图像修复和图像编辑的偏微分方程模型及其求解
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摘要
随着数字图像获取设备和各种图像编辑软件的普及,人们对数字图像处理的要求愈来愈高。相对于常规的增强、滤波、降噪等技术,为达到一定目的而对图像进行特殊处理成为一项备受重视的研究课题。这些处理包括对损伤的珍贵图像进行修补以恢复“原貌”、插入或删除某些特定对象、对图像内容进行合理的编辑、灰度图像的彩色化处理等,统称为图像修复(inpainting)和图像编辑(editing)。研究图像修复和图像编辑对于发展数字图像处理理论,扩大图像处理的应用具有重要意义。
基于偏微分方程(PDE)的图像处理方法因其强大功能和一系列优点在图像处理中的重要性日益受到关注。本论文应用PDE研究图像修复和图像编辑问题,在建立数学模型和寻求高效数值算法方面展开研究,提出了多种有效方法并通过实验进行验证,取得了以下成果。
1.提出一种各向异性热传导方法用于修复图像中损伤的结构信息。通过将热传导模型中的传播方向和强度与图像内容相关联,使得修复图像时两个传播方向分别为等照度线方向及其垂直方向。令沿等照度线方向的传播强度为常数,垂直于等照度线方向的传播强度反比于梯度的模,从而有效地重构了图像结构信息,并且避免了边缘模糊。在数值计算中,根据等照度线曲率大小自适应选择有限差分形式,在获得满意的视觉修复结果同时,大大减少了迭代次数,提高了处理效率。
2.建立一种可传播纹理信息的PDE图像修复模型。根据纹理周期和方向两个参数修正图像修复模型,在一定程度上解决了PDE方法通常不能修复纹理的问题。该方法无需进行图像分层,可同时处理结构层和纹理层,对损伤图像的修复效率较高。
3.在纹理图像修复中为降低搜索相似邻域的计算复杂度,基于空间划分进行最邻近查找,实现了一种快速纹理合成方法。将输入纹理样本看作多维向量构成的搜索空间,通过建立数据结构并更新候选列表来划分搜索空间,快速查找最相似的邻域。与典型的TSVQ等加速算法相比,合成速度更快,得到的视觉效果更接近于穷举搜索结果。
4.针对Poisson方程图像编辑技术存在的颜色失真问题进行改进,提出一种可避免颜色失真的图像编辑方法,并运用在图像克隆处理中。通过增加Dirichlet内边界条件,并线性放大待编辑区域中对应对象轮廓中较大的Laplace值,从而减小了边界对克隆对象颜色的影响,有效地实现了对象的无缝插入。处理中无需精确选定插入对象的轮廓,而只要大致选定其内外边界及插入位置即可自动完成。
5.由于图像局部区域的克隆操作可被用于不正当的内容篡改,本论文还研究了对图像中有无Poisson克隆区域的检测认证技术。通过粗略匹配和精细匹配两步检测是否存在满足Poisson方程的区域,定位对应相似克隆区域及其精确边界。将两个相似区域边界上的Laplace值累加,通过分析边界处的不连续性分别判断出源区和克隆区,该方法对数字图像的被动认证提供了有力手段。
在研究中,针对不同类型的图像损伤和图像编辑的需要进行了大量实验,对一批图像进行了克隆操作和检测判断,实验结果表明上述方法具有良好的处理效果。
Accompanying with the popular use of digital camera and image editing software, people's need for image processing is increasing. At present, compared with the traditional image processing techniques, such as filtering and denoising, image processing techniques for special purposes, for example: repairing the damaged parts of images, seamless inserting objects from different images, replacement of image background and image colorization etc are under more widely research and development in academe and industry.
Recently PDE-based method plays the important role in the field of image processing due to its advantages. In this dissertation, we mainly focus on the problems of image inpainting and editing using PDE-based methods, and propose several new methods in order to contribute to the development of the attractive and useful areas in image processing.
