Skip to main content

Fractal and Lacunarity Analyses: Quantitative Characterization of Hierarchical Surface Topographies

  • Edwin J. Y. Ling (a1), Phillip Servio (a1) and Anne-Marie Kietzig (a1)

Biomimetic hierarchical surface structures that exhibit features having multiple length scales have been used in many technological and engineering applications. Their surface topographies are most commonly analyzed using scanning electron microscopy (SEM), which only allows for qualitative visual assessments. Here we introduce fractal and lacunarity analyses as a method of characterizing the SEM images of hierarchical surface structures in a quantitative manner. Taking femtosecond laser-irradiated metals as an example, our results illustrate that, while the fractal dimension is a poor descriptor of surface complexity, lacunarity analysis can successfully quantify the spatial texture of an SEM image; this, in turn, provides a convenient means of reporting changes in surface topography with respect to changes in processing parameters. Furthermore, lacunarity plots are shown to be sensitive to the different length scales present within a hierarchical structure due to the reversal of lacunarity trends at specific magnifications where new features become resolvable. Finally, we have established a consistent method of detecting pattern sizes in an image from the oscillation of lacunarity plots. Therefore, we promote the adoption of lacunarity analysis as a powerful tool for quantitative characterization of, but not limited to, multi-scale hierarchical surface topographies.

