图像复原算法研究
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摘要
在图像成像、复制、扫描、传输、显示等过程中，不可避免地要造成图像的降质，如图像模糊、噪声干扰等。而在许多应用领域中，又需要清晰的、高质量的图像，因此，图像复原(如去噪、去模糊等)具有重要的意义。图像复原的目的是对降质图像进行处理，使其恢复成原始图像，它是图像处理、模式识别、机器视觉的基础，因而受到广泛的研究，并在天文学、遥感成像、医疗图像等领域获得广泛的应用。
图像复原的传统方法主要是进行图像滤波，由于图像的大部分信息存在于边缘部分，因此要求图像滤波既能去除图像的模糊和噪声，同时又能保持图像的细节。由于图像细节和噪声在频带上混叠，导致图像的平滑和边缘细节的保持成为一对矛盾，传统的滤波方法难以处理这类问题。近年来发展起来的偏微分方程图像处理技术、神经网络技术、小波分析技术以及图割技术，为解决图像复原中的这一矛盾提供了新的方法。
本文以图像复原中的去模糊和去噪问题为主要研究内容，以全变差图像模型为主要研究对象，研究了基于神经网络的图像复原模型和复原算法；分析了小波图像去噪方法与非线性滤波去噪方法之间的关系；并采用图割技术，研究了全变差去噪模型的数值求解方法。研究工作取得了如下创新性成果：
(1)神经网络图像复原算法研究
·提出了两种基于变分PDE的神经网络图像复原模型，并给出了相应的复原算法。将变分偏微分方程，以正则化项的形式引入到传统神经网络图像复原框架中，给出了基于调和方程和全变差方程的神经网络图像复原模型，提出了两种基于改进型Hopfield神经网络的图像复原算法，并分析了算法的收敛性。
·提出了一种基于状态连续改变的快速神经网络复原算法。深入研究了神经网络图像复原算法中的更新规则，将状态连续改变网络模型和最陡下降方法结合，给出了一种状态连续改变的快速神经网络更新规则。进而提出一种基于状态连续改变的快速神经网络复原算法，有效提高了复原算法的收敛速度和复原精度。
·提出了基于调和模型的快速神经网络复原算法以及并行算法。将状态连续改变的快速神经网络更新规则应用到基于调和模型的神经网络复原算法，给出了一种基于调和模型的串行快速神经网络复原算法。将状态连续改变网络模型引入到基于调和模型的神经网络复原算法中，给出了一种基于调和模型的并行神经网络复原算法。
(2)小波图像复原算法研究
·分析了非线性扩散和小波收缩去噪方法之间的关系。分析了二维情况下，Haar小波收缩去噪与非线性扩散之间的关系，给出了非线性扩散滤波在二维Haar小波收缩框架下的解释。从计算角度，说明了非线性扩散方法优于Haar小波收缩去噪方法。
·提出了一种基于图像全变差模型的非线性扩散与二维Haar小波收缩相结合的混合图像去噪算法。该算法对图像经Haar小波分解的低频部分采用全变差扩散进行滤波处理；对高频部分采用小波收缩方法。实验结果表明，混合去噪算法保留了两种去噪方法的优点，在计算复杂度与滤波效果上具有更好的综合性能。
(3)基于图割的图像复原算法研究
·提出了一种基于图割的全变差图像去噪算法。该算法将全变差去噪模型的能量函数最小化问题转化为图的最小割问题，然后采用图割技术(最大流／最小割算法)求得能量函数的全局最优解。并给出了去噪模型中，正则化参数的自适应设定方案。该算法避免了传统能量函数最小化方法中除数为零的不足，并能有效抑制以往最小化方法产生的阶梯效应，获得较优的复原效果。
·提出了一种基于移动空间的全变差图像去噪算法。将基于全变差去噪模型的能量函数最小化问题映射为移动空间中的最优标号问题，采用图割技术来搜索最优标号。给出了基于交换移动空间的全变差去噪模型的求解算法，并讨论了不同正则化参数下的去噪效果。该算法需要较少的额外存贮空间，能够高效逼近全变差能量函数的最小值。
The presence of image degradation, such as noise and blurring, is unavoidable. It may be introduced by the image formation process, image recording, image scanning, image transfer, image showing, etc. However, in many applications, the clear images are needed. The image restoration technique, restoring the original image from the degraded image, is coming to resolve this problem, which is the fundamental problem of image processing, pattern recognition, and computer vision. So, image restoration is widely used in astronomy, remote sensing, medical image, etc.
