Box-counting dimension revisited: Presenting an efficient method of minimizing quantization error and an assessment of the self-similarity of structural root systems

Frontiers in Plant Science

Volumn 7, Issue FEB2016, 2016, Pages

  Bouda, Martin ^a   Caplan, Joshua S ^b   Saiers, James E ^a  

^a Yale University   (United States)

^b RUTGERS UNIVERSITY   (United States)

Author keywords

Code:MATLAB; Fractal dimension; Numerical optimization; Plant root growth; Root architecture; Self similarity

Indexed keywords

EID: 84960082649     PISSN: None     EISSN: 1664462X     Source Type: Journal    
DOI: 10.3389/fpls.2016.00149     Document Type: Article

References (59)

1. Aagaard, K., and Hartvigsen, G. (2014). Assessing spatial patterns of plant communities at varying stages of succession. Appl. Math. 5, 1842-1851. doi: 10.4236/am.2014.512177
2. Audet, C., and Dennis, J. (2003). Analysis of generalized pattern searches. SIAM J. Optim. 13, 889-903. doi: 10.1137/S1052623400378742
3. Bari, A., Ayad, G., Martin, A., Gonzalez-Andujar, J., Nachit, M., and Elouafi, I. (2004). "Fractals and plant water use efficiency," in Thinking in Patterns: Fractals and Related Phenomena in Nature, 8th International Conference on Thinking in Patterns -Fractals and Related Phenomena in Nature, ed M. Novak (Vancouver, BC), 315-316. doi: 10.1142/9789812702746-0029
4. Barthélémy, D., and Caraglio, Y. (2007). Plant architecture: a dynamic, multilevel and comprehensive approach to plant form, structure and ontogeny. Ann. Bot. 99, 375-407. doi: 10.1093/aob/mcl260
5. Barto, E. K., and Cipollini, D. (2009). Garlic mustard (alliaria petiolata) removal method affects native establishment. Invas. Plant Sci. Manag. 2, 230-236. doi: 10.1614/IPSM-09-011.1
6. Berntson, G. (1994). Root systems and fractals - how reliable are calculations of fractal dimensions. Ann. Bot. 73, 281-284. doi: 10.1006/anbo.1994.1033
7. Berntson, G., and Stoll, P. (1997). Correcting for finite spatial scales of self-similarity when calculating the fractal dimensions of real-world structures. Proc. R. Soc. B Biol. Sci. 264, 1531-1537. doi: 10.1098/rspb.1997.0212
8. Boudon, F., Godin, C., Pradal, C., Puech, O., and Sinoquet, H. (2006). Estimating the fractal dimension of plants using the two-surface method: an analysis based on 3d-digitized tree foliage. Fract. Comp. Geometry Patt. Scaling Nat. Soc. 14, 149-163. doi: 10.1142/s0218348x06003179
9. Brodkey, R. S. (1966). "Fluid motion and mixing," in Mixing: Theory and Practice, Vol. 1, eds V. W. Uhl and J. B. Gray (New York, NY: Academic Press), 7-111.
10. Costes, E., Lauri, P. E., Simon, S., and Andrieu, B. (2013). Plant architecture, its diversity and manipulation in agronomic conditions, in relation with pest and pathogen attacks. Eur. J. Plant Pathol. 135, 455-470. doi: 10.1007/s10658-012-0158-3
11. Da Silva, D., Boudon, F., Godin, C., Puech, O., Smith, C., and Sinoquet, H. (2006). "A critical appraisal of the box counting method to assess the fractal dimension of tree crowns," in Advances in Visual Computing, Pt. 1, Vol. 4291 Lecture Notes in Computer Science, 2nd International Symposium on Visual Computing, eds G. Bebis, R. Boyle, B. Parvin, D. Koracin, P. Remagnino, A. Nefian, G. Meenakshisundaram, V. Pascucci, J. Zara, J. Molineros, H. Theisel, and T. Malzbender (Lake Tahoe, NV), 751-760. doi: 10.1007/11919476-75
12. Danjon, F., Caplan, J. S., Fortin, M., and Meredieu, C. (2013). Descendant root volume varies as a function of root type: estimation of root biomass lost during uprooting in pinus pinaster. Front. Plant Sci. 4:402. doi: 10.3389/fpls.2013.00402