Corresponding author
*Corresponding author.
Hide All
Ahmmed, K., Grambow, C. & Kietzig, A.-M. (2014). Fabrication of micro/nano structures on metals by femtosecond laser micromachining. Micromachines 5(4), 12191253.
Ahmmed, K.M.T., Ling, E.J.Y., Servio, P. & Kietzig, A.-M. (2015). Introducing a new optimization tool for femtosecond laser-induced surface texturing on titanium, stainless steel, aluminum and copper. Opt Lasers Eng 66, 258268.
Al-Kadi, O.S. & Watson, D. (2008). Texture analysis of aggressive and nonaggressive lung tumor CE CT images. IEEE Trans Biomed Eng 55(7), 18221830.
Allain, C. & Cloitre, M. (1991). Characterizing the lacunarity of random and deterministic fractal sets. Phys Rev A 44(6), 35523558.
Alvarez, A.C., Passé-Coutrin, N. & Gaspard, S. (2013). Determination of the textural characteristics of carbon samples using scanning electronic microscopy images: Comparison with mercury porosimetry data. Adsorption 19(2–4), 841850.
Barthlott, W. & Neinhuis, C. (1997). Purity of the sacred lotus, or escape from contamination in biological surfaces. Planta 202(1), 18.
Bharati, M.H., Liu, J.J. & MacGregor, J.F. (2004). Image texture analysis: Methods and comparisons. Chemometr Intell Lab Syst 72(1), 5771.
Bhushan, B. (2012). Biomimetics: Bioinspired Hierarchical-Structured Surfaces for Green Science and Technology. Berlin: Springer-Verlag Berlin Heidelberg.
Bhushan, B., Jung, Y.C. & Koch, K. (2009). Micro-, nano- and hierarchical structures for superhydrophobicity, self-cleaning and low adhesion. Philos Trans A Math Phys Eng Sci 367, 16311672.
Butson, C.R. & King, D.J. (2006). Lacunarity analysis to determine optimum extents for sample-based spatial information extraction from high-resolution forest imagery. Int J Remote Sens 27(1), 105120.
Chen, S.S., Keller, J.M. & Crownover, R.M. (1993). On the calculation of fractal features from images. IEEE Trans Pattern Anal Mach Intell 15(10), 10871090.
Cutting, J.E. & Garvin, J.J. (1987). Fractal curves and complexity. Percept Psychophys 42(4), 365370.
Dale, M.R.T. (2000). Lacunarity analysis of spatial pattern: A comparison. Landscape Ecol 15(5), 467478.
Dale, M.R.T., Dixon, P., Fortin, M.-J., Legendre, P., Myers, D.E. & Rosenberg, M.S. (2002). Conceptual and mathematical relationships among methods for spatial analysis. Ecography 25(5), 558577.
Davies, E.R. (2008). Introduction to texture analysis. In Handbook of Texture Analysis, Mirmehdi, M., Xie, X. & Suri, J. (Eds.), pp. 131. London: Imperial College Press.
Demir, A.G., Furlan, V., Lecis, N. & Previtali, B. (2014). Laser surface structuring of AZ31 Mg alloy for controlled wettability. Biointerphases 9(2), 029009.
Dubuisson, M.-P. & Dubes, R.C. (1994). Efficacy of fractal features in segmenting images of natural textures. Pattern Recognit Lett 15(4), 419431.
Ellinas, K., Tserepi, A. & Gogolides, E. (2011). From superamphiphobic to amphiphilic polymeric surfaces with ordered hierarchical roughness fabricated with colloidal lithography and plasma nanotexturing. Langmuir 27(7), 39603969.
Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Applications. West Sussex, England: John Wiley & Sons Ltd.
Feng, J., Tuominen, M.T. & Rothstein, J.P. (2011). Hierarchical superhydrophobic surfaces fabricated by dual-scale electron-beam-lithography with well-ordered secondary nanostructures. Adv Funct Mater 21(19), 37153722.
Gårding, J. (1988). Properties of fractal intensity surfaces. Pattern Recognit Lett 8(5), 319324.
Gefen, Y., Meir, Y., Mandelbrot, B.B. & Aharony, A. (1983). Geometric implementation of hypercubic lattices with noninteger dimensionality by use of low lacunarity fractal lattices. Phys Rev Lett 50(3), 145148.
Gerasopoulos, K., Pomerantseva, E., McCarthy, M., Brown, A., Wang, C., Culver, J. & Ghodssi, R. (2012). Hierarchical three-dimensional microbattery electrodes combining bottom-up self-assembly and top-down micromachining. ACS Nano 6(7), 64226432.
