基于偏微分方程的图像修复及放大算法研究
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摘要
数字图像处理的方法多种多样,其中基于偏微分方程的方法是它的一个重要分支,其基本思想是利用偏微分方程改变图像,然后求解该方程,方程的解就是处理后的结果。偏微分方程的应用几乎覆盖了整个图像处理领域,本文主要以它在数字图像放大和修复技术中的应用为研究对象。
图像放大是指提高一幅图像的分辨率,这一问题的研究历史悠久,解决的方法也很多,其中之一便是基于偏微分方程的方法。本文首先对图像放大技术近些年的发展现状做了总结和分析,在此基础上提出了两种图像放大算法——邻域插值平滑算法和基于扩散率函数的放大算法。前者是传统的邻域扩散方法与本文改进的偏微分方程相结合的一种低复杂度、高放大质量的算法。后者则将各向异性扩散模型中的扩散率函数引入到了图像插值中,可直接对图像进行分数倍放大,并在分数倍放大和整数倍放大时均能得到较理想的结果。另外,本文结合星形插值,将基于扩散率函数的图像放大算法应用到了图像修复中,得到了图像修复的自适应插值算法,并对其做了分析。
图像修复是指对图像中数据完全丢失的区域进行填充,以恢复它的完整性。处理图像修复问题的一类主要方法是建立偏微分方程,把图像修复表述成一个边界值问题,用迭代的方法来求解,其中最具代表性和开创性的算法是BSCB算法。但这类方法往往速度很慢,而且修复的结果边缘模糊。本文在总结了图像修复技术发展现状的基础上,将研究的重点放在了对BSCB模型的改进上,分别针对BSCB模型的两个主要缺点提出了三种算法。其中选择性自适应插值算法是自适应插值算法的改进,它引入了优先值概念,使得运行速度比BSCB模型提高了很多的同时修复的效果也有所改善。而带有预处理的BSCB模型则将选择性自适应插值算法与BSCB模型结合到了一起,在保证提高运算速度的前提下,进一步改善了修复图像的质量。另一方面,在解决修复结果边缘模糊的问题上,本文提出了一个新型的偏微分方程——邻域差值扩散模型。该模型重新定义了图像修复过程中的扩散方向和扩散信息,很大程度上避免了边缘模糊的产生。
The methods for digital image processing are various, among which those based on partial differential equation(PDE) have formed an important branch. The basic idea of them is to inflect an image using a PDE whose solution is the processed image, and then solve it. Although PDEs have been almost applied to the whole field of image processing, this paper is mainly focused on the algorithms of image enlargement and image inpainting.
Image enlargement aimed at image resolution enhancement has been studied for many years. It can be done in various ways including those based on PDEs. In this paper, the development of image enlargement in recent years are summarized and analyzed firstly. Then, two algorithms named adjacent interpolation smoothing algorithm and image enlargement algorithm based on diffusivity function are proposed respectively. The former, which combines th adjacent diffusion method and an amended PDE , has low complexity and high quality. The latter can zoom an image in a fractional factor directly, and performs better at both integer and fractional zooming ratios by introducing the diffusivity function of the anisotropy diffusion model into image interpolation. Moreover, combined with the star-shaped interpolation method, the second algorithm is further applied to image inpainting, and called as adaptive interpolation algorithm for image inpainting.
Image inpainting refers to reconstructing the corrupt regions where the data are all destroyed. A primary class of the technique is to build up a partial differential equation, consider it as a boundary problem, and solve it by some iterative method. The most respresentative and creative one of the inpainting algorithms is BSCB model. After summarizes the development of image inpainting technique, this paper points the research at the improvement on BSCB model, and proposes three algorithms to solve the two drawbacks of this model. The first is selective adaptive interpolation which develops the traditional adaptive interpolation algorithm by introducing a priority value. Besides much faster than BSCB model, it can improve the inpainting effects. The second takes selective adaptive interpolation as a preprocessing step, reduces the operation time and improves the inpainting quality further. The last one called neighborhood difference diffusion model is a new PDE proposed by this paper. To avoid the produce of blurry edge, it redefines the diffusion direction and information during the process of image inpainting, and solves the problem to some extent.
