Computation of Lacunarity from Covariance of Spatial Binary Maps

  • Kassel HingeeEmail author
  • Adrian Baddeley
  • Peter Caccetta
  • Gopalan Nair


We consider a spatial binary coverage map (binary pixel image) which might represent the spatial pattern of the presence and absence of vegetation in a landscape. ‘Lacunarity’ is a generic term for the nature of gaps in the pattern: a popular choice of summary statistic is the ‘gliding-box lacunarity’ (GBL) curve. GBL is potentially useful for quantifying changes in vegetation patterns, but its application is hampered by a lack of interpretability and practical difficulties with missing data. In this paper we find a mathematical relationship between GBL and spatial covariance. This leads to new estimators of GBL that tolerate irregular spatial domains and missing data, thus overcoming major weaknesses of the traditional estimator. The relationship gives an explicit formula for GBL of models with known spatial covariance and enables us to predict the effect of changes in the pattern on GBL. Using variance reduction methods for spatial data, we obtain statistically efficient estimators of GBL. The techniques are demonstrated on simulated binary coverage maps and remotely sensed maps of local-scale disturbance and meso-scale fragmentation in Australian forests. Results show in some cases a fourfold reduction in mean integrated squared error and a twentyfold reduction in sensitivity to missing data.

Supplementary materials accompanying the paper appear online and include a software implementation in the R language.


Forest disturbance Fractal Gliding box Image analysis Random set Spatial statistics 



Our thanks to Michael Small and his research group for generous permission to use their computing resources.


This research was supported by the Australian Research Council (Grant No. DP130104470), Australia’s Commonwealth Scientific and Industrial Research Organisation, and an Australian Government Research Training Program Scholarship.

Supplementary material

13253_2019_351_MOESM1_ESM.pdf (19.8 mb)
Supplementary material 1 (pdf 20316 KB) (36.8 mb)
Supplementary material 2 (zip 37662 KB)


  1. Aghajanzadeh, S., Kashaninejad, M., and Ziaiifar, A. M. (2017), “Cloud stability of sour orange juice as affected by pectin methylesterase during come up time: approached through fractal dimension,” International Journal of Food Properties, 20(S3), S2508–S2519.Google Scholar
  2. Allain, C., and Cloitre, M. (1991), “Characterizing the lacunarity of random and deterministic fractal sets,” Physical Review A, 44(6), 3552–3558.MathSciNetGoogle Scholar
  3. Anderson, J. K., Huang, J. Y., Wreden, C., Sweeney, E. G., Goers, J., Remington, S. J., and Guillemin, K. (2015), “Chemorepulsion from the quorum signal autoinducer-2 promotes helicobacter pylori biofilm dispersal,” mBio, 6(4).Google Scholar
  4. Anovitz, L. M., and Cole, D. R. (2015), “Characterization and analysis of porosity and pore structures,” Reviews in Mineralogy and Geochemistry, 80, 61–164.Google Scholar
  5. Aznarte, J. I., Iglesias-Parro, S., Ibáñez-Molina, A., and Soriano, M. F. (2014), A new computational measure for detection of extrapyramidal symptoms,, in Proceedings of the Second International Work-Conference on Bioinformatics and Biomedical Engineering, University of Grenada, Grenada, pp. 13–22.Google Scholar
  6. Azzaz, N., and Haddad, B. (2017), “Classification of radar echoes using fractal geometry,” Chaos, Solitons & Fractals, 98, 130–144.MathSciNetGoogle Scholar
  7. Baddeley, A., and Jensen, E. B. V. (2005), Stereology for Statisticians, Vol. 103 of Monographs on Statistics & Applied Probability, USA: Chapman and Hall/CRC.Google Scholar
  8. Baddeley, A., Rubak, E., and Turner, R. (2016), Spatial Point Patterns: Methodology and Applications with R, Interdisciplinary Statistics, Boca Raton, FL, USA: Chapman and Hall/CRC.zbMATHGoogle Scholar
  9. Baddeley, A., and Turner, R. (2005), “Spatstat: an R package for analyzing spatial point patterns,” Journal of Statistical Software, 12(6), 1–42.Google Scholar
  10. Baveye, P., Boast, C. W., Gaspard, S., Tarquis, A. M., and Millan, H. (2008), “Introduction to fractal geometry, fragmentation processes and multifractal measures: theory and operational aspects of their application to natural systems,” in Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems, West Sussex, England: John Wiley & Sons, pp. 11–67.Google Scholar
  11. Boer, M. M., Sadler, R. J., Wittkuhn, R. S., McCaw, L., and Grierson, P. F. (2009), “Long-term impacts of prescribed burning on regional extent and incidence of wildfires - evidence from 50 years of active fire management in SW Australian forests,” Forest Ecology and Management, 259(1), 132–142.Google Scholar
  12. Borys, P., Krasowska, M., Grzywna, Z. J., Djamgoz, M. B. A., and Mycielska, M. E. (2008), “Lacunarity as a novel measure of cancer cells behavior,” BioSystems, 94, 276–281.Google Scholar
  13. Caccetta, P., Collings, S., Devereux, A., Hingee, K. L., McFarlane, D., Traylen, A., Wu, X., and Zhou, Z. (2015), “Monitoring land surface and cover in urban and peri-urban environments using digital aerial photography,” International Journal of Digital Earth, pp. 457–475.Google Scholar
  14. Chappard, D., Legrand, E., Haettich, B., Chalès, G., Auvinet, B., Eschard, J., Hamelin, J., Baslé, M., and Audran, M. (2001), “Fractal dimension of trabecular bone: comparison of three histomorphometric computed techniques for measuring the architectural two-dimensional complexity,” Journal of Pathology, 195(4), 515–521.Google Scholar
  15. Cheng, Q. (1997), “Multifractal modeling and lacunarity analysis,” Mathematical Geology, 29(7), 919–932.Google Scholar
  16. Chiu, S. N., Stoyan, D., Kendall, W. S., and Mecke, J. (2013), Stochastic Geometry and Its Applications, 3 edn, Chichester, United Kingdom: John Wiley & Sons.zbMATHGoogle Scholar
  17. Cumbrera, R., Tarquis, A. M., Gascó, G., and Millán, H. (2012), “Fractal scaling of apparent soil moisture estimated from vertical planes of Vertisol pit images,” Journal of Hydrology, 452–453, 205–212.Google Scholar
