基于多尺度几何分析和能量泛函的图像处理算法研究
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摘要
多尺度几何分析和极小化能量泛函方法是当前数学图像处理和计算机视觉等领域最具代表性的两种研究范式,这方面的研究已引起广大学者的普遍关注。一方面新的成果不断涌现,另一方面,已有的成果仍存在许多值得进一步研究的问题。本文基于这两种范式,主要提出并解决了以下问题:
1、针对图像的曲线状奇异性表征问题和Candes单尺度脊波框架的冗余问题,利用局部化原理和Donoho构造的正交脊波,提出了一种单尺度正交脊波紧框架。该框架不仅保留了Candes单尺度脊波的方向性,而且具有正交性。另外,针对边缘具有一定光滑性的图像,讨论了该框架对其的非线性逼近性。实验结果表明,所提单尺度正交脊波在图像压缩、图像恢复及图像去噪中均得到较好地应用。
2、针对Starck分解模型在算法复杂度、纹理表征和噪声约束诸方面所存在的一些不足,提出一种基于基追踪的极小化能量泛函模型。该改进模型利用第二代曲波和波原子,分别表征含噪图像中的结构分量和纹理分量,并采用全变差半范约束分片光滑部分的结构性,同时利用Meyer所建议的广义齐型Besov范数对噪声分量进行约束,最后利用基追踪去噪算法,对新模型进行迭代求解。实验结果表明,所提能量泛函模型不但对噪声具有较强的鲁棒性,而且能使边缘和细小纹理信息保持稳定。
3、为了解决多尺度几何分析在图像抑噪应用中出现的“虚假”效应问题,提出一种带变换域约束条件的极小化全变差能量泛函。该能量泛函将多尺度几何分析和极小化能量泛函方法有机地结合起来。首先对降质图像利用多尺度几何分析进行相应变换和非线性阈值,然后根据保留的变换系数确定可行域,从而建立所提能量泛函模型,最后利用投影梯度算法对其进行求解,并以有限脊波变换和第二代曲波变换为例,进行了图像抑噪仿真实验。实验结果表明,所提能量泛函在有效抑噪和保持边缘的同时,能够有效地抑制伪吉布斯振荡、“卷绕”伪直线和“曲波状”伪曲线等“虚假”效应,取得了较为理想的视觉效果。
4、针对经典极小化能量泛函中平衡参数对图像振荡分量先验信息的过分依赖性,首先讨论了一类更为一般的Meyer分解模型,并证明了解的存在性和唯一性。然后将平衡参数视为尺度参数,提出一种分级多尺度极小化能量泛函,并导出一种图像的多尺度表示方法。同时,对这一多尺度表示方法的收敛性进行了理论分析。最后,利用BV近似W 1,1导出一种新的近似求解算法。数值实验结果表明,所提能量泛函在各类图像处理中均有较好的应用。
5、针对图像恢复应用中,经典全变差正则化方法的阶梯效应和细小纹理信息丢失问题,从两种不同视角提出如下解决方案:
(1)提出一种自适应正则化的极小化能量泛函,将图像分解为结构分量和振荡分量,其中对结构分量的正则化是通过TV光滑化和各向同性光滑化之间的插值得到,即依据图像局部特征进行一种自适应正则化;振荡分量被置于div(BMO)空间中加以讨论。此外,我们对所提能量泛函解的存在唯一性进行了理论证明,并导出其相应的Euler-Lagrange方程。实验结果表明,所提能量泛函在实际图像分解应用中,不但能够较好地保持边缘和细小纹理,而且有效地抑制了阶梯效应。
(2)利用Meyer的振荡模式分解理论,提出了一种磨光流场的全变差正则化抑噪方法。该方法首先引入负指数Hilbert-Sobolev范数来度量逼近项,对图像水平曲线的法向量场进行全变差正则化磨光,然后构造一个曲面拟合能量泛函,对磨光后的流场进行拟合。最后,导出各能量泛函所对应的Euler-Lagrange方程,并利用有限差分法进行数值求解。实验结果表明,该方法在有效去噪的同时,能够较好地保持边缘和纹理信息,并且使阶梯效应也得到有效地抑制。
In mathematical imaging and visual of computer, multi-scale geometric analysis and minimization energy functional are the most representative two paradigms at current. It has attracted attention of researchers widely. A lot of new productions are presented about the two paradigms. On the other hand, the productions proposed still exist some problems which are worthy of researching further. In this dissertation, based on the two paradigms, the following problems are discussed:
1.To represent curve singularity of image and deal with redundancy of Candes’s monoscale ridgelets, a new ridgelets frame, monoscale orthonormal ridgelets frame (MORF), is presented, which use the localization principle and the orthonormal ridgelet constructed by Donoho. The MORF not only remains directionality, but also bears orthonormality. We discuss the nonlinear approximation of the MORF for image which is smooth away from discontinuities across curves. The experiments demonstrate that the new frame have preferable application for image compression, image reconstruction and image denoising.
