基于变分问题和偏微分方程的图像处理技术研究
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摘要
图像处理是一个光学、电子学、数学、成像学和计算机技术学的交叉学科，并且在众多科学与工程领域有重要应用。目前在图像处理领域有随机建模、小波理论和偏微分方程三大类方法。本论文集中探讨了变分偏微分方程方法在图像增强、融合、去噪和图像分解方面的一些关键问题，主要工作和创新成果如下：
图像中的邻域变化对应着一些重要的内容，据此首先设计了一种简单的梯度场线性放大的图像增强方法；其后考虑红外图像噪声较大的特点，针对噪声作了特殊的抑制之后设计了一个随梯度变化而自适应改变的梯度放大系数，这个系数对于大的梯度几乎不放大，防止了梯度场所反映的动态范围过大而带来的伪影效应，同时在重构时加入了图像的全变差约束，进一步抑制了噪声，有效地增强了红外图像；针对电视图像增强中的均值保持的要求，设计了一个均值约束下的最大熵问题，即把均衡的直方图理解成最大熵的直方图，利用变分方法求出了这个直方图的闭式解，然后利用直方图规定化进行变换，在保持均值的条件下对图像进行了有效的增强；另外，关于目标直方图的确定，选择了一个更为直观的平坦性描述，利用凸优化求出了最佳目标直方图，根据这个目标直方图的结果特点，设计了求取目标的简化算法，在对图像做目标直方图的变换时，借用了一种精确的直方图映射算法，在严格保持均值的条件下对图像进行了有效的增强。
将现有的变分偏微分方程图像融合方法从二维推广到三维，定义了多波段三维图像的对比度，再利用变分方法对多波段三维医学图像实现了有效的融合；结合Weber定律中的临界感知变化(JND)概念，将主观对比度的概念引入到现有变分偏微分方程图像融合中来，设计了基于主观对比度的变分图像融合算法；对于不同波段，不同点的梯度反映的信息具有不同的重要性，采用各点自身的重要性对于各波段进行加权处理，求取加权数据的统计量作为融合目标，设计了具备重要特征保持能力的图像融合方法；由于图像中的重要信息是局部变化，所以我们设计了根据源／结果图像梯度的幅度和方向相似性的图像融合质量评估方法，这种方法在理论上可以衡量结果中大部分点的灰度逆序情况，避免现有一些算法的错误评价。
关于椒盐噪声去除问题，我们利用自适应中值滤波检测图像中可能的噪声点，然后采用全变差结构图像修复方法对检测出的噪声点做填充，这种填充本身是与噪声值无关的，可以很好的恢复被椒盐噪声严重污染的图像，恢复效果优于现有算法；对于随机值冲击噪声，设计了自适应噪声图来检测可能的噪声点，然后进行全变差恢复，这种方法可以避免传统迭代式检测-恢复方法所固有的“恶性循环”的弊端，去噪效果也优于现有方法。
根据现有对结构和纹理描述的分析，将Mumford-Shah模型和G空间结合起来，提出了Mumford-Shah-G图像纹理／结构分解算法，该算法分解得到的结构成分在边缘点以外的部分非常光滑，没有TV模型在噪声下的阶梯效应，同时纹理成分也可以得到充分的分离，其分解结果可能被其他的图像处理算法使用；给出了纹理／结构分解后两个成分分别处理的一种改进压缩算法，这种改进算法将结构成分压缩所产生的误差叠加到纹理成分中，减少了误差来源，同时在理论上证明了该算法相对于现有两种成分分别压缩的方法始终具有信噪比增益；考虑到结构型图像中重要的信息是边缘信息，只利用边缘及邻域的信息就可以用图像修复的方法恢复出原始图像，提出了针对边缘及邻域信息的结构图像压缩方法，这种方法结合了边缘跟踪、游程编码和矢量量化技术，有效做到了边缘图像的压缩，对于结构图像最终的压缩结果，具有很好的主观视觉效果，没有低码率下JPEG压缩的块效应，也没有JPEG2000的振铃效应。
Image processing is an interdisciplinary topic, which is connected to photology, electronics, mathematics, imaging and computer techniques. There are lots of important applications about image processing in many scientific areas and engineering fields. In the current stage, image processing approaches can be divided mainly into three classes, i. e., stochastic modelling, wavelets theory, and partial differential equations (PDEs). This thesis focuses on some key issues of PDEs, in the topics of image enhancement, image fusion, impulsive noise removal and structure-texture decomposition. The main work and innovations are listed as follows:
According to the observation that local variation is the important issue in an image, a simple method for image enhancement is designed, which linearly magnifying the gradient; considering the serious noise in infrared images, we design, to the gradient, a new magnification factor, which is adaptive to the gradient itself, such a factor makes the large gradient almost unchanged, which can avoid the halo from large dynamic range, meanwhile, an extra term, TV norm, is involved into the reconstruction to further compress the noise; according to the high demands of brightness preservation of image enhancement in consumer electronics, we interpret the histogram equalization as the maximum entropy, and using the variational perspective, we find the close-form solution for the optimal histogram, and a histogram transform is employed to enhance the image with brightness preservation; furthermore, about the determination of the target histogram, we select a more intuitive way to measure the flatness, and the convex optimization can help us determine such a solution, based on the characteristics of the solution, we design a simplified algorithm for the target histogram, and also an exact histogram specification algorithm is employed to better our enhancement result.