  18. Diaz, S., Casselbrant, I., Piitulainen, E., Magnusson, P., Peterson, B., Pickering, E., Tuthill, T., Ekberg, O., and Akeson, P. (2009), “Progression of emphysema in a 12-month hyperpolarized 3He-MRI study: lacunarity analysis provided a more sensitive measure than standard ADC analysis,” Academic Radiology, 16(6), 700–707.Google Scholar
  19. Diggle, P. J. (1981), “Binary mosaics and the spatial pattern of heather,” Biometrics, 37(3), 531–539.Google Scholar
  20. Dong, P. (2000), “Test of a new lacunarity estimation method for image texture analysis,” International Journal of Remote Sensing, 21(17), 3369–3373.Google Scholar
  21. Dàvila, E., and Parés, D. (2007), “Structure of heat-induced plasma protein gels studied by fractal and lacunarity analysis,” Food Hydrocolloids, 21(2), 147–153.Google Scholar
  22. Feagin, R. A., Wu, X. B., and Feagin, T. (2007), “Edge effects in lacunarity analysis,” Ecological Modelling, 201(3–4), 262–268.Google Scholar
  23. Gould, D. J., Vadakkan, T. J., Poché, R. A., and Dickinson, M. E. (2011), “Multifractal and lacunarity analysis of microvascular morphology and remodeling,” Microcirculation, 18(2), 136–151.Google Scholar
  24. Griffith, J. A. (2004), “The role of landscape pattern analysis in understanding concepts of land cover change,” Journal of Geographical Sciences, 14(1), 3–17.Google Scholar
  25. Hall, P. (1985), “Resampling a coverage pattern,” Stochastic Processes and their Applications, 20(2), 231–246.MathSciNetzbMATHGoogle Scholar
  26. Karperien, A., Ahammer, H., and Jelinek, H. F. (2013), “Quantitating the subtleties of microglial morphology with fractal analysis,” Frontiers in Cellular Neuroscience, 7(3).Google Scholar
  27. Karperien, A. L. (2005), FracLac’s Advanced User Manual, Technical report, Charles Sturt University. Version 2.0f.Google Scholar
  28. Kautz, M., Düll, J., and Ohser, J. (2011), “Spatial dependence of random sets and its application to dispersion of bark beetle infestation in a natural forest,” Image Analysis & Stereology, 30(3), 123–131.zbMATHGoogle Scholar
  29. Kirkpatrick, L. A., and Weishampel, J. F. (2005), “Quantifying spatial structure of volumetric neutral models,” Ecological Modelling, 186(3), 312–325.Google Scholar
  30. Koch, K., Ohser, J., and Schladitz, K. (2003), “Spectral theory for random closed sets and estimating the covariance via frequency space,” Advances in Applied Probability, 35(3), 603–613.MathSciNetzbMATHGoogle Scholar
  31. Kolmogoroff, A., and Leontowitsch, M. (1933), “Zur Berechnung der mittleren Brownschen Fläche,” Physikalische Zeitschrift der Sowjetunion, 4, 1–13.zbMATHGoogle Scholar
  32. Kolmogorov, A., and Leontovitch, M. (1992), “On computing the mean Brownian area,” in Selected works of A.N. Kolmogorov, Volume II: Probability and mathematical statistics, ed. A. Shiryaev, Vol. 26 of Mathematics and its applications (Soviet series), Dordrecht–Boston–London: Kluwer, pp. 128–138.Google Scholar
  33. Lin, B., and Yang, Z. R. (1986), “A suggested lacunarity expression for Sierpinski carpets,” Journal of Physics A: Mathematical and General, 19(2), L49–L52.zbMATHGoogle Scholar
  34. Liu, C., and Zhang, W. (2010), Multiscale study on the spatial heterogeneity of remotely-sensed evapotranspiration in the typical oasis of Tarim Basin,, in Proceedings of the Sixth International Symposium on Digital Earth: Data Processing and Applications, Vol. 7841, International Society for Optics and Photonics, Beijing.Google Scholar
  35. Mandelbrot, B. B. (1983), Fractals and the Geometry of Nature, New York: W. H. Freeman and Company.Google Scholar
  36. Mandelbrot, B. B., and Stauffer, D. (1994), “Antipodal correlations and the texture (fractal lacunarity) in critical percolation clusters,” Journal of Physics A: Mathematical and General, 27(9), L237–L242.MathSciNetzbMATHGoogle Scholar
  37. Matheron, G. (1975), Random Sets and Integral Geometry, USA: John Wiley & Sons.zbMATHGoogle Scholar
  38. Mattfeldt, T., and Stoyan, D. (2000), “Improved estimation of the pair correlation function of random sets,” Journal of Microscopy, 200(2), 158–173.Google Scholar
  39. McIntyre, N. E., and Wiens, J. A. (2000), “A novel use of the lacunarity index to discern landscape function,” Landscape Ecology, 15(4), 313–321.Google Scholar
