交替方向法和TGV正则在图像处理中的应用研究
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摘要
近年来，以变分方法和偏微分方程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两大基本工具。本论文主要应用交替方向法和总广义变分(TGV)正则讨论图像处理中的一些数学模型和算法。主要做了以下几个方面的工作：
1.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的缺陷，提出了两种新的图像去噪模型。一是基于交替迭代的变分去噪模型。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向量场，另一个被用于重构图像。在计算方法上，分别采用对偶方法和分裂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，而且在去噪过程中还能够减缓阶梯效应；二是基于二阶TGV的自适应图像去噪模型。该模型是在二阶TGV中引入边缘指示函数，并利用边缘指示函数引导扩散，使得新模型在去噪的同时不仅能够自适应地对图像的边缘和小结构进行有效的保护，而且还能避免阶梯效应。
2.提出了两种新的图像修复模型。一是改进的TV-Stokes图像修复模型。该模型为包含两个变量的耦合模型，因此新算法首先采用交替迭代策略化原问题为两个去耦的次问题，然后再对两个次问题分别利用对偶方法和分裂Bregman方法进行数值求解。由于新模型对TV-Stokes模型中的两步从构成方式及计算方式上分别进行了改进，因此，新算法相比于TV-Stokes算法不但修复的效果较好，而且修复的速度较快；二是基于二阶TGV的图像修复模型。该模型是以二阶TGV为正则项，因此，新模型不仅能够在修复图像的同时去除噪声，而且还能避免阶梯效应，仿真实验表明，与经典的总变分(TV)图像修复模型相比，提出的新模型在修复结果上有更高的峰值信噪比和更好的视觉效果。
3.围绕着当前图像分解模型中出现的不足，利用变分正则化方法提出了两种自适应的图像分解模型。在提出的新模型中，正则项都是通过一个自适应函数来自动地联立TV和Tikhnov二次TV，因此，两个新模型都具有滤波和变分方法的优点。模型的数值计算是通过交替方向法来实现的。实验结果表明，新模型不仅能够更好地保护卡通成分中的边缘等主要特征，而且还能从原图像中提取更多的纹理或噪声。
4.针对TV正则易导致阶梯效应这一缺陷，利用最近提出的TGV，在变分的框架下提出了两种新的图像分解模型。由于TGV的诸多优点，使得两种新模型在有效进行卡通＋纹理分解的同时还能避免阶梯效应的产生。在数值计算上，分别采用一阶原始－对偶算法、对偶方法和分裂Bregman方法对所提新模型进行求解。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上都能获得较好的结果。
In recent years, variational method and partial differential equation play activeroles in many image processing fields. They have become two basic tools for imageprocessing and computer vision. This dissertation mainly uses alternating directionmethod and total generalized variation (TGV) regularization to study somemathematical models and algorithms in image processing. The main work can besummarized as follows:
1. As the ROF model for additive noise removal tends to produce the staircaseeffect, two novel variational denoising models are proposed. The first is the variationaldenoising model based on alternate iteration. The model can be turned into two simplesubmodels by using the alternative minimization method, one is used to construct thevector field, and the other is used to reconstruct the image. In the numericalcomputation, the dual method and the split Bregman method are used to solve the twosubmodels alternately. The experimental results show that the new algorithm not onlyhas faster convergence rate, but also can alleviate the staircase effect. The second is theadaptive denoising model with the second order TGV. In the new model, an edgeindicator function is introduced in the regularization term of second order TGV toinduct diffusion, which makes the new model can adaptively preserve the edgeinformation while removing noise and avoid the staircase effect.
2. Two image inpainting models are proposed. One is the improved TV-Stokesinpainting model. There are two variables in the new model, so it is firstly turned intotwo simple submodels by using alternating iteration method, and then the twosubproblems are solved by dual formulation and split Bregman method respectively.The new model has improved the TV-Stokes model from the way of its formation andcalculation, so the proposed algorithm can not only get the better inpainting effect, butalso get the faster inpainting speed than the TV-Stokes algorithm. The other is thevariational inpainting model based on the second order TGV. In the model, the secondorder TGV is taken as the regularization term, so the new model can not only repair theimage effectively while removing the noise, but also avoid the staircase effect, thesimulative experiments show that the proposed model is better than the classical totalvariation (TV) model in terms of both peak signal to noise ratio and visual effect.
3. Aiming at the shortages of current image decomposition models, two adaptive image decomposition models are presented by using the method of variationalregularization. In the two new models, the regularization terms can automaticallycombine TV and Tikhnov quadratic TV by an adaptive function, so the proposedmodels can both possess the advantages of filter and the variational method. Thesolutions of the new models can be obtained by using the alternating direction method.Experimental results show that the proposed models can not only well preserve theedge in the cartoon part, but also extract more textures or noises from the originalimage.
4. In order to overcome the drawback that TV tends to produce the staircase effect,two novel image decomposition models are proposed under the variational frameworkbased on the TGV introduced recently, TGV has many advantages, which make thenew models to be more effective for image decomposition while avoiding the staircaseeffect. In the numerical computation, the first-order primal-dual algorithm, the dualmethod and the split Bregman method are respectively used to solve the proposedmodels. The experimental results show that the proposed algorithms can get the betterresults from the visual effect and the peak signal to noise ratio.
