Gray-scale Morphological Operator

Mathematical morphology was born in1960’s, founded by G. Matheron and J. Serra. Inthe beginning, set theory was its main mathematical basis, and nowadays, some modernalgebraic theories, such as lattice theory and group theory etc., started to serve as itsmathematical basis or formats. Mathematical morphology is mainly used in for analyzingimage analysis and image processing and in the characteristics analysis and the systemdesign of morphological filter. In terms of mathematical morphology, the image dataprocessing can be simplified by removing irrelevant structure while not changing theshapes of the graphs. Mathematical morphology algorithm is of a natural structure forparallel implementation, so the image analysis and processing can be greatly speeded up.In the recent decades, mathematical morphology has been widely studied in theinternational academic community and has become a main research topic in imageprocessing. Main subjects in mathematical morphology are the algebraic compositions,properties and applications of morphological operators. There are four basic operators:erosion, dilation, opening and closed which have their own characteristics in the binaryimage and the gray-scale image. In the binary morphology, the binary image is viewed asthe subset of a whole set, so the operations of sets, such as intersection, union com-plementation, inclusion and translation etc., can be used in image analysis and processing.The binary mathematical morphology has been perfectly developed. In gray-scalemathematical morphology, a gray image can be viewed as a function, from the whole set tothe interval [0,1](or any other intervals), so the operations, such as Max, Min. and InfimalConvolution etc., can be used. However, there are still a lot of problems left in this field.In this paper, we study mainly some basic operators of gray-scale morphology.The full text is divided into four parts:Chapter Ⅰ: the origin, development and today’s situation of mathematical morphology isbriefly given. The main contents of this thesis are described.Chapter Ⅱ: basic concepts, definitions, notation and terms, which will be used in thisthesis are collected and listed.Chapter Ⅲ: definitions of gray dilation and erosion are given, which are better thanthose defined in the literature[34] in the sense that no “spillover” problem occurs here. Theadjuction, duality of these dilations and erosions defined here are shown. The commutativeproperties of these operators with other morphological operators are discussed. At the end,the representation problem of morphological operators is considered as well.Chapter Ⅳ: new operators on complete lattices are introduced, and their properties arediscussed.