基于PDE和全变分滤波方法的研究及在多种噪声中的应用
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摘要
基于偏微分方程（PDE）和全变分的图像滤波方法是图像处理领域的重要研究内容，其研究成果广泛地应用在天文，生物，医学，遥感，材料科学等领域。在众多图像滤波算法中，偏微分方程滤波方法以其独特的优势，引人入胜的属性和广阔的应用前景激发了许多研究者的兴趣，至今仍然是较活跃的研究对象。全变分滤波方法由于其与PDE滤波模型的特殊关联性而也吸引许多研究者的注意，不少PDE滤波模型都能通过全变分滤波模型的优化方法而获得快速的求解方法。本文首先研究了偏微分方程图像滤波方法，分别对随机值脉冲噪声，椒盐噪声，大尺寸和高噪声水平的脉冲噪声以及散斑噪声，提出了相应的偏微分滤波方法，然后将PDE滤波模型的思想拓展到全变分模型中，通过全变分的迭代重加权范数法获得了一些PDE滤波模型的快速数值求解方法。主要研究内容如下：
首先，针对传统的均值滤波方法对脉冲噪声和混合噪声滤除的缺陷，介绍了一种改进的非局部加权平均算法，对其引入一个能反映像素点信息损坏程度的统计量来校正权值赋予的过程，实验结果表明这种改进效果非常明显，能获得满意的滤波结果。
其次，针对随机值脉冲噪声，本文引入了一种统计量来区分图像脉冲像素点，内部像素点和边缘像素点，并将该统计量引入到偏微分方程模型的控制函数中去，得到两种新的控制函数用来实现选择性扩散和选择性保真。基于这两种新控制函数提出一类二阶保边的偏微分滤波模型来去除随机值脉冲噪声，实验结果表明不但这两种新的控制函数能将原有偏微分方程模型的去噪能力大为提升，而且图像细节和边缘能得到很好的保持。这两种新控制函数也能扩展到其它PDE滤波模型中。
再次，针对椒盐噪声和尺寸较大的“脉冲”噪声，本文研究了基于噪声检测的偏微分滤波方法。文中针对尺寸较大的噪声提出了一种基于全局信息的检测方法，该方法不仅能探测脉冲噪声，包括随机值脉冲噪声和椒盐噪声，而且也能用于尺寸较大脉冲的检测，如人为污损和斑点等；针对椒盐噪声，通过组合现有的探测方法和全变分修复模型，获得了一个保边效果非常优秀的滤波器。
另外，针对散斑噪声本文介绍了一种自适应的偏微分方程滤波模型。该模型介绍了一种对图像按边缘和纹理特征分块的方法，根据每一块的特征使用相应功能的偏微分方程平滑项，能把不同功能的偏微分方程平滑项进行整合，发挥各自的优势作用，扬长避短。对合成孔径雷达图像的滤波实验表明该自适应滤波模型能有效地去除散斑噪声，同时还能保持和增强合成孔径雷达图像的纹理细节。
最后，将介绍一种拓展了的全变分滤波模型，并用一种形式统一的全变分模型涵盖一些现有偏微分滤除模型，然后针对该统一形式的全变分模型介绍了迭代重加权范数数值求解方法。加速实验表明，在没有降低滤波效果的前提下，该数值算法能有效地缩减原来偏微分方程滤波模型的处理时间。
Partial differential equation (PDE) based and total variation (TV) based imagedenoising techniques are very useful and powerful tool in the field of imageprocessing. Various PDE models are widely applied nowadays in medicine, remotesensing, astronomy, biology, material science and so on. PDE-based denoisingtechniques play a very important role and attract many interests from researchers withtheir unique advantage and prosperous application. As TV-based methods are muchrelated to PDE, they also attract a lot of interested people. Many PDE modelsconcentrate on the removal of non-impulse noise, such as Gaussian noise, and there isvery little work on the suppression of impulse noise with PDE-based methods. In thisdissertation, we focus mainly on the PDE-based methods for image denoising,including the removal of random-value impulse noise, pepper-and-salt noise,superimposed noise and speckle noise, with their fast numerical solution. Thecontents include the following:
1. We propose a fuzzy weighted non-local means filter for improving thenon-local means algorithm, a classical mean-type filter, for the removal of the impulsenoise and the mixed Gaussian and random-valued impulse noise. We introduce a newfuzzy weighting function, which can shut off the impulsive weight effectively, to thenon-local means. According to the new weighting function, the more a pixel iscorrupted, the less it is exploited to reconstruct image information. Experiments showthat the performances of the new filter are surprisingly satisfactory in terms of bothvisual quality and quantitative measurement.
