偏微分方程和小波在图像处理中的建模理论及应用
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摘要
近年来,以小波多分辨分析和偏微分方程(PDEs)方法为代表的数学工具活跃在图像处理的各个研究领域,它们已经成为研究图像处理和计算机视觉的两大基本工具。本论文主要围绕二者在图像处理中的应用进行建模和算法研究。主要做了以下几个方面的工作：
1.小波迭代正则化方法及其在图像去噪中的应用。
建立了基于平移不变小波的迭代正则化和逆尺度空间方法。推导了尺度参数的估计,尺度参数随着迭代次数的变化而更新。这种带自适应尺度参数的小波迭代正则化方法在小波域上能更精确地控制小波阈值的范围。该方法最终是通过全局迭代小波阈值来实现的。我们证明了带变参数小波迭代正则化的收敛性,并得到了迭代过程的停止准则。实验表明,提出的方法在去噪的同时能较好地保留图像的细节特征。
2.图像去噪的改进迭代非局部平均滤波方法。
提出了两种非局部平均滤波的迭代方法。第一种方法可以看作是加权非局部极小化正则泛函的不动点迭代法。该算法保证了迭代过程的稳定性。第二种方法是对第一种算法的推广,我们用迭代更新后的图像灰度值进行图像块相似度的计算,接着对更新后的图像进行加权平均。数值实验表明,提出的算法在去噪的同时能较好地保持图像的纹理结构且处理后的图像对比度更清晰。
3.基于小波和偏微分方程的图像放大建模。
提出了基于小波多分辨分析和尺度型扩散方程的带噪图像放大。利用原图像作为放大图像的小波低频子带,估计高频,小波重构,最后,对得到的放大图像用前向—后向扩散方程离散一步迭代。接着,提出了基于小波和矩阵型扩散方程的图像放大模型。把原图像作为放大图像的小波低频子带,初始高频置零小波重构,接着用矩阵型扩散方程处理,对处理后的图像进行小波分解,得到估计的高频再与原图像重构。实验结果表明,提出的两种图像放大方法在放大的同时既去除了噪声又有效地保持了边缘。
4.基于耦合小波阈值的图像放大模型。
利用单尺度平移不变Haar小波变换和扩散方程离散一步迭代的等价性,建立了一种图像放大增强的新模型。我们把原图像作为放大图像的小波低频子带,估计高频,进行小波重构。对重构后的图像进行平移不变小波分解,用基于扩散方程的耦合小波阈值对得到的高频进行处理,然后小波重构得到放大的图像。该方法简单易实现,数值结果表明,该模型在图像放大的同时能较好地增强图像的边缘部分。
5.结合小波阈值与逆尺度空间理论(ISS)的图像去噪模型。
利用ISS中的曲率项能够消弱小波带来的边缘震荡效应并能很好地保持图像的边缘特征的特点,我们把逆尺度空间理论应用到小波变换上。在小波域上利用Bregman距离,得到迭代的停止准则。实验表明,处理后的图像细节表现更丰富,在去噪的同时抑制小波阈值引起的边缘震荡效应。
综上所述,本文结合小波多分辨分析和偏微分方程(PDEs)在图像去噪和图像放大方面的优点,提出几种有效的变分正则化算法。论文提出的图像去噪和图像放大算法在计算机视觉、图像处理以及模式识别中有广泛的应用前景。
In recent years, wavelet multi-resolution analysis and partial differential equations (PDEs) play active roles in many image processing fields. They have become two basic tools for image processing and computer vision. This dissertation mainly studies the models and applications of wavelet multi-resolution and PDEs in image processing. The main work can be summarized as follows:
1.Wavelet iterative regularization method and its application in image denoising.
Firstly, we generalize the wavelet-based iterative regularization method and the wavelet-based inverse scale space to shift invariant wavelet-based cases. Then, a stepwise parameter is derived from wavelet-based iterative regularization. The wavelet-based iterative regularization with the new parameters, which controls the extent of denoising more precisely in the wavelet domain, leads to iterative global wavelet shrinkage. We provide a proof of the convergence and obtain a stopping criterion for the iterative procedure with the new scale parameter based on wavelet transform. Numerical examples show that the proposed iterative regularized method can well preserve the details of images.
2.The modified iterative non-local means (NLM) for image denoising.
We propose two iterative NLM algorithms for image denoising. The first one can be in-terpreted as a fixed point method for minimizing a weighted non-local regularization functional. The stability of the iterative procedures is guaranteed. The second one generalizes the first one. The computation of the similarity is based on the gray value of the last iterated image, and then the weighted averaging is computed over the last iterated image. Numerical experiments illustrate that the proposed algorithms can preserve the texture structures well, and the contrast of the processed images are much more clearer.
3.Image zooming models based on wavelet and partial differential equations.
We first give a noisy image zooming method which is based on wavelet and diffusion equations with scalar diffusivity. We use the original image as the low pass and estimate its high pass, and then reconstruct to obtain a zoomed image. Finally, to enhance the edges, forward and backward (FAB) diffusion filtering is used. The second model is based on the wavelet and diffusion equation with matrix valued diffusivity. The high pass are set to zeros. The diffusion equation with matrix-valued diffusivity is used to the zoomed image. Then we apply wavelet analysis to estimate high pass again. Numerical examples illustrate that with the proposed two models while the image is zoomed it is denoised and edges are preserved well.
4. Image zooming and enhancing models based on coupled wavelet shrinkage.
A new image zooming and enhancing method is proposed using the relationship between wavelet transform and diffusion equations. We use the wavelet method to obtain the initial zoomed image. We apply the shift invariant wavelet analysis to the initial zoomed image, use the coupled wavelet shrinkage to the high pass, and finally reconstruct to obtain the final zoomed image. The proposed model is easy to implement. Numerical experiments illustrate that it is a valid image zooming and enhancing algorithm.
5. Image denoising based on wavelet shrinkage and inverse scale space method (ISS).
The curvature term of the ISS can suppress the edge artifacts and preserve sharp edges. Based on this property, we apply inverse scale space to wavelet transform. A stopping criterion is obtained by Bregman distance on wavelet domain. Numerical examples show that the pro-posed model can improve the denoising ability. The details of the processed images are well preserved.
In conclusion, based on the advantages of wavelet multi-resolution and partial differential equations (PDEs) in image denoising and image zooming, several variational regularization algorithms are proposed. The proposed image denoising and zooming algorithms are valuable for the study of computer vision, image processing and pattern recognition.
1.小波迭代正则化方法及其在图像去噪中的应用。
建立了基于平移不变小波的迭代正则化和逆尺度空间方法。推导了尺度参数的估计,尺度参数随着迭代次数的变化而更新。这种带自适应尺度参数的小波迭代正则化方法在小波域上能更精确地控制小波阈值的范围。该方法最终是通过全局迭代小波阈值来实现的。我们证明了带变参数小波迭代正则化的收敛性,并得到了迭代过程的停止准则。实验表明,提出的方法在去噪的同时能较好地保留图像的细节特征。
2.图像去噪的改进迭代非局部平均滤波方法。
提出了两种非局部平均滤波的迭代方法。第一种方法可以看作是加权非局部极小化正则泛函的不动点迭代法。该算法保证了迭代过程的稳定性。第二种方法是对第一种算法的推广,我们用迭代更新后的图像灰度值进行图像块相似度的计算,接着对更新后的图像进行加权平均。数值实验表明,提出的算法在去噪的同时能较好地保持图像的纹理结构且处理后的图像对比度更清晰。
3.基于小波和偏微分方程的图像放大建模。
提出了基于小波多分辨分析和尺度型扩散方程的带噪图像放大。利用原图像作为放大图像的小波低频子带,估计高频,小波重构,最后,对得到的放大图像用前向—后向扩散方程离散一步迭代。接着,提出了基于小波和矩阵型扩散方程的图像放大模型。把原图像作为放大图像的小波低频子带,初始高频置零小波重构,接着用矩阵型扩散方程处理,对处理后的图像进行小波分解,得到估计的高频再与原图像重构。实验结果表明,提出的两种图像放大方法在放大的同时既去除了噪声又有效地保持了边缘。
4.基于耦合小波阈值的图像放大模型。
利用单尺度平移不变Haar小波变换和扩散方程离散一步迭代的等价性,建立了一种图像放大增强的新模型。我们把原图像作为放大图像的小波低频子带,估计高频,进行小波重构。对重构后的图像进行平移不变小波分解,用基于扩散方程的耦合小波阈值对得到的高频进行处理,然后小波重构得到放大的图像。该方法简单易实现,数值结果表明,该模型在图像放大的同时能较好地增强图像的边缘部分。
5.结合小波阈值与逆尺度空间理论(ISS)的图像去噪模型。
利用ISS中的曲率项能够消弱小波带来的边缘震荡效应并能很好地保持图像的边缘特征的特点,我们把逆尺度空间理论应用到小波变换上。在小波域上利用Bregman距离,得到迭代的停止准则。实验表明,处理后的图像细节表现更丰富,在去噪的同时抑制小波阈值引起的边缘震荡效应。
综上所述,本文结合小波多分辨分析和偏微分方程(PDEs)在图像去噪和图像放大方面的优点,提出几种有效的变分正则化算法。论文提出的图像去噪和图像放大算法在计算机视觉、图像处理以及模式识别中有广泛的应用前景。