Ho, A.Y.Y., Yeo, L.P., Lam, Y.C. & Rodríguez, I. (2011). Fabrication and analysis of gecko-inspired hierarchical polymer nanosetae. ACS Nano 5(3), 18971906.
Jagdheesh, R., Pathiraj, B., Karatay, E., Römer, G.R.B.E. & Huis in’t Veld, A.J. (2011). Laser-induced nanoscale superhydrophobic structures on metal surfaces. Langmuir 27(13), 84648469.
Jelinek, H.F. & Fernandez, E. (1998). Neurons and fractals: How reliable and useful are calculations of fractal dimensions? J Neurosci Methods 81(1–2), 918.
Jung, Y.C. & Bhushan, B. (2010). Biomimetic structures for fluid drag reduction in laminar and turbulent flows. J Phys Condens Matter 22(3), 035104.
Karperien, A. (2002–2014). FracLac for ImageJ. (accessed April 2014).
Karperien, A., Ahammer, H. & Jelinek, H. (2013). Quantitating the subtleties of microglial morphology with fractal analysis. Front Cell Neurosci 7.
Keller, J.M., Chen, S. & Crownover, R.M. (1989). Texture description and segmentation through fractal geometry. Comput Vis Graph Image Process 45(2), 150166.
Kenkel, N.C. & Walker, D.J. (1993). Fractals and ecology. Abstracta Botanica 17(1–2), 5370.
Khorasani, H., Zheng, Z., Nguyen, C., Zara, J., Zhang, X., Wang, J., Ting, K. & Soo, C. (2011). A quantitative approach to scar analysis. Am J Pathol 178(2), 621628.
Kietzig, A.-M., Hatzikiriakos, S.G. & Englezos, P. (2009). Patterned superhydrophobic metallic surfaces. Langmuir 25(8), 48214827.
Lehr, J. & Kietzig, A.-M. (2014). Production of homogenous micro-structures by femtosecond laser micro-machining. Opt Lasers Eng 57, 121129.
Ling, E.J.Y., Saïd, J., Brodusch, N., Gauvin, R., Servio, P. & Kietzig, A.-M. (2015). Investigating and understanding the effects of multiple femtosecond laser scans on the surface topography of stainless steel 304 and titanium. Appl Surf Sci 353, 512521.
Malhi, Y. & Román-Cuesta, R.M. (2008). Analysis of lacunarity and scales of spatial homogeneity in IKONOS images of Amazonian tropical forest canopies. Remote Sens Environ 112(5), 20742087.
Mandelbrot, B.B. (1983). The Fractal Geometry of Nature. New York, USA: W. H. Freeman and Company.
Manera, M., Dezfuli, B.S., Borreca, C. & Giari, L. (2014). The use of fractal dimension and lacunarity in the characterization of mast cell degranulation in rainbow trout (Oncorhynchus mykiss). J Microsc 256(2), 8289.
Moradi, S., Kamal, S., Englezos, P. & Hatzikiriakos, S.G. (2013). Femtosecond laser irradiation of metallic surfaces: Effects of laser parameters on superhydrophobicity. Nanotechnology 24(41), 415302.
Myint, S.W. & Lam, N. (2005). A study of lacunarity-based texture analysis approaches to improve urban image classification. Comput Environ Urban Syst 29(5), 501523.
Nakayama, K., Tsuji, E., Aoki, Y. & Habazaki, H. (2014). Fabrication of superoleophobic hierarchical surfaces for low-surface-tension liquids. RSC Adv 4(58), 3092730933.
Nayak, B.K. & Gupta, M.C. (2010). Self-organized micro/nano structures in metal surfaces by ultrafast laser irradiation. Opt Lasers Eng 48(10), 940949.
Neumann, G.T. & Hicks, J.C. (2012). Novel hierarchical cerium-incorporated MFI zeolite catalysts for the catalytic fast pyrolysis of lignocellulosic biomass. ACS Catal 2(4), 642646.
Noh, J., Lee, J.-H., Na, S., Lim, H. & Jung, D.-H. (2010). Fabrication of hierarchically micro- and nano-structured mold surfaces using laser ablation for mass production of superhydrophobic surfaces. Jpn J Appl Phys 49, 106502.
Nosonovsky, M. & Bhushan, B. (2008). Multiscale Dissipative Mechanisms and Hierarchical Surfaces: Friction, Superhydrophobicity, and Biomimetics. Berlin and Heidelberg: Springer-Verlag.
PALAZOGLU, A., STROEVE, P. & ROMAGNOLI, J.A. (2010). Wavelet analysis of images from scanning probe and electron microscopy. In Microscopy: Science, Technology, Applications and Education, Méndez-Vilas, A. and Díaz, J. (Eds.), pp. 1251–1262. Badajoz, Spain: Formatex.
Pentland, A.P. (1984). Fractal-based description of natural scenes. IEEE Trans Pattern Anal Mach Intell 6(6), 661674.
