Mathematical Measures for Analyzing Phase Space
Pp. 114-165 (52)
Michael Edward Farmer
To be able to differentiate image sequences exhibiting the complex behavior
of moving objects and contextual change from image sequences simply experiencing
changes in illumination we must be able to characterize the trajectories of these systems
in phase space. The Hausdorff dimension provides a theoretical estimate of the
fractional dimension of any curve in space; however it is quite difficult to calculate.
Fortunately there are a number of approximations to the Hausdorff dimension that will
be defined in this chapter, with one of the most common being the Box Counting
dimension. The Hausdorff dimension is a global measure of the fractional dimension of
a space. One of the goals of many computer vision applications is image segmentation
which will require an estimate of the fractal behavior of each pixel in the image. This
will require local measures of the fractality of the phase space Fortunately there are a
number of local measures that will be available to us. Lastly, fractal dimensional
measures will only differentiate between chaotic and non-chaotic trajectories to
characterize and differentiate various textures we need measures that will be able to also
differentiate between different fractal behaviors. We propose adapting measures used
to analyze the Grey-level Co-occurrence Matrix for this purpose due to the structural
similarity between the GLCM and the phase plot.
Box counting measure, correlation dimension, Hausdorff dimension,
global dimensions, local dimensions, Lyapunov exponent, Renyi spectrum,
Department of Computer Science, Engineering and Physics 207 Murchie Science Building 303 E. Kearsley Street Flint, Michigan USA.