图像稀疏表示理论及其应用研究
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摘要
图像表示是图像处理领域的基本问题。图像内容的有效表示是图像处理应用开展的基础。表示的有效性是指用很少的数据捕获感兴趣目标的重要信息的能力,即稀疏表示的能力。图像稀疏表示研究已成为近年来图像表示研究的热点,尤其是探讨基于人眼视觉的基函数理论模型及构造方法,研究快速、有效的图像稀疏表示算法,将有利于推动图像处理领域研究的开展,为图像表示提供新的理论与方法,具有重要的理论意义。
本文在国家自然科学基金项目“联合基图像稀疏表示理论研究及其应用”的资助下,针对图像稀疏表示理论、方法以及基于稀疏表示的图像处理应用问题,重点围绕多尺度几何分析在图像处理中的应用、基于冗余字典的超完备图像稀疏表示理论及应用、组合变换图像稀疏表示理论及应用等内容进行了探索性的研究。
论文阐述了信号表示的定义、稀疏性度量以及图像稀疏表示的基本理论方法,分析与比较了脊波变换、曲线波变换、Contourlet变换、Bandelet变换各自的非线性逼近性能。
脊波变换作为多尺度几何分析中的一种,能够稀疏地表示图像中的直线状奇异特征。利用脊波变换对直线状奇异特征的稀疏表示特性,提出了一种盲数字图像水印算法。算法能够确定出视觉上重要的图像脊波系数,并将水印自适应地嵌入其中,既保证了水印视觉上的不可见性,又提高了水印的鲁棒性。
针对脊波变换域阈值收缩法图像去噪存在的问题,提出了一种脊波收缩和全变差最小模型相结合的图像去噪算法。算法先对脊波系数进行收缩处理,再采用全变差扩散方式对收缩的脊波系数进行滤波处理。算法保留了两种去噪方法的优点,在计算复杂度与滤波效果上取得了更好的综合性能。
论文分析了自然图像曲线波系数的边缘统计分布,提出了曲线波系数边缘分布的正态反高斯分布建模方法和基于曲线波系数边缘统计模型和最大后验概率估计的图像去噪算法。算法能够减少图像中的噪声,更好地保留图像的边缘、纹理等几何特征,去噪后的图像具有更好的视觉质量。
基于曲线波变换的多方向和各向异性特性,提出了一种曲线波变换域图像融合算法。算法采用局部方向能量比来度量特征的显著程度,采用局部方向能量比的熵来自适应地抑制噪声的干扰。算法充分保留了图像的特征,同时抑制了噪声的干扰,更适用于实际的图像融合系统。
超完备稀疏表示能够获得最可能稀疏的图像表示,能获取比传统的非自适应方法更高分辨率的信息。论文研究了基于冗余字典的超完备图像稀疏表示理论及其应用。(1)对超完备图像稀疏分解算法进行了研究,提出了一种基于冗余字典非相干分解的多原子匹配追踪图像稀疏分解算法。算法通过每次迭代选取若干最匹配原子的方式,实现图像的快速稀疏分解。该算法在保证稀疏分解性能的同时,极大地提高了稀疏分解的速度,为基于超完备图像稀疏表示的实际应用奠定了基础。(2)对超完备图像稀疏表示中冗余字典的构造问题进行研究,提出了冗余多尺度脊波字典的构造方法。新构造出的字典满足人眼视觉的基本特性,具有多分辨率、多尺度、各向异性和多方向选择性等性质,能更稀疏地表示图像。(3)基于多原子匹配追踪算法和冗余多尺度脊波字典,提出了一种静态图像压缩编码方案。该方案通过对稀疏分解系数的自适应量化与编码,实现图像的压缩编码。在低比特率下,该方案获得比JPEG-2000更好的编码性能,非常适用于低比特或甚低比特率下的图像与视频编码。
组合变换图像稀疏表示通过几种变换(或正交基)的级联来构造超完备字典,实现图像的稀疏表示。论文对组合变换图像稀疏表示理论及应用进行了研究,首先,提出了迭代收缩法组合变换图像稀疏表示。该方法利用变换的快速算法,通过迭代收缩的方式实现图像的组合变换稀疏表示,具有运算简单、收敛速度快等优点,适合于大数据环境下的工程应用。其次,提出了组合傅里叶变换与曲线波变换的混合图像复原算法,利用傅里叶变换域收缩法进行去卷积和去除色噪声,并利用曲线波变换域收缩法进行去噪,较好地恢复了降质的图像。最后,提出了组合小波变换与曲线波变换的图像插值算法。利用小波变换与曲线波变换对图像不同内容的稀疏表示特性,将图像插值问题转化为稀疏约束的图像重建问题,采用迭代收缩投影法对问题进行求解,实现了图像分辨率的增强。实验结果表明,该算法显著提高了插值图像的视觉质量。
Image representation is the basic problem in image processing. Efficient representation of image contents lies at the foundation of image processing tasks. Efficiency of a representation refers to the ability to capture significant information of an object of interest in a small description. That is the ability of sparse representation. Recently, image sparse representation has been the hot topic of image representation. The researches of theory models and construction methods of basis based on human visual system, rapid and efficient image sparse representation algorithm will be help to promote the development of image processing. Those researches will provide new theory and method for image representation. It is of great importance in both theory and application.
