变分正则化与Euler-Lagrange方程在图像处理中的应用研究
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摘要
近年来，以变分方法、偏微分方程(PDE)和稀疏表示为代表的数学工具活跃在图像处理的各个研究领域，它们已经成为研究图像处理和计算机视觉的三大基本工具。本论文主要围绕三者在图像处理中的应用进行建模和算法研究。主要做了以下几个方面的工作：
1.提出了两种新的图像修复模型。一是基于非局部扩散的图像修复模型。该模型能够在扩散的过程中充分利用图像的全局信息对损坏的区域进行修复，克服了变分和偏微分方程方法在修复图像时易产生模糊以及不能保持图像纹理信息的不足；二是基于交替迭代的变分修复模型。该模型由于耦合了两个变量，因此新算法中首先采用交替极小化方法对其进行解耦，其次对解耦的两个子模型分别利用分裂Bregman方法进行数值求解。由于分裂Bregman方法的快速收敛性质，使得新算法的修复速度加快，提高了算法的运算效率。实验结果表明，这两种修复模型均能够获得较好的修复效果。
2.针对加性噪声去除的ROF模型在去噪过程中易导致阶梯效应(StaircaseEffect)的缺陷，在研究LOT模型的基础上，提出了一种新的变分去噪模型。该模型可以通过交替极小化方法化为两个简单的子模型，其中一个子模型被用于重构角，另一个被用于重构图像。在计算方法上，我们分别采用分裂Bregman方法对两个子模型进行交替求解。实验结果表明，提出的新算法不但收敛速度较快，而且在去噪过程中能够减缓阶梯效应并能较好地保持图像的边缘信息。
3.针对乘性噪声去除问题，结合变分方法、偏微分方程、稀疏表示以及字典学习这几个数学工具，在对数域提出了一种去除乘性噪声的稀疏正则化方法。该方法主要包括三步：首先，采用稀疏表示和字典学习的方法获得一个较优的log图；其次，对获得的log图利用总变分(TV)模型进行校正；最后，将校正后的结果用一个指数函数和偏差纠正从对数域变回到真实的图像。实验结果表明，新方法在能有效去除乘性噪声的同时还能较好地保持图像的纹理特征。
4. TV正则虽然具有良好的保边性能，但它在去噪过程中容易导致阶梯效应，针对这一缺陷，利用最近提出的总广义变分(TGV)，在对数域建立了一种新的去除乘性噪声的正则化模型，并从数学上证明了该模型解的存在唯一性。由于TGV的诸多优点，使得新模型在有效去除乘性噪声的同时既避免了阶梯效应的产生又较好地保持了图像的细节特征。在数值计算上，我们采用一阶原始－对偶算法和牛顿迭代方法对新模型进行求解。实验结果表明，所提出的算法无论是在视觉效果上还是峰值信噪比(PSNR)上都能获得较好的结果。
5.乘性噪声去除在相干成像系统以及各种各样的图像处理应用中具有十分重要的意义。目前去除乘性噪声的大部分数学模型都是基于正则化方法的，求解这类模型一般都需要知道噪声强度的先验信息，然而，这些信息在某些情况下并不能获得，针对这个缺陷，提出了两种去除乘性噪声的投影方法，并通过利用对偶方法和变量分裂技巧给出了三种快速的数值算法。实验结果表明，所提出的去噪算法不仅收敛速度快，而且在噪声强度先验信息未知的情况下也能有效地滤除乘性噪声。
In recent years, variational method、partial differential equation (PDE) and sparsereprensentations play active roles in many image processing fields. They have becomethree basic tools for image processing and computer vision. This dissertation mainlystudies the models and applications of variation、PDE and sparse reprensentations inimage processing. The main work can be summarized as follows:
1. Two image inpainting models are proposed. One is the image inpainting modelbased on nonlocal diffusion. This model can inpaint the damaged domains by using theglobal information of the image in the diffusion process, it can overcome theshortcomings that the PDE-based models tend to produce some blur and cannotpreserve the texture. The other is the variational inpainting model based on alternateiteration. There are two variables in the new model, so it is firstly turned into twosimple submodels by using the alternative minimization method in the new algorithm,and then the two submodels are solved using split Bregman method respectively. Dueto the fast convergence property of the split Bregman method, the new algorithm canimprove the inpainting speed, thus reducing the runtime. The experimental resultsshow that the two image inpainting models can both obtain the better inpainting effect.
