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 WeierstrassMandelbrot (WM) 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 WM function that permits defining the characterization problem as a high dimensional inverse problem solved by singular value decomposition for the socalled 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 WM 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.