First, we propose an anisotropic heat transfer model to repair the structure information in damaged images. This is achieved by making the propagation direction and intensity relate to the image contents. In the inpainting process, the two components of propagation are the isophote direction and its orthogonal direction. The propagation intensity along the isophote direction is unchanged while intensity along the orthogonal direction of isophote is inversely proportional to the magnitude of gradient. The method can reconstruct the image structure information and avoid edge blurring. In numerical implementation, we adaptively choose the step size of the finite difference according to the curvature. In this way, the proposed method can produce satisfactory image quality and reduce computational complexity.
Second, we develop a partial differential equation model for image inpainting which can propagate texture information. We give an additional term representing texture information using the parameters of periodicity and direction, and add it into the PDE. So it can solve the problem of lacking texture information in most PDE-based methods. The method does not need to decompose the image into different layers, and inpaint both image structure and texture simultaneously with high efficiency.
We also deal with the high complexity problem of searching for similar neighborhood in textural image repairing. By using nearest neighbor searching algorithm with dynamic space partitioning, we propose a fast texture synthesis method. The input sample texture image can be seen as searching space composed by multi-dimension vectors, and we partition the space by establishing data structure and update the candidate list to find the most similar neighborhood. Compared with other typical accelerating algorithms such as TSVQ, our method is faster and has better visual synthesized results which are closer to the results of exhaustive searching.
Third, we propose an improved image editing method without color inconsistency using modified Poisson equation and apply it in image cloning. By adding an additional inner Dirichlet boundary condition and magnify large Laplace values corresponding to the object's true contour in the edited region, the method can insert objects into background of the target image seamlessly, and solve the color inconsistency problem caused by boundary influence. This method does not require precise definition of the object contour, but only needs to choose a rough region of the inserted object and the target location for automatic processing.
We also realize the detection and authentication for standard Poisson editing. By rough and refined matching the regions which satisfy the Poisson equation, the corresponding two cloning regions and their exact boundaries are located and acquired. We sum the Laplace values on these two boundaries and differentiate the original region and cloned region according to the analysis of discontinuity on boundary regions. The method can not only detect the cloning manipulation, but also infer original appearance of the suspicious images.
We have carried out inpainting and editing experiments on the large number of images and various types of damages. Experimental results show the effectiveness and veracity of the proposed methods.
基于偏微分方程(PDE)的图像处理方法因其强大功能和一系列优点在图像处理中的重要性日益受到关注。本论文应用PDE研究图像修复和图像编辑问题,在建立数学模型和寻求高效数值算法方面展开研究,提出了多种有效方法并通过实验进行验证,取得了以下成果。
1.提出一种各向异性热传导方法用于修复图像中损伤的结构信息。通过将热传导模型中的传播方向和强度与图像内容相关联,使得修复图像时两个传播方向分别为等照度线方向及其垂直方向。令沿等照度线方向的传播强度为常数,垂直于等照度线方向的传播强度反比于梯度的模,从而有效地重构了图像结构信息,并且避免了边缘模糊。在数值计算中,根据等照度线曲率大小自适应选择有限差分形式,在获得满意的视觉修复结果同时,大大减少了迭代次数,提高了处理效率。
2.建立一种可传播纹理信息的PDE图像修复模型。根据纹理周期和方向两个参数修正图像修复模型,在一定程度上解决了PDE方法通常不能修复纹理的问题。该方法无需进行图像分层,可同时处理结构层和纹理层,对损伤图像的修复效率较高。
3.在纹理图像修复中为降低搜索相似邻域的计算复杂度,基于空间划分进行最邻近查找,实现了一种快速纹理合成方法。将输入纹理样本看作多维向量构成的搜索空间,通过建立数据结构并更新候选列表来划分搜索空间,快速查找最相似的邻域。与典型的TSVQ等加速算法相比,合成速度更快,得到的视觉效果更接近于穷举搜索结果。
4.针对Poisson方程图像编辑技术存在的颜色失真问题进行改进,提出一种可避免颜色失真的图像编辑方法,并运用在图像克隆处理中。通过增加Dirichlet内边界条件,并线性放大待编辑区域中对应对象轮廓中较大的Laplace值,从而减小了边界对克隆对象颜色的影响,有效地实现了对象的无缝插入。处理中无需精确选定插入对象的轮廓,而只要大致选定其内外边界及插入位置即可自动完成。
5.由于图像局部区域的克隆操作可被用于不正当的内容篡改,本论文还研究了对图像中有无Poisson克隆区域的检测认证技术。通过粗略匹配和精细匹配两步检测是否存在满足Poisson方程的区域,定位对应相似克隆区域及其精确边界。将两个相似区域边界上的Laplace值累加,通过分析边界处的不连续性分别判断出源区和克隆区,该方法对数字图像的被动认证提供了有力手段。
