脊波、曲线波和偏微分方程在图像处理中的算法研究
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摘要
寻求客观事物的“稀疏”表示方法,是计算机视觉、数学、数据压缩等领域的专家学者们致力于的目标。小波分析之所以成功,一个重要的原因是其对范围广泛的函数类的“稀疏”表示能力,尤其表现为对“点奇异”的最优非线性逼近能力。然而,对于含线奇异、面奇异的二维或高维函数,小波的非线性逼近能力却不尽如人意,事实上,此时M项非线性逼近误差ε_n(M)=‖f_M-f‖~2的衰减速度只能达到O(M~(-1))。小波分析的不足,促使人们开始从不同角度试图寻找比小波更好的“稀疏”表示工具。脊波理论由Emmanuel J.Candes在1998年提出,脊波变换对含直线奇异的多变量函数具有最优非线性逼近性能。后来,Candes又提出了曲线波的概念,曲线波变换对含有曲线奇异的多变量函数具有最优的非线性逼近阶。由于脊波、曲线波的这种能力,这使得它们在图像处理中比起小波具有一定的优势,更能保持图像的边缘,所以它们越来越受到广大研究者的重视。在数学图像处理中,相对于计算调和分析方法,另外一种方法就是偏微分方程方法。自从Perona和Malik在1990年引入了各向异性扩散的概念,人们提出了无数的各向异性扩散模型。这些模型在图像去噪、图像增强、边缘检测等方面具有很好的效果。时至今日,偏微分方程方法仍然是图像处理中的一个重要方法,是近年来图像处理的热点。本文将对脊波、曲线波和偏微分方程中有关领域的若干算法进行探讨,得到了一些新的有意义的算法。
(1)有限脊波变换方法是脊波变换的一种数字化实现方法,其核心是有限Radon变换的实现。本文针对有限Radon变换中每一斜率投影的排序问题,利用Besov空间光滑模理论,在时域构造了一种光滑性判别函数,以判别函数值最小的序作为最优序,从而给出了一种新的自适应排序算法。用该算法对图像进行压缩和去噪,得到了很好的效果。
(2)基于真脊函数的数字脊波变换是另一种脊波变换的数字化实现方法。比起有限脊波,它克服了环绕噪声,但是它不具有正交性,而是构成一个紧框架。这样,在对图像做稀疏逼近时并不是最优的。本文给出了数字脊波重构算法的全局对偶框架(GDF)表示,提出了局部对偶框架(LDF)的新概念,并讨论了LDF的性质,在此基础上给出了一种基于(LDF)的新的数字脊波重构算法。该算法减少了脊波重构的冗余,保持了快速计算的特点,提高了逼近的效果。用该算法对图像进行压缩和去噪,得到了很好的结果。
(3)曲线波变换是脊波变换的衍生物,它是由多尺度脊波变换合并带通滤波器得到的,它对具有曲线奇异的图像具有很好的逼近效果。因而对于含有边缘的图像,小波不能很好的探测出线奇异并且会让去噪后的图像边缘变的模糊,然而曲线波能很好的保持图像的边缘,获得很好的去噪效果。本文基于曲线波变换的思想,提出了一种新的数字曲线波变换的重构算法。我们的算法具有更稀疏的表示并且计算量较小。我们用提出的算法对图像进行去噪,得到了比原始算法好得多的结果。
(4)相对于各向同性扩散,各向异性扩散引入了一个关于图像梯度的空间变化的扩散系数。在图象梯度大的地方做弱的扩散,在图象梯度小的地方做强的扩散,因而在去噪的同时很好的保留了图像的边缘,而且不会产生像小波、曲线波引起的Gibbs震荡,是一种很好的图像处理方法。本文提出了一种新的各向异性分数扩散模型,这些方程是图像灰度值函数的分数导数的绝对值的一个增函数的欧拉-拉格朗日方程,所以该模型可以被看成二阶和四阶各向异性扩散模型的推广。我们用离散傅立叶变换计算分数阶导数并且得到了一个在频域的迭代算法。这个算法导致输入图像被看成一个周期图像。为了克服这个问题,我们将图像关于它的边缘对称的延拓(折叠算法)。最后,我们给出了在去噪实际的图像上的数值结果,实验结果表明提出的分数阶各向异性扩散模型产生了很好的视觉效果和信噪比。
(5)将图像分解为卡通部分(有界变差部分)和震荡部分(纹理部分)是近年来图像处理的一个重要问题。图像的卡通部分是由一个有界变差(BV)函数来刻画,相应的将BV罚项合并到变分泛函中需要解偏微分方程。Daubechies用Besov罚项代替BV罚项并且用小波解变分问题。按照这种思想,我们通过设计一种数字曲线波算法和一种依赖于尺度的阈值规则,从而得到了一种有效的基于数字曲线波变换的图像分解算法。我们可以看出该算法对噪声具有很强的鲁棒性并且能使图像边缘保持稳定。
Many image processing tasks take advantages of sparse representations of image data where most information is packed into a small number of samples.Typically, these representations are achieved via invertible and nonredundant transforms. Currently,the most popular choices for this purpose are the wavelet transform.The success of wavelets is mainly due to the good performance for piecewise smooth functions in one dimension.Unfortunately,such is not the case in two dimensions.For two-dimensional linear,surface singularities functions,the wavelet can't approximate them very well.In fact,the decay of the error of the M term approximationε_n(M)=‖f_M-f‖~2 only reach O(M~(-1)).The lack of the wavelet makes reseacher try to seek the other harmonic analysis tools.Emmanuel J.Candes introduced the concept of the ridgelet in 1998.Such system is very well at representing the smooth functions with line singularities.Candes also proposed the concept of the curvelet later.The curvelet has the best ability at representing the smooth functions with curvilinear singularities.Abilities of ridgelet and curvelet make them superior to the wavelet for image processing.They are valued by more and more researchers.Relative to the computational harmonic analysis method in the image