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霍金枝等16名同志被授予“佳木斯市青年五四奖章”荣誉称号，卢丙海等10名同志被授予第九届“佳木斯市十大杰出青年”荣誉称号，周俊龙等7名同志被授予“佳木斯市十大杰出青年”提名奖。赵春晖
论文题目：数字形态滤波器理论及其算法研究
作者简介：赵春晖，男，1965年生，1994年师从哈尔滨工业大学孙圣和教授，于1998年获博士学位。
摘 要
本课题是国家自然科学基金资助和国际合作项目。数字形态滤波器是一种非常重要的非线性滤波器。它在图象分析与处理、计算机视觉和模式识别等领域获得了广泛的应用，是目前非线性数字信号处理学科研究的热点课题。
随着现代数字信号处理技术的发展，非线性数字信号处理方法在信号处理领域中的地位和作用显得越来越重要，因为从自然现象和社会现象中涌现出来的大量信号处理问题是非线性的。线性数字信号处理方法虽然在理论上比较成熟，且实现相对简单，但它对非线性问题的处理结果在大多数情况下是不十分理想的。近二十年来，非线性数字信号处理技术已取得了长足进展，其中包括对非线性数字滤波器的研究。
噪声信号(图象)的滤波是信号(图象)处理的基本任务之一。过去这一任务主要由线性滤波器来完成，但线性滤波器不能有效地抑制各种非加性高斯噪声，且不利于信号边缘等细节特征的保持；因而，近年来的噪声信号恢复问题主要采用非线性滤波器来处理。在诸多种类的非线性滤波器中，形态滤波器是最具代表性和很有发展前途的一种滤波器，因为它是以数学形态学为理论基础，具有并行快速实现的特点，一直受到了国内外学者的普遍关注和广泛研究。形态滤波器是从数学形态学中发展出来的一类新型非线性滤波器。形态滤波理论是由G. Matheron 和J. Serra等人在八十年代初创立的。形态滤波器是基于信号(图象)的几何结构特性，利用预先定义的结构元素(相当于滤波窗)对信号进行匹配或局部修正，以达到提取信号，抑制噪声的目的。
形态滤波器是伴随着数学形态学的发展而发展的，它由最早的二值形态滤波器发展为后来的多值形态滤波器。多值形态滤波器与排序统计滤波器有着密切的联系，它们本质上是层迭滤波器的特殊情况。当采用平结构元素时，多值的膨胀和腐蚀变换就演变为极大和极小滤波。极大滤波器通常能有效地滤除图象中的负脉冲噪声，而极小滤波器可以滤除正脉冲噪声，但两者均对混合型脉冲噪声失效。如果采用两者的各种级联组合，则可达到较全面的脉冲噪声抑制性能。
由于非线性滤波器理论和算法的复杂性、多样性，形态滤波器至今尚未形成系统的设计方法。现有的算法大多是针对某一实际需要提出来的，缺乏深入的理论分析，且应用也存在着局限性。这一领域的研究还很不深入和完善，还有大量工作需要来完成。我国对形态滤波理论和算法的研究起步较晚，其研究水平也较为落后。为了跟踪该领域国际前沿，发展我国非线性数字信号处理技术，满足航天、国防和国民经济对该技术的需求，进一步开展形态滤波理论与应用技术研究是非常必要而有实际意义的。
形态滤波器的性能取决于结构元素和形态变换的类型；如何合理地选择它们，构造性能良好的快速算法，并对其进行较深入的理论分析，是该研究领域中的一个难题。本文从研究形态滤波器的基本理论入手，围绕着结构元素的选取和形态变换的组合这一主题，通过采用串行或并行、线性加权组合和自适应处理等方法，系统地研究了几种形态滤波器的原理、结构和算法。本文的主要研究内容和取得的成果包括如下几个方面：
1. 系统而较全面地总结了数字形态滤波器的基本理论。 基于信号状态模型法和层迭滤波描述法，本文重点研究了传统形态滤波器(包括形态开滤波器、闭滤波器、开—闭滤波器和闭—开滤波器)的根信号特性和输出统计特性，指出了上述各类滤波器根信号间的关系，并阐明了传统形态滤波器的输出存在着严重的输出统计偏倚现象，这是影响它们噪声滤波效果的一个直接原因。另外，通过传统形态滤波器的并行和串行组合，本文将形态滤波方法成功地应用于含噪声心电信号波形的恢复和二维图象物体的提取。
2. 为了减小传统形态开—闭和闭—开滤波器输出统计偏倚，本文采用两个不同尺寸结构元素，提出了一类新型的滤波器—广义形态开—闭(GOC)和闭—开(GCO)滤波器，并证明了这类滤波器满足形态变换的四个基本性质(平移不变性、单调性、对偶性和幂等性)。为了更好地了解它们滤波过程，本文基于上述的信号状态模型法，分析了广义形态滤波器的根信号特性。借助于正布尔函数(PBF)的描述，将广义形态滤波器表示成一种层迭滤波器。通过层迭滤波器的输出统计特性，我们推导出了广义开—闭和闭—开滤波器在一维凸结构元素情况下的输出函数解析表达式，并在三种常见输入噪声分布(均匀、高斯和双指数)条件下，分析了这类滤波器的输出统计规律，同时计算了它们的数字特征(均值和方差)。另外，基于广义开—闭和闭—开滤波器，利用自适应方法，本文提出了一种自适应加权组合广义形态滤波器，并对其进行了一维和二维信号仿真验证，取得了令人满意的结果。
3. 基于多模板匹配方法，本文将线性多结构元素引入到广义开—闭和闭—开滤波器中，定义了一类多结构元素并行复合广义形态滤波器。并利用有约束的最小均方误差算法(CLMS)，研究了一种多结构元素自适应加权平均广义形态滤波算法。仿真结果验证了上述滤波器算法的有效性。
4. 基于全方位结构元素的概念，本文定义了极大开和极小闭运算，在此基础上，通过它们的不同顺序级联组合，构造出了一类全方位多级组合滤波器。最后，采用形态变换的加权平均运算，提出了一种全方位多级加权组合形态滤波方法。并进行了计算机仿真实验，结果表明这种方法在噪声抑制和信号几何特征保持方面有较好的性能。
5. 研究了顺序形态滤波的优化问题。首先分析了顺序形态滤波器的输出统计特性，指出了百分位值和结构元素对滤波性能的影响。针对固定参数滤波器应用的局限性，本文采用自适应LMS算法，在均方误差(MSE)准则和平均绝对误差(MAE)准则下，实现了百分位值和结构元素的自适应处理。这类滤波器可在含噪声(包括混合脉冲噪声)阶跃变化信号的滤波场合获得重要应用。
本论文不仅在理论上有较大的突破，发表了二十多篇有学术价值的论文，且有多篇论文被国际权威检索和刊物收录，具有重要的学术意义；而且其研究成果已在实际工程项目中获得了重要应用，取得了显著的经济效益。
本课题的研究成果对丰富非线性数字信号处理知识宝库有重大的学术价值，对研究和开发其它类型的非线性滤波器具有重要的指导意义和参考价值，特别对图象处理、模式识别和计算机视觉等学科的发展产生积极的影响。其成果将在航空、航天遥感，图象匹配末制导，人工智能，机器人视觉，生物医学、地震和声纳信号处理等领域有广泛的应用前景。
关键词：数学形态学，形态滤波器，结构元素，非线性滤波，图象处理。
RESEARCH ON DIGITAL MORPHOLOGICAL FILTER THEORY AND ITS ALGORITHMS
