摘要
边缘、纹理等几何结构是图像的重要特征。传统的小波变换只能表达点奇异的位置和特性,不能最优地表示图像中的线奇异特征；同时,二维可分离小波变换只有有限个方向,无法有效地捕捉图像的轮廓信息。为克服小波变换的上述局限性,近年逐步发展起来一种多尺度几何分析理论。该理论是一系列分析方法的总称,其目的是设计高维函数的最优表示方法,捕捉图像内在的几何结构如轮廓、边缘和纹理等方向信息,从而更有效地处理图像数据。其中,轮廓波变换(Contourlet变换及其非下采样形式)直接定义在离散域,用类似于线段的基结构逼近图像,是一种真正意义上的二维图像表示方法。轮廓波变换不仅具有多尺度和良好的时(空)频局部特性,还具有多方向和各向异性,能够有效捕捉图像中的高维奇异性,因而在图像处理领域具有广阔的应用前景。
图像的轮廓波系数在尺度间、方向间以及子带的空间邻域内都表现出了较强的相关性。而现有轮廓波系数的统计模型都是由小波统计模型简单推广而来的,没有考虑轮廓波变换与小波变换在各向异性和多方向选择性上的差异。因此,为了充分发挥轮廓波变换的方向捕捉能力,非常有必要在多尺度统计模型的基础上融合图像的几何结构信息,建立具有方向特征的轮廓波系数统计模型。本文以轮廓波系数的统计建模为主要研究问题,通过分析轮廓波系数的内在关联,对构建具有方向特征的轮廓波系数统计模型及其在图像去噪、图像分割中的应用进行了深入、系统的研究,提出了一些有效的理论和方法,可以进一步推广到图像融合、图像重建等其它图像处理领域,具有重要的理论意义和广泛的应用前景。
本文的主要工作和创新点概括如下：
1.从图像的稀疏表示特性和人类视觉系统的反应特点入手,分析了小波变换的局限性,研究并给出了轮廓波变换能够高效表示和处理图像信息的原因,并对当今国内外基于轮廓波变换的图像处理的研究进展进行了总结。
2.研究了图像的轮廓波表示法。
阐述了Contourlet变换的基本原理、滤波器构成和基本性质,利用基函数的形状及非线性逼近效率讨论了Contourlet变换的稀疏性,并用泛函的知识对Contourlet变换所具有的多尺度、多方向框架进行了理论分析；为探讨轮廓波变换的平移不变性,又进一步研究了非下采样Contourlet变换(Nonsubsampled Contourlet Transform,NSCT),讨论了其非下采样滤波器的设计方法。
3.分析了图像轮廓波系数的统计特性,提出了一种基于局部上下文隐马尔可夫模型(Local Contextual Hidden Markov Model,LCHMM)的轮廓波系数统计模型。
根据轮廓波子带具有的非高斯分布统计特性,分别使用广义高斯模型和高斯混合模型来建模子带系数的边缘分布。通过联合分布直方图,定性分析了轮廓波系数分布在尺度间、方向间和空间邻域内的相关性。针对现有的轮廓波系数统计模型无法全面描述上述三种系数相关性以及不符合图像非平稳特征的缺陷,提出了一种具有局部特征的、可全面捕捉上述三种相关性的统计模型——LCHMM模型。LCHMM模型首先使用基于局部数据的高斯混合场来建模轮廓波系数的非高斯分布,与子带同分布的假设相比,更加符合图像的非平稳特征；然后在高斯混合场的基础上,利用上下文来综合相邻尺度间、同一尺度相邻方向间和同一尺度同一方向子带的空间邻域内的系数相关性,结合隐马尔可夫模型,构建了一个全面的相关模型框架,充分表达了轮廓波变换的尺度间持续性、尺度内多方向选择性和邻域内的能量聚集特性,为图像多尺度几何变换域统计建模提供了新的理论思路。本文详细阐述了LCHMM模型框架的数学定义,设计了上述三种相关信息的融合方法。此外,通过合理的设置初始化参数集合,利用迭代最大期望算法,给出了鲁棒、有效的模型训练方法,保证了模型的可行性。
4.提出了一种基于椭圆方向窗信号估计的NSCT域空间自适应Bayes阈值去噪方法。
通过对图像信号在NSCT子带内各向异性的能量聚集的分析,得出NSCT系数在空间邻域内的相关性不具有各向同性,而是满足各向异性和特定方向选择性的结论。为此,抛弃传统的方形窗口,专门设计了能够自适应于各子带分解方向的椭圆方向窗来捕捉存在较强依赖关系的邻域相关系数进行信号方差的估计,结合局部Bayes阈值,提出了一种基于椭圆方向窗信号估计的NSCT域空间自适应Bayes去噪阈值。该阈值不仅自适应于尺度、方向和空间位置,而且能够自动调整椭圆邻域方向以自适应于子带能量的聚集方向,是一个自适应程度更高的Bayes阈值。本文详细阐述了椭圆方向窗的定义、基于椭圆方向窗的信号估计方法和噪声方差的估计方法,给出了基于该阈值的去噪算法的具体步骤,并进行了仿真实验。实验结果表明,与当前一些典型的轮廓波阈值去噪算法相比,该算法的峰值信噪比(Peak Signal-to-noise Ratio,PSNR)和去噪视觉效果都有明显的提高和改善。
5.提出了一种基于NSCT域LCHMM模型的图像去噪方法,适用于去除强噪声。
图像信号经NSCT分解后,其系数在相邻尺度、相邻方向和空间邻域内都有较强的相关性；而高斯白噪声经NSCT变换后仍为白噪声；利用信、噪这种统计相关性上的差异可以实现对图像的去噪处理,尤其适用于去除强噪声。为充分利用NSCT系数的三种相关信息,以信息论中的互信息作为量化工具,确定了LCHMM模型中NSCT系数三种相关性信息具体的融合方案；结合去噪应用背景,制定了融合后相关信息与上下文数据的映射关系；利用LCHMM模型捕捉的相关性先验知识,提出了一种基于NSCT域LCHMM模型的图像去噪算法。本文详细阐述了LCHMM模型中相关信息的融合设计、上下文与相关信息的度量准则以及基于该模型的具体的去噪步骤。通过仿真实验分析,与现有的基于相关性模型的去噪算法相比,该算法显著提高了去噪图像的PSNR值；在视觉效果上,既可以改善平坦区域的伪吉布斯现象,又可以有效地保持边缘细节和纹理信息。
