变分和非凸正则在图像处理中的应用研究
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摘要
近年来，变分、偏微分方程等数学工具被广泛地应用于图像处理、计算机视觉等领域。这些工具一方面为模型的建立提供了有力的理论依据，另一方面方便了人们讨论模型的性能。本论文针对图像处理中的去噪、分割、配准等问题，给出了一些新的基于变分和偏微分方程的图像处理模型，并设计了新模型的快速求解算法。主要包括如下几个方面的工作：
1.提出一个新的加性噪声去除模型和一个新的乘性噪声去除模型。在加性噪声去除模型中，我们利用空间非局部梯度构造了图像的非局部结构张量，通过对非局部结构张量的特征分解得到图像的一个特征空间，依据特征空间的特性设计了非局部扩散张量，建立了基于非局部扩散张量的各向异性扩散模型。该模型和局部各向异性扩散模型的本质不同在于，它在扩散的过程中充分利用了图像的全局信息。数值实验表明：所提的加性噪声去除模型在去噪的同时，能够更好地保护图像的边缘，尤其是纹理等重要细节特征。在乘性噪声去除模型中，我们采用非凸形式的正则项度量恢复图像的光滑性。与经典的总变差正则项相比，该非凸正则项具有更稀疏的性质，并且能够更好地保持恢复图像的边缘几何结构。为了快速求解新的乘性噪声去除模型，原始无约束的模型首先被转化为含约束的等价模型，然后采用分裂Bregman算法进行求解。为了克服模型中正则项的非凸性，分裂Bregman算法中还嵌入了一个迭代重加权过程。新算法不仅可以输出恢复图像，而且可以获得图像的边缘指示子。数值实验表明：所提乘性噪声去除模型无论在视觉效果上还是在量化指标上都能获得较好的去噪效果。
2.将图像分割技术引入图像颜色迁移过程中，提出一种基于图像分割和颜色主题的织物图像颜色迁移新方法。该方法包括三个过程。第一个过程是分割彩色织物图像，通过引入图像局部偏场的概念，该过程不仅能够较好地解决亮度不一致的彩色图像分割问题，而且能够估计出给定织物图像各个区域的颜色均值，这些颜色均值的组合被定义为该织物图像的颜色主题。第二个过程是在颜色主题数据库上检索和原始织物图像颜色主题相似的并且分值较高的颜色主题。第三个过程是将检索到的颜色主题迁移到原始织物图像上。上述三个过程中，图像分割过程是所提颜色迁移方法的关键和主要创新部分。数值实验表明：所提颜色迁移方法能够为设计师们提供一种有效的织物设计工具。
3.对多区域模糊区域竞争(MFRC)图像分割模型进行改进，得到一类新的图像分割模型，其中包括灰度图像分割和自然(彩色－纹理)图像分割两个模型。两个模型的共同点是采用非凸形式的正则项代替MFRC模型中的总变差正则项度量模糊隶属度函数的光滑性，较好地保持了各个隶属度函数的边缘几何结构。在灰度图像分割模型中，各个同性区域的亮度分布并非假定为MFRC模型中的固定形式的高斯分布，而是采用核估计的方法估计得到，从而能够分割更复杂的图像。在彩色图像分割模型中，针对原始MFRC模型很难分割自然图像的问题，通过引入PCA描述子的方法，使自然图像分割问题能够被合理地纳入到模糊区域竞争框架下讨论。数值实验表明：所提的两个新分割模型与经典的水平集模型和MFRC模型相比，能够获得更好的分割结果。所提的第二个分割模型与当前最好的基于图割的自然图像分割模型相比，具有更高的运算效率。
4.提出一类基于变分方法的非刚性图像配准模型，其中包括单模态和多模态两个模型。在单模态模型中，正则项采用加权的L2范数度量，一方面克服了迭代收敛不同步的问题，另一方面使新模型既能保持图像的边缘几何结构，又能避免块效应的产生。在多模态模型中，不同模态的图像被转化为同一模态进行处理，提高了配准的效率。在模型求解方面，利用算子分裂和交替迭代的方法，原问题被转化为阈值和加性算子分裂的迭代格式进行求解。数值实验表明：所提配准方法对含噪以及变形较大的图像都能实现较好的配准。
5.针对传统的多模态配准中平均互信息不能很好地配准局部非刚性形变以及位移场在迭代过程中被过度磨光的问题，我们给出了一个新的配准模型。新模型采用局部联合熵作为数据项来度量浮动图像和模板图像之间的相似性，以及采用加权的Horn型正则项来避免位移场被过度磨光。数值实验表明：所提模型与经典的配准模型相比，能够更好地配准局部形变。
In recent years, some mathematical tools such as variational calculus (VC) andpartial differential equation (PDE) are widely used in the fields of image processingand computer vision. These tools not only provide very theoretial supports for aproposed model, but also facilitate the discussion on the performance of the model. Inthis thesis, we do some research on some basic but important problems in imageprocessing, including denoising, segmentation, and registration. Some novel models aswell as their corresponding efficient algorithms are proposed. The main work issummarized as follows.
