Complete High Dimensional Inverse Characterization of Fractal Surfaces and Volumes

Abstract

In the present paper, we are describing a methodology for the determination of the complete set of parameters associated with the Weierstrass-Mandelbrot (W-M) function that can describe a fractal scalar field distribution defined by measured or computed data distributed on a surface or in a volume. Our effort is motivated not only by the need for accurate fractal surface and volume reconstruction but also by the need to be able to describe analytically a scalar field quantity distribution on a surface or in a volume that corresponds to various material properties distributions for engineering and science applications. Our method involves utilizing a refactoring of the W-M function that permits defining the characterization problem as a high dimensional inverse problem solved by singular value decomposition for 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 participating in the definition of the W-M function. Numerical applications of the proposed method on both synthetic and actual surface and volume 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.