Abstract
Different summarized shape indices, like mean shape index (MSI) and area weighted mean shape index (AWMSI) can change over multiple size scales. This variation is important to describe scale heterogeneity of landscapes, but the exact mathematical form of the dependence is rarely known. In this paper, the use of fractal geometry (by the perimeter and area Hausdorff dimensions) made us able to describe the scale dependence of these indices. Moreover, we showed how fractal dimensions can be deducted from existing MSI and AWMSI data. In this way, the equality of a multiscale tabulated MSI and AWMSI dataset and two scale-invariant fractal dimensions has been demonstrated.
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References
Angiolini C, Maccherini S, Chiarucci A, Gabellini A, De Dominicis V (1999) Memoria illustrativa alla carta della vegetazione della Riserva Naturale “Poggio all’Olmo” (Grosseto, Toscana Meridionale). Atti Mus Stor Nat Maremma 19:29–47
Avnir D, Biham O, Lidar D, Malcai O (1998) Is the geometry of nature fractal? Science 279:39–40. doi:10.1126/science.279.5347.39
Bogaert J, Myneni RB, Knyazikhin Y (2002) A mathematical comment on the formulae for the aggregation index and the shape index. Landsc Ecol 17:87–90. doi:10.1023/A:1015204923187
Burrough PA (1981) Fractal dimensions of landscapes and other environmental data. Nature 294:240–242. doi:10.1038/294240a0
Carranza ML, Acosta A, Ricotta C (2007) Analyzing landscape diversity in time: the use of Rényi’s generalized entropy function. Ecol Indic 7:505–510. doi:10.1016/j.ecolind.2006.05.005
Cracknell AP (1998) Synergy in remote sensing: what’s in a pixel? Int J Remote Sens 19:2025–2047. doi:10.1080/014311698214848
DiBari JN (2003) Scaling exponent and rank-size distributions as indicators of landscape character and change. Ecol Indic 3:275–284. doi:10.1016/j.ecolind.2003.11.006
Feder J (1988) Fractals. Plenum Press, New York
Fisher P (1997) The pixel: a snare and a delusion. Int J Remote Sens 18:679–685. doi:10.1080/014311697219015
Frohn RC, Hao Y (2006) Landscape metric performance in analyzing two decades of deforestation in the Amazon Basin of Rondonia, Brazil. Remote Sens Environ 100:237–251. doi:10.1016/j.rse.2005.10.026
Heegaard E, Okland H, Bratli H, Dramstad E, Engan G, Pedersen O, Solstad H (2007) Regularity of species richness relationships to patch size and shape. Ecography 30:589–597
Imre A (1992) Problems of measuring the fractal dimension by the slit-island method. Scr Metall 27:1713–1716. doi:10.1016/0956-716X(92)90007-2
Imre AR (2006) Artificial fractal dimension obtained by using perimeter-area relationship on digitalized images. Appl Math Comput 173:443–449. doi:10.1016/j.amc.2005.04.042
Imre AR, Bogaert J (2004) The fractal dimension as a measure of the quality of habitats. Acta Biotheor 52:41–56. doi:10.1023/B:ACBI.0000015911.56850.0f
Imre AR, Bogaert J (2006) The Minkowski–Bouligand dimension and the interior-to-edge ratio of habitats. Fractals 14:49–53. doi:10.1142/S0218348X06003027
Jelinski DE, Wu J (1996) The modifiable areal unit problem and implications for landscape ecology. Landsc Ecol 11:129–140. doi:10.1007/BF02447512
Johnson DD, Howarth PJ (1987) The effects of spatial resolution on land cover/land use theme extraction from airborne digital data. Can J Rem Sens 13:68–75
Krummel JR, Gardner RH, Sugihara G, O’Neill RV, Coleman PR (1987) Landscape patterns in disturbed environment. Oikos 48:321–324. doi:10.2307/3565520
Laba M, Downs R, Smith S, Welsh S, Neider C, White S, Richmond M, Philpot W, Baveye P (2008) Mapping invasive wetland plants in the Hudson River National Estuarine Research Reserve using quickbird satellite imagery. Remote Sens Environ 112:286–300. doi:10.1016/j.rse.2007.05.003
LaGro J Jr (1991) Assessing patch shape in landscape mosaics. Photogramm Eng Remote Sens 57:285–293