1. A new additive noise removal model and a new multiplicative noise removalmodel are proposed. In the additive noise removal model, the nonlocal structure tensorof images is defined by using the nonlocal spatial gradients. The eigenvectors of thenonlocal structure tensor consist in a characteristic space for the image, based on whichthe nonlocal diffusion tensor is constructed. Using the nonlocal diffusion tensor, wepropose the nonlocal anisotropic diffusion model for image denoising. This model isdifferent from the local anisotropic diffusion in that, not only neighboring pixels butalso pixels faraway with similar intensities are concerned in our model. The mainadvantage of taking those pixels faraway but with similar intensities into considerationis that that model protects edges and textures much better than the local model. In themultiplicative noise removal model a nonconvex regularization term is used tomeasure the smoothness of the restored image. Different from the classical totalvariation (TV) regularization term, the proposed nonconvex regularization term has amuch sparser property, and it can preserve the geometrical structure of the image better.In order to obtain a more efficient algorithm to solve the model, we first take use of thesplitting technology to convert the original model into an equivalent one. Then, thesplit Bregman algorithm is used to solve the equivalent model. To overcome thenonconvexity of our model, an iteratively reweighting process is incorporated into thesplit Bregman algorithm. From the well-designed algorithm, we can obtain the restoredimage as well as the edge indicator of the image. Comprehensive experiments areconducted to measure the performance of the proposed denoising methods in terms ofvisual evaluation and a variety of quantitative indices.
2. Image segmentation is introduced into the process of color transfer, and a new image-segmentation and color-theme-based color transfer method for textile images isproposed. The method contains three phases. The first phase is to segment the inputtextile image into several regions. As the introduction of a bias field function, thisphase can not only partition images with nonhomogeneous illumination, but alsooutput color means of different regions of the image. The combination of these colormeans is considered as the color theme of the input image. The second phase is toretrieve the relevant color themes from a database of color themes. The third phase isto reconstruct new images with different appearances from the input image by usingthe retrieved color themes. In the three phases mentioned above, the most importantphase is the image segmentation phase which is the main innovation of the proposedcolor transfer method. Numerical results indicate that the proposed color transfermethod can provide a powerful tool for designers to generate textile patterns.
3. To improve the segmentation performance of the so called multiphase fuzzyregion competition (MFRC) model, we propose two new image segmentation models,one for gray-scale images, and the other one for color-scale images. In both of the twomodels, we introduce a nonconvex regularization term on the fuzzy membershipfunctions. This regularization term performs better than the usual convex TV used inthe MFRC model in that it can protect edges of the fuzzy membership functions fromover-smoothing. In addition, in the gray-scale image segmentation model, the intensitydistribution of each homogeneous region is not assumed to be the Gaussian distributionas in the MFRC model, but estimated from the kernal estimation method. For thisreason, our model can be used to partition more complicated images. In the color-scaleimage segmentation model, to overcome the difficulty that the classical MFRC modelcannot be simply used to partition natural (color-texture) images, we introduce thePCA descriptors so that the natural image segmentation problem can be discussed inthe framework of region competition. Numerical results demonstrate that the proposedtwo models achieve better segmentation results compared with some other well-knownmodels, such as the level-set (LS) model and the MFRC model, while the algorithm ofour natural image segmentation model is much more efficient than the state-of-the-artgraph cut-based algorithm.
4. Two variational calculus-based nongird registration models are proposed, one ofwhich is the one-modality model, and the other is the multi-modality model. In theone-modality model, the weighted L2norm is used as the regularization term, whichbrings out two advantages. Firstly, it avoids the imbalance problem of the convergingspeed in different regions. Secondly, it preserves the important geometric structures of an image while restrains the staircase effect. In the multi-modality model, imagesobtained from different modalities are converted into the one-modality ones. Then themethods which are used to handle the one-modality problems can be used to deal withthe multi-modality problems. By exploiting the techniques of the operator splitting andthe alternative iteration minimization, we solve the models by shrinking and additiveoperator splitting (AOS). Numerical results demonstrate that the proposed registrationmethods perform well for noisy images and images with large deformation.
