الفهرس 
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Abstract
Digital topology is the study of the topological properties of the digital images. Over the last decades, digital topology has proved to be a very strong concept in image analysis and image processing. Rosenfeld [26] introduced the fundamentals of digital topology, which provides a sound mathematical basis for image processing operations such as image thinning, border following, contour filling, object counting, and signal processing [12, 24]. Whenever spatial relations are modeled on a computer, a digital topology is needed. Digital topology aims to transfer concepts from classical topology to digital spaces such as: connectivity, boundary, neighborhood, and continuity which are used to model computer images.
The problem of finding a topology for the digital plane and the digital 3-space is of importance in image processing and more generally in all situations where spatial relations are modeled on a computer. In these applications it is essential to have a data structure on the computer which shares as many as possible features with the real topological situation.
The points of the digital (2- or 3- ) space are assigned some gray-level or color values. Based on these values and adjacency relations, the points in the digital space are grouped or segmented into connected components which are used in computer graphics to display and computer vision to recognize the parts of the real objects. The grouping processmust consistent with human spatial intuition about nearness of points in the digital space. Construction of a mathematical model to the digital space is not easy, because the theoretical framework of epistemological must be consistent with the human spatial intuition.
There are mainly three approaches for defining a digital analog of the well-known ”nature” topology of the Euclidean space they are:
• Graph-theoretical approach: A very elementary structure which can be handled easily on computer is graph. A graph is obtained when a neighborhood relation is introduced into the digital set. Such a structure allows investigating connectivity of sets.
• Imbedding approach: The discrete structure is imbedded into a known continuous structure, usually into an Euclidean space. Topological properties of discrete objects are then defined by means of their continuous images.
• Axiomatic approach: Certain subsets of the underlying digital structure are declared to be ”open sets” and are required to fulfill certain axioms. These axioms have to be chosen in such a way that the digital structure gets properties which are as close as possible to the properties of usual topology.
All three approaches have advantages and disadvantages. The axiomatic approach is mathematically very elegant but it does not directly provide the language which is wanted in applications. The objects which are practically investigated are not open sets but rather connected sets, sets which are contained in other sets ect. The graph theoretic approach yields directly connectedness but it becomes very difficult to handle more complicated concepts of topology such as continuity, homotopy ect. Theembedding approach is of course only adequate for structures which can be related to an Euclidean space. The problem is that one has to find for each question an appropriate embedding. Because of we are researcher of pure mathematics, we concentrated on the axiomatic approach in this thesis but we stated some graph-theoretical concepts that are needed.
The notion of supra-topology was introduced by Abd El-Monsef et al. [2] in 1986. It is a generalization of the topological space by omitting the condition of finite intersection. Note that the notion of the supra-topology is the same of the semi- topology that Longin Latecki [17] introduced in 1992 but we keep it supra-topology to avoid the Confusion between the notion of semi-topology and the notion of semi-open sets. The standard supra-topologies generated by the neighborhoods of the points are compatible with the connectedness of the graph.
Filters are quite popular tools in image processing and the signal theory. Also, they are often used to de-noise images. Median filters were introduced by Tuckey [28] in 1977. There are three median filters in the digital plane ℤ2, denoted by 𝑀𝑒𝑑4, 𝑀𝑒𝑑6, and 𝑀𝑒𝑑8. The median filter 𝑀𝑒𝑑4 is called the cross median filter in the sequel. In the digital space ℤ3, there are another three median filters, denoted by 𝑀𝑒𝑑6, 𝑀𝑒𝑑18, and 𝑀𝑒𝑑26 [4].
Root image is a digital picture which left unchanged by applying the median filter. The root image is introduced by Döhler [7] in 1989.Root images play a prominent role in the signal theory [12, 24], because they help us to understand the properties of the median filter. For example, let we want to detect an object of a given shape in a noisy digital picture. In order to de-noise this picture, we apply a median filter. If this filter does not only remove the noise, but also it removes a part of the object of interest in image or changes the shape, then it is notadvisable to apply this filter. The best choice of a median filter would be the median filter where our object of interest is a root image of this filter.
In 2002, Andreas Alpers [4] introduced the relationship between the semi-topology and the root image of the cross median filter 𝑀𝑒𝑑4 by showing that the regular open sets in the 8-supra-topology (8-semi-topology) is root images.
In 2003, Ulrich Eckhardt and Longin Latecki deduced two different topologies on ℤ2, they are Khalimsky [13] and Marcus-Wyse [20] topologies. They also introduced five different topologies on ℤ3.
In this thesis, we use the notions of semi-open sets and 𝜆-open sets to introduce a semi-topology finer than the standard semi-topology. We found a relation between these notions and the root images of the median filters.
This thesis contains three chapters:
• The introductory Chapter I contains the basic concepts and notations that are needed such as the topological space, supra-topology, semi-open set, 𝜆-open set, median filter and its root image.
• In Chapter II, we give a summary about the digital topology and how can deduce it. Also, we show some properties of the digital topology such as the separation axioms, dense set, point-wise dense set, cut point, extremely disconnected, open and closed subsets.
• The aim of Chapter III is to find a relation between the digital topology and the root images of median filters. We use the concepts of semi-open and 𝜆-open sets in Marcus-Wyse and Khalimsky topologies to introduce a class finer than the standard semi-topology and deduced a relationbetween these types of open sets and root images of median filters.
We ended to the following results:
If 𝐵⊆ℤ2 is a regular open set in the 6-supra-topology, then 𝐵 is a root image of the cross median filter 𝑀𝑒𝑑4.
If 𝐵⊆ℤ2 is a regular open set in the 6-supra-topology, then 𝐵 is a root image of the median filter 𝑀𝑒𝑑6.
If 𝐵⊆ℤ3 is a regular open set in the 26-supra-topology, then 𝐵 is a root image of the median filter 𝑀𝑒𝑑6.
If 𝐵⊆ℤ2 is a root image of the cross median filter 𝑀𝑒𝑑4, then 𝐵 is a regular semi-open set in Marcus-Wyse topology on ℤ2.
If 𝐵⊆ℤ2 is a root image of the median filter 𝑀𝑒𝑑8, then 𝐵 is a regular semi-open set in Marcus-Wyse topology on ℤ2.
If 𝐵⊆ℤ2 is a root image of the cross median filter 𝑀𝑒𝑑4, then 𝐵 is a regular 𝜆-open set in Marcus-Wyse topology on ℤ2.
If 𝐵⊆ℤ2 is a root image of the median filter 𝑀𝑒𝑑8, then 𝐵 is a regular 𝜆-open set in Marcus-Wyse topology on ℤ2.
If 𝐵⊆ℤ2 is a root image of the cross median filter 𝑀𝑒𝑑4, then 𝐵 is a 𝜆-closed set in Khalimky topology on ℤ2.
If 𝐵⊆ℤ3 is a root image of the median filter 𝑀𝑒𝑑6, then 𝐵 is a regular semi-open set in Marcus-Wyse topology on ℤ3.
If 𝐵⊆ℤ3 is a root image of the median filter 𝑀𝑒𝑑6, then 𝐵 is a regular 𝜆-open set in Marcus-Wyse topology on ℤ3.
If 𝐵⊆ℤ3 is a root image of the median filter 𝑀𝑒𝑑26, then 𝐵 is a regular 𝜆-open set in Marcus-Wyse topology on ℤ3.
The results of this chapter:
Some of them are accepted for ”Bulletin faculty of science, Zagazig University, Egypt, 2011(submitted) [8].
The other some are preparing to submission.