2. We derive the general way to reduce impulse noise for the PDE diffusion. Thebasic idea is that we attenuate impulses and simultaneously preserve image featuressuch as edges and details by selective diffusion and selective fidelity. Especially forthe random-valued impulse noise we introduce a new notion of ENI that issignificantly different for edge pixels, noisy pixels, and interior pixels. We redefinethe controlling speed function and the controlling fidelity function to depend on ENI.According to the new controlling functions, the diffusion and fidelity process at edgepixels, noisy pixels, and interior pixels can be selectively carried out. Furthermore, aclass of second-order improved and edge-preserving PDE denoising models based onthe two new controlling functions testifies that the proposed method can deal with random-valued impulse noise reliably.
3. We also introduce the noise detection process for the PDE methods to removeimpulses. To reduce random-valued impulse and other superimposed artifacts whichhave a large or long size, we derive a homogeneous amount based filter. The filteridentifies impulses without local window restriction which is very different fromother detection methods, and can recognize and remove impulses and othersuperimposed artifacts efficiently. Combining a more efficient salt-and-pepperdetector with TV inpainting method, we demonstrate the hybrid method is verysatisfactory in noise reduction and edge preservation.
4. For the removal of speckle noise, we also propose an adaptive PDE model. Anew statistic is introduced to partition the noisy image into blocks based on imageedges, and then the oriented regularization is applied within the blocks which containimage details and edges, while non-oriented regularization within the blocks whichonly has smooth regions. The experiments on the synthetic aperture radar imagesshow the hybrid PDE model can smooth noise, meanwhile, preserve and enhanceimage textures and edges satisfactorily.
5. We extend the model of TV regularization to accelerate the proposedPDE-based filters, and also form a unified TV-based model with its fast numericalalgorithm is proposed. The iterative reweighted norm approach can solve the extendedmodel fast. Experiments show that the extended model is feasible and the numericalsolution is faster than that of the time matching method.
首先，针对传统的均值滤波方法对脉冲噪声和混合噪声滤除的缺陷，介绍了一种改进的非局部加权平均算法，对其引入一个能反映像素点信息损坏程度的统计量来校正权值赋予的过程，实验结果表明这种改进效果非常明显，能获得满意的滤波结果。
其次，针对随机值脉冲噪声，本文引入了一种统计量来区分图像脉冲像素点，内部像素点和边缘像素点，并将该统计量引入到偏微分方程模型的控制函数中去，得到两种新的控制函数用来实现选择性扩散和选择性保真。基于这两种新控制函数提出一类二阶保边的偏微分滤波模型来去除随机值脉冲噪声，实验结果表明不但这两种新的控制函数能将原有偏微分方程模型的去噪能力大为提升，而且图像细节和边缘能得到很好的保持。这两种新控制函数也能扩展到其它PDE滤波模型中。
再次，针对椒盐噪声和尺寸较大的“脉冲”噪声，本文研究了基于噪声检测的偏微分滤波方法。文中针对尺寸较大的噪声提出了一种基于全局信息的检测方法，该方法不仅能探测脉冲噪声，包括随机值脉冲噪声和椒盐噪声，而且也能用于尺寸较大脉冲的检测，如人为污损和斑点等；针对椒盐噪声，通过组合现有的探测方法和全变分修复模型，获得了一个保边效果非常优秀的滤波器。
另外，针对散斑噪声本文介绍了一种自适应的偏微分方程滤波模型。该模型介绍了一种对图像按边缘和纹理特征分块的方法，根据每一块的特征使用相应功能的偏微分方程平滑项，能把不同功能的偏微分方程平滑项进行整合，发挥各自的优势作用，扬长避短。对合成孔径雷达图像的滤波实验表明该自适应滤波模型能有效地去除散斑噪声，同时还能保持和增强合成孔径雷达图像的纹理细节。
最后，将介绍一种拓展了的全变分滤波模型，并用一种形式统一的全变分模型涵盖一些现有偏微分滤除模型，然后针对该统一形式的全变分模型介绍了迭代重加权范数数值求解方法。加速实验表明，在没有降低滤波效果的前提下，该数值算法能有效地缩减原来偏微分方程滤波模型的处理时间。
