散数字滤波过程的斑点噪声，提出一种结合曲线波变换和新型反应扩散滤波器模型的图像去噪算法。实验结果表明，所提算法在有效去噪和保持边缘的同时，一定程度上也克服了曲波变换本身的伪Gibbs效应和类曲波伪曲线现象，视觉效果较好。
2、讨论了波原子变换的基本特征、构造及数值实现，研究了波原子在图像处理中的应用。提出两种基于波原子变换的图像去噪算法。
一种是基于波原子系数全变差最小的图像去噪算法。由于硬阈值算子的不连续性和波原子变换的FFT周期化过程，在去噪图像的不连续点附近产生了新方向性纹理失真和伪Gibbs震荡，而全变差正则化可以抑制这些震荡。将超小波分析和变分方法有机地结合起来，提出了基于波原子系数全变差最小化的图像去噪算法。该算法首先对降质图像利用波原子变换和非线性阈值，然后根据保留的变换系数确定可行域建立模型，最后利用投影梯度算法对其进行求解。实验结果表明，所提算法在有效抑噪和保持边缘的同时，能够有效地抑制伪吉布斯震荡，取得较为理想的视觉效果。
另一种是基于Cycle Spinning思想的波原子变换图像去噪算法。由于波原子变换不具有平移不变性，对系数阈值后会产生伪Gibbs现象，而Cycle Spinning可以很好地避免这些失真。将波原子变换和Cycle Spinning有效结合，提出了基于CycleSpinning的波原子变换图像去噪算法。实验结果表明，所提算法可以很好地减少图像在波原子阈值去噪过程中出现的伪Gibbs现象，且能更多地保留图像的纹理细节，视觉效果明显地较曲线波、小波方法要好。
3、基于对偶树复小波变换，提出一种图像质量评价的结构相似性指标。对图像采用小波变换进行多分辨率分析，可以将图像分解为不同尺度下的子带图像，这样图像的边缘结构就表达成不同尺度下的小波系数。对偶树复小波变换具有平移不变性和更好的方向选择性，可以更好地表达图像的细节与边缘信息。针对SSIM指标对平移、尺度变化和旋转非常敏感的缺点，提出了基于对偶树复小波变换的结构相似度指标。通过几个数值实验验证了所提指标对平移、尺度和旋转变化的鲁棒性，在图像去噪的应用说明了所提指标的有效性。
4、通过考虑不同的图像空间，详细分析了图像调和修补、全变差修补算法的误差。重点讨论了基于小波的图像修补问题的三个模型：图像的全变差小波修补、基于曲率驱动的小波域图像修补及小波域图像修补的空域实现算法，并对这三种算法进行仿真实验。数值仿真实验表明，TV模型选择全变差的最小化来推进图像修补的进程，也能系统性地抑制图像中的噪声；曲率驱动的小波域图像修补利用曲率正则化标准惩罚了边缘线的长度和曲率沿边缘线的积分，保证了边缘线曲率的连续性；小波域图像修补的空域实现算法是利用空域的全变差修补技术完成小波域丢失系数的图像修补。
In the field of mathematical imaging and computer visual, beyond wavelet analysisand variation method are the most representative methods at current．Beyond waveletanalysis is a new development based on wavelet theory, which come from the conceptand method in the application of modern harmonic analysis and theory of grouprepresentation. Beyond wavelet analysis aims to test, represent, and process the data insome high dimension space, but some of the important characteristics of these datafocus on low-dimensional subspace. For two or higher dimensional functions includingline or surface singular, beyond wavelet analysis shows the more sparseness of therepresentation ability than the wavelet. Difference from the calculation harmonicanalysis method, variation method is another effective tool in image processing. Therehave found a lot of successful applications in the image denoising and imageenhancement, edge detection and so on. For the background of image processingapplication, this dissertation has made some useful exploration and research around thebeyond wavelet analysis and variation methods and got some preliminary researchachevements.The main research results are as follows:
1. Curvelet transform and reaction diffusion equation are discussed in detail.Twoimage denoising algorithms based on curvelet transform and reaction diffusion equationare presented.