processing,another method is partial differential equations.Since the work of Perona and Malik,which replaced the isotropic diffusion by an anisotropic difuusion,reserchers have proposed many anisotropic diffusion models which preserve important structures in images,while removing noise.These models make good effects in image denoising,image enhancement,edge detetion et al.Until now,the partial differential equation(PDE) method is still an important method and a hotspot in image processing.In this paper we will study some aspects of algorithms of ridgelet,curvelet and PDE and obtain some new instructive algorithms.
(1)The finite ridgelet transform is one of the digital methods of ridgelet transform. The key is the implementation of the finite Radon transform.The sequencing problem of each slope projection in finite Radon transform is studied on this paper.We use the theory of modulus of smoothness of the Besov space and construct a smooth discriminant function in time domain.The optimal sequence has the smallest functional value,so a new adaptive sequencing algorithm is proposed.When applying it to the image compression and denoising,the good results are obtained.
(2)The other is the digital ridgelet transform based on true ridge fuction.Compare with the finite ridgelet transform,it eliminates the "wrap around" and consists a frame. Thus it can't provide the most sparer representations of the image.A global dual frame (GDF)representation for the digital ridgelet reconstruction algorithm is discussed,then a novel concept of the local dual frame(LDF)is presented.Based on the properties of LDF,we propose a new digital ridgelet reconstruction algorithm.The method reduces the redundancy in the digital ridgelet reconstruction while keeping the characteristics of low compute cost.When applying it to the image compression and denoising,the good results are obtained.
(3)For an anisotropic image,wavelets lose their effects on singularity detection because discontinuities across edges are spatially distributed.Based on the idea of the curvelet,a new digital curvelet reconstruction algorithm is proposed.Our algorithm provides sparser representations and keeps low computational complexity.When applying it to the image denoising,much better results than the original algorithm are obtained.
(4)This paper introduces a new class of fractional-order anisotropic diffusion equations for noise removal.These equations are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function,so the proposed equations can be seen as generalizations of second-order and fourth-order anisotropic diffusion equations.We use the discrete Fourier transform(DFT)to implement the numerical algorithm and give an iterative scheme in the frequency domain.It is one important aspect of the algorithm that it considers the input image as a periodic image.To overcome this problem,we use a folded algorithm by extending the image symmetrically about its borders.And finally,we list various numerical results on denoising real images. Experiments show that the proposed fractional-order anisotropic diffusion equations yield good visual effects and better signal-to-noise ratio.