2. As the ROF model for additive noise removal tends to produce the staircaseeffect, a novel variational denoising model is proposed based on the study of the LOTmodel. The model can be turned into two simple submodels by using the alternativeminimization method, one is used to construct an angle, and the other is used toreconstruct the image. In the numerical computation, we apply the split Bregmanmethod to solve the two submodels alternately. The experimental results show that thenew algorithm not only has faster convergence rate, but also can alleviate the staircaseeffect and preserve the edge information better.
3. Aiming at the problem of multiplicative noise removal, combining thevariational method、PDE、sparse reprensentations and dictionary learning, a sparsityregularization method for multiplicative noise removal is proposed in log-domain. Theproposed method mainly contains three steps. Firstly, a better log-image is obtained byusing the sparse representation based on the dictionary learning. Secondly, the totalvariation (TV) model is used to amend the log-image. Finally, via an exponentialfunction and bias correction, the result is transformed back from the log-image domain to the real one. The experimental results show that the new method is more effective tofilter out the multiplicative noise while well preserving the texture.
4. TV can preserve the edge information better, but it tends to produce the staircaseeffect. In order to overcome the drawback, a novel variational regularization model formultiplicative noise removal is proposed in log-domain based on the total generalizedvariation (TGV) introduced recently, and the existence and uniqueness of a minimizerfor the proposed model are proven. TGV has many advantages, which make the newmodel to be more effective for multiplicative noise removal while avoiding thestaircase effect and well preserving the feature and details. In the numericalcomputation, we use the first-order primal-dual algorithm and the Newton method tosolve the new model. The experimental results show that the proposed algorithm canget the better results from the visual effect and the peak signal to noise ratio (PSNR).
5. Multiplicative noise removal is of momentous significance in coherent imagingsystems and various image processing applications. So far, most multiplicative noiseremoval models focus on the regularization method, these models require knowing theprior level of noise beforehand, however, the information isn't obtained in some case,to overcome the drawback, two projection methods for multiplicative noise removalmodels are presented, and three fast numerical algorithms are given by using theduality technique and the variable-splitting. The experimental results show that theproposed algorithms not only have faster convergence rate, but also can effectivelyfilter out the multiplicative noise when the prior of noise is unknown.
1.提出了两种新的图像修复模型。一是基于非局部扩散的图像修复模型。该模型能够在扩散的过程中充分利用图像的全局信息对损坏的区域进行修复，克服了变分和偏微分方程方法在修复图像时易产生模糊以及不能保持图像纹理信息的不足；二是基于交替迭代的变分修复模型。该模型由于耦合了两个变量，因此新算法中首先采用交替极小化方法对其进行解耦，其次对解耦的两个子模型分别利用分裂Bregman方法进行数值求解。由于分裂Bregman方法的快速收敛性质，使得新算法的修复速度加快，提高了算法的运算效率。实验结果表明，这两种修复模型均能够获得较好的修复效果。
2.针对加性噪声去除的ROF模型在去噪过程中易导致阶梯效应(StaircaseEffect)的缺陷，在研究LOT模型的基础上，提出了一种新的变分去噪模型。该模型可以通过交替极小化方法化为两个简单的子模型，其中一个子模型被用于重构角，另一个被用于重构图像。在计算方法上，我们分别采用分裂Bregman方法对两个子模型进行交替求解。实验结果表明，提出的新算法不但收敛速度较快，而且在去噪过程中能够减缓阶梯效应并能较好地保持图像的边缘信息。
3.针对乘性噪声去除问题，结合变分方法、偏微分方程、稀疏表示以及字典学习这几个数学工具，在对数域提出了一种去除乘性噪声的稀疏正则化方法。该方法主要包括三步：首先，采用稀疏表示和字典学习的方法获得一个较优的log图；其次，对获得的log图利用总变分(TV)模型进行校正；最后，将校正后的结果用一个指数函数和偏差纠正从对数域变回到真实的图像。实验结果表明，新方法在能有效去除乘性噪声的同时还能较好地保持图像的纹理特征。
4. TV正则虽然具有良好的保边性能，但它在去噪过程中容易导致阶梯效应，针对这一缺陷，利用最近提出的总广义变分(TGV)，在对数域建立了一种新的去除乘性噪声的正则化模型，并从数学上证明了该模型解的存在唯一性。由于TGV的诸多优点，使得新模型在有效去除乘性噪声的同时既避免了阶梯效应的产生又较好地保持了图像的细节特征。在数值计算上，我们采用一阶原始－对偶算法和牛顿迭代方法对新模型进行求解。实验结果表明，所提出的算法无论是在视觉效果上还是峰值信噪比(PSNR)上都能获得较好的结果。
5.乘性噪声去除在相干成像系统以及各种各样的图像处理应用中具有十分重要的意义。目前去除乘性噪声的大部分数学模型都是基于正则化方法的，求解这类模型一般都需要知道噪声强度的先验信息，然而，这些信息在某些情况下并不能获得，针对这个缺陷，提出了两种去除乘性噪声的投影方法，并通过利用对偶方法和变量分裂技巧给出了三种快速的数值算法。实验结果表明，所提出的去噪算法不仅收敛速度快，而且在噪声强度先验信息未知的情况下也能有效地滤除乘性噪声。