6.提出了一种基于Contourlet域隐马尔可夫树(Hidden Markov Tree,HMT)模型和改进上下文结构的图像分割技术。
研究了多尺度贝叶斯图像分割的基本原理,分析了基于小波HMT模型的图像分割存在边缘模糊和奇异性扩散现象的原因。一方面,小波变换不能有效的表达边缘等高维特征；另一方面,该方法使用的上下文仅考虑了较粗尺度上的类标相关性,虽然保证了分割主体轮廓的可靠性,但是缺乏尺度内相邻节点类标的依赖关系,使得边缘部位的分割不够平滑和准确。为此,本文提出了一种基于Contourlet域HMT模型和改进上下文结构的图像分割技术。该分割技术首先利用Contourlet域HMT模型得到与图像各个尺度上数据块相对应的不同纹理的相似度,通过相似度大小的比较获得各个尺度上的初始分割；在多尺度贝叶斯分割的基础上,设计了一种新的上下文结构来捕捉尺度间和尺度内相邻节点的类标相关性,通过简单投票推举,为相邻尺度间的初始分割融合提供先验知识：利用该先验知识逐步由粗尺度到细尺度融合相邻两尺度的初始分割结果,最终得到像素级的分割结果。由于新设计的上下文综合考虑了尺度间和尺度内初始分割的依赖关系,因而,在获取可靠的主体轮廓的基础上,更为有效地保留了局部边缘细节,提高了边缘分割的准确性。本文详细阐述了上下文的定义,给出了具体的算法实现。仿真实验表明,与小波HMT分割技术相比较,本文算法有效减少了均质区域的孤立误分类现象,提高了不同纹理区域边缘的分割精度,获得了很好的分割效果。
Geometrical structures such as edges and textures are key features in natural images. The traditional wavelet transform only represents point-singularities efficiently, but is less efficient for line-singularities and curve-singularities that always exist in images. Moreover, the separable two-dimensional wavelet cannot capture contour information of images accurately due to the very limited directionality. These limitations of wavelet have led to the development of multiscale geometric analysis (MGA) theory, which includes a serial of MGA tools. The goal of MGA theory is to find the optimal representation for multidimensional signals to capture and process the intrinsic geometrical structures such as contours, edges and textures of images efficiently. Among these tools, contourlet transform and the nonsubsampled contourlet transform (NSCT) distinguish themselves with their excellent efficiencies and flexible structures. In this thesis, both contourlet transform and NSCT are referred as contourlets transforms. The contourlets transforms are directly defined in discrete-domain and use rectangular-shapes for their basis elements to follow the contours, thus forming "true" two-dimensional sparse representations. Contourlets transforms not only possess the features of multiscale and time-frequency localization, but also offer a high degree of directionality and anisotropy, which have proven to be very promising in various image processing applications.