We extend the variational image fusion from 2D to 3D, define the contrast of the multi-channel 3D image, and a variational method help us effectively fuse the multichannel 3D medical images; with the concept of Just-Noticeable-Differences in the Weber's law, we introduce the perceptual contrast into the variational PDE image fusion, and derive the perceptual contrast based variational image fusion method; according to the different importance of each channel, we set different weights to each pixel in different channels, and the statistics are extracted to make the result preserve the salience well; since local variation is very important in image fusion, we design a measure to evaluate the quality of the fusion results, which employs the similarity of the gradient amplitude and direction between the source and the result, such a method has some desired property in differentiating the inverse direction.
About the two kinds of impulse noise removal, we employ a two-steps approach, i.e., detection and restoration, among which, the restoration is achieved by the edge preserving TV image inpainting technique. As for the salt-and-pepper noise, since its value has some distinct characteristics, an adaptive median filter is employed to identify them, and the result is better than the state-of-the-art methods; for the random valued impulse noise, we design an adaptive neighborhood noise map to identify the corrupted pixels, with a TV inpainting followed, it can overcome some weakness of existing methods and the performance is better.
According to the description ability of the existing model for characterizing structure and texture, we combine the Mumford-Shah model in modelling structure and G-space in modelling texture, and advance a structure-texture decomposition method, namely Mumford-Shah-G method, it can make the resultant structure component piece-wise smooth except some simple edges without the staircase effect from the TV-type model, while the texture can be well separated from the source image; when we consider to compress the structure and texture component of an image respectively, we propose an improved method, which adds the error from structure compression to the texture component to reduce the error source, such a scheme has a theoretical boost in performance, comparing to the existing method; for a structure image, edge is important, we can restore the whole image using only edge together with its neighborhood by structure inpainting, upon which we propose a compression method to such edge-like information, it combines edge tracking, running length encoding and vector quantization technique, and effectively compress the structure image, the result has a perceptually good performance.
图像中的邻域变化对应着一些重要的内容，据此首先设计了一种简单的梯度场线性放大的图像增强方法；其后考虑红外图像噪声较大的特点，针对噪声作了特殊的抑制之后设计了一个随梯度变化而自适应改变的梯度放大系数，这个系数对于大的梯度几乎不放大，防止了梯度场所反映的动态范围过大而带来的伪影效应，同时在重构时加入了图像的全变差约束，进一步抑制了噪声，有效地增强了红外图像；针对电视图像增强中的均值保持的要求，设计了一个均值约束下的最大熵问题，即把均衡的直方图理解成最大熵的直方图，利用变分方法求出了这个直方图的闭式解，然后利用直方图规定化进行变换，在保持均值的条件下对图像进行了有效的增强；另外，关于目标直方图的确定，选择了一个更为直观的平坦性描述，利用凸优化求出了最佳目标直方图，根据这个目标直方图的结果特点，设计了求取目标的简化算法，在对图像做目标直方图的变换时，借用了一种精确的直方图映射算法，在严格保持均值的条件下对图像进行了有效的增强。
将现有的变分偏微分方程图像融合方法从二维推广到三维，定义了多波段三维图像的对比度，再利用变分方法对多波段三维医学图像实现了有效的融合；结合Weber定律中的临界感知变化(JND)概念，将主观对比度的概念引入到现有变分偏微分方程图像融合中来，设计了基于主观对比度的变分图像融合算法；对于不同波段，不同点的梯度反映的信息具有不同的重要性，采用各点自身的重要性对于各波段进行加权处理，求取加权数据的统计量作为融合目标，设计了具备重要特征保持能力的图像融合方法；由于图像中的重要信息是局部变化，所以我们设计了根据源／结果图像梯度的幅度和方向相似性的图像融合质量评估方法，这种方法在理论上可以衡量结果中大部分点的灰度逆序情况，避免现有一些算法的错误评价。
关于椒盐噪声去除问题，我们利用自适应中值滤波检测图像中可能的噪声点，然后采用全变差结构图像修复方法对检测出的噪声点做填充，这种填充本身是与噪声值无关的，可以很好的恢复被椒盐噪声严重污染的图像，恢复效果优于现有算法；对于随机值冲击噪声，设计了自适应噪声图来检测可能的噪声点，然后进行全变差恢复，这种方法可以避免传统迭代式检测-恢复方法所固有的“恶性循环”的弊端，去噪效果也优于现有方法。
根据现有对结构和纹理描述的分析，将Mumford-Shah模型和G空间结合起来，提出了Mumford-Shah-G图像纹理／结构分解算法，该算法分解得到的结构成分在边缘点以外的部分非常光滑，没有TV模型在噪声下的阶梯效应，同时纹理成分也可以得到充分的分离，其分解结果可能被其他的图像处理算法使用；给出了纹理／结构分解后两个成分分别处理的一种改进压缩算法，这种改进算法将结构成分压缩所产生的误差叠加到纹理成分中，减少了误差来源，同时在理论上证明了该算法相对于现有两种成分分别压缩的方法始终具有信噪比增益；考虑到结构型图像中重要的信息是边缘信息，只利用边缘及邻域的信息就可以用图像修复的方法恢复出原始图像，提出了针对边缘及邻域信息的结构图像压缩方法，这种方法结合了边缘跟踪、游程编码和矢量量化技术，有效做到了边缘图像的压缩，对于结构图像最终的压缩结果，具有很好的主观视觉效果，没有低码率下JPEG压缩的块效应，也没有JPEG2000的振铃效应。