在研究中,针对不同类型的图像损伤和图像编辑的需要进行了大量实验,对一批图像进行了克隆操作和检测判断,实验结果表明上述方法具有良好的处理效果。
Accompanying with the popular use of digital camera and image editing software, people's need for image processing is increasing. At present, compared with the traditional image processing techniques, such as filtering and denoising, image processing techniques for special purposes, for example: repairing the damaged parts of images, seamless inserting objects from different images, replacement of image background and image colorization etc are under more widely research and development in academe and industry.
Recently PDE-based method plays the important role in the field of image processing due to its advantages. In this dissertation, we mainly focus on the problems of image inpainting and editing using PDE-based methods, and propose several new methods in order to contribute to the development of the attractive and useful areas in image processing.
First, we propose an anisotropic heat transfer model to repair the structure information in damaged images. This is achieved by making the propagation direction and intensity relate to the image contents. In the inpainting process, the two components of propagation are the isophote direction and its orthogonal direction. The propagation intensity along the isophote direction is unchanged while intensity along the orthogonal direction of isophote is inversely proportional to the magnitude of gradient. The method can reconstruct the image structure information and avoid edge blurring. In numerical implementation, we adaptively choose the step size of the finite difference according to the curvature. In this way, the proposed method can produce satisfactory image quality and reduce computational complexity.
Second, we develop a partial differential equation model for image inpainting which can propagate texture information. We give an additional term representing texture information using the parameters of periodicity and direction, and add it into the PDE. So it can solve the problem of lacking texture information in most PDE-based methods. The method does not need to decompose the image into different layers, and inpaint both image structure and texture simultaneously with high efficiency.
We also deal with the high complexity problem of searching for similar neighborhood in textural image repairing. By using nearest neighbor searching algorithm with dynamic space partitioning, we propose a fast texture synthesis method. The input sample texture image can be seen as searching space composed by multi-dimension vectors, and we partition the space by establishing data structure and update the candidate list to find the most similar neighborhood. Compared with other typical accelerating algorithms such as TSVQ, our method is faster and has better visual synthesized results which are closer to the results of exhaustive searching.
Third, we propose an improved image editing method without color inconsistency using modified Poisson equation and apply it in image cloning. By adding an additional inner Dirichlet boundary condition and magnify large Laplace values corresponding to the object's true contour in the edited region, the method can insert objects into background of the target image seamlessly, and solve the color inconsistency problem caused by boundary influence. This method does not require precise definition of the object contour, but only needs to choose a rough region of the inserted object and the target location for automatic processing.
We also realize the detection and authentication for standard Poisson editing. By rough and refined matching the regions which satisfy the Poisson equation, the corresponding two cloning regions and their exact boundaries are located and acquired. We sum the Laplace values on these two boundaries and differentiate the original region and cloned region according to the analysis of discontinuity on boundary regions. The method can not only detect the cloning manipulation, but also infer original appearance of the suspicious images.
We have carried out inpainting and editing experiments on the large number of images and various types of damages. Experimental results show the effectiveness and veracity of the proposed methods.
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