Supported by the National Science Foundation of China, this thesis concentrates on the image sparse representation theories, methods and the image processing applications based on sparse representation. We make deeply study for the applications based on multiscale geometry analysis, theories of overcomplete sparse image representation and its applications, theories of sparse image representation via combined transforms and its applications.
Firstly, the definition of image representation, measure of sparsity, and the basic methods of sparse representation are reviewed. And then, we analyze the performance of nonlinear approximation of Ridgelet, Curvelet, Contourlet and Bandelet transform.
Ridgelet transform as one of the multiscale geometry analysis methods can sparsely represent the line singularities. Based on the property of Ridgelet sparse representation to line singularities, a blind digital image watermarking algorithm is proposed. The algorithm can exactly determine the important coefficients of image in vision and adaptively embed watermarks into the original image. Experimental results show the proposed watermarking algorithm ensures the invisibility of watermarks and improves the robustness.
Aiming at the problem of image denoising based on shrinkage in Ridgelet domain, a new hybrid denoising algorithm via combined Ridgelet shrinkage and total variation minimization model is proposed. Ridgelet coefficients are firstly thresholded, and then the thresholded coefficients are filtered by total variation diffusion. The hybrid denoising algorithm preserves the advantages of these two image denoising methods and has better general performance.
The marginal statistical model of Curvelet coefficients is studied. And a new image denoising algorithm based on marginal statistical model of Curvelet coefficients and maximum a posteriori (MAP) estimator is proposed, where the normal inverse Gaussian (NIG) distribution is used as the prior model of curvelet coefficients of images. Under this prior, a Bayesian Curvelet estimator is derived by using the MAP rule. The proposed denoising algorithm can reduce the noise efficiently and keep the details meanwhile.
Based on the multidirection and anisotropy of Curvelet transform, a new image fusion algorithm in Curvelet domain is proposed. The algorithm uses local direction energy ratio to measure the significance of features and uses local direction energy entropy to adaptively restrain noise disturb. The new fusion algorithm can sufficiently preserve image features and restrain noise disturb. It is more suited to real image fusion system.
Overcomplete sparse representation can obtain the sparsest possible representation of the object and a resolution of sparse objects that is much higher-resolution than that possible with traditional non-adaptive approaches. We study the theories of overcomplete sparse image representation based on redundant dictionary and its application. (1) The spare decomposition method of overcomplete image sparse representation is studied. And a multi-atom matching pursuit method based on incoherent decomposition of redundant dictionary is proposed. In this method, the image is decomposed by several the best matching atoms selected at each iteration. The performances of the proposed method are comparable with those of the matching pursuit. And the speed for image sparse decomposition is greatly improved. It will be help to the research of applications based on overcomplete sparse representation. (2) The problems of redundant dictionary construction in overcomplete sparse representation are studied. A method of multiscale Ridgelet dictionary construction is proposed. The new constructed dictionary satisfies the human visual system characteristics. It is multiresolution, multiscale, anisotropic and multidirectional. It can provide sparser representation for images. (3) A still image coding scheme based on multi-atom matching pursuit and redundant multiscale Ridgelet dictionary is proposed. In this scheme, the image is firstly sparsely decomposed, and then the decomposed coefficients are adaptively quantized and encoded. The performances of the new coding scheme are shown to compare favorably against those of the state of the art JPEG-2000 scheme at low bit rate. The new coding scheme is more suitable to the image and video coding at low or very low bit rate.
Image sparse representation via combined transforms means that an image is sparsely represented by overcomplete dictionary combined by several transforms. In this dissertation, we study the theories of sparse representation via combined transforms and its applications. Firstly, we study the methods of sparse representation via combined transforms and propose an iterative shrinkage algorithm. The iterative shrinkage can use the rapid algorithm of each transform. It is simple and rapidly convergent. It is more suited to use in large data. And then, a hybrid image restoration algorithm via combined Fourier and Curvelet transform is proposed. The Fourier shrinkage and Curvelet shrinkage are used to reduce the colored noise and the remaining noise separately. The hybrid restoration algorithm can restore the degraded images very well. In the end, an image interpolation algorithm via combined wavelet and Curvelet transform is proposed. The new interpolation algorithm exploits wavelet and Curvelet transforms’sparse representation of different kind of image contents. The image interpolation problem is turned to be image restoration problem enforced a sparsity constraint on the coefficients. We use an iterative shrinkage projection process to drive the solution towards an improved high-resolution image. Experimental results show the new interpolation algorithm substantially improves the subjective quality of interpolated images.