Li B-L (2000) Fractal geometry applications in description and analysis of patch patterns and patch dynamics. Ecol Model 132:33–50. doi:10.1016/S0304-3800(00)00303-3
Maccherini S, Chiarucci A, Selvi F, De Dominicis V (2001) Flora vascolare della Riserva Naturale di Poggio all’Olmo (Cinigiano, Grosseto). Atti Soc Tosc Sci Nat Mem Ser B 108:27–41
Mandelbrot BB (1982) The fractal geometry of Nature. Freeman, New York
Mandelbrot BB, Passoja DE, Paullay AJ (1984) Fractal character of fracture surfaces of metals. Nature 308:721–722. doi:10.1038/308721a0
Marignani M, Rocchini D, Torri D, Chiarucci A, Maccherini S (2008) Planning restoration in a cultural landscape in Italy using an object-based approach and historical analysis. Landsc Urban Plan 84:28–37. doi:10.1016/j.landurbplan.2007.06.005
Millington J, Perry G, Romero-Calcerrada R (2007) Regression techniques for examining land use/cover change: a case study of a mediterranean landscape. Ecosystems (N Y., Print) 10:562–578. doi:10.1007/s10021-007-9020-4
Milne BT (1988) Measuring the fractal geometry of landscapes. Appl Math Comput 27:67–79. doi:10.1016/0096-3003(88)90099-9
Moser D, Zechmeister HG, Plutzar C, Sauberer N, Wrbka T, Grabherr G (2002) Landscape patch shape complexity as an effective measure for plant species richness in rural landscapes. Landsc Ecol 17:657–669. doi:10.1023/A:1021513729205
O’Neill RV, Krummel JR, Gardner RH, Sugihara G, Jackson B, DeAngelis DL, Milne BT, Turner MG, Zygmunt B, Christensen SW, Dale VH, Graham RL (1988) Indices of landscape pattern. Landsc Ecol 1:153–162. doi:10.1007/BF00162741
Olsen ER, Ramsey RD, Winn DS (1993) A modified fractal dimension as a measure of landscape diversity. Photogramm Eng Remote Sens 59:1517–1520
Rocchini D (2005) Resolution problems in calculating landscape metrics. J Spat Sci 50:25–36
Rocchini D, Perry GLW, Salerno M, Maccherini S, Chiarucci A (2006) Landscape change and the dynamics of open formations in a natural reserve. Landsc Urban Plan 77:167–177. doi:10.1016/j.landurbplan.2005.02.008
Romero-Calcerrada R, Perry GLW (2004) The role of land abandonment in landscape dynamics in the SPA Encinares del rio Alberche y Cofio, Central Spain, 1984–1999. Landsc Urban Plan 66:217–232. doi:10.1016/S0169-2046(03)00112-9
Russ JC (1994) Fractal surfaces. Plenum Press, New York
Saura S, Castro S (2007) Scaling functions for landscape pattern metrics derived from remotely sensed data: are their subpixel estimates really accurate? ISPRS J Photogramm Remote Sens 62:201–216. doi:10.1016/j.isprsjprs.2007.03.004
Sugihara G, May RM (1990) Applications of fractals in ecology. Trends Ecol Evol 5:79–86. doi:10.1016/0169-5347(90)90235-6
Uuemaa E, Roosaare J, Mander Ü (2005) Scale dependence of landscape metrics and their indicatory value for nutrient and organic matter losses from catchments. Ecol Indic 5:350–369. doi:10.1016/j.ecolind.2005.03.009
Vicsek T (1989) Fractal growth phenomena. World Scientific, Singapore
Wu J, Jelinski DE, Luck M, Tueller PT (2000a) Multiscale analysis of landscape heterogeneity: scale variance and pattern metrics. Geogr Inf Sci 6:6–19
Wu XB, Thurow TL, Whisenant SG (2000b) Fragmentation and changes in hydrologic function of tiger bush landscapes, South–West Niger. J Ecol 88:790–800. doi:10.1046/j.1365-2745.2000.00491.x
Zurlini G, Zaccarelli N, Petrosillo I (2006) Indicating retrospective resilience of multi-scale patterns of real habitats in a landscape. Ecol Indic 6:184–204. doi:10.1016/j.ecolind.2005.08.013
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Imre, A.R., Rocchini, D. Explicitly Accounting for Pixel Dimension in Calculating Classical and Fractal Landscape Shape Metrics. Acta Biotheor 57, 349–360 (2009). https://doi.org/10.1007/s10441-009-9069-x
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DOI: https://doi.org/10.1007/s10441-009-9069-x