Determination of Heterogeneity for Manganese Dendrites Using Lacunarity Analysis

The surface patterns of natural and experimental deposits are important as they result from the internal microstructure. For this purpose, lacunarity analysis is applied to determine the heterogeneous nature of deposit surface patterns. In this study, images were digitally moved onto the square mesh to determine the heterogeneous situation of manganese dendrite patterns on the natural magnesite surface. The relation between the lacunarity values of the images and the box size was examined. The lacunarity values corresponding to the box size values were estimated using the gliding-box algorithm. This relation was determined numerically as a power-law function using nonlinear regression method. It has been shown that the system examined with the generated numerical model function can be defined with three specific parameters. As a result, it has been shown that it is possible to describe the relationship between numerical solution-based lacunarity-box size and a third-order nonlinear differential equation. With this study, the lacunarity-box size value on different system images can be determined by using the gliding box algorithm and calculating the coefficient value from the power-law relationship.

Keywords:

Fractal and geometric patterns Lacunarity, Mathematical model,

PDF

___

[1] T. G. Smith, G. D. Lange, W. B. Marks, Fractal methods and results in cellular morphology—dimensions, lacunarity and multifractals, J. Neurosci. Methods, 69(2) (1996), 123-136.
[2] R. E. Plotnick, R. H. Gardner, W. W. Hargrove, K. Prestegaard, M. Perlmutter, Lacunarity analysis: a general technique for the analysis of spatial patterns, Phys. Rev. E, 53 (1996), 5461.
[3] R. E. Plotnick, R. H. Gardner, R. V. O’Neill, Lacunarity indices as measures of landscape texture, Landsc. Ecol., 8(3) (1993), 201-211.
[4] B. B. Mandelbrot, The Fractal Geometry of Nature, Times Books, 1983.
[5] V. Mesev, Remotely Sensed Cities, CRC Press, London, 1-13, 2003.
[6] A. Balay-Karperien, Defining Microglial Morphology: Form, Function, and Fractal Dimension, Charles Sturt University, Ph.D. Thesis, 86, 2004.
[7] M. Bayirli, The geometrical approach of the manganitise compound deposition on the surface of manganisite ore, Phys. A: Stat. Mech. Appl., 353 (2005), 1-8.
[8] C. Allain, M. Cloitre, Characterizing the lacunarity of random and deterministic fractal sets, Phys. Rev. A, 44 (1991), 3552.
[9] Z. Merdan, M. Bayirli, Computation of the fractal pattern in manganese dendrites, Chin. Phys. Lett., 22(8) (2005), 2112.
[10] Y. Gefen, Y. Meir, B. B. Mandelbrot, A. Aharony, Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices, Phys. Rev. Lett., 50(3) (1983), 145.
[11] A. Roy, E. Perfect, W. M. Dunne, N. Odling, J.W. Kim, Lacunarity analysis of fracture networks: Evidence for scale-dependent clustering, J. Struct. Geol., 32(10) (2010), 1444-1449.
[12] C. R. Butson, D. J. King, Lacunarity analysis to determine optimum extents for sample based spatial information extraction from high resolution forest imagery, Int. J. Remote Sens., 27(1) (2006), 105-120.
[13] L. Wan, D. Xie, X. Hu, Study of local mineralized intensity using rescaled range analysis and lacunarity analisis, Engineering Science and Technology Review, 6 (2013), 105-109.
[14] C.A. Scheneider, W.S. Rasband, K.W. Eliceiri, NIH Image to ImageJ: 25 years of image analysis, Nat. Methods, 9(7) (2012), 671-675.