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This research concerns fuzzy inclusion and entropy measures and their use on image thresholding. It is composed of two basic parts which involve our theoretical study on these inclusion and entropy measures (first part) and our experimentations on their application on image binarization (second part). Fuzzy inclusion and entropy measurements are important to fuzzy set theory and they are used in a wide range of applications where fuzziness may be involved (e.g. fuzzy controllers, feature selection, fuzzy clustering, image processing). Several authors and researchers have dealt with such measures. Some of them have tried to axiomatize these indicators while others have introduced such measures based on specific desired properties. Significant results have been obtained; results that have led to several alternative approaches and solutions of various problems. Apart from these interesting and innovative ideas, in these studies, open matters of further discussion and research have occurre ...
This research concerns fuzzy inclusion and entropy measures and their use on image thresholding. It is composed of two basic parts which involve our theoretical study on these inclusion and entropy measures (first part) and our experimentations on their application on image binarization (second part). Fuzzy inclusion and entropy measurements are important to fuzzy set theory and they are used in a wide range of applications where fuzziness may be involved (e.g. fuzzy controllers, feature selection, fuzzy clustering, image processing). Several authors and researchers have dealt with such measures. Some of them have tried to axiomatize these indicators while others have introduced such measures based on specific desired properties. Significant results have been obtained; results that have led to several alternative approaches and solutions of various problems. Apart from these interesting and innovative ideas, in these studies, open matters of further discussion and research have occurred as well. In the theoretical part of our research we tried to produce new possible fuzzy inclusion measures based on a specific formula. These indicators are formed by using in our formula binary operations on [0,1] and they include some already known and widely used measures. At the same time, we tried to subsume these inclusion and entropy measures to a more general theoretical context by setting specific rules in order for them to satisfy an axiomatic base. The axiomatization we followed, concerning fuzzy inclusion, was Young’s (1996) axiomatization. Young was skeptical about Sihna and Dougherty’s (1993) axioms and tried to include these measures in the context of Kosko’s (1986, 1990, 1992, 1993) general research. Her axiomatic properties were chosen so that a specific theorem, which connects fuzzy inclusion measures with fuzzy entropy measures, could be valid. According to this theorem, the latter are directly produced by the former by using a specific math formula. The binary operations which we used in our formula were the usual fuzzy intersections (t-norms) along with the most common fuzzy implications. This was mainly because we wanted a connection and a consistency with crisp logic. However, this doesn’t mean that we are restricted to these operations since we saw that our formula is still effective while being used with other binary operations on [0,1] as well. As far as fuzzy intersections and implications are concerned, the initial difficulty in our study lay in finding such proper couples (intersection – implication) which could return new fuzzy inclusion measures satisfying Young’s axioms. In fact, what we obtained was only Kosko’s measures. Then we observed that this was due to a certain part of Young’s third axiom which was stricter than needed in the context in which it was proposed. By “loosing” this specific part (in two phases) and reformulating our propositions, we were able to produce several new possible fuzzy inclusion indicators. This interference didn’t invalidate Young’s theorem which means that these measures returned their respective entropy measures as well. While we were studying these indicators and their behavior in fuzzy measurements, we observed that they have – at least some of them – some very interesting (or even unique) attributes. Our next step was towards the evaluation of the reliability of these measures and their capability of being used in specific applications.We believe that these measures could be used in applications which require or properly exploit fuzzy inclusion and entropy measurements. Possibly they could offer us more information or lead to alternative ways of solving specific problems of these areas of research. In order to support these arguments, during the second part of this research, we introduce an open, general and adaptive method of global and local image thresholding which effectively uses some of these measures. Image thresholding or image binarization deals with its foreground – background segmentation by turning the image into binary and it’s the first step of many advanced techniques of image processing or computer vision processes. This may look like a simple procedure, but in fact, it is a difficult problem – considering the vast variety of images of different characteristics – whose research is still open despite the large number of techniques which have been proposed. Our proposition significantly varies from the usual thresholding methods and it always (regardless of our input) relies on specific measurements obtained by our fuzzy inclusion and entropy measures. As far as global thresholding is concerned (here our image is binarized using a single threshold), our method doesn’t depend on histogram analysis nor does it rely on optimizing some statistical measure (e.g. variance minimization) of the gray – level information. It only needs specific attributes of the image which are measured by some of our fuzzy inclusion and entropy indicators. These attributes can be connected with the proper threshold mathematically (e.g. using some proper function) or automatically (e.g. using some neuro – fuzzy network). It’s more of an open and adjustable process than a strict mathematical method which can be easily adapted to the demands of different domains or fields of research. So, our transition to local thresholding was immediate. Local thresholding methods calculate a different threshold for each pixel based on the information of its adjoining pixels and they are used in cases of “difficult” images where global thresholding is inadequate. They are far more complicated and they demand more attention during their development and their testing. Our main area of interest, regarding local thresholding, were degraded or unevenly illuminated document images. The recognition of the content of such images is often achieved through their binarization and it’s fundamental for document analysis systems or OCR (Optical Character Recognition) processes. This doesn’t mean that we didn’t experiment with other categories of images as well, seizing on the flexibility of our method. During our experimentations, we tried to maintain the automatic character of our method by avoiding any sensitivity or bias parameters.The results of our algorithms on global and local thresholding were very encouraging and led us to the conclusion that our fuzzy inclusion – entropy measurements are definitely connected with the threshold of the image. This drove us to our most recent part of our research which includes the automatic thresholding of an image using a corresponding ANFIS (Adaptive Neuro – Fuzzy Inference System). These systems were trained with the aid of our measures and they returned a lot of good and promising results during their testing on global and local thresholding. This way, our work is led to a new level and we believe that in the long term and after much additional study and experimentation, we will have some safe and definite propositions which will become a constitutive and useful part of this field of research. All of these would be much easier if there were corresponding, specialized databases of the form image – proper threshold, however, at this point we are obligated to collect our own essential data in order to proceed to our experimentations. Of course, something like this demands a whole lot of time and study. So, this thesis is separated in nine chapters and our research is divided in two basic parts. The first four chapters are introductory and they contain information and preliminaries concerning fuzzy logic area and fuzzy systems as well as image binarization. Chapter 5 is about our aforementioned theoretical study while chapters 6, 7 and 8 contain our algorithms and experimental results on image thresholding. Chapters 6 and 7 are about our inquiry for proper math functions which connect our measures with the desired threshold while chapter 8 contains our conclusions of the automatic connection achieved with ANFIS. In chapter 9, which is our conclusion, besides the summary of this thesis, we also include the most important aspects of our future research concerning image thresholding and fuzzy inclusion and entropy measures. Finally, there, we also mention further applications which could possibly use our measures as well as various studies which could be linked or even incorporated with ours.
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