13. Danjon, F., Sinoquet, H., Godin, C., Colin, F., and Drexhage, M. (1999). Characterisation of structural tree root architecture using 3d digitising and amapmod software. Plant Soil 211, 241-258. doi: 10.1023/A:1004680824612
14. Dannowski, M., and Block, A. (2005). Fractal geometry and root system structures of heterogeneous plant communities. Plant Soil 272, 61-76. doi: 10.1007/s11104-004-3981-2
15. Dibble, E. D., and Thomaz, S. M. (2009). Use of fractal dimension to assess habitat complexity and its influence on dominant invertebrates inhabiting tropical and temperate macrophytes. J. Freshwater Ecol. 24, 93-102. doi: 10.1080/02705060.2009.9664269
16. Dupuy, L. X., Fourcaud, T., Lac, P., and Stokes, A. (2007). A generic 3d finite element model of tree anchorage integrating soil mechanics and real root system architecture.Am. J. Bot. 94, 1506-1514. doi: 10.3732/ajb.94.9.1506
17. Dutilleul, P., Han, L., and Smith, D. L. (2008). Plant light interception can be explained via computed tomography scanning: demonstration with pyramidal cedar (thuja occidentalis, fastigiata). Ann. Bot. 101, 19-23. doi: 10.1093/aob/mcm273
18. Dutilleul, P., Han, L., Valladares, F., and Messier, C. (2015). Crown traits of coniferous trees and their relation to shade tolerance can differ with leaf type: a biophysical demonstration using computed tomography scanning data. Front. Plant Sci. 6:172. doi: 10.3389/fpls.2015.00172
19. Dzierzon, H., Sievanen, R., Kurth, W., Perttunen, J., and Sloboda, B. (2003). Enhanced possibilities for analyzing tree structure as provided by an interface between different modelling systems. Silva Fennica 37, 31-44. doi: 10.14214/sf.510
20. Eghball, B., Settmi, J., Maranville, J., and Parkhurst, A. (1993). Fractal analysis for morphological description of corn roots under nitrogen stress. Agron. J. 85, 287-289. doi: 10.2134/agronj1993.00021962008500020023x
21. Eshel, A. (1998). On the fractal dimensions of a root system. Plant Cell Environ. 21, 247-251. doi: 10.1046/j.1365-3040.1998.00252.x
22. Falconer, K. (2003). Fractal Geometry: Mathematical Foundaitons and Applications. Hoboken, NJ: John Wiley & Sons. doi: 10.1002/0470013850
23. Ferreiro, N., Giorgi, A., and Feijoo, C. (2013). Effects of macrophyte architecture and leaf shape complexity on structural parameters of the epiphytic algal community in a pampean stream. Aquat. Ecol. 47, 389-401. doi: 10.1007/s10452-013-9452-1
24. Foroutan-pour, K., Dutilleul, P., and Smith, D. (1999). Advances in the implementation of the box-counting method of fractal dimension estimation. Appl. Math. Comput. 105, 195-210. doi: 10.1016/S0096-3003(98)10096-6
25. Games, P. A., and Howell, J. F. (1976). Pairwise multiple comparison procedures with unequal n's and/or variances: a monte carlo study. J. Educ. Behav. Stat. 1, 113-125. doi: 10.3102/10769986001002113
26. Gibbons, J., and Chakraborti, S. (2003). Nonparametric Statistical Inference, Fourth Edition: Revised and Expanded, Statistics: A Series of Textbooks and Monographs. New York, NY: Taylor & Francis.
27. Godin, C., Costes, E., and Sinoquet, H. (1999). A method for describing plant architecture which integrates topology and geometry. Ann. Bot. 84, 343-357. doi: 10.1006/anbo.1999.0923
28. Gonzato, G., Mulargia, F., and Ciccotti, M. (2000). Measuring the fractal dimensions of ideal and actual objects: implications for application in geology and geophysics.Geophys. J. Int. 142, 108-116. doi: 10.1046/j.1365-246x.2000.00133.x
29. Grift, T. E., Novais, J., and Bohn, M. (2011). High-throughput phenotyping technology for maize roots. Biosyst. Eng. 110, 40-48. doi: 10.1016/j.biosystemseng.2011.06.004
30. Halley, J., Hartley, S., Kallimanis, A., Kunin, W., Lennon, J., and Sgardelis, S. (2004). Uses and abuses of fractal methodology in ecology. Ecol. Lett. 7, 254-271. doi: 10.1111/j.1461-0248.2004.00568.x
