联合X-let和变分的图像恢复模型及其算法研究
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摘要
X-let是所有经典的小波及其新进展,甚至将来被定义的小波的总称,涵盖了本文所涉及到的高维连续小波、脊波、第二代曲线波、波原子、Log-Gabor小波等。其理论框架已经比较成熟,将其应用于图像处理是当前研究的热点问题。数学图像处理的另外一个重要方法是变分偏微分方程方法,它已被成功地应用于图像处理和计算机视觉的许多方面,并且取得了很好的效果。
图像恢复是图像处理的一个基本问题,这类问题在Hadamard意义下一般是不适定的反问题。解决这类病态问题的关键是如何利用先验信息,建立相应的数学模型,从而获得恢复问题的适定解。
本论文结合X-let和变分思想,对图像恢复问题进行探讨和研究,得到了一系列有意义的结果。主要工作如下:
1、以高维连续小波和脊波框架为研究背景,得到两个理论结果。首先从理论上分析高维连续小波阈值和伪微分方程的关系,利用高维Fourier变换的性质和高维连续小波的算子表示结论,得到新的高阶非线性扩散方程。其次,基于曲线波型紧框架的构造思想,给出脊波型结构覆盖,结合Fourier基得到了脊波型紧框架;进一步,针对Candes的脊波框架没有显式对偶框架的不足,组合正交脊波和所构造的脊波型覆盖,构造了三种新的框架。
2、以第二代曲线波为主线,从理论和算法上给出新的变分图像去噪模型和去模糊模型。首先提出曲线波型分解空间光滑性约束的变分正则化图像去噪模型,利用曲线波型分解空间半范和第二代曲线波加权系数的等价模关系,得到依赖于曲线波分解尺度和光滑参数的图像阈值算法;进一步,从图像分解的角度出发,证明了采用负指数的曲线波型分解空间约束的分解模型和采用L2范数约束的去噪模型在满足相应的条件下是等价的,然而偏微分方程的分解模型却不具有这样的结论。其次利用Bregman距离提出曲线波迭代正则化模型和逆尺度空间方法。最后提出曲线波型分解空间光滑性约束的图像去模糊模型,利用广义条件梯度法,给出迭代阈值算法和自适应的迭代停止准则。上述模型和算法拓宽和发展了小波Besov空间方法,并且实验结果验证了将结构丰富的图像约束在曲线波型分解空间用曲线波来刻画是恰当的。
3、以波原子为研究对象,提出几种新的模型和算法。首先基于波原子的构造特点,利用Besov空间,得到了新的依赖于波原子尺度和光滑性指标的纹理图像去噪模型和阈值算法。其次,利用波原子对纹理图像表示的稀疏性,给出了完全离散的波原子迭代正则化模型和逆尺度空间模型。最后,利用波原子阈值代替通常高斯卷积的运算,推广Perona-Malik模型,得到了波原子正则化的非线性扩散模型。
4、分别以相位一致性和主成份分析为工具,提出新的张量扩散模型和变分去模糊模型。首先利用相位信息的重要性和相位一致性对图像特征检测的优势,设计了一种新的散度矩阵,提出基于相位一致性的张量扩散模型。其次,提出基于主成份分析的变分图像去模糊模型,并通过替代泛函的思想给出主成份分析域的迭代学习算法;进一步,针对该算法收敛速度慢的缺陷,给出一种快速算法。
X-let is a general definition of wavelets, including all the classical wavelets andtheir new progress, even those defined in future,which covers the high-dimensionalcontinuous wavelets, ridgelets, the second-generation curvelets, wave atoms and Log-Gabor wavelets et al related in this dissertation. The theory framework of X-let isrelatively mature and its application to image processing is the highlight of currentresearch. Another important method in mathematical image processing is variationalpartial differential equations, which has been applied to many fields in image process-ing and computer vision with good results.
Image restoration is a kind of fundamental problem in image processing, which isan ill-posed inverse problem in the sense of Hadamard. The key point of solving thisproblem is how to use the priori knowledge to establish the corresponding mathemati-cal model, and obtain the well-posed solution.
In this dissertation, we combine X-let with variational idea to explore and studyproblem of image restoration, and obtain a series of meaningful results. The main workcan be summarized as follows:
1 Two theoretical results related to high-dimensional continuous wavelet trans-form and ridgelet frame are obtained. Firstly, the relationship of high-dimensionalcontinuous wavelet thresholding with pseudo-differential equation is theoretically ana-lyzed, which results in a new high-order nonlinear diffusion equation by using both theproperties of Fourier transform and operator representation theory in high-dimensionalcase. Secondly, a ridgelet-type structured admissible covering and associated framesusing Fourier basis are designed based on the ideology of constructing curvelet-typeframe. Moreover, we obtain three novel ridgelet-type frames by associating orthonor-mal ridgelet with the ridgelet-type covering.