本文在国家自然科学基金项目“联合基图像稀疏表示理论研究及其应用”的资助下,针对图像稀疏表示理论、方法以及基于稀疏表示的图像处理应用问题,重点围绕多尺度几何分析在图像处理中的应用、基于冗余字典的超完备图像稀疏表示理论及应用、组合变换图像稀疏表示理论及应用等内容进行了探索性的研究。
论文阐述了信号表示的定义、稀疏性度量以及图像稀疏表示的基本理论方法,分析与比较了脊波变换、曲线波变换、Contourlet变换、Bandelet变换各自的非线性逼近性能。
脊波变换作为多尺度几何分析中的一种,能够稀疏地表示图像中的直线状奇异特征。利用脊波变换对直线状奇异特征的稀疏表示特性,提出了一种盲数字图像水印算法。算法能够确定出视觉上重要的图像脊波系数,并将水印自适应地嵌入其中,既保证了水印视觉上的不可见性,又提高了水印的鲁棒性。
针对脊波变换域阈值收缩法图像去噪存在的问题,提出了一种脊波收缩和全变差最小模型相结合的图像去噪算法。算法先对脊波系数进行收缩处理,再采用全变差扩散方式对收缩的脊波系数进行滤波处理。算法保留了两种去噪方法的优点,在计算复杂度与滤波效果上取得了更好的综合性能。
论文分析了自然图像曲线波系数的边缘统计分布,提出了曲线波系数边缘分布的正态反高斯分布建模方法和基于曲线波系数边缘统计模型和最大后验概率估计的图像去噪算法。算法能够减少图像中的噪声,更好地保留图像的边缘、纹理等几何特征,去噪后的图像具有更好的视觉质量。
基于曲线波变换的多方向和各向异性特性,提出了一种曲线波变换域图像融合算法。算法采用局部方向能量比来度量特征的显著程度,采用局部方向能量比的熵来自适应地抑制噪声的干扰。算法充分保留了图像的特征,同时抑制了噪声的干扰,更适用于实际的图像融合系统。
超完备稀疏表示能够获得最可能稀疏的图像表示,能获取比传统的非自适应方法更高分辨率的信息。论文研究了基于冗余字典的超完备图像稀疏表示理论及其应用。(1)对超完备图像稀疏分解算法进行了研究,提出了一种基于冗余字典非相干分解的多原子匹配追踪图像稀疏分解算法。算法通过每次迭代选取若干最匹配原子的方式,实现图像的快速稀疏分解。该算法在保证稀疏分解性能的同时,极大地提高了稀疏分解的速度,为基于超完备图像稀疏表示的实际应用奠定了基础。(2)对超完备图像稀疏表示中冗余字典的构造问题进行研究,提出了冗余多尺度脊波字典的构造方法。新构造出的字典满足人眼视觉的基本特性,具有多分辨率、多尺度、各向异性和多方向选择性等性质,能更稀疏地表示图像。(3)基于多原子匹配追踪算法和冗余多尺度脊波字典,提出了一种静态图像压缩编码方案。该方案通过对稀疏分解系数的自适应量化与编码,实现图像的压缩编码。在低比特率下,该方案获得比JPEG-2000更好的编码性能,非常适用于低比特或甚低比特率下的图像与视频编码。
组合变换图像稀疏表示通过几种变换(或正交基)的级联来构造超完备字典,实现图像的稀疏表示。论文对组合变换图像稀疏表示理论及应用进行了研究,首先,提出了迭代收缩法组合变换图像稀疏表示。该方法利用变换的快速算法,通过迭代收缩的方式实现图像的组合变换稀疏表示,具有运算简单、收敛速度快等优点,适合于大数据环境下的工程应用。其次,提出了组合傅里叶变换与曲线波变换的混合图像复原算法,利用傅里叶变换域收缩法进行去卷积和去除色噪声,并利用曲线波变换域收缩法进行去噪,较好地恢复了降质的图像。最后,提出了组合小波变换与曲线波变换的图像插值算法。利用小波变换与曲线波变换对图像不同内容的稀疏表示特性,将图像插值问题转化为稀疏约束的图像重建问题,采用迭代收缩投影法对问题进行求解,实现了图像分辨率的增强。实验结果表明,该算法显著提高了插值图像的视觉质量。
Image representation is the basic problem in image processing. Efficient representation of image contents lies at the foundation of image processing tasks. Efficiency of a representation refers to the ability to capture significant information of an object of interest in a small description. That is the ability of sparse representation. Recently, image sparse representation has been the hot topic of image representation. The researches of theory models and construction methods of basis based on human visual system, rapid and efficient image sparse representation algorithm will be help to promote the development of image processing. Those researches will provide new theory and method for image representation. It is of great importance in both theory and application.
