- freely available
Fractal Fract 2019, 3(2), 17; https://doi.org/10.3390/fractalfract3020017
- D = 0: Zero represents the Euclidean dimension of a point; this is appropriate, since, as n→∞, the remaining square becomes a point (Figure 1a). In other words, the studied phenomenon has a negligible distribution in the region.
- D = 1: When the squares are converted into a line, then Euclidean dimension gives a line (Figure 1c).
- D = 2: This value, which represents the Euclidean dimension of a plan, is expected, since all squares are assigned at each level (Figure 1e). In other words, the examined phenomenon has a very high distribution throughout the region.
2. DIPT of Rock Media
3. The Process of Determining the FD Using DIPT
- Digital image preparation from the face:The brightness of the environment was one of the most important points during the photography. For better results, photography should be done perpendicularly to the face.
- Converting a color image to a monochrome photo:Although the images taken by digital cameras are usually colored, their grayscale-equivalent versions (consisting of a matrix of gray-level pixels) were used for the ease of image processing.
- Pre-processing of digital rock mass photos using histogram equalization:In general, the aim of the pre-processing was to reduce noise and eliminate unwanted details, such as the gap among the lines in the image. Here, the noise refers to any undesirable information in the image . In this research, histogram equalization was employed to adjust image intensities in order to enhance the image’s contrast. For more details, refer to Reference .In order to reveal the discontinuities in the image, it was essential to mark and interconnect the represented pixels as discontinuities. These pixels are known as edges in the image . Some algorithms have been proposed for improving these methods by considering some factors, such as image noise and the nature of the edges. The Canny algorithm is one of the best edge detection algorithms. In this algorithm, edge detection was considered as an optimization problem . In this research, we set one optimum threshold for the Canny edge detection algorithm via a trial and error method. Therefore, we utilized the same threshold to perform the edge detection procedure on different examples. More details are presented in the reference .
- Calculation of the FD of the FNS.
4. Results and Discussion
Conflicts of Interest
- McCarter, M. Application of Plane Table Photogrammetry to Open Pit Mapping. In Proceedings of the US Symposium on Rock Mechanics, Snowbird, UT, USA, 25–27 August 1976; pp. 93–98. [Google Scholar]
- Reid, T.R.; Harrison, J.P. A semi-automated methodology for discontinuity trace detection in digital images of rock mass exposures. Rock Mech. Min. Sci. 2000, 37, 1073–1089. [Google Scholar] [CrossRef]
- Kemeny, J.; Post, R. Estimating three-dimensional rock discontinuity orientation from digital images of fracture traces. Comput. Geosci. 2003, 29, 65–77. [Google Scholar] [CrossRef]
- Otoo, J.N.; Maerz, N.H.; Xiaoling, L.; Duan, Y. 3-D Discontinuity Orientations Using Combined Optical Imaging and LIDAR Techniques. In Proceedings of the 45th U.S. Rock Mechanics/Geomechanics Symposium, San Francisco, CA, USA, 26–29 June 2011. [Google Scholar]
- Hong, K.; Han, E.; Kang, K. Determination of geological strength index of jointed rock mass based on image processing. J. Rock Mech. Geotech. Eng. 2017, 9, 702–708. [Google Scholar] [CrossRef]
- Mohebbi, M.; Yarahmadi Bafghi, A.R.; Fatehi Marji, M.; Gholamnejad, J. Rock mass structural data analysis using image processing techniques (Case study: Choghart iron ore mine northern slopes). J. Min. Environ. 2017, 8, 61–74. [Google Scholar]
- Barton, C.; La Pointe, P. Fractals in the Earth Sciences; Plenum Press: New York, NY, USA, 1995. [Google Scholar]
- Turcotte, D.; Huang, J. Fractal Distributions in Geology, Scale Invariance, and Deterministic Chaos. In Fractals in the Earth Sciences; Barton, C.C., Ed.; Springer: Boston, MA, USA, 1995; pp. 1–40. [Google Scholar]
- Odling, E.N. Network properties of a two-dimensional natural fracture pattern. Pure Appl. Geophys. 1992, 138, 95–114. [Google Scholar] [CrossRef]
- Berkowitz, B.; Hadad, A. Fractal and multifractal measure of natural and synthetic fracture networks. J. Geophys. Res. 1997, 1021, 12205–12218. [Google Scholar] [CrossRef]
- Bour, O.; Davy, P.; Darcel, C.; Odling, N. A statistical scaling model for fracture network geometry, with validation on a multiscale mapping of a joint network (Hornelen Basin, Norway). J. Geophys. Res. 2002, 107. [Google Scholar] [CrossRef][Green Version]
- Lei, Q.; Latham, J.P.; Tsang, C.F.; Xiang, J.; Lang, P. A new approach to upscaling fracture network models while preserving geostatistical and geomechanical characteristics. Geophys. Res. Solid Earth 2015, 120, 4784–4807. [Google Scholar] [CrossRef][Green Version]
- Mandelbrot, B. The Fractal Geometry of Nature; Freeman: New York, NY, USA, 1984. [Google Scholar]
- Raines, L.G. Are Fractal dimensions of the spatial distribution of mineral deposits meaningful? Nat. Resour. 2008, 17, 87–97. [Google Scholar] [CrossRef]
- Carranza, E.J.; Wibowo, H.; Barritt, S.D.; Sumintadireja, P. Spatial data analysis and integration for regional-scale geothermal potential mapping, West Java, Indonesia. Geothermics 2008, 37, 267–299. [Google Scholar] [CrossRef]
- Carranza, E.J. Controls on mineral deposit occurrence inferred from analysis of their spatial pattern and spatial association with geological features. Ore Geol. Rev. 2009, 35, 383–400. [Google Scholar] [CrossRef]
- Turcotte, L.D. Fractals and Chaos in Geology and Geophysics; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Li, X.; Zhao, G.; Fang, S.; Ren, C.; Yang, K.; Wang, F. Fractal characterization of nanopore structure in shale, tight sandstone and mudstone from the ordos basin of china using nitrogen adsorption. Energies 2019, 12. [Google Scholar] [CrossRef]
- Deng, J.; Ma, C.; Yu, H. Different soil particle-size classification systems for calculating volume fractal dimension—A case study of Pinus sylvestris var. Mongolica in Mu Us sandy land, China. Appl. Sci. 2018, 8, 1872. [Google Scholar] [CrossRef]
- Zhong, W.; Yue, F.; Ciancio, A. Fractal behavior of particle size distribution in the rare earth tailings crushing process under high stress condition. Appl. Sci. 2018, 8, 1058. [Google Scholar] [CrossRef]
- Gong, L.; Fu, X.; Gao, S.; Zhao, P.; Luo, Q.; Zeng, L.; Yue, W.; Zhang, B.; Liu, B. Characterization and prediction of complex natural fractures in the tight conglomerate reservoirs: A fractal method. Energies 2018, 11. [Google Scholar] [CrossRef]
- Chen, X.; He, P.; Qin, Z. Damage to the microstructure and strength of altered granite under wet-dry cycles. Symmetry 2018, 10, 716. [Google Scholar] [CrossRef]
- Bagde, M.; Raina, A.; Chakraborty, A.K.; Jethwa, J. Rock mass characterization by fractal dimension. Eng. Geol. 2002, 63, 141–155. [Google Scholar] [CrossRef]
- Kulatilake, H.S.W.P.; Fiedler, R.; Panda, B. Box fractal dimension as a measure of statistical homogeneity of jointed rock masses. Eng. Geol. 1997, 48, 217–229. [Google Scholar] [CrossRef]
- Pavičić, I.; Dragičević, I.; Vlahović, T.; Grgasović, T. Fractal analysis of fracture systems in upper Triassic dolomites in Žumberak Mountain, Croatia. Min. Geol. Pet. Eng. Bull. 2017, 32, 1–13. [Google Scholar] [CrossRef]
- Gonzalez, R.C.; Woods, R.E. Digital Image Processing, 2nd ed.; Prentice Hall: Upper Saddle River, NJ, USA, 2002. [Google Scholar]
- Canny, J. A Computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 1986, 8, 679–698. [Google Scholar] [CrossRef]
- Oskoei, M.; Hu, H. A Survey on Edge Detection Methods; University of Essex: Colchester, UK, 2010. [Google Scholar]
- Falconer, K. Fractal Geometry: Mathematical Foundations and Applications/K. Falconer; Wiley: Chichester, UK, 1990. [Google Scholar]
- Du, S.; Van Wyk, B.; Tu, C.; Zhang, X. An improved hough transform neighborhood map for straight line segments. IEEE Trans. Image Process. 2009, 19, 573–585. [Google Scholar]
- Bies, A.J.; Boydston, C.R.; Taylor, R.P.; Sereno, M.E. Relationship between fractal dimension and spectral scaling decay rate in computer-generated fractals. Symmetry 2016, 8. [Google Scholar] [CrossRef]
- Lei, Q.; Wang, X. Tectonic interpretation of the connectivity of a multiscale fracture system in limestone. Geophys. Res. Lett. 2016, 43, 1551–1558. [Google Scholar] [CrossRef][Green Version]
|Image Index||FD: Reference Value||FD: M1||FD: M2||FD: M3||Error M1 (%)||Error M2 (%)||Error M3 (%)|
|Average Relative Error (%)||3.023||1.086||1.694|
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).