2 The novel variational models and algorithms for both image denoising and de-blurring are given with the second-generation curvelets. Firstly, we apply constraintof curvelet-type decomposition spaces as a regularizing term to variational model forimage denoising. Based on the equivalent modulus relationship between semi-norm ofcurvelet-type decomposition spaces and the weighted curvelet coefficients, solution ofthe proposed model equals to different curvelet shrinkages. Moreover, we prove that the decomposition model with constraint of semi-norm of curvelet-type decompositionspaces with negative degree of differentiability is equivalent to a denoising model withconstraint of L2 norm to the fidelity term on a given condition. Secondly, an iterativeregularization method and inverse scale space method of curvelets are introduced byapplying Bregman distance. Finally, a model for image deblurring is considered byemploying the sparse constraint of curvelet-type decomposition spaces. Especially, aniterative hard shrinkage algorithm and an adaptive stopping criterion are obtained bythe generalized conditional gradient method. These models and algorithms describedabove extend the wavelet method in Besov spaces. Moreover, experimental resultsshow that the use of curvelets in curvelet-type decomposition spaces is appropriate forcharacterizing image with rich structures.
3 Several models and algorithms are presented with wave atoms. Firstly, a noveldenoising model for texture images is proposed by using Besov spaces, which results ina soft thresholding algorithm depending on both the scale of wave atoms and smoothingparameter. Secondly, considering the sparse representation of wave atoms for texturalimage, an iterative regularization moethod and an inverse scale spaces method for fullydiscrete wave atom coefficients are designed. Finally, as an extension of Perona-Malikmodel, we develop a nonlinear diffusion model by replacing Gaussian regularizationwith wave atoms regularization.
4 A novel tensor diffusion model and a variational deblurring model arepresented with two tools of phase congruency and principal component analysis.Firstly,taking advantage of both the importance of phase and the merit of phase con-gruency for image feature detection, we design a new scatter matrix, then propose atensor diffusion model based on phase congruency. Secondly, a model for image de-blurring using principal component analysis is given. Then an iterative shrinkage al-gorithm is obtained in principal component analysis domain by employing the idea ofsurrogate functional. Due to the slow speed of convergence, we further offer a newaccelerated algorithm.
图像恢复是图像处理的一个基本问题,这类问题在Hadamard意义下一般是不适定的反问题。解决这类病态问题的关键是如何利用先验信息,建立相应的数学模型,从而获得恢复问题的适定解。
本论文结合X-let和变分思想,对图像恢复问题进行探讨和研究,得到了一系列有意义的结果。主要工作如下:
1、以高维连续小波和脊波框架为研究背景,得到两个理论结果。首先从理论上分析高维连续小波阈值和伪微分方程的关系,利用高维Fourier变换的性质和高维连续小波的算子表示结论,得到新的高阶非线性扩散方程。其次,基于曲线波型紧框架的构造思想,给出脊波型结构覆盖,结合Fourier基得到了脊波型紧框架;进一步,针对Candes的脊波框架没有显式对偶框架的不足,组合正交脊波和所构造的脊波型覆盖,构造了三种新的框架。
2、以第二代曲线波为主线,从理论和算法上给出新的变分图像去噪模型和去模糊模型。首先提出曲线波型分解空间光滑性约束的变分正则化图像去噪模型,利用曲线波型分解空间半范和第二代曲线波加权系数的等价模关系,得到依赖于曲线波分解尺度和光滑参数的图像阈值算法;进一步,从图像分解的角度出发,证明了采用负指数的曲线波型分解空间约束的分解模型和采用L2范数约束的去噪模型在满足相应的条件下是等价的,然而偏微分方程的分解模型却不具有这样的结论。其次利用Bregman距离提出曲线波迭代正则化模型和逆尺度空间方法。最后提出曲线波型分解空间光滑性约束的图像去模糊模型,利用广义条件梯度法,给出迭代阈值算法和自适应的迭代停止准则。上述模型和算法拓宽和发展了小波Besov空间方法,并且实验结果验证了将结构丰富的图像约束在曲线波型分解空间用曲线波来刻画是恰当的。
3、以波原子为研究对象,提出几种新的模型和算法。首先基于波原子的构造特点,利用Besov空间,得到了新的依赖于波原子尺度和光滑性指标的纹理图像去噪模型和阈值算法。其次,利用波原子对纹理图像表示的稀疏性,给出了完全离散的波原子迭代正则化模型和逆尺度空间模型。最后,利用波原子阈值代替通常高斯卷积的运算,推广Perona-Malik模型,得到了波原子正则化的非线性扩散模型。
4、分别以相位一致性和主成份分析为工具,提出新的张量扩散模型和变分去模糊模型。首先利用相位信息的重要性和相位一致性对图像特征检测的优势,设计了一种新的散度矩阵,提出基于相位一致性的张量扩散模型。其次,提出基于主成份分析的变分图像去模糊模型,并通过替代泛函的思想给出主成份分析域的迭代学习算法;进一步,针对该算法收敛速度慢的缺陷,给出一种快速算法。
X-let is a general definition of wavelets, including all the classical wavelets andtheir new progress, even those defined in future,which covers the high-dimensionalcontinuous wavelets, ridgelets, the second-generation curvelets, wave atoms and Log-Gabor wavelets et al related in this dissertation. The theory framework of X-let isrelatively mature and its application to image processing is the highlight of currentresearch. Another important method in mathematical image processing is variationalpartial differential equations, which has been applied to many fields in image process-ing and computer vision with good results.
