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Research Papers

Complete High Dimensional Inverse Characterization of Fractal Surfaces and Volumes

[+] Author and Article Information
John G. Michopoulos

Computational Multiphysics Systems Laboratory,
Center of Computational Material Science,
Naval Research Laboratory,
Washington, DC 20375

Athanasios Iliopoulos

Computational Materials Science Center,
George Mason University,
Fairfax, VA 22030

1Resident at: Center of Computational Material Science, Naval Research Laboratory, Washington DC, 20375.

Contributed by the Computers and Information Division of ASME for publication in the Journal of Computers and Information Division in Engineering. Manuscript received October 17, 2012; final manuscript received October 25, 2012; published online December 19, 2012. Assoc. Editor: Bahram Ravani.

J. Comput. Inf. Sci. Eng 13(1), 011001 (Dec 19, 2012) (9 pages) Paper No: JCISE-12-1189; doi: 10.1115/1.4007987 History: Received October 17, 2012; Revised October 25, 2012

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.

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References

Figures

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Fig. 1

Uniform sampling of a sphere (dots) and four typical sampling vectors

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Fig. 2

Systemic view of the forward and inverse characterization problem

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Fig. 3

Pseudo-colored density plots of synthetic and inversely identified surfaces of a W-M fractal

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Fig. 4

Pseudo-colored density plots of absolute difference between reference and identified surfaces

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Fig. 17

The mean absolute error of the inversion as a function of γ

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Fig. 16

Original and inversely identified microtomographic volumes

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Fig. 15

8 of the 243 microtomography slices

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Fig. 14

Result of noise in the error of the fractal inverse identification from synthetic data

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Fig. 13

Noisy data (a)–(c) and respective identified fractal volumes (d)–(f)

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Fig. 12

Mean error of inverse identification through SVD versus parameter γ

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Fig. 11

Pseudo-colored combined isosurface and density plots of synthetic (a) and inversely identified volumes of a w-m fractal (b)

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Fig. 10

Mean error of inverse identification through SVD versus parameter γ for the aluminum surface and m × n = 50 × 50

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Fig. 9

Mean error of inverse identification through SVD versus parameter γ for the aluminum surface and m × n = 25 × 25

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Fig. 8

Pseudo-colored density plots of percentage difference between the surfaces of Figs. 6(a) and 6(c)

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Fig. 7

Pseudo-colored density plots of percentage difference between the surfaces of Figs. 6(a) and 6(b)

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Fig. 6

Pseudo-colored density plots of synthetic and inversely identified elevation of aluminum 6061-T6 surface data

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Fig. 5

Mean error of inverse identification through SVD versus parameter γ

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