The traditional image restoration methods rest on the image filter. Since image edges contain lots of image information, and human being is sensitive to these high frequency parts, image filter technique need deblurring image and suppressing the noise, while preserving image edges. However, both image edges and the noise are high frequency part of the image. So, image smoothing is in contradiction to with preserving image edges during image restoration. The traditional filter methods can not deal with this. In recent years, some image processing techniques, such as variational PDEs (Partial Differential Equations) methods, neural network, wavelet analysis, and graph cut techique, are emerging to solve this contradiction.
This dissertation is mainly focus on the image deblurring and de-noising problem, and total variation restoration model. In this dissertation, we research into the image restoration models and algorithms based on neural network; analyze the relationship between wavelet image de-noising method and nonlinear filter de-noising method; study the computation methods of total variation de-noising model using graph cut technique. The main original contributions of this dissertation are summarized as follows:
(1) The study on image restoration algorithm based on neural network
◆Two image restoration models and algorithms based on a modified Hopfield neural network and variational PDEs are proposed. Two variational PDEs as the regualarization terms are proposed to the image restoration model based on the modified Hopfield neural network. One is based on a harmonic model and the other is based on a total variation model. The performance of these regularization terms is analyzed from the viewpoint of nonlinear diffusion. It can be shown that the two proposed restoration models have superior edge preserving performance than the traditional restoration model. Two algorithms have been proposed based on the harmonic model and the total variation model.
◆A fast neural network restoration algorithm based on MHNN (modified Hopfield neural network) of continuous state change is presented. This algorithm uses the MHNN based on continuous state change, and maximal energy descent in the update rule. The convergence of the algorithm is proofed. Experimental results show that the algorithm could converge to a stable point with higher speed, and give more precise restoration results.
◆Two improved restoration algorithms based on the harmonic model and MHNN are given. One is a fast sequential algorithm. The other is a parallel algorithm based on MHNN of continuous state change. Experiment results show that the fast algorithm can restore the degraded image while preserving the edge in a fast speed, and that the parallel algorithm based on the harmonic model is superior to the existing parallel algorithms based on Laplace operator in the performance of preserving image edges.
(2) The study on image restoration algorithm based on wavelet analysis
◆Studying the relationship between nonlinear diffusion and wavelet shrinkage denoising method. We study the relations and differences between nonlinear diffusion and 2D Haar wavelet shrinkage de-noising, explain the characteristic of nonlinear diffusion in wavelet shrinkage framework, and show that nonlinear diffusion is superior to Haar wavelet shrinkage.
◆A hybrid de-noising algorithm, based on 2D Haar wavelet shrinkage and total variation (TV) diffusion, is proposed. The hybrid algorithm is a tradeoff between restoration quality of images and computing complexity. This algorithm applies TV diffusion to low frequency part of image decomposed by Haar wavelet, and shrinks the wavelet coefficient. Some experiment results show that this hybrid algorithm preserves the advantages of these two image de-noising methods, and has the better general performance.
(3) The study on image restoration algorithm based on graph cut technique
◆A TV image de-noising algorithm based on graph cut is proposed. In this algorithm, the minimum of the total variation image de-noising energy function is transformed to a minimum cut of a certain graph. Then, some maximum flow/minimum cut algorithms could solve this problem, and get the global minimum of the TV energy function. In addition, an adaptive method of the proportion coefficient is given. Experiment results show that the algorithm proposed could avoid the staircase effect occurred in some classical total variation minimization methods, and has the better restored effect.
◆A TV image de-noising method based on move space is presented. In this method, the minimum problem of energy function based on TV model is mapped to a the optimal label problem in move space, which could be solved by minmum cut/maximum flow algorithm. This method avoids the computing trouble occurred in classical minimization methods. In addition, the regulative parameter can be adaptively set according to the local character of the noised image. In this way, the minimization method proposed avoids the staircase effect and over smoothing occurred in some classical total variation minimization methods. Experimental results show that the proposed algorithm is preferable to classical total variation minimization method in image denoising performance.
