A Study On The Fractal Theory And The Real-time Computing Of The Fractal Dimension Of Images

Abstract The fractal theory is a very active branch of modem mathematics and non-lineal science. In the last decade, it has played an increasingly important role in image processing and analysis. Many reports have been made about the applications of fractal theory in the fields of natural image simulation, image texture analysis, pattern recognition and etc. Combined with the image texture processing, this paper has made some remarkable and fruitful results by going deep into the detailed study about the .1 fractal mathematical theory, especially the fractal dimension theory and fractal dimension computing. The fractal dimension is a basic mathematical conception in the fractal theory and it is one of the most important factors in the applications of the fractal theory. In mathematics, there are several definitions about fractal dimension, including Hausdorfi? dimension, Box Counting dimension, Minkowski-Bouligand dimension, etc.. And even based on the same definition, there will be some differences in computing methods caused by different cover, such as globe cover, cube cover, center cover and so on. This paper has done some profound research on the commonness and differences between them. Especially the paper proved the equivalence between Boxing Counting dimension and Minkowski-Bouligand dimension, which has laid a theoretical foundation for the application of Minkowski-Bouligand Dimension in computation of fractal dimension of images discussed in Chapter Four. In practice, it has been proved that the method utilizing this definition in computation of fractal dimension of images is fast and convenient, can be widely adopted and can easily meet the needs of real-time computation. The fractal dimension of images measures the degree of irregularity of images. In practical applications, the problem of real-time computing of this irregularity is very important in the research on the intelligent operating-robot. This paper applied the mathematical morphology to put forward a fuzzy morphology based on the threshold decomposition. We defined fuzzy morphological operators based on the threshold decomposition and fully analyzed and proved the properties of those operators. According to the definition of Minkowski-Bouligand dimension, we proposed an 4 algorithm and implementation architecture for real-time parallel computing of the fractal dimension of images by using morphological cover based on fuzzy morphology and 4. threshold decomposition. The VLSI implementation of this algorithm is given. Finally, we did some examples computing and software simulation. The results showed that this method of computation had ~æ…¹ry good stability and easy accessibility.