(5)Recent years,decomposing an image into cartoon component(bounded variation component)and oscillating component(texture component)is an important problem in the field of image processing.The cartoon component of an image is modeled by a bounded variation(BV)function;the corresponding incorporation of BV penalty terms in the variational functional leads to solve PDE equations.Daubechies replaced the BV penalty term by a Besov term and wrote the problem in a wavelet framework. Following this idea,we propose a new image decomposition algorithm based on the digital curvelet transform.By designing a digital curvelet transform algorithm and a scale-dependent thresholding rule,elegant and numerically efficient schemes are obtained.We can see that this approach is very robust to additive noise and can keep the image edges stable.
(1)有限脊波变换方法是脊波变换的一种数字化实现方法,其核心是有限Radon变换的实现。本文针对有限Radon变换中每一斜率投影的排序问题,利用Besov空间光滑模理论,在时域构造了一种光滑性判别函数,以判别函数值最小的序作为最优序,从而给出了一种新的自适应排序算法。用该算法对图像进行压缩和去噪,得到了很好的效果。
(2)基于真脊函数的数字脊波变换是另一种脊波变换的数字化实现方法。比起有限脊波,它克服了环绕噪声,但是它不具有正交性,而是构成一个紧框架。这样,在对图像做稀疏逼近时并不是最优的。本文给出了数字脊波重构算法的全局对偶框架(GDF)表示,提出了局部对偶框架(LDF)的新概念,并讨论了LDF的性质,在此基础上给出了一种基于(LDF)的新的数字脊波重构算法。该算法减少了脊波重构的冗余,保持了快速计算的特点,提高了逼近的效果。用该算法对图像进行压缩和去噪,得到了很好的结果。
(3)曲线波变换是脊波变换的衍生物,它是由多尺度脊波变换合并带通滤波器得到的,它对具有曲线奇异的图像具有很好的逼近效果。因而对于含有边缘的图像,小波不能很好的探测出线奇异并且会让去噪后的图像边缘变的模糊,然而曲线波能很好的保持图像的边缘,获得很好的去噪效果。本文基于曲线波变换的思想,提出了一种新的数字曲线波变换的重构算法。我们的算法具有更稀疏的表示并且计算量较小。我们用提出的算法对图像进行去噪,得到了比原始算法好得多的结果。
(4)相对于各向同性扩散,各向异性扩散引入了一个关于图像梯度的空间变化的扩散系数。在图象梯度大的地方做弱的扩散,在图象梯度小的地方做强的扩散,因而在去噪的同时很好的保留了图像的边缘,而且不会产生像小波、曲线波引起的Gibbs震荡,是一种很好的图像处理方法。本文提出了一种新的各向异性分数扩散模型,这些方程是图像灰度值函数的分数导数的绝对值的一个增函数的欧拉-拉格朗日方程,所以该模型可以被看成二阶和四阶各向异性扩散模型的推广。我们用离散傅立叶变换计算分数阶导数并且得到了一个在频域的迭代算法。这个算法导致输入图像被看成一个周期图像。为了克服这个问题,我们将图像关于它的边缘对称的延拓(折叠算法)。最后,我们给出了在去噪实际的图像上的数值结果,实验结果表明提出的分数阶各向异性扩散模型产生了很好的视觉效果和信噪比。
(5)将图像分解为卡通部分(有界变差部分)和震荡部分(纹理部分)是近年来图像处理的一个重要问题。图像的卡通部分是由一个有界变差(BV)函数来刻画,相应的将BV罚项合并到变分泛函中需要解偏微分方程。Daubechies用Besov罚项代替BV罚项并且用小波解变分问题。按照这种思想,我们通过设计一种数字曲线波算法和一种依赖于尺度的阈值规则,从而得到了一种有效的基于数字曲线波变换的图像分解算法。我们可以看出该算法对噪声具有很强的鲁棒性并且能使图像边缘保持稳定。
Many image processing tasks take advantages of sparse representations of image data where most information is packed into a small number of samples.Typically, these representations are achieved via invertible and nonredundant transforms. Currently,the most popular choices for this purpose are the wavelet transform.The success of wavelets is mainly due to the good performance for piecewise smooth functions in one dimension.Unfortunately,such is not the case in two dimensions.For two-dimensional linear,surface singularities functions,the wavelet can't approximate them very well.In fact,the decay of the error of the M term approximationε_n(M)=‖f_M-f‖~2 only reach O(M~(-1)).The lack of the wavelet makes reseacher try to seek the other harmonic analysis tools.Emmanuel J.Candes introduced the concept of the ridgelet in 1998.Such system is very well at representing the smooth functions with line singularities.Candes also proposed the concept of the curvelet later.The curvelet has the best ability at representing the smooth functions with curvilinear singularities.Abilities of ridgelet and curvelet make them superior to the wavelet for image processing.They are valued by more and more researchers.Relative to the computational harmonic analysis method in the image processing,another method is partial differential equations.Since the work of Perona and Malik,which replaced the isotropic diffusion by an anisotropic difuusion,reserchers have proposed many anisotropic diffusion models which preserve important structures in images,while removing noise.These models make good effects in image denoising,image enhancement,edge detetion et al.Until now,the partial differential equation(PDE) method is still an important method and a hotspot in image processing.In this paper we will study some aspects of algorithms of ridgelet,curvelet and PDE and obtain some new instructive algorithms.