2.A new minimization energy functional based on the basis pursuit is presented to improve on Starck’s decomposition model. In this new functional, the second generation curvelets and wave atoms are used to represent structure and texture respectively, and the total variational semi-norm is added for restricting structure parts. In addition, the generalized homogeneous Besov norm proposed by Meyer is used to constrain noisy components. Finally, the basis pursuit denoisiing algorithm is used to solve the new model. Experiments show that the new model is very robust for noise, and that can keep edges and textures stably.
3. To reduce the“aliasing effects”resulted from using multi-scale geometric analysis for image denoising, a minimization total variational energy functional with constraint condition on transform domain is presented. Firstly, the nonlinear thresholding strategy associated with certain multi-scale geometric analysis is applied to the transform coefficients of noisy image. And then, the feasible domain of the proposed model is determined by the coefficients remained. Finally, the projected gradient algorithm is used to solve the proposed model. Experiments show that the presented model can remove noisy and remain edges, while the“aliasing effects”such as psudo-Gibbs effects,“wrap around”effects and the“curvelet like”aliased curves are suppressed efficiently, when the finite ridgelet transform and the curvelet transform are applied respectively.
4.In the classical minimization energy functional, the balance parameter often depends on the priori information of the oscillatory component. To reduce this dependence, Firstly, a more general decomposition model is discussed, while the existence and uniqueness of solution about this model are proved. And then, the balance parameter is taken as scale. Instead of L2 space, we discuss the oscillatory component in distribution space W ?1 ,∞. To this end, a multiscale hierarchical decomposition model is established, which still obtains a multiscale representation of image, at the same time, its convergence is analyzed. Finally, a new algorithm is proposed by applying BV to approximate W 1,1. The numerical experiments show that the multiscale hierarchical decomposition model has well application in many image processing.
5.In the two different viewpoints, we investigate staircasing effect and losing of small scale texture which are caused by TV regularization in the classical minimization energy functional, and present two solutions as follows:
(1) A new minimization energy functional with adaptive regularization is proposed, which decomposes image into structure component and oscillatory component, where the regularization for the structure component is obtained by interpolation of TV regularization and isotropic smoothing, namely, take adaptive regularization in terms of the local feature of image; the oscillatory component is investigated in div(BMO) space. In addition, we proof the existence and uniqueness of the presented model. Finally, the corresponding Euler-Lagrange equations are derived to numerically implement. Experimental results and comparisons demonstrate that the proposed model has an advantage of improving visual effect of image decomposition, namely, both edge and texture are well remained, while the staircasing effect is avoided efficiently.
(2) Using oscillating patterns theory in image processing proposed by Meyer, a total variation image denoising method is presented, which based on smoothing flow field. Firstly, through applying Hilbert-Sobolev norm to measure fidelity term, a total variation filter is used to smooth the normal vectors of the level curves in noise image. And then, a model is constructed to find a surface which fit smoothed normal vectors. Finally, finite difference schemes are used to solve the Euler-Lagrange functions derived from above models. The experiments show that the approach not only can remove noisy efficiently, but also can retain edges and texture and the staircasing effect also is avoided.