Supported by the National Science Foundation of China, this thesis concentrates on the image sparse representation theories, methods and the image processing applications based on sparse representation. We make deeply study for the applications based on multiscale geometry analysis, theories of overcomplete sparse image representation and its applications, theories of sparse image representation via combined transforms and its applications.
Firstly, the definition of image representation, measure of sparsity, and the basic methods of sparse representation are reviewed. And then, we analyze the performance of nonlinear approximation of Ridgelet, Curvelet, Contourlet and Bandelet transform.
Ridgelet transform as one of the multiscale geometry analysis methods can sparsely represent the line singularities. Based on the property of Ridgelet sparse representation to line singularities, a blind digital image watermarking algorithm is proposed. The algorithm can exactly determine the important coefficients of image in vision and adaptively embed watermarks into the original image. Experimental results show the proposed watermarking algorithm ensures the invisibility of watermarks and improves the robustness.
Aiming at the problem of image denoising based on shrinkage in Ridgelet domain, a new hybrid denoising algorithm via combined Ridgelet shrinkage and total variation minimization model is proposed. Ridgelet coefficients are firstly thresholded, and then the thresholded coefficients are filtered by total variation diffusion. The hybrid denoising algorithm preserves the advantages of these two image denoising methods and has better general performance.
The marginal statistical model of Curvelet coefficients is studied. And a new image denoising algorithm based on marginal statistical model of Curvelet coefficients and maximum a posteriori (MAP) estimator is proposed, where the normal inverse Gaussian (NIG) distribution is used as the prior model of curvelet coefficients of images. Under this prior, a Bayesian Curvelet estimator is derived by using the MAP rule. The proposed denoising algorithm can reduce the noise efficiently and keep the details meanwhile.
Based on the multidirection and anisotropy of Curvelet transform, a new image fusion algorithm in Curvelet domain is proposed. The algorithm uses local direction energy ratio to measure the significance of features and uses local direction energy entropy to adaptively restrain noise disturb. The new fusion algorithm can sufficiently preserve image features and restrain noise disturb. It is more suited to real image fusion system.
Overcomplete sparse representation can obtain the sparsest possible representation of the object and a resolution of sparse objects that is much higher-resolution than that possible with traditional non-adaptive approaches. We study the theories of overcomplete sparse image representation based on redundant dictionary and its application. (1) The spare decomposition method of overcomplete image sparse representation is studied. And a multi-atom matching pursuit method based on incoherent decomposition of redundant dictionary is proposed. In this method, the image is decomposed by several the best matching atoms selected at each iteration. The performances of the proposed method are comparable with those of the matching pursuit. And the speed for image sparse decomposition is greatly improved. It will be help to the research of applications based on overcomplete sparse representation. (2) The problems of redundant dictionary construction in overcomplete sparse representation are studied. A method of multiscale Ridgelet dictionary construction is proposed. The new constructed dictionary satisfies the human visual system characteristics. It is multiresolution, multiscale, anisotropic and multidirectional. It can provide sparser representation for images. (3) A still image coding scheme based on multi-atom matching pursuit and redundant multiscale Ridgelet dictionary is proposed. In this scheme, the image is firstly sparsely decomposed, and then the decomposed coefficients are adaptively quantized and encoded. The performances of the new coding scheme are shown to compare favorably against those of the state of the art JPEG-2000 scheme at low bit rate. The new coding scheme is more suitable to the image and video coding at low or very low bit rate.
Image sparse representation via combined transforms means that an image is sparsely represented by overcomplete dictionary combined by several transforms. In this dissertation, we study the theories of sparse representation via combined transforms and its applications. Firstly, we study the methods of sparse representation via combined transforms and propose an iterative shrinkage algorithm. The iterative shrinkage can use the rapid algorithm of each transform. It is simple and rapidly convergent. It is more suited to use in large data. And then, a hybrid image restoration algorithm via combined Fourier and Curvelet transform is proposed. The Fourier shrinkage and Curvelet shrinkage are used to reduce the colored noise and the remaining noise separately. The hybrid restoration algorithm can restore the degraded images very well. In the end, an image interpolation algorithm via combined wavelet and Curvelet transform is proposed. The new interpolation algorithm exploits wavelet and Curvelet transforms’sparse representation of different kind of image contents. The image interpolation problem is turned to be image restoration problem enforced a sparsity constraint on the coefficients. We use an iterative shrinkage projection process to drive the solution towards an improved high-resolution image. Experimental results show the new interpolation algorithm substantially improves the subjective quality of interpolated images.
引文
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