(1)The finite ridgelet transform is one of the digital methods of ridgelet transform. The key is the implementation of the finite Radon transform.The sequencing problem of each slope projection in finite Radon transform is studied on this paper.We use the theory of modulus of smoothness of the Besov space and construct a smooth discriminant function in time domain.The optimal sequence has the smallest functional value,so a new adaptive sequencing algorithm is proposed.When applying it to the image compression and denoising,the good results are obtained.
(2)The other is the digital ridgelet transform based on true ridge fuction.Compare with the finite ridgelet transform,it eliminates the "wrap around" and consists a frame. Thus it can't provide the most sparer representations of the image.A global dual frame (GDF)representation for the digital ridgelet reconstruction algorithm is discussed,then a novel concept of the local dual frame(LDF)is presented.Based on the properties of LDF,we propose a new digital ridgelet reconstruction algorithm.The method reduces the redundancy in the digital ridgelet reconstruction while keeping the characteristics of low compute cost.When applying it to the image compression and denoising,the good results are obtained.
(3)For an anisotropic image,wavelets lose their effects on singularity detection because discontinuities across edges are spatially distributed.Based on the idea of the curvelet,a new digital curvelet reconstruction algorithm is proposed.Our algorithm provides sparser representations and keeps low computational complexity.When applying it to the image denoising,much better results than the original algorithm are obtained.
(4)This paper introduces a new class of fractional-order anisotropic diffusion equations for noise removal.These equations are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function,so the proposed equations can be seen as generalizations of second-order and fourth-order anisotropic diffusion equations.We use the discrete Fourier transform(DFT)to implement the numerical algorithm and give an iterative scheme in the frequency domain.It is one important aspect of the algorithm that it considers the input image as a periodic image.To overcome this problem,we use a folded algorithm by extending the image symmetrically about its borders.And finally,we list various numerical results on denoising real images. Experiments show that the proposed fractional-order anisotropic diffusion equations yield good visual effects and better signal-to-noise ratio.
(5)Recent years,decomposing an image into cartoon component(bounded variation component)and oscillating component(texture component)is an important problem in the field of image processing.The cartoon component of an image is modeled by a bounded variation(BV)function;the corresponding incorporation of BV penalty terms in the variational functional leads to solve PDE equations.Daubechies replaced the BV penalty term by a Besov term and wrote the problem in a wavelet framework. Following this idea,we propose a new image decomposition algorithm based on the digital curvelet transform.By designing a digital curvelet transform algorithm and a scale-dependent thresholding rule,elegant and numerically efficient schemes are obtained.We can see that this approach is very robust to additive noise and can keep the image edges stable.
引文
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