Partial differential equation (PDE) based and total variation (TV) based imagedenoising techniques are very useful and powerful tool in the field of imageprocessing. Various PDE models are widely applied nowadays in medicine, remotesensing, astronomy, biology, material science and so on. PDE-based denoisingtechniques play a very important role and attract many interests from researchers withtheir unique advantage and prosperous application. As TV-based methods are muchrelated to PDE, they also attract a lot of interested people. Many PDE modelsconcentrate on the removal of non-impulse noise, such as Gaussian noise, and there isvery little work on the suppression of impulse noise with PDE-based methods. In thisdissertation, we focus mainly on the PDE-based methods for image denoising,including the removal of random-value impulse noise, pepper-and-salt noise,superimposed noise and speckle noise, with their fast numerical solution. Thecontents include the following:
1. We propose a fuzzy weighted non-local means filter for improving thenon-local means algorithm, a classical mean-type filter, for the removal of the impulsenoise and the mixed Gaussian and random-valued impulse noise. We introduce a newfuzzy weighting function, which can shut off the impulsive weight effectively, to thenon-local means. According to the new weighting function, the more a pixel iscorrupted, the less it is exploited to reconstruct image information. Experiments showthat the performances of the new filter are surprisingly satisfactory in terms of bothvisual quality and quantitative measurement.
2. We derive the general way to reduce impulse noise for the PDE diffusion. Thebasic idea is that we attenuate impulses and simultaneously preserve image featuressuch as edges and details by selective diffusion and selective fidelity. Especially forthe random-valued impulse noise we introduce a new notion of ENI that issignificantly different for edge pixels, noisy pixels, and interior pixels. We redefinethe controlling speed function and the controlling fidelity function to depend on ENI.According to the new controlling functions, the diffusion and fidelity process at edgepixels, noisy pixels, and interior pixels can be selectively carried out. Furthermore, aclass of second-order improved and edge-preserving PDE denoising models based onthe two new controlling functions testifies that the proposed method can deal with random-valued impulse noise reliably.
3. We also introduce the noise detection process for the PDE methods to removeimpulses. To reduce random-valued impulse and other superimposed artifacts whichhave a large or long size, we derive a homogeneous amount based filter. The filteridentifies impulses without local window restriction which is very different fromother detection methods, and can recognize and remove impulses and othersuperimposed artifacts efficiently. Combining a more efficient salt-and-pepperdetector with TV inpainting method, we demonstrate the hybrid method is verysatisfactory in noise reduction and edge preservation.
4. For the removal of speckle noise, we also propose an adaptive PDE model. Anew statistic is introduced to partition the noisy image into blocks based on imageedges, and then the oriented regularization is applied within the blocks which containimage details and edges, while non-oriented regularization within the blocks whichonly has smooth regions. The experiments on the synthetic aperture radar imagesshow the hybrid PDE model can smooth noise, meanwhile, preserve and enhanceimage textures and edges satisfactorily.
5. We extend the model of TV regularization to accelerate the proposedPDE-based filters, and also form a unified TV-based model with its fast numericalalgorithm is proposed. The iterative reweighted norm approach can solve the extendedmodel fast. Experiments show that the extended model is feasible and the numericalsolution is faster than that of the time matching method.
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