The contourlets coefficients of images show strong dependencies across scales, directions and locations (or space). However, the current statistical models of contourlets coefficients are mostly simple extension of those of wavelet coefficients, disregarding the difference in multi-directionality and anisotropy. To take advantage of captured directional information of contourlets transforms, exploring how to integrate the geometrical information into the multiscale statistical model to statistically characterize contourlets coefficients is greatly in need. Therefore, the thesis focuses on the contourlets-based statistical modeling for images. By investigating the statistical correlation of contourlets coefficients, the thesis studies the directional multiscale statistical modeling based on contourlets transforms and develops some new denosing and segmentation method based on the proposed statistical model. In addition, the proposed statistical model can be further extended to image fusion, image reconstruction and other image processing areas, which provides important theoretical significance and wide application prospects.
The main contributions of the thesis are as follows:
1. The limitation of wavelet transform is first discussed from the views of image sparse representation and receptive characteristics of human visual system. Then the capability of contourlets transforms in capturing the directional information of the natural images is analyzed. Moreover, the current progresses of image processing based on the contourlets transforms are summarized.
2. The contourlets representations for images are studied.
After reviewing the contourlet transform theory, including its basic principle, filter bank structure and basic characteristics, the thesis discusses the sparsity of contourlet transform by analyzing the shapes of basis functions and their nonlinear approximation rate, and sets up the directional multiresolution frame by functional analysis. To discuss the shift-invariant property, the NSCT transform especially the design of the nonsubsampled filter is studied.
3. The statistics of the contourlets coefficients is discussed in detail and described by a newly local contextual hidden Markov model (LCHMM).
The thesis adopts the generalized Gaussian distributed (GGD) model and Gaussian mixture model (GMM) respectively to approximate the highly non-Gaussian marginal distributions of contourlets subbands. By joint distribution histogram, the strong interlocation, interscale and interdirection dependencies of contourlets coefficients are analyzed qualitatively. However, the current statistical models of contourlets coefficients do not consider all three dependencies. Moreover, those methods cannot fit the nonstationary properties of images due to the lack of spatial adaptability. To overcome these problems, a new LCHMM framework that statistically characterizes contourlets coefficients is proposed. In the LCHMM model, the Gaussian mixture field (GMF) where each coefficient follows a local Gaussian mixture distribution determined by their neighborhoods is introduced to approximate the non-Gaussian density. Compared to the GMM, the GMF is more suitable for the non-stationary properties of images. To integrate all interscale, interdirection and interlocation dependencies of these corresponding coefficients, a context variable is associated with each coefficient. Based on the GMF, the HMM model combined by context, which called LCHMM model, may describe the complex dependencies between contourlets coefficients effectively and completely. The LCHMM model, which reveals the properties of persistence across scales, multi-directional selectivity within scales and energy clustering in the subbands of contourlets transforms to the full, provides a new theoretical idea for the MGA-domain statistical modeling. The thesis gives the mathematical description of LCHMM model in detail and develops the fusion method of three dependencies. To obtain a LCHMM from an image, typically a training procedure based on the iterative expectation maximization algorithm is utilized. The robustness of training process is achieved by providing a good initial parameter setting.
