复数小波理论及其在图像去噪与增强中的应用研究
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摘要
非平稳信号的稀疏表示和高效处理算法是数学和信息科学研究的重要内容,其中,近年来建立起来的小波理论与算法已经成为信号稀疏表示的有效方法。但是,传统小波变换在处理信号和图像时存在平移敏感性和方向选择性弱等缺陷,因此,研究具有更好的近似平移不变性和奇异特征表示能力的新型小波变换,成为当前小波理论发展以及图像处理中非常重要的课题。由于图像获取方式的限制或在传输过程中受到干扰,通常导致观测的图像质量过低或被各种噪声所污染。图像去噪的主要目的是在保留图像原有重要信息的前提下降低或消除噪声,获得高质量的为人类视觉所接受的图像,从而为下一步的图像处理奠定基础。图像增强的目的是通过处理凸显原图像不够清晰的细节信息,使得处理后的图像更加便于人眼理解或机器识别。图像去噪和增强都是目前计算机视觉和图像处理领域最基本的且仍未很好解决的挑战性课题。
针对传统离散小波变换(DWT,Discrete Wavelet Transform)的局限,本文深入研究了二元树复数小波变换(DT-CWT,Dual-Tree Complex Wavelet Transform)的相关性质,包括近似平移不变性、方向性和实现问题等,并在此基础上提出了构造二元树复数小波滤波器组的新算法;提出了一种新型复数小波变换—高密二元树离散小波变换(HD-DT DWT,Higher-Density Dual-Tree DWT),研究了其相关的性质及满足各种约束条件的滤波器组的构造方法;为更好的处理非平稳信号,初步研究了基于全变差模型和优化方法的信号和图像自适应分解问题;进一步深入研究了新型复数小波变换在图像去噪和增强中的应用,获得了比现有方法有显著改进的实验结果。
本文的主要工作和创新如下:
■研究了二元树复数小波中双正交Hilbert变换对的构造。对线性相位双正交小波的构造和二元树复数小波变换的相关性质进行了充分而详尽的研究,在此基础上提出了利用参数化技术和最优化方法构造二元树复数小波变换中的Hilbert变换对的方法。这种滤波器设计的优点在于,对参数作适当的调节就能得到有理系数的二元树复数小波滤波器组,对于提高变换速度和效率、降低计算复杂度都有显著意义。
■针对传统DWT的缺陷,提出了高密度二元树离散小波变换这一新型复数小波变换的概念,系统深入的研究了高密度二元树离散小波变换的性质和构造方法,利用分数阶延迟滤波器、谱因子分解等技术构造出了具有紧支撑、消失矩、较高阶的光滑性、近似Hilbert变换对关系、中间尺度等优良性质的小波函数,为信号和图像等高维数据的分析提供了一种新的变换方法。
■作为用小波变换对信号和图像进行分解的一种推广,本文还初步研究了基于优化方法的信号和图像自适应分解问题,根据信号自适应的得到其低分辨率近似和重构滤波器,使得重构信号与原信号之间的误差最小。为提高所得近似图像的视觉质量,我们进一步将全变差模型引入自适应分解方法中,为对信号或图像进行自适应分解提供了一种新思路。
■基于理论研究的结果,进一步深入探讨了新型复数小波变换在图像去噪和增强中的应用,提出了三种基于DT-CWT的图像去噪新算法:(ⅰ)复数小波变换域利用系数尺度间和尺度内相关性的图像去噪算法;(ⅱ)基于局部参数的二元树复数小波域隐马尔可夫树(HMT,Hidden Markov Tree)模型图像去噪;(ⅲ)复数小波域高斯尺度混合(GSM,Gaussian Scale Mixture)模型去噪。这些方法充分利用了复数小波变换的优良性质及其系数分布的统计规律,实验表明,在简化计算复杂度、提高计算效率的同时获得了比现有相关去噪算法有显著改进的的去噪效果。另外,我们还提出了一种基于尺度间和尺度内相关性SURE方法的正交小波阈值去噪方法,解决了最近提出的正交小波域去噪算法对含较多纹理的图像处理效果不佳的缺陷,成为目前非冗余小波变换域效果最好的去噪算法。
■最后,我们还探讨了结合新型复数小波变换和最优视觉表示的统计特性的图像增强问题,提出了两种图像增强算法:(ⅰ)基于双密度二元树离散小波变换(DD-DT DWT,Double-Density Dual-Tree DWT)和视觉表示的图像增强算法,取得了非常好的视觉效果;(ⅱ)基于二元树复数小波和视觉表示的噪声图像增强算法,较好的缓解了带噪声图像增强中噪声抑制和细节保护之间的矛盾。
Efficient sparse representation and processing of unstable signal are the main contents in mathematics and information science. Recently, the Discrete Wavelet Transform (DWT) has become efficient in the sparse representation of unstable signal and also is a powerful tool for signal and image processing. It, however, has some disadvanges, including, (1) It is shift sensitive because the input signal shift generates unpredictable changes in DWT coefficients; (2) It suffers from poor directionality because DWT coefficients reveal merely three spatial orientations; (3) It lacks of the phase information that accurately dscribes non-stationary signal behavior; that undermine its usage in many applications. Therefore, there is a strong motivation to study new types of wavelet transforms with better shift invariance and directionality. Due to the imperfection of image acquisition systems and transmission channels, the observed images are often in low-quality or degraded by noise. The goal of image denoising is to remove the noise while retaining as much as possible the important features (edges) and obtain acceptable image for vision. The image enhancement algorithms are to process a given image so the results are better than original image for their applications or objectives. Noise elimination and image enhancement are still the most fundamental, widely studied, and largely unsolved problems in computer vision and image processing.