Nordstr m energy functional minimization problem is discussed in detail. From itsEuler equation, a new reaction diffusion filter model and related algorithm are given forimage denoising by new definition of appropriate control function. Our digitizedformulation leads to nonlinear algebraic equations instead of PDE’s and the analysisand application of the digital method needs no knowledge on numerical approximationof PDE’s. The advantages and disadvantages of the model are discussed. A smoothoperator and a new diffusion function are introduced to improve the model andeffectively avoid ill-posed diffusion coefficients and some spots in the process of usingthe reaction diffusion digital filter. The iterative process and the properties of the filtermodel are analyzed. The convergence of the iterative algorithm is proposed theoretically.The experiments demonstrate that the proposed model have preferable application forimage denoising with different types and degree noise．
In order to avoid the "false" effect in the curvelet transform and reduce the spots inthe process of using the reaction diffusion digital filter, the denoising algorithm based on the proposed reaction diffusion filter model and the curvelet transform is presented.Experimental results show that the proposed algorithms reduce noise effectively andkeep edges well. At the same time, more important is that the algorithm can overcomethe Pseudo-Gibbs effect and “class curvelet” Pseudo curve phenomenon in the curvelettransform with the ideal visual effect.
2. Basic characteristics, construction and digital realization of wave atoms arediscussed in detail, and its applications in image processing are introduced. Twoalgorithms for image denoising are proposed based on wave atoms transform.
One algorithm is based on total variation minimization of wave atoms coefficients.Due to discontinuity of hard threshold operators and FFT cycle process of wave atoms,Pseudo Gibbs and new orientation texture in the neighborhood of discontinuous pointwas found in the denoised images. These distortions can be seen as a total oscillatoryand TV regularization can better suppress the oscillatory. Combined the minimizationof total variation, the algorithm based on wave atoms is presented for image denoising.Firstly, the nonlinear threshold strategy associated with wave atoms is applied to thetransform coefficients of noisy image．And then，the feasible domain of the proposedmodel is determined by the coefficients remained. Finally, the projected gradientalgorithm is used to solve the proposed model. Experiments results show that theproposed algorithm can reduce noise efficiently and preserve edges well while thePseudo-Gibbs aritifacts was suppressed with better visual effects.
Another algorithm for image denoising is presented， which combine theadvantages of wave atoms transform and Cycle Spinning．Due to lack of translationinvariance of the wave atoms transform, image denoising method by coefficientthreshold would lead to Pseudo-Gibbs phenomena．Cycle Spinning is employed toavoid these artifacts．Experimental results show that the proposed algorithm can removenoise and remain edges，while Pseudo-Gibbs phenomena are suppressed efficiently withbetter visual effect and PSNR gains, compared with the methods like simple waveatoms or wavelet denoising using Cycle Spinning．
4. Based on the dual tree complex wavelet transform, the structural similarity index(CW-SSIM) is proposed. Using multi-resolution analysis, the image can be decomposedinto sub-band images in different wavelet scales and image edges and structure isexpressed into wavelet coefficients. The dual tree complex wavelet transform has theadvantages of translation invariance and good direction selectivity, and it can respondmore detail information and edges of images. The index SSIM is very sensitive totranslation, scale change, rotating, so the structural similarity index based on dual tree complex wavelet transform is presented. The index is robust for small range oftranslation and rotation. When evaluating the image similarity, the image preprocessingis not needed. The effectiveness of the proposed index is amply illustrated on a varietyof examples.
5. Some detailed discussions are made for image harmonic inpainting and totalvariation inpainting by considering some different image space. Three models based onwavelet transform and algorithms are discussed in image inpainting, which are totalvariation wavelet inpainting and image inpainting model based on curvature driven, aswell as the image domain realization for wavelet domain images inpainting. The TVwavelet inpainting model promotes the process of image repairing by total variationminimization and the minimization can also inhibit the noise of the image. Usingcurvature regularization standard, image inpainting model based on curvature driven haspunished the length of the edge line and the integral of curvature along the edge line,thus ensured the continuity of the curvature edge. The image domain realization forwavelet domain image inpainting is to use all inpainting technology in image domain inorder to finish the repairing of wavelet domain. Their effectiveness is illustrated on avariety of examples.
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