5. To overcome the problem that local deformation are not aligned well by mutualinformation, we propose a new registration model in which local joint entropy is usedto measure the similarity between the moving image and the template image. In themodel, the weighted Horn-type regularizer is used to protect the displacement fieldfrom over-smoothing. Numerical results demonstrate that the proposed model has theadvantage of aligning local edges of the images better than classical models.
1.提出一个新的加性噪声去除模型和一个新的乘性噪声去除模型。在加性噪声去除模型中，我们利用空间非局部梯度构造了图像的非局部结构张量，通过对非局部结构张量的特征分解得到图像的一个特征空间，依据特征空间的特性设计了非局部扩散张量，建立了基于非局部扩散张量的各向异性扩散模型。该模型和局部各向异性扩散模型的本质不同在于，它在扩散的过程中充分利用了图像的全局信息。数值实验表明：所提的加性噪声去除模型在去噪的同时，能够更好地保护图像的边缘，尤其是纹理等重要细节特征。在乘性噪声去除模型中，我们采用非凸形式的正则项度量恢复图像的光滑性。与经典的总变差正则项相比，该非凸正则项具有更稀疏的性质，并且能够更好地保持恢复图像的边缘几何结构。为了快速求解新的乘性噪声去除模型，原始无约束的模型首先被转化为含约束的等价模型，然后采用分裂Bregman算法进行求解。为了克服模型中正则项的非凸性，分裂Bregman算法中还嵌入了一个迭代重加权过程。新算法不仅可以输出恢复图像，而且可以获得图像的边缘指示子。数值实验表明：所提乘性噪声去除模型无论在视觉效果上还是在量化指标上都能获得较好的去噪效果。
2.将图像分割技术引入图像颜色迁移过程中，提出一种基于图像分割和颜色主题的织物图像颜色迁移新方法。该方法包括三个过程。第一个过程是分割彩色织物图像，通过引入图像局部偏场的概念，该过程不仅能够较好地解决亮度不一致的彩色图像分割问题，而且能够估计出给定织物图像各个区域的颜色均值，这些颜色均值的组合被定义为该织物图像的颜色主题。第二个过程是在颜色主题数据库上检索和原始织物图像颜色主题相似的并且分值较高的颜色主题。第三个过程是将检索到的颜色主题迁移到原始织物图像上。上述三个过程中，图像分割过程是所提颜色迁移方法的关键和主要创新部分。数值实验表明：所提颜色迁移方法能够为设计师们提供一种有效的织物设计工具。
3.对多区域模糊区域竞争(MFRC)图像分割模型进行改进，得到一类新的图像分割模型，其中包括灰度图像分割和自然(彩色－纹理)图像分割两个模型。两个模型的共同点是采用非凸形式的正则项代替MFRC模型中的总变差正则项度量模糊隶属度函数的光滑性，较好地保持了各个隶属度函数的边缘几何结构。在灰度图像分割模型中，各个同性区域的亮度分布并非假定为MFRC模型中的固定形式的高斯分布，而是采用核估计的方法估计得到，从而能够分割更复杂的图像。在彩色图像分割模型中，针对原始MFRC模型很难分割自然图像的问题，通过引入PCA描述子的方法，使自然图像分割问题能够被合理地纳入到模糊区域竞争框架下讨论。数值实验表明：所提的两个新分割模型与经典的水平集模型和MFRC模型相比，能够获得更好的分割结果。所提的第二个分割模型与当前最好的基于图割的自然图像分割模型相比，具有更高的运算效率。
4.提出一类基于变分方法的非刚性图像配准模型，其中包括单模态和多模态两个模型。在单模态模型中，正则项采用加权的L2范数度量，一方面克服了迭代收敛不同步的问题，另一方面使新模型既能保持图像的边缘几何结构，又能避免块效应的产生。在多模态模型中，不同模态的图像被转化为同一模态进行处理，提高了配准的效率。在模型求解方面，利用算子分裂和交替迭代的方法，原问题被转化为阈值和加性算子分裂的迭代格式进行求解。数值实验表明：所提配准方法对含噪以及变形较大的图像都能实现较好的配准。
5.针对传统的多模态配准中平均互信息不能很好地配准局部非刚性形变以及位移场在迭代过程中被过度磨光的问题，我们给出了一个新的配准模型。新模型采用局部联合熵作为数据项来度量浮动图像和模板图像之间的相似性，以及采用加权的Horn型正则项来避免位移场被过度磨光。数值实验表明：所提模型与经典的配准模型相比，能够更好地配准局部形变。
In recent years, some mathematical tools such as variational calculus (VC) andpartial differential equation (PDE) are widely used in the fields of image processingand computer vision. These tools not only provide very theoretial supports for aproposed model, but also facilitate the discussion on the performance of the model. Inthis thesis, we do some research on some basic but important problems in imageprocessing, including denoising, segmentation, and registration. Some novel models aswell as their corresponding efficient algorithms are proposed. The main work issummarized as follows.