图像放大是指提高一幅图像的分辨率,这一问题的研究历史悠久,解决的方法也很多,其中之一便是基于偏微分方程的方法。本文首先对图像放大技术近些年的发展现状做了总结和分析,在此基础上提出了两种图像放大算法——邻域插值平滑算法和基于扩散率函数的放大算法。前者是传统的邻域扩散方法与本文改进的偏微分方程相结合的一种低复杂度、高放大质量的算法。后者则将各向异性扩散模型中的扩散率函数引入到了图像插值中,可直接对图像进行分数倍放大,并在分数倍放大和整数倍放大时均能得到较理想的结果。另外,本文结合星形插值,将基于扩散率函数的图像放大算法应用到了图像修复中,得到了图像修复的自适应插值算法,并对其做了分析。
图像修复是指对图像中数据完全丢失的区域进行填充,以恢复它的完整性。处理图像修复问题的一类主要方法是建立偏微分方程,把图像修复表述成一个边界值问题,用迭代的方法来求解,其中最具代表性和开创性的算法是BSCB算法。但这类方法往往速度很慢,而且修复的结果边缘模糊。本文在总结了图像修复技术发展现状的基础上,将研究的重点放在了对BSCB模型的改进上,分别针对BSCB模型的两个主要缺点提出了三种算法。其中选择性自适应插值算法是自适应插值算法的改进,它引入了优先值概念,使得运行速度比BSCB模型提高了很多的同时修复的效果也有所改善。而带有预处理的BSCB模型则将选择性自适应插值算法与BSCB模型结合到了一起,在保证提高运算速度的前提下,进一步改善了修复图像的质量。另一方面,在解决修复结果边缘模糊的问题上,本文提出了一个新型的偏微分方程——邻域差值扩散模型。该模型重新定义了图像修复过程中的扩散方向和扩散信息,很大程度上避免了边缘模糊的产生。
The methods for digital image processing are various, among which those based on partial differential equation(PDE) have formed an important branch. The basic idea of them is to inflect an image using a PDE whose solution is the processed image, and then solve it. Although PDEs have been almost applied to the whole field of image processing, this paper is mainly focused on the algorithms of image enlargement and image inpainting.
Image enlargement aimed at image resolution enhancement has been studied for many years. It can be done in various ways including those based on PDEs. In this paper, the development of image enlargement in recent years are summarized and analyzed firstly. Then, two algorithms named adjacent interpolation smoothing algorithm and image enlargement algorithm based on diffusivity function are proposed respectively. The former, which combines th adjacent diffusion method and an amended PDE , has low complexity and high quality. The latter can zoom an image in a fractional factor directly, and performs better at both integer and fractional zooming ratios by introducing the diffusivity function of the anisotropy diffusion model into image interpolation. Moreover, combined with the star-shaped interpolation method, the second algorithm is further applied to image inpainting, and called as adaptive interpolation algorithm for image inpainting.
Image inpainting refers to reconstructing the corrupt regions where the data are all destroyed. A primary class of the technique is to build up a partial differential equation, consider it as a boundary problem, and solve it by some iterative method. The most respresentative and creative one of the inpainting algorithms is BSCB model. After summarizes the development of image inpainting technique, this paper points the research at the improvement on BSCB model, and proposes three algorithms to solve the two drawbacks of this model. The first is selective adaptive interpolation which develops the traditional adaptive interpolation algorithm by introducing a priority value. Besides much faster than BSCB model, it can improve the inpainting effects. The second takes selective adaptive interpolation as a preprocessing step, reduces the operation time and improves the inpainting quality further. The last one called neighborhood difference diffusion model is a new PDE proposed by this paper. To avoid the produce of blurry edge, it redefines the diffusion direction and information during the process of image inpainting, and solves the problem to some extent.
引文
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