The Study Of Belief Function’s Fractal Dimension In Pattern Recognition

Pattern recognition was developed from the early twentieth century to the present, playing an increasingly important role in people’s daily lives and attracting researchers in many fields to explore its theory and application. Pattern recognition is a scientific technology for the theory and methods of classification and recognition. It’s an important part of information science and artificial intelligence, mainly used in image analysis and processing, voice recogni-tion, computer-aided diagnosis, data mining and other fields. Pattern recognition, which we usually talking about, means machine identification. It analyzes the studying objects’char-acteristics and does identification judgment based on the machine systems and some analysis algorithms. The theory of evidence was proposed by Dempster and developed by his student Shafer. It is a relatively mature method of uncertainty reasoning and a generalization of the Bayesian theory. And the theory provides a very important way to present and fuse the uncer-tainty information, widely used in the fields of expert systems, data fusion, medical diagnosis, pattern recognition, decision analysis and so on. Fractal theory was put forward by Benoit B. Mandelbrot, specializing in the fractal nature of its application. It can well describe the nature of complex, irregular things and phenomena. The most basic characteristic of frac-tal theory is the perspective and mathematical methods of fractional dimension are used to describe and research objective things, that is, with a fractal dimension mathematical tools to describe the studying objects. Fractal dimension can reflect the effectiveness of complex objects occupy the space, which is a measure of the complexity of the irregular form and structure, with scale invariance.For the pattern recognition problems in practice, lots of information is incomplete, seg-mentary or not that reliable. How to effectively deal with these uncertain information is a significant point in solving the problem of pattern recognition. As the generalization of proba-bility theory and with great capabilities to deal with uncertainty, incompleteness information, DS theory of evidence provides a ideal technological route for solving the pattern recognition problem. The fractal dimension is a parameters to quantitatively describe the features and geometric complexity of the fractal collection in fractal geometry. It can demonstrate the inherent characteristics and attributes of a system or collection.In this paper, we will discuss the DS theory of evidence with the fractal theory, hoping to use the concepts of fractal dimension and interaction dimension to describe the nature of belief function and the relationship between belief functions, as well as the relationship between a belief function and a set of belief functions. Finally, we want to apply them to pattern recognition situation.On the basis of previous studies, a new method combining DS theory of evidence and fractal theory is proposed to solve pattern recognition problems. First, all patterns, no mat-ter the description of observed object or prototype objects of the various categories in a knowledge base, will be characterized by belief functions. Second, learning from the work of others, we present a new way to measure the divergence among the focal elements in a belief function. Sequentially, the calculations of a belief function’s fractal dimension and the interaction dimension between belief functions are improved. Named the belief function de-scribing observed object as input evidence. Then the interaction dimension, which can be considered as the matching degree, between input evidence and each prototype object in the knowledge base is obtained. Next, the method to calculate the interaction dimension between a belief function and a set of belief functions is proposed. So the matching degree between a input evidence and a pattern category, which includes no less than one prototype object, can be measured by this interaction dimension. Due to the interaction dimension, each input evidence corresponding to output a belief function in the discernment frame of all pattern categories in the knowledge base. Finally, a comprehensive and aggregated belief function based on the discernment frame of all pattern categories can be obtained for the judgment of which category dose the observed object mostly probable belong to. The experimental results show that the proposed method is effective and feasible.As an active new theory and discipline nowadays, fractal theory is bound to arouse the interest of many researchers. The researches and applications combined DS theory of evidence and fractal theory will have broad prospects. The research on the properties of belief functions in DS theory of evidence through fractal theory is a good direction to explore.