1、针对图像的曲线状奇异性表征问题和Candes单尺度脊波框架的冗余问题,利用局部化原理和Donoho构造的正交脊波,提出了一种单尺度正交脊波紧框架。该框架不仅保留了Candes单尺度脊波的方向性,而且具有正交性。另外,针对边缘具有一定光滑性的图像,讨论了该框架对其的非线性逼近性。实验结果表明,所提单尺度正交脊波在图像压缩、图像恢复及图像去噪中均得到较好地应用。
2、针对Starck分解模型在算法复杂度、纹理表征和噪声约束诸方面所存在的一些不足,提出一种基于基追踪的极小化能量泛函模型。该改进模型利用第二代曲波和波原子,分别表征含噪图像中的结构分量和纹理分量,并采用全变差半范约束分片光滑部分的结构性,同时利用Meyer所建议的广义齐型Besov范数对噪声分量进行约束,最后利用基追踪去噪算法,对新模型进行迭代求解。实验结果表明,所提能量泛函模型不但对噪声具有较强的鲁棒性,而且能使边缘和细小纹理信息保持稳定。
3、为了解决多尺度几何分析在图像抑噪应用中出现的“虚假”效应问题,提出一种带变换域约束条件的极小化全变差能量泛函。该能量泛函将多尺度几何分析和极小化能量泛函方法有机地结合起来。首先对降质图像利用多尺度几何分析进行相应变换和非线性阈值,然后根据保留的变换系数确定可行域,从而建立所提能量泛函模型,最后利用投影梯度算法对其进行求解,并以有限脊波变换和第二代曲波变换为例,进行了图像抑噪仿真实验。实验结果表明,所提能量泛函在有效抑噪和保持边缘的同时,能够有效地抑制伪吉布斯振荡、“卷绕”伪直线和“曲波状”伪曲线等“虚假”效应,取得了较为理想的视觉效果。
4、针对经典极小化能量泛函中平衡参数对图像振荡分量先验信息的过分依赖性,首先讨论了一类更为一般的Meyer分解模型,并证明了解的存在性和唯一性。然后将平衡参数视为尺度参数,提出一种分级多尺度极小化能量泛函,并导出一种图像的多尺度表示方法。同时,对这一多尺度表示方法的收敛性进行了理论分析。最后,利用BV近似W 1,1导出一种新的近似求解算法。数值实验结果表明,所提能量泛函在各类图像处理中均有较好的应用。
5、针对图像恢复应用中,经典全变差正则化方法的阶梯效应和细小纹理信息丢失问题,从两种不同视角提出如下解决方案:
(1)提出一种自适应正则化的极小化能量泛函,将图像分解为结构分量和振荡分量,其中对结构分量的正则化是通过TV光滑化和各向同性光滑化之间的插值得到,即依据图像局部特征进行一种自适应正则化;振荡分量被置于div(BMO)空间中加以讨论。此外,我们对所提能量泛函解的存在唯一性进行了理论证明,并导出其相应的Euler-Lagrange方程。实验结果表明,所提能量泛函在实际图像分解应用中,不但能够较好地保持边缘和细小纹理,而且有效地抑制了阶梯效应。
(2)利用Meyer的振荡模式分解理论,提出了一种磨光流场的全变差正则化抑噪方法。该方法首先引入负指数Hilbert-Sobolev范数来度量逼近项,对图像水平曲线的法向量场进行全变差正则化磨光,然后构造一个曲面拟合能量泛函,对磨光后的流场进行拟合。最后,导出各能量泛函所对应的Euler-Lagrange方程,并利用有限差分法进行数值求解。实验结果表明,该方法在有效去噪的同时,能够较好地保持边缘和纹理信息,并且使阶梯效应也得到有效地抑制。
In mathematical imaging and visual of computer, multi-scale geometric analysis and minimization energy functional are the most representative two paradigms at current. It has attracted attention of researchers widely. A lot of new productions are presented about the two paradigms. On the other hand, the productions proposed still exist some problems which are worthy of researching further. In this dissertation, based on the two paradigms, the following problems are discussed:
1.To represent curve singularity of image and deal with redundancy of Candes’s monoscale ridgelets, a new ridgelets frame, monoscale orthonormal ridgelets frame (MORF), is presented, which use the localization principle and the orthonormal ridgelet constructed by Donoho. The MORF not only remains directionality, but also bears orthonormality. We discuss the nonlinear approximation of the MORF for image which is smooth away from discontinuities across curves. The experiments demonstrate that the new frame have preferable application for image compression, image reconstruction and image denoising.
