Algorithm and a method for characterizing surfaces with fractal nature

摘要

A computer implemented method for directly determining parameters defining a Weierstrass-Mandelbrot (W-M) analytical representation of a rough surface scalar field with fractal character, embedded in a three dimensional space, utilizing pre-existing measured elevation data of a rough surface in the form of a discrete collection of data describing a scalar field at distinct spatial coordinates, is carried out by applying an inverse algorithm to the elevation data to thereby determine the parameters that define the analytical and continuous W-M representation of the rough surface. The invention provides a comprehensive approach for identifying all parameters of the W-M function including the phases and the density of the frequencies that must greater than 1. This enables the infinite-resolution analytical representation of any surface or density array through the W-M fractal function.

主权项

1. A computer software product for directly determining parameters defining a Weierstrass-Mandelbrot (W-M) analytical representation of a rough surface scalar field with fractal character, embedded in a three dimensional space, utilizing pre-existing measured elevation data of a rough surface in the form of a discrete collection of data describing a scalar field at distinct spatial coordinates, the product comprising a non-transitory physical computer-readable medium including stored instructions that, when executed by a computer, cause the computer to apply an inverse algorithm to the elevation data to thereby determine the parameters that define the analytical and continuous W-M representation of the rough surface, and wherein the surface has an array of elevation or height measurements zije, i, j=1 . . . K over a square region of size L×L and the parameters are γ and φmn of the surface z(x, y) and the surface is represented by the complex Weierstrass-Mandelbrot (W-M) function W as:W⁡(x)=∑n=-∞∞⁢γ(D-2)⁢n⁡(1-ⅇⅈγn⁢x)⁢ⅇⅈϕn(1) where x is a real variable, and where a two dimensional profile obtained from the real part of Eq. 1 is:z⁡(x)=Re⁡[W⁡(x)]=∑n=-∞∞⁢γ(D-2)⁢n⁡[cos⁢⁢ϕn-cos⁡(γn⁢x+ϕn)].(2) and where D is the fractal dimension (1