图像复原的传统方法主要是进行图像滤波，由于图像的大部分信息存在于边缘部分，因此要求图像滤波既能去除图像的模糊和噪声，同时又能保持图像的细节。由于图像细节和噪声在频带上混叠，导致图像的平滑和边缘细节的保持成为一对矛盾，传统的滤波方法难以处理这类问题。近年来发展起来的偏微分方程图像处理技术、神经网络技术、小波分析技术以及图割技术，为解决图像复原中的这一矛盾提供了新的方法。
本文以图像复原中的去模糊和去噪问题为主要研究内容，以全变差图像模型为主要研究对象，研究了基于神经网络的图像复原模型和复原算法；分析了小波图像去噪方法与非线性滤波去噪方法之间的关系；并采用图割技术，研究了全变差去噪模型的数值求解方法。研究工作取得了如下创新性成果：
(1)神经网络图像复原算法研究
·提出了两种基于变分PDE的神经网络图像复原模型，并给出了相应的复原算法。将变分偏微分方程，以正则化项的形式引入到传统神经网络图像复原框架中，给出了基于调和方程和全变差方程的神经网络图像复原模型，提出了两种基于改进型Hopfield神经网络的图像复原算法，并分析了算法的收敛性。
·提出了一种基于状态连续改变的快速神经网络复原算法。深入研究了神经网络图像复原算法中的更新规则，将状态连续改变网络模型和最陡下降方法结合，给出了一种状态连续改变的快速神经网络更新规则。进而提出一种基于状态连续改变的快速神经网络复原算法，有效提高了复原算法的收敛速度和复原精度。
·提出了基于调和模型的快速神经网络复原算法以及并行算法。将状态连续改变的快速神经网络更新规则应用到基于调和模型的神经网络复原算法，给出了一种基于调和模型的串行快速神经网络复原算法。将状态连续改变网络模型引入到基于调和模型的神经网络复原算法中，给出了一种基于调和模型的并行神经网络复原算法。
(2)小波图像复原算法研究
·分析了非线性扩散和小波收缩去噪方法之间的关系。分析了二维情况下，Haar小波收缩去噪与非线性扩散之间的关系，给出了非线性扩散滤波在二维Haar小波收缩框架下的解释。从计算角度，说明了非线性扩散方法优于Haar小波收缩去噪方法。
·提出了一种基于图像全变差模型的非线性扩散与二维Haar小波收缩相结合的混合图像去噪算法。该算法对图像经Haar小波分解的低频部分采用全变差扩散进行滤波处理；对高频部分采用小波收缩方法。实验结果表明，混合去噪算法保留了两种去噪方法的优点，在计算复杂度与滤波效果上具有更好的综合性能。
(3)基于图割的图像复原算法研究
·提出了一种基于图割的全变差图像去噪算法。该算法将全变差去噪模型的能量函数最小化问题转化为图的最小割问题，然后采用图割技术(最大流／最小割算法)求得能量函数的全局最优解。并给出了去噪模型中，正则化参数的自适应设定方案。该算法避免了传统能量函数最小化方法中除数为零的不足，并能有效抑制以往最小化方法产生的阶梯效应，获得较优的复原效果。
·提出了一种基于移动空间的全变差图像去噪算法。将基于全变差去噪模型的能量函数最小化问题映射为移动空间中的最优标号问题，采用图割技术来搜索最优标号。给出了基于交换移动空间的全变差去噪模型的求解算法，并讨论了不同正则化参数下的去噪效果。该算法需要较少的额外存贮空间，能够高效逼近全变差能量函数的最小值。
The presence of image degradation, such as noise and blurring, is unavoidable. It may be introduced by the image formation process, image recording, image scanning, image transfer, image showing, etc. However, in many applications, the clear images are needed. The image restoration technique, restoring the original image from the degraded image, is coming to resolve this problem, which is the fundamental problem of image processing, pattern recognition, and computer vision. So, image restoration is widely used in astronomy, remote sensing, medical image, etc.
