【作者】 蔡超；

【导师】 丁明跃；

【作者基本信息】 华中科技大学 ， 模式识别与智能系统， 2005， 博士

【摘要】 图像科学是一门集多学科于一体的交叉学科,与相关学科(特别是数学学科)的基础理论在该学科的成功应用密不可分。在图像处理中,无论是图像模型的建立,图像特征的描述,图像处理算子的设计,还是图像优化处理中的泛函极小化,最终都可归结为一个数学理论问题。特别是近年来,以小波分析(Wavelet analysis)和偏微分方程(Partial derivative equation, PDE)为代表的数学工具活跃在图像处理的各个研究领域,“图像科学”正在形成,并逐步为人们所接受。该文旨在以小波分析和偏微分方程为主要工具,对底层图像处理中图像恢复、图像分割、边缘提取等问题展开研究,并探讨它们在图像处理中的联合应用问题。小波分析,作为一种新的数学分析工具,是泛函分析、傅立叶分析、样条分析、调和分析以及数值分析理论的完美结合,已被广泛地应用于计算机视觉、图像处理以及目标检测等领域,并在理论和方法上取得了重大进展。特别是在图像压缩方面,以Shaprio 为代表提出的嵌入式零树小波编码方法(Embedded zerotree wavelet,EZW)获得了巨大成功。以小波变换(Wavelet transforms)为理论基础的图像(视频)压缩算法已经在多个国际图像压缩标准中采用。在图像恢复、特征提取、目标识别等领域,小波分析同样发挥了巨大作用。如何更好地利用小波的特长,发展更有效的图像处理新方法仍然是图像处理领域的一个研究热点。作为论文的一部分,该文重点探讨了小波分析在数字图像处理中的应用问题,包括基于小波分析的磁共振图像恢复; 小波域图像平滑; 多小波理论及图像特征提取。在磁共振图像恢复方面,利用小波分析了磁共振图像直流伪影的产生机理,提出了一种基于小波分析的伪影消除办法; 在分析了小波域图像平滑现有方法基础上,提出了小波变换域双边滤波和基于相位一致性约束的多小波图像去噪新方法; 针对图像复合边缘特征提取问题,该论文进一步研究了基于图像复合边缘模型的多小波构造方法。提出了基于零系统定位误差约束的多小波图像边缘特征提取方法。从理论上这种新的边缘检测算子提取复合边缘可以达到任意高的定位精度。实验表明这种新方法对于复合边缘的检测无论从定位精度还是从检测能力上来看都优于Canny 算子、Prewitt 算子以及Mallat-Zhong 提出的单小波边缘检测算子。人们对偏微分方程的研究已经有近300 年的历史。早期的偏微分方程问题产生于力学、几何、物理等理论学科和实际工程中。近年来,在生命科学、经济学中也更多还原

【Abstract】 Image science is an inter-discipline subject and has a great relationship with the successful applications of multi-subject, especially mathematics. In image processing, no matter image modeling, representation of image contents, description of image feature, design of image processing operator, or energy functional minimization of image optimization procedure, all can be upgraded to a mathematic problem. Especially in recent years, as a representation of mathematic tools, wavelet analysis and partial differential equation (PDE) are active in many image-processing fields, making the name of “Image Science”gradually being accepted by people. The aim of this thesis is to study the applications of wavelet and partial differential equation in image processing, and try to set up their relationship both in application and theory。As a new mathematics tool, wavelet analysis is a perfect integration of functional analysis, Fourier analysis, spline analysis, harmonic analysis and numerical analysis. It has been applied in computer vision, image processing, object recognition and other fields and a great number of advances have been made both in theories and techniques. For instance, the Embedded Zertree wavelet, ETW, proposed by Shaprio etc. has been widely used in image compression and the video coding algorithm based on wavelet transform has been used in several international standards about image compression. It also plays an important role in image restoration, feature extraction, as well as object recognition. However, in other fields of image processing, there is still a big gap between people’s expectation about wavelet analysis and its applications. As an important part of this thesis, it focuses on the applications of wavelet transforms, including wavelet analysis based medical MRI image restoration, image smoothing in the wavelet domain, multi-wavelets and image features extraction. In the medical image processing application, based on the tool of wavelets, we analyzed the causes producing DC artifacts, and the suppression methods were given; after introducing several traditional denoising methods, a new bilateral filter in wavelet domain was proposed; based on the constraint of phase congruency, a multi-wavelet based image denoising method was put forward; for the composite edge detection, we first constructed a new multi-wavelet function, and then, with this function, a new composite edge detector was proposed. Theoretically, this detector can reach any detection accuracy for composite edges. Experiments demonstrated that our method had a higher accuracy for the detection of composite edges than Canny’s operator, Prewitt’s operator and the single wavelet edge operator proposed by Mallet-Zhong. It has been more than 300 years since people first did the research on partial differential equation (PDE). The earliest partial differential equation problem was introduced in mechanics, geometry and physics. In recent years, many partial differential equation problems are also appeared in life science and economics. How to apply partial differential equation in image processing is another important part of the thesis. One advantage using partial differential equation in image processing is there are many numerical computational methods available. Experiments demonstrated that partial differential equation plays an important role in the development of image science. For example, the total variation model can represent a large number of image types; the thermal diffusion equation can simulate image degeneration phenomena, and the reverse equation can be successfully used in image restoration; level set can track the evolving process of wave front and used in image segmentation and target recognition. This thesis addressed image smoothing and segmentation, proposed a new morphological erosion operator, and with this operator, a new edge detector was proposed. Based on the eroding results of gradient data, a new anisotropic diffusion scheme was designed. To solve the parameter selection problem in C-V image segmentation, a new level-set based segmentation method was proposed, which can segment image automatically while its parameter selection is quite easy. The combination of PDE and wavelet analysis will provide a solid base for the establishment of image science. Currently their theoretical relationship is very weak. Whether it can play the same role as Fourier analysis did is unknown. This thesis discussed the correlation between wavelet analysis and PDE in image processing based on the researches on them. As a brief combination of them, a new multi-resolution analysis based level set image segmentation scheme was proposed. Furthermore, wavelet based numerical homogenization and the heatlet decomposing of linear PDE were studied. Finally, the summary and further research directions were given.更多还原