2.A new minimization energy functional based on the basis pursuit is presented to improve on Starck’s decomposition model. In this new functional, the second generation curvelets and wave atoms are used to represent structure and texture respectively, and the total variational semi-norm is added for restricting structure parts. In addition, the generalized homogeneous Besov norm proposed by Meyer is used to constrain noisy components. Finally, the basis pursuit denoisiing algorithm is used to solve the new model. Experiments show that the new model is very robust for noise, and that can keep edges and textures stably.
3. To reduce the“aliasing effects”resulted from using multi-scale geometric analysis for image denoising, a minimization total variational energy functional with constraint condition on transform domain is presented. Firstly, the nonlinear thresholding strategy associated with certain multi-scale geometric analysis is applied to the transform coefficients of noisy image. And then, the feasible domain of the proposed model is determined by the coefficients remained. Finally, the projected gradient algorithm is used to solve the proposed model. Experiments show that the presented model can remove noisy and remain edges, while the“aliasing effects”such as psudo-Gibbs effects,“wrap around”effects and the“curvelet like”aliased curves are suppressed efficiently, when the finite ridgelet transform and the curvelet transform are applied respectively.
4.In the classical minimization energy functional, the balance parameter often depends on the priori information of the oscillatory component. To reduce this dependence, Firstly, a more general decomposition model is discussed, while the existence and uniqueness of solution about this model are proved. And then, the balance parameter is taken as scale. Instead of L2 space, we discuss the oscillatory component in distribution space W ?1 ,∞. To this end, a multiscale hierarchical decomposition model is established, which still obtains a multiscale representation of image, at the same time, its convergence is analyzed. Finally, a new algorithm is proposed by applying BV to approximate W 1,1. The numerical experiments show that the multiscale hierarchical decomposition model has well application in many image processing.
5.In the two different viewpoints, we investigate staircasing effect and losing of small scale texture which are caused by TV regularization in the classical minimization energy functional, and present two solutions as follows:
(1) A new minimization energy functional with adaptive regularization is proposed, which decomposes image into structure component and oscillatory component, where the regularization for the structure component is obtained by interpolation of TV regularization and isotropic smoothing, namely, take adaptive regularization in terms of the local feature of image; the oscillatory component is investigated in div(BMO) space. In addition, we proof the existence and uniqueness of the presented model. Finally, the corresponding Euler-Lagrange equations are derived to numerically implement. Experimental results and comparisons demonstrate that the proposed model has an advantage of improving visual effect of image decomposition, namely, both edge and texture are well remained, while the staircasing effect is avoided efficiently.
(2) Using oscillating patterns theory in image processing proposed by Meyer, a total variation image denoising method is presented, which based on smoothing flow field. Firstly, through applying Hilbert-Sobolev norm to measure fidelity term, a total variation filter is used to smooth the normal vectors of the level curves in noise image. And then, a model is constructed to find a surface which fit smoothed normal vectors. Finally, finite difference schemes are used to solve the Euler-Lagrange functions derived from above models. The experiments show that the approach not only can remove noisy efficiently, but also can retain edges and texture and the staircasing effect also is avoided.
引文
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