  40. Molchanov, I. S. (1997), Statistics of the Boolean Model for Practitioners and Mathematicians, Chichester: John Wiley & Sons.zbMATHGoogle Scholar
  41. — (2005), Theory of Random Sets, Probability and its Applications, USA: Springer.zbMATHGoogle Scholar
  42. Myint, S. W., and Lam, N. (2005), “A study of lacunarity-based texture analysis approaches to improve urban image classification,” Computers, Environment and Urban Systems, 29(5), 501–523.Google Scholar
  43. Niemelä, J. (1999), “Management in relation to disturbance in the boreal forest,” Forest Ecology and Management, 115(2), 127–134.Google Scholar
  44. Nordman, D. J., and Lahiri, S. N. (2004), “On optimal spatial subsample size for variance estimation,” The Annals of Statistics, 32(5), 1981–2027.MathSciNetzbMATHGoogle Scholar
  45. Nott, D. J., and Wilson, R. J. (2000), “Multi-phase image modelling with excursion sets,” Signal Processing, 80(1), 125–139.zbMATHGoogle Scholar
  46. Owen, K. K. (2011), Settlement indicators of wellbeing and economic status - lacunarity and vegetation,, in Proceedings of the Pecora 18 Symposium, American Society for Photogrammetry and Remote Sensing, Virginia, USA.Google Scholar
  47. — (2012), Geospatial and Remote Sensing-based Indicators of Settlement Type – Differentiating Informal and Formal Settlements in Guatemala City, PhD thesis, George Mason University.Google Scholar
  48. Pendleton, D. E., Dathe, A., and Baveye, P. (2005), “Influence of image resolution and evaluation algorithm on estimates of the lacunarity of porous media,” Physical Review E, 72(4).Google Scholar
  49. Picka, J. D. (1997), Variance-reducing modifications for estimators of dependence in random sets, Ph.D., The University of Chicago, Illinois, USA.Google Scholar
  50. — (2000), “Variance reducing modifications for estimators of standardized moments of random sets,” Advances in Applied Probability, 32(3), 682–700.MathSciNetzbMATHGoogle Scholar
  51. Pintilii, R., Andronache, I., Diaconu, D. C., Dobrea, R. C., Zeleňáková, M., Fensholt, R., Peptenatu, D., Drăghici, C., and Ciobotaru, A. (2017), “Using fractal analysis in modeling the dynamics of forest areas and economic impact assessment: Maramureş County, Romania, as a case study,” Forests, 8(1).Google Scholar
  52. Plotnick, R. E., Gardner, R. H., Hargrove, W. W., Prestegaard, K., and Perlmutter, M. (1996), “Lacunarity analysis: a general technique for the analysis of spatial patterns,” Physical Review E, 53(5), 5461–5468.Google Scholar
  53. Plotnick, R. E., Gardner, R. H., and O’Neill, R. V. (1993), “Lacunarity indices as measures of landscape texture,” Landscape Ecology, 8(3), 201–211.Google Scholar
  54. Quintanilla, J. A. (1999), “Microstructure functions for random media with impenetrable particles,” Physical Review E, 60(5), 5788–5794.Google Scholar
  55. — (2008), “Necessary and sufficient conditions for the two-point phase probability function of two-phase random media,” Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 464(2095), 1761–1779.MathSciNetzbMATHGoogle Scholar
  56. Quintanilla, J. A., Chen, J. T., Reidy, R. F., and Allen, A. J. (2007), “Versatility and robustness of Gaussian random fields for modelling random media,” Modelling and Simulation in Materials Science and Engineering, 15(4), S337.Google Scholar
  57. R Core Team (2018), “R: A Language and Environment for Statistical Computing”. URL: Accessed: 5 January 2019.
  58. Rankey, E. C. (2016), “On facies belts and facies mosaics: Holocene isolated platforms, South China Sea,” Sedimentology, 63(7), 2190–2216.Google Scholar
  59. Reiss, M. (2016), “frac2D: Fractal and Lacunarity methods for 2D digital images”. URL: Accessed: 31 July 2018.Google Scholar
  60. Reiss, M. A., Lemmerer, B., Hanslmeier, A., and Ahammer, H. (2016), “Tug-of-war lacunarity - a novel approach for estimating lacunarity,” Chaos, 26(11).Google Scholar
  61. Robbins, H. (1947), “Acknowledgement of priority,” Annals of Mathematical Statistics, 18, 297.MathSciNetGoogle Scholar
  62. Robbins, H. E. (1944), “On the measure of a random set,” The Annals of Mathematical Statistics, 15(1), 70–74.MathSciNetzbMATHGoogle Scholar