Image restoration is a kind of fundamental problem in image processing, which isan ill-posed inverse problem in the sense of Hadamard. The key point of solving thisproblem is how to use the priori knowledge to establish the corresponding mathemati-cal model, and obtain the well-posed solution.
In this dissertation, we combine X-let with variational idea to explore and studyproblem of image restoration, and obtain a series of meaningful results. The main workcan be summarized as follows:
1 Two theoretical results related to high-dimensional continuous wavelet trans-form and ridgelet frame are obtained. Firstly, the relationship of high-dimensionalcontinuous wavelet thresholding with pseudo-differential equation is theoretically ana-lyzed, which results in a new high-order nonlinear diffusion equation by using both theproperties of Fourier transform and operator representation theory in high-dimensionalcase. Secondly, a ridgelet-type structured admissible covering and associated framesusing Fourier basis are designed based on the ideology of constructing curvelet-typeframe. Moreover, we obtain three novel ridgelet-type frames by associating orthonor-mal ridgelet with the ridgelet-type covering.
2 The novel variational models and algorithms for both image denoising and de-blurring are given with the second-generation curvelets. Firstly, we apply constraintof curvelet-type decomposition spaces as a regularizing term to variational model forimage denoising. Based on the equivalent modulus relationship between semi-norm ofcurvelet-type decomposition spaces and the weighted curvelet coefficients, solution ofthe proposed model equals to different curvelet shrinkages. Moreover, we prove that the decomposition model with constraint of semi-norm of curvelet-type decompositionspaces with negative degree of differentiability is equivalent to a denoising model withconstraint of L2 norm to the fidelity term on a given condition. Secondly, an iterativeregularization method and inverse scale space method of curvelets are introduced byapplying Bregman distance. Finally, a model for image deblurring is considered byemploying the sparse constraint of curvelet-type decomposition spaces. Especially, aniterative hard shrinkage algorithm and an adaptive stopping criterion are obtained bythe generalized conditional gradient method. These models and algorithms describedabove extend the wavelet method in Besov spaces. Moreover, experimental resultsshow that the use of curvelets in curvelet-type decomposition spaces is appropriate forcharacterizing image with rich structures.
3 Several models and algorithms are presented with wave atoms. Firstly, a noveldenoising model for texture images is proposed by using Besov spaces, which results ina soft thresholding algorithm depending on both the scale of wave atoms and smoothingparameter. Secondly, considering the sparse representation of wave atoms for texturalimage, an iterative regularization moethod and an inverse scale spaces method for fullydiscrete wave atom coefficients are designed. Finally, as an extension of Perona-Malikmodel, we develop a nonlinear diffusion model by replacing Gaussian regularizationwith wave atoms regularization.
4 A novel tensor diffusion model and a variational deblurring model arepresented with two tools of phase congruency and principal component analysis.Firstly,taking advantage of both the importance of phase and the merit of phase con-gruency for image feature detection, we design a new scatter matrix, then propose atensor diffusion model based on phase congruency. Secondly, a model for image de-blurring using principal component analysis is given. Then an iterative shrinkage al-gorithm is obtained in principal component analysis domain by employing the idea ofsurrogate functional. Due to the slow speed of convergence, we further offer a newaccelerated algorithm.
引文
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