Plotnick, R.E., Gardner, R.H., Hargrove, W.W., Prestegaard, K. & Perlmutter, M. (1996). Lacunarity analysis: A general technique for the analysis of spatial patterns. Phys Rev E 53(5), 54615468.
Plotnick, R.E., Gardner, R.H. & O’Neill, R.V. (1993). Lacunarity indices as measures of landscape texture. Landscape Ecol 8(3), 201211.
Rasband, W. (1997–2014). ImageJ. (accessed December 2009).
Rivera-Virtudazo, R., Tapia, A., Valenzuela, J., Cruz, L., Mendoza, H. & Castriciones, E. (2009). Lacunarity analysis of TEM images of heat-treated hybrid organosilica materials. In Innovations in Chemical Biology, Şener, B. (Ed.), pp. 397403. Dordrecht, The Netherlands: Springer.
Sarkar, N. & Chaudhuri, B.B. (1994). An efficient differential box-counting approach to compute fractal dimension of image. IEEE Trans Syst Man Cybern 24(1), 115120.
Saunders, S.C., Chen, J., Drummer, T.D., Gustafson, E.J. & Brosofske, K.D. (2005). Identifying scales of pattern in ecological data: A comparison of lacunarity, spectral and wavelet analyses. Ecol Complexity 2(1), 87105.
Shao, F., Sun, J., Gao, L., Yang, S. & Luo, J. (2011). Template-free synthesis of hierarchical TiO2 structures and their application in dye-sensitized solar cells. ACS Appl Mater Interfaces 3(6), 21482153.
Smith, T.G. Jr, Lange, G.D. & Marks, W.B. (1996). Fractal methods and results in cellular morphology—dimensions, lacunarity and multifractals. J Neurosci Methods 69(2), 123136.
Tuceryan, M. & Jain, A.K. (1998). Texture analysis. In The Handbook of Pattern Recognition and Computer Vision Chen, C.H. & Pau, L.F. (Eds.), pp. 207248. Singapore: World Scientific Publishing Co. Pte. Ltd.
Updike, S.X. & Nowzari, H. (2008). Fractal analysis of dental radiographs to detect periodontitis-induced trabecular changes. J Periodontal Res 43(6), 658664.
Uppal, S.O., Voronine, D.V., Wendt, E. & Heckman, C.A. (2010). Morphological fractal analysis of shape in cancer cells treated with combinations of microtubule-polymerizing and -depolymerizing agents. Microsc Microanal 16(4), 472477.
Utrilla-Coello, R.G., Bello-Pérez, L.A., Vernon-Carter, E.J., Rodriguez, E. & Alvarez-Ramirez, J. (2013). Microstructure of retrograded starch: Quantification from lacunarity analysis of SEM micrographs. J Food Eng 116(4), 775781.
Wang, H., Zhou, H., Gestos, A., Fang, J. & Lin, T. (2013). Robust, superamphiphobic fabric with multiple self-healing ability against both physical and chemical damages. ACS Appl Mater Interfaces 5(20), 1022110226.
Workman, M.J., Serov, A., Halevi, B., Atanassov, P. & Artyushkova, K. (2015). Application of the discrete wavelet transform to SEM and AFM micrographs for quantitative analysis of complex surfaces. Langmuir 31(17), 49244933.
Yaşar, F. & Akgünlü, F. (2005). Fractal dimension and lacunarity analysis of dental radiographs. Dentomaxillofac Radiol 34(5), 261267.
Zhang, J., Huang, W. & Han, Y. (2006). Wettability of zinc oxide surfaces with controllable structures. Langmuir 22(7), 29462950.
Zhang, M., Shao, C., Guo, Z., Zhang, Z., Mu, J., Zhang, P., Cao, T. & Liu, Y. (2011). Highly efficient decomposition of organic dye by aqueous-solid phase transfer and in situ photocatalysis using hierarchical copper phthalocyanine hollow spheres. ACS Appl Mater Interfaces 3(7), 25732578.
Zhu, T., Li, J. & Wu, Q. (2011). Construction of TiO2 hierarchical nanostructures from nanocrystals and their photocatalytic properties. ACS Appl Mater Interfaces 3(9), 34483453.
Zou, M., Beckford, S., Wei, R., Ellis, C., Hatton, G. & Miller, M.A. (2011). Effects of surface roughness and energy on ice adhesion strength. Appl Surf Sci 257, 37863792.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Microscopy and Microanalysis
  • ISSN: 1431-9276
  • EISSN: 1435-8115
  • URL: /core/journals/microscopy-and-microanalysis
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Type Description Title
Supplementary materials

Ling supplementary material
Figure S1

 Unknown (45.3 MB)
45.3 MB


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed