基于分数阶变分PDE的图像建模与去噪算法研究
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摘要
图像处理过程中,保持图像的边缘、纹理等重要结构特征是非常重要的。要保持这重要的图像信息,在图像处理时就要对图像中的卡通、边缘、纹理及噪声等不同成分进行区分和处理,也就需要对图像的不同成分进行合适的数学建模。
本论文结合图像去噪问题的研究,利用小波分析理论与方法、函数空间图像建模理论与方法以及新的数学工具如分数阶导数等,探索对图像中的不同成分的新的建模方法,并在此基础上利用正则化理论和方法、算子分裂法、变分法等提出新的图像去噪变分PDE模型和算法。论文结合图像去噪提出的基于分数阶变分PDE的图像建模理论和方法,可以进一步推广到图像分割、超分辨率重建等其它图像处理领域,具有重要的理论意义和广泛的实际应用前景。
本文取得的主要成果及创新点包括：
(1)利用负指数Sobolev空间对图像的纹理进行多尺度建模,并用正交小波变换下Sobolev空间中的多尺度范数来进行刻画,在此基础上提出了负指数Sobolev空间中的多尺度变分PDE图像去噪模型。针对提出的多尺度变分模型,提出了三种数值算法,从理论上分析并证明了算法收敛的充分条件。数值实验表明,利用负指数Sobolev空间对纹理和进行多尺度建模,可以更好地对图像中不同尺度的纹理和噪声进行区分和刻画,所提出的多尺度变分PDE图像去噪模型能在去有效提高峰值信噪比的同时更好地保持图像纹理等细节信息,在本文所提出的三种去噪算法中,基于算子分裂思想提出的交替投影算法是最快速稳定的算法。
(2)利用分数阶导数对图像进行建模,在此基础上提出了分数阶变分PDE图像去噪模型及算法。
首先,利用Fourier变换域定义的分数阶导数对图像进行建模,提出一种连续形式的分数阶变分PDE图像去噪模型,并设计了求解模型的离散梯度下降算法。
其次,利用空域中的Grumwald-Letnikov分数阶导数对图像进行建模,分别从连续和离散两个角度提出新的分数阶图像去噪变分模型及算法。从连续建模角度,利用Grumwald-Letnikov分数阶导数和特定卷积积分等价关系,建立了一种连续形式的基于卷积积分的分数阶图像去噪变分模型,并设计了相应的离散梯度下降算法。从离散建模角度,将离散意义下的总变分和有界变差空间进行了推广,构造新的离散意义下的分数阶总变分和分数阶有界变差空间(BVα空间),研究了分数阶总变分及其共轭算子的性质,利用分数阶有界变差空间对图像进行建模,建立了离散意义下的分数阶图像去噪变分模型。对所建立的离散分数阶变分模型,设计了空间投影算法,并对算法的收敛性进行了分析,给出并证明了算法收敛的充分条件。
数值实验表明,利用分数阶导数对图像进行建模,可以在有效改善峰值信噪比同时,较好地抑制“阶梯效应”,保持图像的纹理细节。在所提出的三种算法中,针对离散模型提出的空间投影算法是计算量最小、速度最快的算法。
(3)将负指数Sobolev空间中的多尺度图像建模和基于分数阶导数的图像建模进行耦合,提出了统一的分数阶多尺度变分图像去噪模型,设计了求解分数阶多尺度变分模型的交替投影算法,对算法的收敛性进行了分析,并给出了算法收敛的充分条件。数值实验表明,耦合模型综合了两种建模方法的优点,在峰值信噪比改善、纹理保持以及“阶梯效应”抑制方面比耦合前的两种模型都具有更好的效果。
在统一的多尺度分数阶变分模型基础上,对模型进行进一步改进,并利用图像的局部方差等统计信息,对图像的“纹理区域”和“非纹理区域”进行区分,利用小波系数的幅值和函数正则性之间的关系分析图像各个尺度的正则性,在此基础上提出了参数自适应选择方法以及自适应交替投影算法。数值实验表明,自适应算法在图像的“非纹理区域”具有良好的去噪效果和“阶梯效应”抑制能力,在“纹理区域”具有良好的纹理保持能力,是一种快速高效的图像去噪方法。
(4)基于f=u+v+w和f=u+uv两种图像分解形式,利用提出的分数阶导数图像建模和负指数Sobolev空间中的多尺度图像建模方法,分别对加性噪声去噪和乘性噪声去噪问题,提出新的分数阶图像去噪变分模型及算法。
首先,基于f=u+v+w的图像分解形式,利用分数阶导数,构造了新的Gμα空间,并在此基础上提出了针对加性噪声去噪的两种分数阶变分模型：第一,利用BV_α、G_μ~α空间和负指数Sobolev空间分别对图像的卡通、纹理及噪声进行建模,提出了新的分数阶变分模型及算法；第二,利用BVα空间、Gμα空间以及齐次BesOv空间B∞-1.∞，分别对图像卡通、纹理以及噪声分别进行建模,提出新的分数阶变分模型及算法。数值实验表明,利用分数阶、多尺度函数空间对图像卡通、纹理及噪声分别建模,使得对图像不同成分的建模更加精细,所提出的分数阶多尺度噪模型在峰值信噪比改善、纹理保持以及“阶梯效应”的抑制方面都有较好的效果。
其次,基于f=u+uv的图像分解形式,针对噪声分别分服从Gauss分布和Gamma分布的乘性噪声去噪问题,利用BV_α空间和负指数Sobolev空间对图像的不同成分进行建模,建立了相应的分数阶多尺度图像去噪变分模型,提出了模型参数的自适应选择方法,并设计了相应的数值算法。数值实验表明,对于乘性噪声去噪问题,本文提出的分数阶多尺度变分模型及算法在峰值信噪比改善、图像细节保持以及“阶梯效应”的抑制方面同样具有较好的效果。
In image processing, it is important to preserving useful structures such as edges and textures. To preserve these important structures, we need to distinguish and deal with these different components of image such as cartoon, edge, texture and noise separately, and then we need modeling these components appropriately.
In this paper, using wavelet analysis theory and methods, Function space image mod-eling methods and fractional calculus, new methods for modeling different components of image are proposed. Based on these works and using regularization theory and methods and variational methods, new fractional-order variational PDE models and algorithms are proposed for image denoising. The fractional-order variational PDE based image model-ing theory and methods can be applied in other field of image processing such as image segmentation, super-resolution reconstruction, have important theoretical significance and application values.
The main results achieved and innovations include:
(1)Based on multi-scale modeling the texture of image in negative Sobolev space and using the orthogonal wavelet transform based multi-scale norm in negative Sobolev space to characterize the norm of texture, a new multi-scale variational PDE model in negative Sobolev space is proposed. For solving the proposed model, this paper gives three kinds of numerical algorithms, and the sufficient conditions for convergence of these algorithms are theoretically analyzed and proved. Numerical experiments show that using the multi-scale modeling method in negative Sobolev space for the texture of image, we can distinguish and describe the texture and noise better at different scales. The proposed multi-scale variational PDE model can preserve texture efficiently while improve the PSNR of image, and the alternating projection algorithm based on the operator splitting method is is the most rapid and stable one of the three proposed algorithms.
(2) Modeling the image by using fractional-order derivative, Fractional-order Varia-tion PDE models and algorithms are proposed for image denoising.
First of all, using the fractional derivative defined in Fourier transform domain, a fractional-order variation model is established and the corresponding discrete gradient de-scent algorithm is proposed。
Secondly, based on Grumwald-Letnikov fractional-order derivative, this paper pro-pose two models from the point of view continuous modeling and discrete modeling. From the point of view continuous modeling, using the relation between Griimwald-Letnikov fractional-order derivative and certain convolution integral, a convolution integral based fractional-order variation model and the corresponding algorithm are proposed. From the point of view Continuous modeling, based on the Grumwald-Letnikov fractional-order difference, this paper extend the discrete function space of bounded variation (BV space) to the discrete fractional-order function space of bounded variation BV_a at first and then using BV_a space for image modeling, a new fractional-order variation model and the corresponding projection algorithm are proposed. For the convergence of the projection algorithm, an efficient condition is proved.
Numerical experiments show that using the fractional-order variation to model the image, we can preserve the finer scale details such as textures and edges, restrain the "blocky effect" and improve the PSNR effectively. In the proposed three kinds of algo-rithms, the the cost of computation of projection algorithm is minimum and the speed of taht is the fastest.
(3) Coupling the multi-scale modeling method in negative Sobolev space and the fractional-order derivative based image modeling method, a uniform multi-scale fractional-order variational PDE model for image denoising and the corresponding alter-nating projection algorithm are propoesd. The convergence of the alternating projection algorithm is analyzed and an efficient condition is proved in this paper. Numerical experi-ments show that the coupled model combines the advantages of both models, it have better results in the improvement of the PSNR, preserving of textures, as well as restraining of the "blocky effect".
Based on the uniform multi-scale fractional-order variational PDE model, by using image statistical information such as local variance of image to distinguish of "texture area" and "non-texture area" of image and the relationship between wavelet confidences and the regularity of function, the adaptive multi-scale fractional-order variational PDE model and the corresponding adaptive alternating projection algorithm for image denoising are proposed. Numerical results show the adaptive fractional-order model can not only remove noise and restrain "blocky effect" in the "non-texture area" effectively, but also preserve the finer scale details such as texture in the "texture area" efficiently and is a fast and effective method for image denoising.
(4) Based on two kinds of image decomposition of iamge such as f= u+v+w and f= u+uv, making use of the fractional-order derivative based image modeling and multi-scale modeling method in negative Sobolev spaces, new fractional-order variational denoising models and algorithms are proposed for additive and multiplicative noise respectively.
First of all, based on image decomposition form as f= u+v+w,using fractional-order derivative, this paper constructs a new space denote G(_μ~α), and then proposes two models: first, modeling cartoon, texture and noise of image with BV_a space, G(_μ~α) and Multi-scale negative Sobolev space respectively, and propose a new fractional-order variation PDE model and algorithm. Secondly, modeling cartoon, texture and noise with BV_a space, G(_μ~α) and Besov space B(_-1,∞~∞) respectively, and propose a new fractional-order variation model and algorithm. Numerical results show that the new models are effective for improving PSNR, preserving texture and restraining "blocky effect"
Secondly, based on image decomposition form as f= u+uv, multiplicative noise re-moving problems whose noise follow the Gauss law and Gamma Law are considered. New multi-scale fractional-order multi-scale models are propose. This paper gives the adaptive parameter selection method and designs the corresponding numerical algorithms. Numer-ical experiments show that for multiplicative noise removing, fractional-order multi-scale models are effective for improving PSNR, preserving texture and restraining "blocky ef-fect" also.
本论文结合图像去噪问题的研究,利用小波分析理论与方法、函数空间图像建模理论与方法以及新的数学工具如分数阶导数等,探索对图像中的不同成分的新的建模方法,并在此基础上利用正则化理论和方法、算子分裂法、变分法等提出新的图像去噪变分PDE模型和算法。论文结合图像去噪提出的基于分数阶变分PDE的图像建模理论和方法,可以进一步推广到图像分割、超分辨率重建等其它图像处理领域,具有重要的理论意义和广泛的实际应用前景。
本文取得的主要成果及创新点包括：
(1)利用负指数Sobolev空间对图像的纹理进行多尺度建模,并用正交小波变换下Sobolev空间中的多尺度范数来进行刻画,在此基础上提出了负指数Sobolev空间中的多尺度变分PDE图像去噪模型。针对提出的多尺度变分模型,提出了三种数值算法,从理论上分析并证明了算法收敛的充分条件。数值实验表明,利用负指数Sobolev空间对纹理和进行多尺度建模,可以更好地对图像中不同尺度的纹理和噪声进行区分和刻画,所提出的多尺度变分PDE图像去噪模型能在去有效提高峰值信噪比的同时更好地保持图像纹理等细节信息,在本文所提出的三种去噪算法中,基于算子分裂思想提出的交替投影算法是最快速稳定的算法。
(2)利用分数阶导数对图像进行建模,在此基础上提出了分数阶变分PDE图像去噪模型及算法。
首先,利用Fourier变换域定义的分数阶导数对图像进行建模,提出一种连续形式的分数阶变分PDE图像去噪模型,并设计了求解模型的离散梯度下降算法。
其次,利用空域中的Grumwald-Letnikov分数阶导数对图像进行建模,分别从连续和离散两个角度提出新的分数阶图像去噪变分模型及算法。从连续建模角度,利用Grumwald-Letnikov分数阶导数和特定卷积积分等价关系,建立了一种连续形式的基于卷积积分的分数阶图像去噪变分模型,并设计了相应的离散梯度下降算法。从离散建模角度,将离散意义下的总变分和有界变差空间进行了推广,构造新的离散意义下的分数阶总变分和分数阶有界变差空间(BVα空间),研究了分数阶总变分及其共轭算子的性质,利用分数阶有界变差空间对图像进行建模,建立了离散意义下的分数阶图像去噪变分模型。对所建立的离散分数阶变分模型,设计了空间投影算法,并对算法的收敛性进行了分析,给出并证明了算法收敛的充分条件。
数值实验表明,利用分数阶导数对图像进行建模,可以在有效改善峰值信噪比同时,较好地抑制“阶梯效应”,保持图像的纹理细节。在所提出的三种算法中,针对离散模型提出的空间投影算法是计算量最小、速度最快的算法。
(3)将负指数Sobolev空间中的多尺度图像建模和基于分数阶导数的图像建模进行耦合,提出了统一的分数阶多尺度变分图像去噪模型,设计了求解分数阶多尺度变分模型的交替投影算法,对算法的收敛性进行了分析,并给出了算法收敛的充分条件。数值实验表明,耦合模型综合了两种建模方法的优点,在峰值信噪比改善、纹理保持以及“阶梯效应”抑制方面比耦合前的两种模型都具有更好的效果。
在统一的多尺度分数阶变分模型基础上,对模型进行进一步改进,并利用图像的局部方差等统计信息,对图像的“纹理区域”和“非纹理区域”进行区分,利用小波系数的幅值和函数正则性之间的关系分析图像各个尺度的正则性,在此基础上提出了参数自适应选择方法以及自适应交替投影算法。数值实验表明,自适应算法在图像的“非纹理区域”具有良好的去噪效果和“阶梯效应”抑制能力,在“纹理区域”具有良好的纹理保持能力,是一种快速高效的图像去噪方法。
(4)基于f=u+v+w和f=u+uv两种图像分解形式,利用提出的分数阶导数图像建模和负指数Sobolev空间中的多尺度图像建模方法,分别对加性噪声去噪和乘性噪声去噪问题,提出新的分数阶图像去噪变分模型及算法。
首先,基于f=u+v+w的图像分解形式,利用分数阶导数,构造了新的Gμα空间,并在此基础上提出了针对加性噪声去噪的两种分数阶变分模型：第一,利用BV_α、G_μ~α空间和负指数Sobolev空间分别对图像的卡通、纹理及噪声进行建模,提出了新的分数阶变分模型及算法；第二,利用BVα空间、Gμα空间以及齐次BesOv空间B∞-1.∞，分别对图像卡通、纹理以及噪声分别进行建模,提出新的分数阶变分模型及算法。数值实验表明,利用分数阶、多尺度函数空间对图像卡通、纹理及噪声分别建模,使得对图像不同成分的建模更加精细,所提出的分数阶多尺度噪模型在峰值信噪比改善、纹理保持以及“阶梯效应”的抑制方面都有较好的效果。
其次,基于f=u+uv的图像分解形式,针对噪声分别分服从Gauss分布和Gamma分布的乘性噪声去噪问题,利用BV_α空间和负指数Sobolev空间对图像的不同成分进行建模,建立了相应的分数阶多尺度图像去噪变分模型,提出了模型参数的自适应选择方法,并设计了相应的数值算法。数值实验表明,对于乘性噪声去噪问题,本文提出的分数阶多尺度变分模型及算法在峰值信噪比改善、图像细节保持以及“阶梯效应”的抑制方面同样具有较好的效果。
In image processing, it is important to preserving useful structures such as edges and textures. To preserve these important structures, we need to distinguish and deal with these different components of image such as cartoon, edge, texture and noise separately, and then we need modeling these components appropriately.
In this paper, using wavelet analysis theory and methods, Function space image mod-eling methods and fractional calculus, new methods for modeling different components of image are proposed. Based on these works and using regularization theory and methods and variational methods, new fractional-order variational PDE models and algorithms are proposed for image denoising. The fractional-order variational PDE based image model-ing theory and methods can be applied in other field of image processing such as image segmentation, super-resolution reconstruction, have important theoretical significance and application values.
