变分模型和稀疏冗余表示在图像恢复中的应用研究
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摘要
图像恢复是以观测图像为已知数据,根据对图像退化模型以及图像本身先验知识的理解来还原反映客观世界真实场景的原始图像,它是保障人们能正确理解图像中所蕴含的信息以及图像后处理是否有效的重要前端技术。建立变分恢复模型是解决该问题的有效途径,当图像退化模型事先已知或已估计时,图像先验知识的表达直接关系到变分模型是否准确,进而决定了图像恢复的性能。分段光滑性和变换系数的稀疏性是图像信号的两个重要属性,对变分模型中正则项的设计起到了重要的帮助作用。近些年来,随着人们对图像理解的不断深入,稀疏冗余表示在图像处理领域中被广泛应用起来,它可以根据观测数据自适应地学习出稀疏表示所用的冗余字典,从而更真实地表达出待处理图像的先验知识。
本文利用变分模型和稀疏冗余表示技术,选取图像恢复中的噪声污染图像和低分辨率图像的恢复问题进行了详细研究,主要工作和创新点如下：
1.研究了高比率冲击噪声污染图像的恢复算法,重点针对随机值冲击噪声的情况进行了分析,提出了一种基于冲击噪声分级决策的细节保持变分恢复方法。首先,将变窗和序贯检测的策略融入到自适应中心加权中值滤波噪声检测技术中,不仅以较低的错误率甄选出了可能的冲击噪声,还根据噪声点与周围近邻点间的灰度差异给每个噪声候选点标记了“等级”。然后,设计了一个正则化参数值随噪声等级变化而变化的细节保持变分模型,利用Jacobi松弛算法最小化该模型,将噪声候选点进行更新,恢复出隐藏在这些点之下的本真信息。同时,保持非噪声候选点的像素值不变,图像恢复即完成。受益于噪声检测的准确性以及L1-norm数据保真项和边缘保持平滑正则项在处理冲击噪声上的优势,我们的算法在噪声比率较高的情况下仍然可以有效地去除冲击噪声,并且极大程度地恢复出图像原本的边缘和细节。此外,我们还提出了一种能粗略估计冲击噪声比率的方法,当噪声比率未知时,可以利用这种方法进行大致估计。
2.研究了混合噪声污染图像的恢复算法,重点针对加性高斯噪声和随机值冲击噪声混合的情况进行了分析。为了准确地构建图像恢复变分模型,我们事先对图像中的两种噪声进行分类并标识,采用基于绝对差图像和贝叶斯决策的噪声分类法将高斯噪声和冲击噪声有效地分离。然后,从含噪图像本身中抽取一定量的样本图块进行自适应冗余字典的学习,结合噪声点的类型标识矩阵并利用"Masked K-SVD"字典学习算法从样本集的有效信息中获取能稀疏表示目标图像块的冗余字典。接着,构建一个较准确的变分恢复模型,其中数据保真项的形式由噪声类型来决定,而正则项的设计是根据图像块在自适应冗余字典下能稀疏表示的先验知识,通过块坐标下降法最小化该变分模型而获得图像恢复结果。受益于噪声分类的准确以及变分模型的准确,我们所得到的图像恢复结果较好地还原了图像本真的边缘和纹理信息,尤其对于纹理丰富的被污染图像,恢复效果令人满意。此外,我们将所构建的变分恢复框架扩展至高斯和冲击噪声的任意组合污染形式以及图像修复问题,结果表明所提算法能“对付”多种情况,并且效果良好。由此验证了所构建的变分恢复模型对于噪声图像具有一定的通用性。
3.研究了低分辨率图像的恢复算法,重点针对单幅图像的超分辨率重建问题进行了分析。在已知低分辨率图像退化模型的前提下,通过对外部高分辨率图像集的学习来帮助分辨率恢复性能的提升。在基于样本学习和稀疏冗余表示的图像恢复框架下,我们将样本图像块的分类和边缘样本块的扩展巧妙地融入其中,使得分类训练出的冗余字典有了更丰富的原子类型以及更“专”的稀疏表示性能。在保证字典质量的同时,采用了一种快速字典对的构建方法,大大降低了字典训练的时耗。在图像恢复时,输入的低分辨率图像首先被裂解为若干相互重叠的图像块,每一个图像块送入分类器中进行类别标识.以此指导字典对的选择,再在所选择的低分辨率冗余字典下通过稀疏度可调的直角匹配追踪算法进行稀疏编码,进而重建出高分辨率图块。所有高分辨率图像块再经过合理重组并修正后生成最终的高分辨率恢复图像。通过分析发现,上述分辨率重建的过程实际上也是基于稀疏冗余表示的变分恢复方法的一种特例,从而进一步证明了变分模型在图像恢复中的普适性。
Image restoration is to revert the original image of the real scene about the objective world from the observed image based on the prior knowledge of degradation model and image itself. It is an important prior technology to guarantee the right understanding of the information contained in the image and the effectiveness of the subsequent image processes. The construction of a variational restoration model is an effective approach to solve the problem of image restoration. When the image degradation model is known or estimated in advance, the prior knowledge of image itself is directly related to the accuracy of the variational restoration model and thus determines the image restoration performance. The piecewise smooth and the sparsity of the transform coefficients are the two important characteristics about the image signal. They help to design the regularization term of the variational model. In recent years, with the deeper understanding of the image prior knowledge, the technology of sparse and redundant representation is widely used in image processing. It can learn adaptively the redundant dictionary from the observed data, so the representation of the image patches in concern based on the adaptive dictionary is more effective.
In the thesis, we employ the techniques of variational model and sparse and redundant representation to study the typical problems in image restoration. The main work and innovations are listed as follows:
1. The restoration of images contaminated by impulse noise with high noise ratio has been studied. We focus on the random-valued impulse noise and propose a hierarchical decision-based detail-preserving variational restoration method. Firstly, the ideas of window variation and sequential detection are merged into the adaptive centre weighted median filter based noise detection technology. We not only select the possible noisy pixels with a low error rate, but also label them with different noise marks according to the differences in grey levels between a noisy pixel and its neighbours. Secondly, a detail-preserving variational model with a variable