4. A spatially adaptive BayesShrink thresholding algorithm with elliptic directional windows'estimation in the NSCT domain is proposed for image denoising.
The anisotropic energy clusters in NSCT subbands reveal that the distribution of correlated coefficients in the neighborhood is not isotropic, but exhibits anisotropic and specific directional features. Therefore, instead of the traditional square windows, the elliptic directional windows, which match the direction of the energy clusters' distribution in each subband, are used to select the neighboring coefficients with strong dependency for signal estimation. Reasonably, based on Bayesian estiamtion, the spatially adaptive BayesShrink thresholding algorithm with elliptic directional windows' estimation in the NSCT domain is proposed for image denoising. The proposed threshold that explores the anisotropic and directional features of NSCT deeply obtains the adaptability in anisotropic neighborhood as well as in scale and direction. The thesis discusses the definition of elliptic directional windows, the signal estimation method based on elliptic directional windows and noise variance estimation in detail. Experimental results show that, compared with some current outstanding denoising algorithms based on the contourelts transforms, the proposed algorithm gains the better denoising performance in both the value of peak signal-to-noise ratio (PSNR) and the visual quality.
5. An image denoising algorithm based on the proposed LCHMM model in the NSCT domain is developed, which is particularly used to remove the noise with high level.
The NSCT coefficients of an image display strong dependencies not only across scales, but also across directions and space. By contrast, the NSCT transform of zero-mean white Gaussian noise is still zero-mean white Gaussian noise of the same power. Thus, the statistical correlation difference between NSCT coefficients of signal and those of noise can be used to remove noise, especially for the noise with high level. To take full use of three correlation information, the weight combination for integrating three dependencies of NSCT coefficients is studied by using mutual information as a measuring tool. Then the mapping function from the correlation information to context variable is determined for the denoising application. Based on the prior information captured by LCHMM model, the denoised image is obtained by the Bayesian estimation. The thesis presents the fusion method of correlation information, the context construction method and the denoising steps based on the LCHMM model in detail. Simulation experiments show that the proposed algorithm outperforms the denoising algorithms based on both the wavelet-domain contextual hidden Markov model and the contourlet-domain hidden Markov tree model (HMT) in terms of PSNR value and visual appearance. The proposed algorithm reduces the Gibbs artifacts obviously and preserves edges and texture information of original images effectively.
6. A new texture image segmentation algorithm based on the contourlet-domain HMT model with a modified context scheme is proposed.
The thesis studies the multiscale image segmentation based on Bayesian criterion, especially the image segmentation based on the wavelet-domain HMT. However, the blurry edges and singular diffusion often occur in wavelet-domain HMT segmentation results. By analyzed, the reason lies in that (1) the wavelet can not represents contours and edges efficiently; and (2) the context model mainly describe interscale dependencies and encourage the formation of large uniformly classified regions with less consideration on directional and texture characters. To solve these problems, the context scheme is modified for the contourlet-domain HMT and applied to the texture image segmentation. Firstly, the likelihoods of all dyadic squares of the image given the different texture classes are obtained during the model training of contourlet-domain HMT. According to the maximum likelihood classification, direct block-by-block comparison of the likelihoods yields the multiscale raw segmentations. Then, to capture the interscale and neighboring dependencies of class labels, the new contextual design is proposed by voting the correlated class labels. Using the contextual prior, the interscale fusions of multiscale raw segmentations proceed in a multiscale, coarse-to-fine manner and achieve the final pixel-level segmentation. Since the proposed contextual scheme strengthens the boundary dependency, it can improve the accuracy of segmentation around boundaries, as well as provides the reliable outline segmentations. The thesis discusses the context construction and gives the detail segmentation steps. Experiments show that the proposed algorithm outperforms the wavelet-domain HMT segmentation and produces an accurate segmentation of texture images,by improving misclassification and boundary localization.
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