1. A new additive noise removal model and a new multiplicative noise removalmodel are proposed. In the additive noise removal model, the nonlocal structure tensorof images is defined by using the nonlocal spatial gradients. The eigenvectors of thenonlocal structure tensor consist in a characteristic space for the image, based on whichthe nonlocal diffusion tensor is constructed. Using the nonlocal diffusion tensor, wepropose the nonlocal anisotropic diffusion model for image denoising. This model isdifferent from the local anisotropic diffusion in that, not only neighboring pixels butalso pixels faraway with similar intensities are concerned in our model. The mainadvantage of taking those pixels faraway but with similar intensities into considerationis that that model protects edges and textures much better than the local model. In themultiplicative noise removal model a nonconvex regularization term is used tomeasure the smoothness of the restored image. Different from the classical totalvariation (TV) regularization term, the proposed nonconvex regularization term has amuch sparser property, and it can preserve the geometrical structure of the image better.In order to obtain a more efficient algorithm to solve the model, we first take use of thesplitting technology to convert the original model into an equivalent one. Then, thesplit Bregman algorithm is used to solve the equivalent model. To overcome thenonconvexity of our model, an iteratively reweighting process is incorporated into thesplit Bregman algorithm. From the well-designed algorithm, we can obtain the restoredimage as well as the edge indicator of the image. Comprehensive experiments areconducted to measure the performance of the proposed denoising methods in terms ofvisual evaluation and a variety of quantitative indices.
2. Image segmentation is introduced into the process of color transfer, and a new image-segmentation and color-theme-based color transfer method for textile images isproposed. The method contains three phases. The first phase is to segment the inputtextile image into several regions. As the introduction of a bias field function, thisphase can not only partition images with nonhomogeneous illumination, but alsooutput color means of different regions of the image. The combination of these colormeans is considered as the color theme of the input image. The second phase is toretrieve the relevant color themes from a database of color themes. The third phase isto reconstruct new images with different appearances from the input image by usingthe retrieved color themes. In the three phases mentioned above, the most importantphase is the image segmentation phase which is the main innovation of the proposedcolor transfer method. Numerical results indicate that the proposed color transfermethod can provide a powerful tool for designers to generate textile patterns.
3. To improve the segmentation performance of the so called multiphase fuzzyregion competition (MFRC) model, we propose two new image segmentation models,one for gray-scale images, and the other one for color-scale images. In both of the twomodels, we introduce a nonconvex regularization term on the fuzzy membershipfunctions. This regularization term performs better than the usual convex TV used inthe MFRC model in that it can protect edges of the fuzzy membership functions fromover-smoothing. In addition, in the gray-scale image segmentation model, the intensitydistribution of each homogeneous region is not assumed to be the Gaussian distributionas in the MFRC model, but estimated from the kernal estimation method. For thisreason, our model can be used to partition more complicated images. In the color-scaleimage segmentation model, to overcome the difficulty that the classical MFRC modelcannot be simply used to partition natural (color-texture) images, we introduce thePCA descriptors so that the natural image segmentation problem can be discussed inthe framework of region competition. Numerical results demonstrate that the proposedtwo models achieve better segmentation results compared with some other well-knownmodels, such as the level-set (LS) model and the MFRC model, while the algorithm ofour natural image segmentation model is much more efficient than the state-of-the-artgraph cut-based algorithm.
4. Two variational calculus-based nongird registration models are proposed, one ofwhich is the one-modality model, and the other is the multi-modality model. In theone-modality model, the weighted L2norm is used as the regularization term, whichbrings out two advantages. Firstly, it avoids the imbalance problem of the convergingspeed in different regions. Secondly, it preserves the important geometric structures of an image while restrains the staircase effect. In the multi-modality model, imagesobtained from different modalities are converted into the one-modality ones. Then themethods which are used to handle the one-modality problems can be used to deal withthe multi-modality problems. By exploiting the techniques of the operator splitting andthe alternative iteration minimization, we solve the models by shrinking and additiveoperator splitting (AOS). Numerical results demonstrate that the proposed registrationmethods perform well for noisy images and images with large deformation.
5. To overcome the problem that local deformation are not aligned well by mutualinformation, we propose a new registration model in which local joint entropy is usedto measure the similarity between the moving image and the template image. In themodel, the weighted Horn-type regularizer is used to protect the displacement fieldfrom over-smoothing. Numerical results demonstrate that the proposed model has theadvantage of aligning local edges of the images better than classical models.
引文
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