The Dissertation of Harbin Institute of Technology
Author: Zhao Chunhui
Abstract
This subject is supported by national science foundation and an international collaboration project. Digital morphological filter is an important nonlinear filter. It has found wide applications in many research fields, such as image analysis and processing, computer vision and mode recognition. At present, it is a hot subject of nonlinear research in digital signal processing.
With the developments of modern digital signal processing techniques, the methods of nonlinear digital signal processing are increasingly important in the field of signal processing because of a majority of nonlinear problems existing in natural and social phenomena. Although the methods of linear digital signal processing are mature in principle and they are easily implemented, the processing results to nonlinear problems are not very prefect in most cases. In recent 20 years, the techniques of nonlinear digital signal processing have made great progress such as researching on nonlinear digital filters.
The filtering of noisy signals (or images) is one of basic signal processing tasks. In the past, the task is mainly finished by linear filters. But linear filters cannot suppress various Gauss white noises and preserve fine-details features such as the edges of signals effectively. Thus, the restoration problems of noisy signals are solved by nonlinear filters in recent years. Morphological filter is a representative and hopeful nonlinear filter in all kinds of them. Because it is based on Mathematical Morphology (MM), and it can be realized in parallel. It has been noticed and researched widely by some scholars from home or oversea. Morphological filter is a new type of nonlinear filter stemming from Mathematical Morphology. Morphological filtering theory is founded by G. Matheron and J. Serra in the early eighties. It is based on the geometrical-structural features of signals (or images). In order to achieve the aims of collecting signals and suppressing noises, Morphological filter is matched with or modifies locally the signals by the structuring elements (filtering window) defined in advance.
Morphological filter is developing with Mathematical Morphology. It is evaluated from binary one to grayscale one. The grayscale morphological filter has close connection with order statistic filter. Both of them are one of stack filters in essence. When adopted flat structuring elements, grayscale dilation and erosion evolve to maximum and minimum filtering. Usually the maximum filter can remove negative impulsive noise and the minimum filter can remove positive impulsive noise effectively. However, both of them invalidate to mixed impulsive noise. If using the all serial combinations of them, we can obtain the capability to suppress all kinds of impulsive noises.
