霍金枝等16名同誌被授予“佳木斯市青年五四獎章”榮譽稱號，盧丙海等10名同誌被授予第九屆“佳木斯市十大傑出青年”榮譽稱號，周俊龍等7名同誌被授予“佳木斯市十大傑出青年”提名獎。趙春暉
論文題目：數字形態濾波器理論及其算法研究
作者簡介：趙春暉，男，1965年生，1994年師從哈爾濱工業大學孫聖和教授，於1998年獲博士學位。
摘 要
本課題是國家自然科學基金資助和國際合作項目。數字形態濾波器是壹種非常重要的非線性濾波器。它在圖象分析與處理、計算機視覺和模式識別等領域獲得了廣泛的應用，是目前非線性數字信號處理學科研究的熱點課題。
隨著現代數字信號處理技術的發展，非線性數字信號處理方法在信號處理領域中的地位和作用顯得越來越重要，因為從自然現象和社會現象中湧現出來的大量信號處理問題是非線性的。線性數字信號處理方法雖然在理論上比較成熟，且實現相對簡單，但它對非線性問題的處理結果在大多數情況下是不十分理想的。近二十年來，非線性數字信號處理技術已取得了長足進展，其中包括對非線性數字濾波器的研究。
噪聲信號(圖象)的濾波是信號(圖象)處理的基本任務之壹。過去這壹任務主要由線性濾波器來完成，但線性濾波器不能有效地抑制各種非加性高斯噪聲，且不利於信號邊緣等細節特征的保持；因而，近年來的噪聲信號恢復問題主要采用非線性濾波器來處理。在諸多種類的非線性濾波器中，形態濾波器是最具代表性和很有發展前途的壹種濾波器，因為它是以數學形態學為理論基礎，具有並行快速實現的特點，壹直受到了國內外學者的普遍關註和廣泛研究。形態濾波器是從數學形態學中發展出來的壹類新型非線性濾波器。形態濾波理論是由G. Matheron 和J. Serra等人在八十年代初創立的。形態濾波器是基於信號(圖象)的幾何結構特性，利用預先定義的結構元素(相當於濾波窗)對信號進行匹配或局部修正，以達到提取信號，抑制噪聲的目的。
形態濾波器是伴隨著數學形態學的發展而發展的，它由最早的二值形態濾波器發展為後來的多值形態濾波器。多值形態濾波器與排序統計濾波器有著密切的聯系，它們本質上是層叠濾波器的特殊情況。當采用平結構元素時，多值的膨脹和腐蝕變換就演變為極大和極小濾波。極大濾波器通常能有效地濾除圖象中的負脈沖噪聲，而極小濾波器可以濾除正脈沖噪聲，但兩者均對混合型脈沖噪聲失效。如果采用兩者的各種級聯組合，則可達到較全面的脈沖噪聲抑制性能。
由於非線性濾波器理論和算法的復雜性、多樣性，形態濾波器至今尚未形成系統的設計方法。現有的算法大多是針對某壹實際需要提出來的，缺乏深入的理論分析，且應用也存在著局限性。這壹領域的研究還很不深入和完善，還有大量工作需要來完成。我國對形態濾波理論和算法的研究起步較晚，其研究水平也較為落後。為了跟蹤該領域國際前沿，發展我國非線性數字信號處理技術，滿足航天、國防和國民經濟對該技術的需求，進壹步開展形態濾波理論與應用技術研究是非常必要而有實際意義的。
形態濾波器的性能取決於結構元素和形態變換的類型；如何合理地選擇它們，構造性能良好的快速算法，並對其進行較深入的理論分析，是該研究領域中的壹個難題。本文從研究形態濾波器的基本理論入手，圍繞著結構元素的選取和形態變換的組合這壹主題，通過采用串行或並行、線性加權組合和自適應處理等方法，系統地研究了幾種形態濾波器的原理、結構和算法。本文的主要研究內容和取得的成果包括如下幾個方面：
