Fractal Characterization of Dynamic Systems Sectional Images

Abstract

Image characterizations play vital roles in several disciplines of human endeavour and engineering education applications in particular. It can provide pre-failure warning for engineering systems; predict complex diffusion or seeping of radioactive substances and early detection of defective human body tissues among others. It is therefore the object of this study to simulate three selected known surfaces that are of engineering interest, pass section plane through these surfaces arbitrarily and use fractal disk dimension to characterize the resulting image on the sectioned plane. Three carefully selected surfaces based on their engineering education and application worth’s were simulated with respective relevant set of equations. In each case studied, the simulation was driven either by random number generation with seed value of 9876 coupled with relevant set of equations or by numerical integration based on Runge-Kutta fourth order algorithms or combination of both. However, all simulations were coded in FORTRAN 90 Language. Section plane was passed through each simulated surface arbitrarily and in two hundred (200) different times for the purpose of obtaining reliable results only. Image obtained at less or equal to four percent (4%) tolerance level by sectioning was characterised by optimum disks counting algorithms implemented over ten (10) scales of observations and five (5) different iteration each. The estimated disk dimension was obtained by implementing the least square regression procedures on optimum disks counted at corresponding scales of observations. A visual and fractal disk dimension characterization of selected images on sectioned plane form cases studied validated algorithms coded in FORTRAN 90 computer language. The surface of Case-III is the most rough with disk dimension of 2.032 and 1.6% relative error above the dimension of smooth surface (2.0). This is followed by Case-II with disk dimension of 1.905 and 4.8% relative error below the dimension of smooth surface. Case-I has the least disk dimension of 1.897 and with 5.2% relative error below smooth surface. Case-I and case-II that suffered negative relative error originated from set of linear systems while Case-III that suffered positive relative error originated from set of non linear systems. Non linearity manifested in graphical display of disk distribution by frequency in Case-III by multiple peaks and substantial shift above disk dimension of 1.0.This study has demonstrated the high potentiality of fractal disk dimension as characterising tool for images. The coded algorithms can serve well as instruction material for students of linear and non linear dynamic systems.

Keywords: Fractal, Sectional Images, Fractal Disk Dimension, Dynamic Systems and Algorithms