There is not a systematic method for morphological filter designing because of the complexion and diversity of nonlinear filter theory and algorithms. Most of existing algorithms are proposed in accordance with a certain practical requirement, lacking penetrating theoretical analysis and having limitation in their applications. The research is not deeper yet and there are a lot of work to do in this field. Our country initiate late in researching on morphological filter theories and it algorithms. In order to trace with international forward position and develop our country’ techniques of nonlinear digital signal processing, it is very necessary and realistic that we research further on morphological filter theory and practical techniques to meet the needs of spaceflight, national defense and economy.
The performances of morphological filters depend on the types of their structural elements and morphological transforms. The reasonable choice of morphological filters, the constructing of their good-functioned fast algorithms and deep theoretical analyzing for them are still difficult problems. Starting from the investigation of the fundamental theory of morphological filters and concentrating on the choice of structuring elements and the combinations of morphological transforms, this dissertation systematically researches the principles, structures and algorithms of morphological filters by using the methods of serial/parallel processing, linear weighted combination and adaptive processing. The main contents and contributions of this dissertation are as follows:
1. The fundamental theory of digital morphological filters is systematically and completely summarized in this dissertation. On the basis of the methods of signal state modeling and stack filter description, this dissertation researches the root signal characteristics and output statistical properties of traditional morphological filters (including opening, closing, open-closing and clos-opening), and illustrates the relationship between various root signals of above filters and points out that the phenomena of statistical biasing existing in the outputs of traditional morphological filters is the direct reason for their noise-removing efficiencies. In addition, the morphological filtering methods have successfully applied in the waveform restoration of noisy ECG signal and the extraction of geometrical shapes of objects in two-dimensional images.
2. In order to reduce the statistical bias in the output of traditional morphological open-closing and clos-opening filters, this dissertation presents a new class of generalized open-closing (GOC) and clos-opening (GCO) morphological filters by using two different sized of structuring elements and proves that this class of filters possess the four fundamental properties (translation invariance, monotonically, duality and idempotence). In order to understand their filtering processes well, this dissertation analyzes the root signal characteristics of generalized morphological filters based on above method of signal state modeling. With the aid of positive Boolean function(PBF) descriptions, the generalized morphological filters are expressed as a kind of stack filters. The analytic expressions of output functions of GOC and GCO filters with one-dimensional convex structuring elements are derived form the statistical properties of stack filters. We analyze the statistical regularities of this kinds of filters and calculate the numerical features(means and variances). In addition, an adaptive weighted combination generalized morphological filter is proposed on the basis of GOC/GCO filters and adaptive processing method. The simulation verifications of one-dimensional and two-dimensional signals give satisfying results.
3. On the basis of multitemplate-matching method, this dissertation introduces linear multiple structuring elements to GOC and GCO filters, and gives the definitions of a parallel-complex generalized morphological filter. Moreover, an adaptive weighted averaging generalized algorithm is investigated by using the constrained least-mean-squared(CLMS)error method. Simulation results have shown that filtering algorithm above is efficient.
4. In this dissertation, we define a class of maximum-opening and minimum-closing operations based on the omnidirectional structuring elements, and further construct a class of omnidrectional multistages combination morphological filters by their difficult orders cascading. Finally, an omnidrectional multistages weighted combination morphological filtering algorithm is presented by means of weighted averaging operations of morphological transformations. The computer simulation results show that this algorithm has an excellent performance on noise-suppressing and geometrical features-preserving.
5. The optimizing problems of ranked-order morphological filtering are investigated. We point out that structuring elements and percentiles have the influences on the filtering results. According to the limitation of the filters with fixed parameters, Under the mean-square-error (MSE) and mean-absolute-error (MAE) criteria, adaptive processing of percentiles and structuring elements in ranked-order morphological filtering are implemented by using the adaptive LMS algorithm in this dissertation. These filters have the important applications in the cases of the filtering to noisy step-variance signals.
The results obtained in this dissertation have published more than 20 papers with academic values. Furthermore, many papers have been recorded by the authority international indexes and abstracts. They have the important academic meanings. This dissertation not only has made major progress in principle, but also has the important applications in the practical engineering projects. There are many economical benefits to be obtained remarkably.
The achievements of the dissertation have the significant academic values to enrich the thesaurus of nonlinear digital signal processing and have the instructional meanings to develop the other nonlinear filters. Especially, they have the active influences on accelerating the subjects of image processing, computer vision and mode recognition. They have the wide applications in many fields such as aviation and space remote sensing, image mated guide, artificial intelligence, robot vision, biologic medicine, seism and sonar signal processing.
Key words: Mathematical morphology, morphological filters, structuring elements, nonlinear filtering, image processing
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