On The Gray-Scale Dilation(Erosion) Equations And Related Problems

Mathematical morphology was born in 1960’s, it is a mathematical theory based on the lattice theory, group theory and other analysis theories as well, trying to realize image analysis and processing through constructing the morphological operators such as dilations, erosions, openings, closings and their compositions.G. Matheron and J. Serra created and studied the mathematical morphology for binary image analysis. The binary images are viewed as subsets, and so the most morphological operators are constructed in terms of compositions for sets such as the formations of union, intersection, translation, taking complement etc. The relevant mathematical theory for binary image analysis has been perfectly developed. Then, D. Digabel and C. Lzatuejoul in 1978 proposed the gray-scale image’s mathematical morphology, and mathematicians and some experts in computing started to study the corresponding mathematical theory, defining and studying the corresponding gray-scale morphological operators. Up to now, there are still a lot of problems to be solved in this field.In this paper, we study the gray-scale dilation equation ? ?g?f ?h(*) and the gray-scale erosion equation ? ?g?f ?h(**) where g, h are given gray-scale functions, f is the unknown variable, g? and g? are gray-scale dilation and erosion operations. More precisely, we are looking for the necessary and sufficient conditions for the existence of solutions and the structures of all solutions. The main achievements in this thesis are the followings:Theorem 4.1(*)has a solution if and only if ? ?g?h is its solution.Theorem 4.3 Suppose that(*)has a solution, then f is a solution if and only if ? ?g f h?? ? ?? for some ?g h??? ??,(see the thesis for the definition of ??g,h?).Theorem 4.4(**)has a solution if and only if()g?h is its solution.Theorem 4.6 Suppose that(**)has a solution, then f is a solution if and only if ? ?g h f?? ? ?? for some ?g,h??? ??(see the thesis for the definition of ??g,h?).In addition, we discussed the particular gray-scale flat dilation and flat erosion equations.