  63. Roces-Díaz, J. V., Díaz-Varela, E. R., and Ávarez Álvarez, P. (2014), “Analysis of spatial scales for ecosystem services: Application of the lacunarity concept at landscape level in Galicia (NW Spain),” Ecological Indicators, 36, 495–507.Google Scholar
  64. Schneider, R., and Weil, W. (2008), Stochastic and Integral Geometry, Probability and Its Applications, Germany: Springer-Verlag.Google Scholar
  65. Serra, J. P. (1982), Image Analysis and Mathematical Morphology, Norwich, Great Britain: Academic Press.zbMATHGoogle Scholar
  66. Shah, R. G., Salafia, C. M., Girardi, T., and Merz, G. S. (2016), “Villus packing density and lacunarity: markers of placental efficiency?,” Placenta, 48, 68–71.Google Scholar
  67. Shearer, B. L., Crane, C. E., Barrett, S., and Cochrane, A. (2007), “Phytophthora cinnamomi invasion, a major threatening process to conservation of flora diversity in the South-west Botanical Province of Western Australia,” Australian Journal of Botany, 55(3), 225–238.Google Scholar
  68. Sherman, M. (1996), “Variance Estimation for Statistics Computed from Spatial Lattice Data,” Journal of the Royal Statistical Society. Series B (Methodological), 58(3), 509–523.MathSciNetzbMATHGoogle Scholar
  69. Stoyan, D., and Stoyan, H. (1994), Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics, Chichester, United Kingdom: John Wiley & Sons.zbMATHGoogle Scholar
  70. Sui, D. Z., and Wu, X. B. (2006), “Changing patterns of residential segregation in a prismatic metropolis: a lacunarity-based study in Houston, 1980–2000,” Environment and Planning B: Planning and Design, 33(4), 559–579.Google Scholar
  71. Sun, W., Xu, G., Gong, P., and Liang, S. (2006), “Fractal analysis of remotely sensed images: a review of methods and applications,” International Journal of Remote Sensing, 27(22), 4963–4990.Google Scholar
  72. Sung, C. Y., Yi, Y.-j., and Li, M. (2013), “Impervious surface regulation and urban sprawl as its unintended consequence,” Land Use Policy, 32, 317–323.Google Scholar
  73. Torquato, S. (2002), Random Heterogeneous Materials: Microstructure and Macroscopic Properties, number 16, New York, USA: Springer Science+Business Media.zbMATHGoogle Scholar
  74. U. S. Geological Survey (2016), “SLC-off Products: Background”. URL: Accessed: 18th of April 2017.
  75. Valous, N. A., Mendoza, F., Sun, D., and Allen, P. (2009), “Texture appearance characterization of pre-sliced pork ham images using fractal metrics: Fourier analysis dimension and lacunarity,” Food Research International, 42(3), 353–362.Google Scholar
  76. Velazquez-Camilo, O., Bolaños-Reynoso, E., Rodriguez, E., and Alvarez-Ramirez, J. (2010), “Characterization of cane sugar crystallization using image fractal analysis,” Journal of Food Engineering, 100(1), 77–84.Google Scholar
  77. Vere-Jones, D. (1999), “On the fractal dimensions of point patterns,” Advances in Applied Probability, 31(3), 643–663.MathSciNetzbMATHGoogle Scholar
  78. Voss, R. F. (1986), “Characterization and measurement of random fractals,” Physica Scripta, (T13), 27–32. Proceedings of the 6th General Conference of the Condensed Matter Division of the European Physical Society.Google Scholar
  79. Wallace, J., Behn, G., and Furby, S. (2006), “Vegetation condition assessment and monitoring from sequences of satellite imagery,” Ecological Management & Restoration, 7(s1), S31–S36.Google Scholar
  80. Xue, S., Wang, H., Yang, H., Yu, X., Bai, Y., Tendu, A. A., Xu, X., Ma, H., and Zhou, G. (2017), “Effects of high-pressure treatments on water characteristics and juiciness of rabbit meat sausages: role of microstructure and chemical interactions,” Innovative Food Science & Emerging Technologies, 41, 150–159.Google Scholar

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© International Biometric Society 2019

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of Western AustraliaPerthAustralia
  2. 2.Data61CSIROPerthAustralia
  3. 3.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

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