To overcome the disadvantages of the traditional DWT, this thesis mainly focus on two new types of complex wavelet transform: the dual-tree complex wavelet transform (DT-CWT) and the higher density dual tree DWT. The properties of the DT-CWT such as approximate shift-invariance, directionality and implementation issue are carefully investigigated. Furthermore, a new algorithm to construct wavelet filterbank of the DT-CWT is presented. At the same time, a new complex wavelet transform - the higher denstiy dual tree DWT is introduced and the corresponding characteristics are studies and a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed is developed. To better process the non-stational signal, the total variation and optimization based schemes for signal and image adaptive decomposition are preliminarily studied. Some classical applications of the proposed complex wavelet transforms are also further studies such as image denoising and enhancement. Results of experiments show that the proposed new algorithms perform better than the now existing methods.
The main achievements in this paper are as follow:
First, an approach for designing biorthogonal DT-CWT filters is proposed; where the two related wavelets pairs form approximate Hilbert transform pairs. Different from the existing design techniques, the two wavelet filterbanks obtained here are both of linear phases. By adjusting the parameters, wavelet fitlers with rational coefficients may be achieved, which can speed up the DT-CWT effectively.
Then, to overcome the disadavanges of the DWT, we introduce the higher-density dual-tree DWT, which is a DWT that combines the higher-density DWT and the DT-CWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. We develop a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractinal-delay allpass filters, spectral factorization and the solutions have vanishing moments, compact support, a high degree of smootheness, intermediate scales, approximate Hilbert transform properties, and are nearly shift-invariant.
In addition, we investigate the problem of adaptive deocompositon of the signal and image, the optimization method and total variation model are employed in the process. Experimental results show that the proposed methods are effective to a wide range of signals and images; when compared to the fixed wavelet bases method, the produced reconstruct images with our adaptive method are with better PSNR and visual quality.
Based on the theory above, to address the problems of the image denoising and enhancement, we investigate the image denosing and enhancement in the new types of complex wavelet transform domain in detail. Three new image denoising algorithms based on the DT-CWT are proposed: (i) a new locally adaptive image denoising method, which exploits the intra-scale and inter-scale depencencies in the DT-CWT domain; (ii) a new non-tranining compelx wavelet Hidden Markov Tree (CHMT) model, which is based on the DT-CWT and a fast parameters estimation technique; (iii) a new denoising algorithm based on the Gaussian scale mixture (GSM) of the coefficients of the DT-CWT. These methods exploit the properties of the DT-CWT and the statistics of the coefficients and the obtained better denoising performances while reducing the computational complexity. At the same time, we introduce an effective integration of the intrascale correlations within the interscale SURE based orthonormal wavelet thresholding, which can solve the problem of the interscale method that is not very effective for those images that have substantial high-frequency contents.