1. 系統而較全面地總結了數字形態濾波器的基本理論。 基於信號狀態模型法和層叠濾波描述法，本文重點研究了傳統形態濾波器(包括形態開濾波器、閉濾波器、開—閉濾波器和閉—開濾波器)的根信號特性和輸出統計特性，指出了上述各類濾波器根信號間的關系，並闡明了傳統形態濾波器的輸出存在著嚴重的輸出統計偏倚現象，這是影響它們噪聲濾波效果的壹個直接原因。另外，通過傳統形態濾波器的並行和串行組合，本文將形態濾波方法成功地應用於含噪聲心電信號波形的恢復和二維圖象物體的提取。
2. 為了減小傳統形態開—閉和閉—開濾波器輸出統計偏倚，本文采用兩個不同尺寸結構元素，提出了壹類新型的濾波器—廣義形態開—閉(GOC)和閉—開(GCO)濾波器，並證明了這類濾波器滿足形態變換的四個基本性質(平移不變性、單調性、對偶性和冪等性)。為了更好地了解它們濾波過程，本文基於上述的信號狀態模型法，分析了廣義形態濾波器的根信號特性。借助於正布爾函數(PBF)的描述，將廣義形態濾波器表示成壹種層叠濾波器。通過層叠濾波器的輸出統計特性，我們推導出了廣義開—閉和閉—開濾波器在壹維凸結構元素情況下的輸出函數解析表達式，並在三種常見輸入噪聲分布(均勻、高斯和雙指數)條件下，分析了這類濾波器的輸出統計規律，同時計算了它們的數字特征(均值和方差)。另外，基於廣義開—閉和閉—開濾波器，利用自適應方法，本文提出了壹種自適應加權組合廣義形態濾波器，並對其進行了壹維和二維信號仿真驗證，取得了令人滿意的結果。
3. 基於多模板匹配方法，本文將線性多結構元素引入到廣義開—閉和閉—開濾波器中，定義了壹類多結構元素並行復合廣義形態濾波器。並利用有約束的最小均方誤差算法(CLMS)，研究了壹種多結構元素自適應加權平均廣義形態濾波算法。仿真結果驗證了上述濾波器算法的有效性。
4. 基於全方位結構元素的概念，本文定義了極大開和極小閉運算，在此基礎上，通過它們的不同順序級聯組合，構造出了壹類全方位多級組合濾波器。最後，采用形態變換的加權平均運算，提出了壹種全方位多級加權組合形態濾波方法。並進行了計算機仿真實驗，結果表明這種方法在噪聲抑制和信號幾何特征保持方面有較好的性能。
5. 研究了順序形態濾波的優化問題。首先分析了順序形態濾波器的輸出統計特性，指出了百分位值和結構元素對濾波性能的影響。針對固定參數濾波器應用的局限性，本文采用自適應LMS算法，在均方誤差(MSE)準則和平均絕對誤差(MAE)準則下，實現了百分位值和結構元素的自適應處理。這類濾波器可在含噪聲(包括混合脈沖噪聲)階躍變化信號的濾波場合獲得重要應用。
本論文不僅在理論上有較大的突破，發表了二十多篇有學術價值的論文，且有多篇論文被國際權威檢索和刊物收錄，具有重要的學術意義；而且其研究成果已在實際工程項目中獲得了重要應用，取得了顯著的經濟效益。
本課題的研究成果對豐富非線性數字信號處理知識寶庫有重大的學術價值，對研究和開發其它類型的非線性濾波器具有重要的指導意義和參考價值，特別對圖象處理、模式識別和計算機視覺等學科的發展產生積極的影響。其成果將在航空、航天遙感，圖象匹配末制導，人工智能，機器人視覺，生物醫學、地震和聲納信號處理等領域有廣泛的應用前景。
關鍵詞：數學形態學，形態濾波器，結構元素，非線性濾波，圖象處理。
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