regularization parameter decided by the value of noise mark is designed. The Jacobi-type relaxation algorithm is employed to minimize the variational model. Then, the noise candidates are all updated and the true information hidden under the noisy pixels is restored. At the same time, the noise-free pixels are kept unchanged and a restored image is obtained. Benifitted from the good performance of noise detection and the advantages of-f,-norm data-fidelity combined with edge-preserving smooth regularization term for impulse noise, we can not only remove the impulse noise well, but also revert the real information about edges and details greatly, even in the situation of high noise ratio. Besides, we offer a method to estimate the noise ratio of a corrupted image roughly. So, if the noise ratio is unknown beforehand, it can be used to do estimation.
2. The restoration of images contaminated by mixed noise has been studied. We focus on the problem of additive Gaussian noise plus random-valued impulse noise. In order to construct an accurate variational restoration model, we classify the noisy pixels into Gaussian or impulse type in advance based on the absolute difference image and Bayesian decision theory. Next, we extract a number of signal examples from the noisy image itself and utilize the Masked K-SVD dictionary learning method to train an adaptive redundant dictionary from the effective information contained in the examples. Then, an accurate variational model which includes an optional data-fidelity term decided by noise type and a regularization term respecting sparse representation of every image patch over the adaptive redundant dictionary is constructed. The restored image is obtained through the block-coordinate descent algorithm to minimize the variational model. Benifitted from the accuracy of noise classification and variational model, the real information about edges and textures can be reverted well in our restored image, especially for the corrupted images with rich textures. Besides, we extend the designed variational model into the restoration of image contaminated by the arbitrary combination of Gaussian and impulse noise and the problem of image inpainting. The results show that this model can cope with various situations and the restored images are satisfactory. It validates the universal application of the designed variational model in the noisy image.
3. The restoration of low resolution (LR) image has been studied. We focus on the problem of single image super resolution. In the condition of the known LR image degradation model, we learn from an external training set containing various high resolution (HR) images to help promote the performance of resolution recovery. Under the restoration framework based on example learning and sparse representation, we merge the classification of image patches and the extension of edge patches into it skillfully. It makes the learned redundant dictionaries have the richer atom types and the better abilities to represent sparsely the image patches. In the premise of assurance of good dictionary quality, a fast way to learn the dictionary-pair is employed to accelerate the training process. In restoration, the input LR image is firstly split into the overlapped patches and then each patch is put into a classifier which guides to choose the dictionary-pair. Then, the sparse representation coefficient of the LR signal over the selected LR dictionary is inferred using a sparsity-adjustable OMP algorithm and the HR image patch can be reconstructed. After combining all the reconstructed patches rationally and further modification, we can get the final restored HR image. The analysis reveals that the above HR image reconstruction is actually a special case of the sparse and redundant representation based variational restoration method which is universal in some problems of image restoration.
