Algorithm and a method for characterizing surfaces with fractal nature

Inventors

Michopoulos, John G. • Iliopoulos, Athanasios

Assignees

US Department of Navy

Abstract

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.

Core Innovation

The invention provides 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 three-dimensional space. This is done by applying an inverse algorithm to pre-existing measured elevation data of a rough surface presented as a discrete collection of scalar field data at distinct spatial coordinates. The method determines the parameters that define the analytical and continuous W-M representation of the rough surface, including all critical parameters such as the phases and the density of the frequencies greater than 1.

The problem being addressed arises from limitations in current fractal surface modeling, which typically only determine a single parameter—the fractal dimension—via methods like the power spectrum method. These approaches are inaccurate, limited to identifying only one parameter, and require restrictive assumptions. Furthermore, previous efforts to identify the complete set of parameters controlling the W-M function, including the phases, have been computationally infeasible due to the high dimensionality of the inverse problem and have relied on optimization approaches impractical for phase identification.

The invention overcomes these issues by reformulating the surface representation to decouple the phases from other variables, enabling a direct and computationally efficient inverse identification of all parameters defining the W-M surface function. This allows for an infinite-resolution analytical representation of any surface or density array through the W-M fractal function, facilitating comprehensive surface characterization. The approach employs a least squares solution using Singular Value Decomposition (SVD) to solve an overdetermined linear system derived from the elevation data, allowing identification of both the phases and the frequency density parameter γ while recognizing that fractal dimension and roughness parameters can be held fixed as their effects can be captured by adjusting phases.

Claims Coverage

The patent contains one independent claim focused on a computer software product implementing an inverse algorithm for determining parameters of the Weierstrass-Mandelbrot representation of fractal surfaces. The main inventive features include direct parameter determination from elevation data and the application of a specific matrix decomposition technique.

The claims cover a computer software product which implements an inverse method to determine the parameters of a W-M fractal surface from measured elevation data, specifically leveraging a matrix decomposition approach (SVD) to efficiently solve for all parameters including phases and frequency density, enabling a continuous fractal surface representation.

Stated Advantages

Documented Applications