In addition, we also investigate the image enhancement based on the new types of wavelet transform and the statistical characters of visual representation and propose two new method of image enhancement: (i) a novel method for image enhancement, which exploits the properties of the double-density dual-tree DWT and the statistical characters of visual representation; (ii) a new method for noisy image enhancement, which is based on the GSM model of the DT-CWT coefficients and the combination of the DT-CWT and the statistical characters of visual representation, and can optimize the contrast of image features of while minimizing image denoise.
针对传统离散小波变换(DWT,Discrete Wavelet Transform)的局限,本文深入研究了二元树复数小波变换(DT-CWT,Dual-Tree Complex Wavelet Transform)的相关性质,包括近似平移不变性、方向性和实现问题等,并在此基础上提出了构造二元树复数小波滤波器组的新算法;提出了一种新型复数小波变换—高密二元树离散小波变换(HD-DT DWT,Higher-Density Dual-Tree DWT),研究了其相关的性质及满足各种约束条件的滤波器组的构造方法;为更好的处理非平稳信号,初步研究了基于全变差模型和优化方法的信号和图像自适应分解问题;进一步深入研究了新型复数小波变换在图像去噪和增强中的应用,获得了比现有方法有显著改进的实验结果。
本文的主要工作和创新如下:
■研究了二元树复数小波中双正交Hilbert变换对的构造。对线性相位双正交小波的构造和二元树复数小波变换的相关性质进行了充分而详尽的研究,在此基础上提出了利用参数化技术和最优化方法构造二元树复数小波变换中的Hilbert变换对的方法。这种滤波器设计的优点在于,对参数作适当的调节就能得到有理系数的二元树复数小波滤波器组,对于提高变换速度和效率、降低计算复杂度都有显著意义。
■针对传统DWT的缺陷,提出了高密度二元树离散小波变换这一新型复数小波变换的概念,系统深入的研究了高密度二元树离散小波变换的性质和构造方法,利用分数阶延迟滤波器、谱因子分解等技术构造出了具有紧支撑、消失矩、较高阶的光滑性、近似Hilbert变换对关系、中间尺度等优良性质的小波函数,为信号和图像等高维数据的分析提供了一种新的变换方法。
■作为用小波变换对信号和图像进行分解的一种推广,本文还初步研究了基于优化方法的信号和图像自适应分解问题,根据信号自适应的得到其低分辨率近似和重构滤波器,使得重构信号与原信号之间的误差最小。为提高所得近似图像的视觉质量,我们进一步将全变差模型引入自适应分解方法中,为对信号或图像进行自适应分解提供了一种新思路。
■基于理论研究的结果,进一步深入探讨了新型复数小波变换在图像去噪和增强中的应用,提出了三种基于DT-CWT的图像去噪新算法:(ⅰ)复数小波变换域利用系数尺度间和尺度内相关性的图像去噪算法;(ⅱ)基于局部参数的二元树复数小波域隐马尔可夫树(HMT,Hidden Markov Tree)模型图像去噪;(ⅲ)复数小波域高斯尺度混合(GSM,Gaussian Scale Mixture)模型去噪。这些方法充分利用了复数小波变换的优良性质及其系数分布的统计规律,实验表明,在简化计算复杂度、提高计算效率的同时获得了比现有相关去噪算法有显著改进的的去噪效果。另外,我们还提出了一种基于尺度间和尺度内相关性SURE方法的正交小波阈值去噪方法,解决了最近提出的正交小波域去噪算法对含较多纹理的图像处理效果不佳的缺陷,成为目前非冗余小波变换域效果最好的去噪算法。
■最后,我们还探讨了结合新型复数小波变换和最优视觉表示的统计特性的图像增强问题,提出了两种图像增强算法:(ⅰ)基于双密度二元树离散小波变换(DD-DT DWT,Double-Density Dual-Tree DWT)和视觉表示的图像增强算法,取得了非常好的视觉效果;(ⅱ)基于二元树复数小波和视觉表示的噪声图像增强算法,较好的缓解了带噪声图像增强中噪声抑制和细节保护之间的矛盾。
Efficient sparse representation and processing of unstable signal are the main contents in mathematics and information science. Recently, the Discrete Wavelet Transform (DWT) has become efficient in the sparse representation of unstable signal and also is a powerful tool for signal and image processing. It, however, has some disadvanges, including, (1) It is shift sensitive because the input signal shift generates unpredictable changes in DWT coefficients; (2) It suffers from poor directionality because DWT coefficients reveal merely three spatial orientations; (3) It lacks of the phase information that accurately dscribes non-stationary signal behavior; that undermine its usage in many applications. Therefore, there is a strong motivation to study new types of wavelet transforms with better shift invariance and directionality. Due to the imperfection of image acquisition systems and transmission channels, the observed images are often in low-quality or degraded by noise. The goal of image denoising is to remove the noise while retaining as much as possible the important features (edges) and obtain acceptable image for vision. The image enhancement algorithms are to process a given image so the results are better than original image for their applications or objectives. Noise elimination and image enhancement are still the most fundamental, widely studied, and largely unsolved problems in computer vision and image processing.