1.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的缺陷，提出了两种新的图像去噪模型。一是基于交替迭代的变分去噪模型。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向量场，另一个被用于重构图像。在计算方法上，分别采用对偶方法和分裂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，而且在去噪过程中还能够减缓阶梯效应；二是基于二阶TGV的自适应图像去噪模型。该模型是在二阶TGV中引入边缘指示函数，并利用边缘指示函数引导扩散，使得新模型在去噪的同时不仅能够自适应地对图像的边缘和小结构进行有效的保护，而且还能避免阶梯效应。
2.提出了两种新的图像修复模型。一是改进的TV-Stokes图像修复模型。该模型为包含两个变量的耦合模型，因此新算法首先采用交替迭代策略化原问题为两个去耦的次问题，然后再对两个次问题分别利用对偶方法和分裂Bregman方法进行数值求解。由于新模型对TV-Stokes模型中的两步从构成方式及计算方式上分别进行了改进，因此，新算法相比于TV-Stokes算法不但修复的效果较好，而且修复的速度较快；二是基于二阶TGV的图像修复模型。该模型是以二阶TGV为正则项，因此，新模型不仅能够在修复图像的同时去除噪声，而且还能避免阶梯效应，仿真实验表明，与经典的总变分(TV)图像修复模型相比，提出的新模型在修复结果上有更高的峰值信噪比和更好的视觉效果。
3.围绕着当前图像分解模型中出现的不足，利用变分正则化方法提出了两种自适应的图像分解模型。在提出的新模型中，正则项都是通过一个自适应函数来自动地联立TV和Tikhnov二次TV，因此，两个新模型都具有滤波和变分方法的优点。模型的数值计算是通过交替方向法来实现的。实验结果表明，新模型不仅能够更好地保护卡通成分中的边缘等主要特征，而且还能从原图像中提取更多的纹理或噪声。
4.针对TV正则易导致阶梯效应这一缺陷，利用最近提出的TGV，在变分的框架下提出了两种新的图像分解模型。由于TGV的诸多优点，使得两种新模型在有效进行卡通＋纹理分解的同时还能避免阶梯效应的产生。在数值计算上，分别采用一阶原始－对偶算法、对偶方法和分裂Bregman方法对所提新模型进行求解。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上都能获得较好的结果。
In recent years, variational method and partial differential equation play activeroles in many image processing fields. They have become two basic tools for imageprocessing and computer vision. This dissertation mainly uses alternating directionmethod and total generalized variation (TGV) regularization to study somemathematical models and algorithms in image processing. The main work can besummarized as follows:
1. As the ROF model for additive noise removal tends to produce the staircaseeffect, two novel variational denoising models are proposed. The first is the variationaldenoising model based on alternate iteration. The model can be turned into two simplesubmodels by using the alternative minimization method, one is used to construct thevector field, and the other is used to reconstruct the image. In the numericalcomputation, the dual method and the split Bregman method are used to solve the twosubmodels alternately. The experimental results show that the new algorithm not onlyhas faster convergence rate, but also can alleviate the staircase effect. The second is theadaptive denoising model with the second order TGV. In the new model, an edgeindicator function is introduced in the regularization term of second order TGV toinduct diffusion, which makes the new model can adaptively preserve the edgeinformation while removing noise and avoid the staircase effect.
2. Two image inpainting models are proposed. One is the improved TV-Stokesinpainting model. There are two variables in the new model, so it is firstly turned intotwo simple submodels by using alternating iteration method, and then the twosubproblems are solved by dual formulation and split Bregman method respectively.The new model has improved the TV-Stokes model from the way of its formation andcalculation, so the proposed algorithm can not only get the better inpainting effect, butalso get the faster inpainting speed than the TV-Stokes algorithm. The other is thevariational inpainting model based on the second order TGV. In the model, the secondorder TGV is taken as the regularization term, so the new model can not only repair theimage effectively while removing the noise, but also avoid the staircase effect, thesimulative experiments show that the proposed model is better than the classical totalvariation (TV) model in terms of both peak signal to noise ratio and visual effect.
3. Aiming at the shortages of current image decomposition models, two adaptive image decomposition models are presented by using the method of variationalregularization. In the two new models, the regularization terms can automaticallycombine TV and Tikhnov quadratic TV by an adaptive function, so the proposedmodels can both possess the advantages of filter and the variational method. Thesolutions of the new models can be obtained by using the alternating direction method.Experimental results show that the proposed models can not only well preserve theedge in the cartoon part, but also extract more textures or noises from the originalimage.
4. In order to overcome the drawback that TV tends to produce the staircase effect,two novel image decomposition models are proposed under the variational frameworkbased on the TGV introduced recently, TGV has many advantages, which make thenew models to be more effective for image decomposition while avoiding the staircaseeffect. In the numerical computation, the first-order primal-dual algorithm, the dualmethod and the split Bregman method are respectively used to solve the proposedmodels. The experimental results show that the proposed algorithms can get the betterresults from the visual effect and the peak signal to noise ratio.
引文
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