本文利用变分模型和稀疏冗余表示技术,选取图像恢复中的噪声污染图像和低分辨率图像的恢复问题进行了详细研究,主要工作和创新点如下：
1.研究了高比率冲击噪声污染图像的恢复算法,重点针对随机值冲击噪声的情况进行了分析,提出了一种基于冲击噪声分级决策的细节保持变分恢复方法。首先,将变窗和序贯检测的策略融入到自适应中心加权中值滤波噪声检测技术中,不仅以较低的错误率甄选出了可能的冲击噪声,还根据噪声点与周围近邻点间的灰度差异给每个噪声候选点标记了“等级”。然后,设计了一个正则化参数值随噪声等级变化而变化的细节保持变分模型,利用Jacobi松弛算法最小化该模型,将噪声候选点进行更新,恢复出隐藏在这些点之下的本真信息。同时,保持非噪声候选点的像素值不变,图像恢复即完成。受益于噪声检测的准确性以及L1-norm数据保真项和边缘保持平滑正则项在处理冲击噪声上的优势,我们的算法在噪声比率较高的情况下仍然可以有效地去除冲击噪声,并且极大程度地恢复出图像原本的边缘和细节。此外,我们还提出了一种能粗略估计冲击噪声比率的方法,当噪声比率未知时,可以利用这种方法进行大致估计。
2.研究了混合噪声污染图像的恢复算法,重点针对加性高斯噪声和随机值冲击噪声混合的情况进行了分析。为了准确地构建图像恢复变分模型,我们事先对图像中的两种噪声进行分类并标识,采用基于绝对差图像和贝叶斯决策的噪声分类法将高斯噪声和冲击噪声有效地分离。然后,从含噪图像本身中抽取一定量的样本图块进行自适应冗余字典的学习,结合噪声点的类型标识矩阵并利用"Masked K-SVD"字典学习算法从样本集的有效信息中获取能稀疏表示目标图像块的冗余字典。接着,构建一个较准确的变分恢复模型,其中数据保真项的形式由噪声类型来决定,而正则项的设计是根据图像块在自适应冗余字典下能稀疏表示的先验知识,通过块坐标下降法最小化该变分模型而获得图像恢复结果。受益于噪声分类的准确以及变分模型的准确,我们所得到的图像恢复结果较好地还原了图像本真的边缘和纹理信息,尤其对于纹理丰富的被污染图像,恢复效果令人满意。此外,我们将所构建的变分恢复框架扩展至高斯和冲击噪声的任意组合污染形式以及图像修复问题,结果表明所提算法能“对付”多种情况,并且效果良好。由此验证了所构建的变分恢复模型对于噪声图像具有一定的通用性。
3.研究了低分辨率图像的恢复算法,重点针对单幅图像的超分辨率重建问题进行了分析。在已知低分辨率图像退化模型的前提下,通过对外部高分辨率图像集的学习来帮助分辨率恢复性能的提升。在基于样本学习和稀疏冗余表示的图像恢复框架下,我们将样本图像块的分类和边缘样本块的扩展巧妙地融入其中,使得分类训练出的冗余字典有了更丰富的原子类型以及更“专”的稀疏表示性能。在保证字典质量的同时,采用了一种快速字典对的构建方法,大大降低了字典训练的时耗。在图像恢复时,输入的低分辨率图像首先被裂解为若干相互重叠的图像块,每一个图像块送入分类器中进行类别标识.以此指导字典对的选择,再在所选择的低分辨率冗余字典下通过稀疏度可调的直角匹配追踪算法进行稀疏编码,进而重建出高分辨率图块。所有高分辨率图像块再经过合理重组并修正后生成最终的高分辨率恢复图像。通过分析发现,上述分辨率重建的过程实际上也是基于稀疏冗余表示的变分恢复方法的一种特例,从而进一步证明了变分模型在图像恢复中的普适性。
Image restoration is to revert the original image of the real scene about the objective world from the observed image based on the prior knowledge of degradation model and image itself. It is an important prior technology to guarantee the right understanding of the information contained in the image and the effectiveness of the subsequent image processes. The construction of a variational restoration model is an effective approach to solve the problem of image restoration. When the image degradation model is known or estimated in advance, the prior knowledge of image itself is directly related to the accuracy of the variational restoration model and thus determines the image restoration performance. The piecewise smooth and the sparsity of the transform coefficients are the two important characteristics about the image signal. They help to design the regularization term of the variational model. In recent years, with the deeper understanding of the image prior knowledge, the technology of sparse and redundant representation is widely used in image processing. It can learn adaptively the redundant dictionary from the observed data, so the representation of the image patches in concern based on the adaptive dictionary is more effective.
