High Dimensional Full Inverse Characterization of Fractal Volumes

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 fractal scalar field distribution of measured data defined within a volume. Our effort is motivated by the need to be able to describe a scalar field quantity distribution in a volume in order to be able to represent analytically various non-homogeneous 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 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. Numerical applications of the proposed method on both synthetic and actual 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 and generalizes the approach developed by the authors for fractal surfaces to that of fractal volumes.