In recent years, wavelet multi-resolution analysis and partial differential equations (PDEs) play active roles in many image processing fields. They have become two basic tools for image processing and computer vision. This dissertation mainly studies the models and applications of wavelet multi-resolution and PDEs in image processing. The main work can be summarized as follows:
1.Wavelet iterative regularization method and its application in image denoising.
Firstly, we generalize the wavelet-based iterative regularization method and the wavelet-based inverse scale space to shift invariant wavelet-based cases. Then, a stepwise parameter is derived from wavelet-based iterative regularization. The wavelet-based iterative regularization with the new parameters, which controls the extent of denoising more precisely in the wavelet domain, leads to iterative global wavelet shrinkage. We provide a proof of the convergence and obtain a stopping criterion for the iterative procedure with the new scale parameter based on wavelet transform. Numerical examples show that the proposed iterative regularized method can well preserve the details of images.
2.The modified iterative non-local means (NLM) for image denoising.
We propose two iterative NLM algorithms for image denoising. The first one can be in-terpreted as a fixed point method for minimizing a weighted non-local regularization functional. The stability of the iterative procedures is guaranteed. The second one generalizes the first one. The computation of the similarity is based on the gray value of the last iterated image, and then the weighted averaging is computed over the last iterated image. Numerical experiments illustrate that the proposed algorithms can preserve the texture structures well, and the contrast of the processed images are much more clearer.
3.Image zooming models based on wavelet and partial differential equations.
We first give a noisy image zooming method which is based on wavelet and diffusion equations with scalar diffusivity. We use the original image as the low pass and estimate its high pass, and then reconstruct to obtain a zoomed image. Finally, to enhance the edges, forward and backward (FAB) diffusion filtering is used. The second model is based on the wavelet and diffusion equation with matrix valued diffusivity. The high pass are set to zeros. The diffusion equation with matrix-valued diffusivity is used to the zoomed image. Then we apply wavelet analysis to estimate high pass again. Numerical examples illustrate that with the proposed two models while the image is zoomed it is denoised and edges are preserved well.
4. Image zooming and enhancing models based on coupled wavelet shrinkage.
A new image zooming and enhancing method is proposed using the relationship between wavelet transform and diffusion equations. We use the wavelet method to obtain the initial zoomed image. We apply the shift invariant wavelet analysis to the initial zoomed image, use the coupled wavelet shrinkage to the high pass, and finally reconstruct to obtain the final zoomed image. The proposed model is easy to implement. Numerical experiments illustrate that it is a valid image zooming and enhancing algorithm.
5. Image denoising based on wavelet shrinkage and inverse scale space method (ISS).
The curvature term of the ISS can suppress the edge artifacts and preserve sharp edges. Based on this property, we apply inverse scale space to wavelet transform. A stopping criterion is obtained by Bregman distance on wavelet domain. Numerical examples show that the pro-posed model can improve the denoising ability. The details of the processed images are well preserved.
In conclusion, based on the advantages of wavelet multi-resolution and partial differential equations (PDEs) in image denoising and image zooming, several variational regularization algorithms are proposed. The proposed image denoising and zooming algorithms are valuable for the study of computer vision, image processing and pattern recognition.
引文
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