The main results achieved and innovations include:
(1)Based on multi-scale modeling the texture of image in negative Sobolev space and using the orthogonal wavelet transform based multi-scale norm in negative Sobolev space to characterize the norm of texture, a new multi-scale variational PDE model in negative Sobolev space is proposed. For solving the proposed model, this paper gives three kinds of numerical algorithms, and the sufficient conditions for convergence of these algorithms are theoretically analyzed and proved. Numerical experiments show that using the multi-scale modeling method in negative Sobolev space for the texture of image, we can distinguish and describe the texture and noise better at different scales. The proposed multi-scale variational PDE model can preserve texture efficiently while improve the PSNR of image, and the alternating projection algorithm based on the operator splitting method is is the most rapid and stable one of the three proposed algorithms.
(2) Modeling the image by using fractional-order derivative, Fractional-order Varia-tion PDE models and algorithms are proposed for image denoising.
First of all, using the fractional derivative defined in Fourier transform domain, a fractional-order variation model is established and the corresponding discrete gradient de-scent algorithm is proposed。
Secondly, based on Grumwald-Letnikov fractional-order derivative, this paper pro-pose two models from the point of view continuous modeling and discrete modeling. From the point of view continuous modeling, using the relation between Griimwald-Letnikov fractional-order derivative and certain convolution integral, a convolution integral based fractional-order variation model and the corresponding algorithm are proposed. From the point of view Continuous modeling, based on the Grumwald-Letnikov fractional-order difference, this paper extend the discrete function space of bounded variation (BV space) to the discrete fractional-order function space of bounded variation BV_a at first and then using BV_a space for image modeling, a new fractional-order variation model and the corresponding projection algorithm are proposed. For the convergence of the projection algorithm, an efficient condition is proved.
Numerical experiments show that using the fractional-order variation to model the image, we can preserve the finer scale details such as textures and edges, restrain the "blocky effect" and improve the PSNR effectively. In the proposed three kinds of algo-rithms, the the cost of computation of projection algorithm is minimum and the speed of taht is the fastest.
(3) Coupling the multi-scale modeling method in negative Sobolev space and the fractional-order derivative based image modeling method, a uniform multi-scale fractional-order variational PDE model for image denoising and the corresponding alter-nating projection algorithm are propoesd. The convergence of the alternating projection algorithm is analyzed and an efficient condition is proved in this paper. Numerical experi-ments show that the coupled model combines the advantages of both models, it have better results in the improvement of the PSNR, preserving of textures, as well as restraining of the "blocky effect".
Based on the uniform multi-scale fractional-order variational PDE model, by using image statistical information such as local variance of image to distinguish of "texture area" and "non-texture area" of image and the relationship between wavelet confidences and the regularity of function, the adaptive multi-scale fractional-order variational PDE model and the corresponding adaptive alternating projection algorithm for image denoising are proposed. Numerical results show the adaptive fractional-order model can not only remove noise and restrain "blocky effect" in the "non-texture area" effectively, but also preserve the finer scale details such as texture in the "texture area" efficiently and is a fast and effective method for image denoising.
(4) Based on two kinds of image decomposition of iamge such as f= u+v+w and f= u+uv, making use of the fractional-order derivative based image modeling and multi-scale modeling method in negative Sobolev spaces, new fractional-order variational denoising models and algorithms are proposed for additive and multiplicative noise respectively.
First of all, based on image decomposition form as f= u+v+w,using fractional-order derivative, this paper constructs a new space denote G(_μ~α), and then proposes two models: first, modeling cartoon, texture and noise of image with BV_a space, G(_μ~α) and Multi-scale negative Sobolev space respectively, and propose a new fractional-order variation PDE model and algorithm. Secondly, modeling cartoon, texture and noise with BV_a space, G(_μ~α) and Besov space B(_-1,∞~∞) respectively, and propose a new fractional-order variation model and algorithm. Numerical results show that the new models are effective for improving PSNR, preserving texture and restraining "blocky effect"
Secondly, based on image decomposition form as f= u+uv, multiplicative noise re-moving problems whose noise follow the Gauss law and Gamma Law are considered. New multi-scale fractional-order multi-scale models are propose. This paper gives the adaptive parameter selection method and designs the corresponding numerical algorithms. Numer-ical experiments show that for multiplicative noise removing, fractional-order multi-scale models are effective for improving PSNR, preserving texture and restraining "blocky ef-fect" also.
引文
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