In the thesis, we employ the techniques of variational model and sparse and redundant representation to study the typical problems in image restoration. The main work and innovations are listed as follows:
1. The restoration of images contaminated by impulse noise with high noise ratio has been studied. We focus on the random-valued impulse noise and propose a hierarchical decision-based detail-preserving variational restoration method. Firstly, the ideas of window variation and sequential detection are merged into the adaptive centre weighted median filter based noise detection technology. We not only select the possible noisy pixels with a low error rate, but also label them with different noise marks according to the differences in grey levels between a noisy pixel and its neighbours. Secondly, a detail-preserving variational model with a variable regularization parameter decided by the value of noise mark is designed. The Jacobi-type relaxation algorithm is employed to minimize the variational model. Then, the noise candidates are all updated and the true information hidden under the noisy pixels is restored. At the same time, the noise-free pixels are kept unchanged and a restored image is obtained. Benifitted from the good performance of noise detection and the advantages of-f,-norm data-fidelity combined with edge-preserving smooth regularization term for impulse noise, we can not only remove the impulse noise well, but also revert the real information about edges and details greatly, even in the situation of high noise ratio. Besides, we offer a method to estimate the noise ratio of a corrupted image roughly. So, if the noise ratio is unknown beforehand, it can be used to do estimation.
2. The restoration of images contaminated by mixed noise has been studied. We focus on the problem of additive Gaussian noise plus random-valued impulse noise. In order to construct an accurate variational restoration model, we classify the noisy pixels into Gaussian or impulse type in advance based on the absolute difference image and Bayesian decision theory. Next, we extract a number of signal examples from the noisy image itself and utilize the Masked K-SVD dictionary learning method to train an adaptive redundant dictionary from the effective information contained in the examples. Then, an accurate variational model which includes an optional data-fidelity term decided by noise type and a regularization term respecting sparse representation of every image patch over the adaptive redundant dictionary is constructed. The restored image is obtained through the block-coordinate descent algorithm to minimize the variational model. Benifitted from the accuracy of noise classification and variational model, the real information about edges and textures can be reverted well in our restored image, especially for the corrupted images with rich textures. Besides, we extend the designed variational model into the restoration of image contaminated by the arbitrary combination of Gaussian and impulse noise and the problem of image inpainting. The results show that this model can cope with various situations and the restored images are satisfactory. It validates the universal application of the designed variational model in the noisy image.
3. The restoration of low resolution (LR) image has been studied. We focus on the problem of single image super resolution. In the condition of the known LR image degradation model, we learn from an external training set containing various high resolution (HR) images to help promote the performance of resolution recovery. Under the restoration framework based on example learning and sparse representation, we merge the classification of image patches and the extension of edge patches into it skillfully. It makes the learned redundant dictionaries have the richer atom types and the better abilities to represent sparsely the image patches. In the premise of assurance of good dictionary quality, a fast way to learn the dictionary-pair is employed to accelerate the training process. In restoration, the input LR image is firstly split into the overlapped patches and then each patch is put into a classifier which guides to choose the dictionary-pair. Then, the sparse representation coefficient of the LR signal over the selected LR dictionary is inferred using a sparsity-adjustable OMP algorithm and the HR image patch can be reconstructed. After combining all the reconstructed patches rationally and further modification, we can get the final restored HR image. The analysis reveals that the above HR image reconstruction is actually a special case of the sparse and redundant representation based variational restoration method which is universal in some problems of image restoration.
引文
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