To overcome the disadvantages of the traditional DWT, this thesis mainly focus on two new types of complex wavelet transform: the dual-tree complex wavelet transform (DT-CWT) and the higher density dual tree DWT. The properties of the DT-CWT such as approximate shift-invariance, directionality and implementation issue are carefully investigigated. Furthermore, a new algorithm to construct wavelet filterbank of the DT-CWT is presented. At the same time, a new complex wavelet transform - the higher denstiy dual tree DWT is introduced and the corresponding characteristics are studies and a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed is developed. To better process the non-stational signal, the total variation and optimization based schemes for signal and image adaptive decomposition are preliminarily studied. Some classical applications of the proposed complex wavelet transforms are also further studies such as image denoising and enhancement. Results of experiments show that the proposed new algorithms perform better than the now existing methods.
The main achievements in this paper are as follow:
First, an approach for designing biorthogonal DT-CWT filters is proposed; where the two related wavelets pairs form approximate Hilbert transform pairs. Different from the existing design techniques, the two wavelet filterbanks obtained here are both of linear phases. By adjusting the parameters, wavelet fitlers with rational coefficients may be achieved, which can speed up the DT-CWT effectively.
Then, to overcome the disadavanges of the DWT, we introduce the higher-density dual-tree DWT, which is a DWT that combines the higher-density DWT and the DT-CWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. We develop a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractinal-delay allpass filters, spectral factorization and the solutions have vanishing moments, compact support, a high degree of smootheness, intermediate scales, approximate Hilbert transform properties, and are nearly shift-invariant.
In addition, we investigate the problem of adaptive deocompositon of the signal and image, the optimization method and total variation model are employed in the process. Experimental results show that the proposed methods are effective to a wide range of signals and images; when compared to the fixed wavelet bases method, the produced reconstruct images with our adaptive method are with better PSNR and visual quality.
Based on the theory above, to address the problems of the image denoising and enhancement, we investigate the image denosing and enhancement in the new types of complex wavelet transform domain in detail. Three new image denoising algorithms based on the DT-CWT are proposed: (i) a new locally adaptive image denoising method, which exploits the intra-scale and inter-scale depencencies in the DT-CWT domain; (ii) a new non-tranining compelx wavelet Hidden Markov Tree (CHMT) model, which is based on the DT-CWT and a fast parameters estimation technique; (iii) a new denoising algorithm based on the Gaussian scale mixture (GSM) of the coefficients of the DT-CWT. These methods exploit the properties of the DT-CWT and the statistics of the coefficients and the obtained better denoising performances while reducing the computational complexity. At the same time, we introduce an effective integration of the intrascale correlations within the interscale SURE based orthonormal wavelet thresholding, which can solve the problem of the interscale method that is not very effective for those images that have substantial high-frequency contents.
In addition, we also investigate the image enhancement based on the new types of wavelet transform and the statistical characters of visual representation and propose two new method of image enhancement: (i) a novel method for image enhancement, which exploits the properties of the double-density dual-tree DWT and the statistical characters of visual representation; (ii) a new method for noisy image enhancement, which is based on the GSM model of the DT-CWT coefficients and the combination of the DT-CWT and the statistical characters of visual representation, and can optimize the contrast of image features of while minimizing image denoise.
引文
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