Complete High Dimensional Inverse Characterization of Fractal Surfaces

The present paper describes a methodology for the inverse identification of the complete set of parameters associated with the Weirstrass-Mandelbrot (W-M) function that can describe any rough surface known by its profilometric or topographic data. Our effort is motivated by the need to determine the mechanical, electrical and thermal properties of contact surfaces between deformable materials that conduct electricity and heat and require an analytical representation of the surfaces involved. Our method involves utilizing a refactoring of the W-M function that permits defining the characterization problem as a high dimensional singular value decomposition problem for the determination of the so-called phases of the function. Coupled with this process is a second level exhaustive search that enables the determination of the density of the frequencies involved in defining the trigonometric functions involved in the definition of the W-M function. Our approach proves that this is the only additional parameter that needs to be determined for full characterization of the W-M function as the rest can be selected arbitrarily. Numerical applications of the proposed method on both synthetic and actual elevation data, validate the efficiency and the accuracy of the proposed approach. This approach constitutes a radical departure from the traditional fractal dimension characterization studies and opens the road for a very large number of applications.