In recent years, variational method、partial differential equation (PDE) and sparsereprensentations play active roles in many image processing fields. They have becomethree basic tools for image processing and computer vision. This dissertation mainlystudies the models and applications of variation、PDE and sparse reprensentations inimage processing. The main work can be summarized as follows:
1. Two image inpainting models are proposed. One is the image inpainting modelbased on nonlocal diffusion. This model can inpaint the damaged domains by using theglobal information of the image in the diffusion process, it can overcome theshortcomings that the PDE-based models tend to produce some blur and cannotpreserve the texture. The other is the variational inpainting model based on alternateiteration. There are two variables in the new model, so it is firstly turned into twosimple submodels by using the alternative minimization method in the new algorithm,and then the two submodels are solved using split Bregman method respectively. Dueto the fast convergence property of the split Bregman method, the new algorithm canimprove the inpainting speed, thus reducing the runtime. The experimental resultsshow that the two image inpainting models can both obtain the better inpainting effect.
2. As the ROF model for additive noise removal tends to produce the staircaseeffect, a novel variational denoising model is proposed based on the study of the LOTmodel. The model can be turned into two simple submodels by using the alternativeminimization method, one is used to construct an angle, and the other is used toreconstruct the image. In the numerical computation, we apply the split Bregmanmethod to solve the two submodels alternately. The experimental results show that thenew algorithm not only has faster convergence rate, but also can alleviate the staircaseeffect and preserve the edge information better.
3. Aiming at the problem of multiplicative noise removal, combining thevariational method、PDE、sparse reprensentations and dictionary learning, a sparsityregularization method for multiplicative noise removal is proposed in log-domain. Theproposed method mainly contains three steps. Firstly, a better log-image is obtained byusing the sparse representation based on the dictionary learning. Secondly, the totalvariation (TV) model is used to amend the log-image. Finally, via an exponentialfunction and bias correction, the result is transformed back from the log-image domain to the real one. The experimental results show that the new method is more effective tofilter out the multiplicative noise while well preserving the texture.
4. TV can preserve the edge information better, but it tends to produce the staircaseeffect. In order to overcome the drawback, a novel variational regularization model formultiplicative noise removal is proposed in log-domain based on the total generalizedvariation (TGV) introduced recently, and the existence and uniqueness of a minimizerfor the proposed model are proven. TGV has many advantages, which make the newmodel to be more effective for multiplicative noise removal while avoiding thestaircase effect and well preserving the feature and details. In the numericalcomputation, we use the first-order primal-dual algorithm and the Newton method tosolve the new model. The experimental results show that the proposed algorithm canget the better results from the visual effect and the peak signal to noise ratio (PSNR).
5. Multiplicative noise removal is of momentous significance in coherent imagingsystems and various image processing applications. So far, most multiplicative noiseremoval models focus on the regularization method, these models require knowing theprior level of noise beforehand, however, the information isn't obtained in some case,to overcome the drawback, two projection methods for multiplicative noise removalmodels are presented, and three fast numerical algorithms are given by using theduality technique and the variable-splitting. The experimental results show that theproposed algorithms not only have faster convergence rate, but also can effectivelyfilter out the multiplicative noise when the prior of noise is unknown.
引文
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