Fractional fractal geometry for image processing

In this dissertation, an image processing system based on fractional fractal dimension and progressive fractal blanket is presented. Fractal dimension is an important measure of the properties of scaling and roughness of an image. Such information can be used to analyze the key features of an image or video frame and therefore be incorporated into image processing and video compression systems. Because of the pivotal role of the estimation of fractal dimension in these systems, accurate and efficient dimension estimation is essential. In this dissertation, we first propose a novel approach to accurate fractal dimension estimation. We then introduce a new method to locate the optimal scales for the precise estimation of fractal dimension.; Among the various approaches to fractal dimension estimation the most popular one is the box-counting method. However, the partition and counting methods used in the regular box-counting schemes produce inaccurate results. In this dissertation, a more accurate fractional box-counting approach is proposed to estimate the fractal dimension in an image. The crux of the new approach is the separation of the concepts of base scale from counting scale. Any physical measurement starts with a predetermined resolution. All the errors brought about in the measurement are also related to such a resolution. The predetermined resolution is called the base scale. When measuring the roughness of a surface, boxes of different sizes are used. The sizes of the boxes here are called counting scales. By using fractional box counting to capture the fractal property at some predetermined resolution, we achieve more accurate results.; After knowing how to estimate the fractal dimension, the next step is to know how to select the appropriate counting scales. Traditional methods simply increase the counting scales by a fixed amount. Since a fractal set may not reproduce itself at such a rate, estimation may be performed at scales where the least amount of information is available. In this dissertation, a progressive extraction method is developed. After establishing the boundaries of the targeted surface by enclosing it with internal and external blankets, the new method determines the features of the surface by calculating the characteristics of such blankets. It then analyzes the spatial relationships of the changes in gray levels on these surfaces to obtain a quantitative measure of the concentrations of such relationships. Because a fractal surface statistically duplicates itself at different scales, there exist distinctive concentrations of such spatial relationships. By determining the concentration of the duplications, the algorithm is able to progressively extract information on dimensionality and other crucial information about surface properties and thus able to capture fractal features precisely. Preliminary results generated from the fractional fractal box counting approach and the progressive fractal blanket approach on satellite, wild life, medical and range images have proved to be effective.