The traditional image restoration methods rest on the image filter. Since image edges contain lots of image information, and human being is sensitive to these high frequency parts, image filter technique need deblurring image and suppressing the noise, while preserving image edges. However, both image edges and the noise are high frequency part of the image. So, image smoothing is in contradiction to with preserving image edges during image restoration. The traditional filter methods can not deal with this. In recent years, some image processing techniques, such as variational PDEs (Partial Differential Equations) methods, neural network, wavelet analysis, and graph cut techique, are emerging to solve this contradiction.
This dissertation is mainly focus on the image deblurring and de-noising problem, and total variation restoration model. In this dissertation, we research into the image restoration models and algorithms based on neural network; analyze the relationship between wavelet image de-noising method and nonlinear filter de-noising method; study the computation methods of total variation de-noising model using graph cut technique. The main original contributions of this dissertation are summarized as follows:
(1) The study on image restoration algorithm based on neural network
◆Two image restoration models and algorithms based on a modified Hopfield neural network and variational PDEs are proposed. Two variational PDEs as the regualarization terms are proposed to the image restoration model based on the modified Hopfield neural network. One is based on a harmonic model and the other is based on a total variation model. The performance of these regularization terms is analyzed from the viewpoint of nonlinear diffusion. It can be shown that the two proposed restoration models have superior edge preserving performance than the traditional restoration model. Two algorithms have been proposed based on the harmonic model and the total variation model.
◆A fast neural network restoration algorithm based on MHNN (modified Hopfield neural network) of continuous state change is presented. This algorithm uses the MHNN based on continuous state change, and maximal energy descent in the update rule. The convergence of the algorithm is proofed. Experimental results show that the algorithm could converge to a stable point with higher speed, and give more precise restoration results.
◆Two improved restoration algorithms based on the harmonic model and MHNN are given. One is a fast sequential algorithm. The other is a parallel algorithm based on MHNN of continuous state change. Experiment results show that the fast algorithm can restore the degraded image while preserving the edge in a fast speed, and that the parallel algorithm based on the harmonic model is superior to the existing parallel algorithms based on Laplace operator in the performance of preserving image edges.
(2) The study on image restoration algorithm based on wavelet analysis
◆Studying the relationship between nonlinear diffusion and wavelet shrinkage denoising method. We study the relations and differences between nonlinear diffusion and 2D Haar wavelet shrinkage de-noising, explain the characteristic of nonlinear diffusion in wavelet shrinkage framework, and show that nonlinear diffusion is superior to Haar wavelet shrinkage.
◆A hybrid de-noising algorithm, based on 2D Haar wavelet shrinkage and total variation (TV) diffusion, is proposed. The hybrid algorithm is a tradeoff between restoration quality of images and computing complexity. This algorithm applies TV diffusion to low frequency part of image decomposed by Haar wavelet, and shrinks the wavelet coefficient. Some experiment results show that this hybrid algorithm preserves the advantages of these two image de-noising methods, and has the better general performance.
(3) The study on image restoration algorithm based on graph cut technique
◆A TV image de-noising algorithm based on graph cut is proposed. In this algorithm, the minimum of the total variation image de-noising energy function is transformed to a minimum cut of a certain graph. Then, some maximum flow/minimum cut algorithms could solve this problem, and get the global minimum of the TV energy function. In addition, an adaptive method of the proportion coefficient is given. Experiment results show that the algorithm proposed could avoid the staircase effect occurred in some classical total variation minimization methods, and has the better restored effect.
◆A TV image de-noising method based on move space is presented. In this method, the minimum problem of energy function based on TV model is mapped to a the optimal label problem in move space, which could be solved by minmum cut/maximum flow algorithm. This method avoids the computing trouble occurred in classical minimization methods. In addition, the regulative parameter can be adaptively set according to the local character of the noised image. In this way, the minimization method proposed avoids the staircase effect and over smoothing occurred in some classical total variation minimization methods. Experimental results show that the proposed algorithm is preferable to classical total variation minimization method in image denoising performance.
引文
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