Image processing is an interdisciplinary topic, which is connected to photology, electronics, mathematics, imaging and computer techniques. There are lots of important applications about image processing in many scientific areas and engineering fields. In the current stage, image processing approaches can be divided mainly into three classes, i. e., stochastic modelling, wavelets theory, and partial differential equations (PDEs). This thesis focuses on some key issues of PDEs, in the topics of image enhancement, image fusion, impulsive noise removal and structure-texture decomposition. The main work and innovations are listed as follows:
According to the observation that local variation is the important issue in an image, a simple method for image enhancement is designed, which linearly magnifying the gradient; considering the serious noise in infrared images, we design, to the gradient, a new magnification factor, which is adaptive to the gradient itself, such a factor makes the large gradient almost unchanged, which can avoid the halo from large dynamic range, meanwhile, an extra term, TV norm, is involved into the reconstruction to further compress the noise; according to the high demands of brightness preservation of image enhancement in consumer electronics, we interpret the histogram equalization as the maximum entropy, and using the variational perspective, we find the close-form solution for the optimal histogram, and a histogram transform is employed to enhance the image with brightness preservation; furthermore, about the determination of the target histogram, we select a more intuitive way to measure the flatness, and the convex optimization can help us determine such a solution, based on the characteristics of the solution, we design a simplified algorithm for the target histogram, and also an exact histogram specification algorithm is employed to better our enhancement result.
We extend the variational image fusion from 2D to 3D, define the contrast of the multi-channel 3D image, and a variational method help us effectively fuse the multichannel 3D medical images; with the concept of Just-Noticeable-Differences in the Weber's law, we introduce the perceptual contrast into the variational PDE image fusion, and derive the perceptual contrast based variational image fusion method; according to the different importance of each channel, we set different weights to each pixel in different channels, and the statistics are extracted to make the result preserve the salience well; since local variation is very important in image fusion, we design a measure to evaluate the quality of the fusion results, which employs the similarity of the gradient amplitude and direction between the source and the result, such a method has some desired property in differentiating the inverse direction.
About the two kinds of impulse noise removal, we employ a two-steps approach, i.e., detection and restoration, among which, the restoration is achieved by the edge preserving TV image inpainting technique. As for the salt-and-pepper noise, since its value has some distinct characteristics, an adaptive median filter is employed to identify them, and the result is better than the state-of-the-art methods; for the random valued impulse noise, we design an adaptive neighborhood noise map to identify the corrupted pixels, with a TV inpainting followed, it can overcome some weakness of existing methods and the performance is better.
According to the description ability of the existing model for characterizing structure and texture, we combine the Mumford-Shah model in modelling structure and G-space in modelling texture, and advance a structure-texture decomposition method, namely Mumford-Shah-G method, it can make the resultant structure component piece-wise smooth except some simple edges without the staircase effect from the TV-type model, while the texture can be well separated from the source image; when we consider to compress the structure and texture component of an image respectively, we propose an improved method, which adds the error from structure compression to the texture component to reduce the error source, such a scheme has a theoretical boost in performance, comparing to the existing method; for a structure image, edge is important, we can restore the whole image using only edge together with its neighborhood by structure inpainting, upon which we propose a compression method to such edge-like information, it combines edge tracking, running length encoding and vector quantization technique, and effectively compress the structure image, the result has a perceptually good performance.
引文
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