Foundations of Chaos and Fractals
Pp. 15-50 (36)
Michael Edward Farmer
Chaos theory has been found to be a very effective tool for analyzing
complex non-linear systems with its beginnings being traced to weather systems
prediction by Lorenz. Chaotic systems have also been tied to information theory, and
systems that exhibit chaotic behavior have been shown to be sources of information.
Definitively identifying systems as being chaotic, however, is extremely difficult.
Fortunately simply identifying a system as being chaos-like can be beneficial and even
adequate for certain analysis, and this is the approach we will adopt in this text for
addressing computer vision applications using chaos theoretic techniques. Chaos and
fractals are interconnected because systems that behave chaotically have phase space
trajectories that are fractal in nature. Thus analyzing the fractal dimension of the phase
space trajectories of systems will allow us to determine how chaotic they are and more
importantly for computer vision how strongly they are creating information. The
strength of information generation will prove effective in developing robust motion and
change detection algorithms even in the face of complex spatio-temporally varying
illumination in later chapters.
Eckmann-Ruelle conjecture, ergodicity, embedded dimension,
information, fovea, phase plot, noise, thermal equilibrium.
Department of Computer Science, Engineering and Physics 207 Murchie Science Building 303 E. Kearsley Street Flint, Michigan USA.