31. Hamburger, D., Biham, O., and Avnir, D. (1996). Apparent fractality emerging from models of random distributions. Phys. Rev. E 53, 3342-3358. doi: 10.1103/PhysRevE.53.3342
32. Han, L., Dutilleul, P., Prasher, S. O., Beaulieu, C., and Smith, D. L. (2008). Assessment of common scab-inducing pathogen effects on potato underground organs via computed tomography scanning. Phytopathology 98, 1118-1125. doi: 10.1094/PHYTO-98-10-1118
33. Izumi, Y., and Iijima, M. (2002). Fractal and multifractal analysis of cassava root system grown by the root-box method. Plant Prod. Sci. 5, 146-151. doi: 10.1626/pps.5.146
34. Karperien, A. (2007-2012). FracLac for ImageJ Version 2.5. Available online at:http://rsb.info.nih.gov/ij/plugins/fraclac/FLHelp/Introduction.htm.
35. Ketipearachchi, K. W., and Tatsumi, J. (2000). Local fractal dimensions and multifractal analysis of the root system of legumes. Plant Prod. Sci. 3, 289-295. doi: 10.1626/pps.3.289
36. Lontoc-Roy, M., Dutilleul, P., Prasher, S., Han, L., and Smith, D. (2005). Computed tomography scanning for three-dimensional imaging and complexity analysis of developing root systems. Can. J. Bot. Revue Can. Bot. 83, 1434-1442. doi: 10.1139/b05-118
37. Lontoc-Roy, M., Dutilleul, P., Prasher, S. O., Han, L., Brouillet, T., and Smith, D. L. (2006). Advances in the acquisition and analysis of ct scan data to isolate a crop root system from the soil medium and quantify root system complexity in 3-d space.Geoderma 137, 231-241. doi: 10.1016/j.geoderma.2006.08.025
38. Lynch, J., and Vanbeem, J. (1993). Growth and architecture of seedling roots of common bean genotypes. Crop Sci. 33, 1253-1257. doi: 10.2135/cropsci1993.0011183X003300060028x
39. Mandelbrot, B. (1967). How long is coast of britain - statistical self-similarity and fractional dimension. Science 156, 636. doi: 10.1126/science.156.3775.636
40. Mandelbrot, B. (1983). Fractal Geometry of Nature. New York, NY: W.H. Freeman & Co. doi: 10.1119/1.13295
41. Manschadi, A. M., Hammer, G. L., Christopher, J. T., and deVoil, P. (2008). Genotypic variation in seedling root architectural traits and implications for drought adaptation in wheat (triticum aestivum L.). Plant Soil 303, 115-129. doi: 10.1007/s11104-007-9492-1
42. Masi, C., and Maranville, J. (1998). Evaluation of sorghum root branching using fractals.J. Agric. Sci. 131, 259-265. doi: 10.1017/S0021859698005826
43. Mathworks (2014). Matlab and Statistics Toolbox Release 2014b. Natick, MA: Mathworks.
44. Nielsen, K., Lynch, J., and Weiss, H. (1997). Fractal geometry of bean root systems: correlations between spatial and fractal dimension. AM. J. Bot. 84, 26-33. doi: 10.2307/2445879
45. Nielsen, K., Miller, C., Beck, D., and Lynch, J. (1999). Fractal geometry of root systems: field observations of contrasting genotypes of common bean (phaseolus vulgaris L.) grown under different phosphorus regimes. Plant Soil 206, 181-190. doi: 10.1023/A:1004499224717
46. Oppelt, A., Kurth, W., Dzierzon, H., Jentschke, G., and Godbold, D. (2000). Structure and fractal dimensions of root systems of four co-occurring fruit tree species from botswana. Ann. For. Sci. 57, 463-475. doi: 10.1051/forest:2000135
47. Ozier-Lafontaine, H., Lecompte, F., and Sillon, J. (1999). Fractal analysis of the root architecture of gliricidia sepium for the spatial prediction of root branching, size and mass: model development and evaluation in agroforestry. Plant Soil 209, 167-180. doi: 10.1023/A:1004461130561
48. Pierce, S. C., Koontz, M. B., Pezeshki, S. R., and Kroeger, R. (2013). Response of salix nigra [Marsh.] cuttings to horizontal asymmetry in soil saturation. Environ. Exp. Bot.87, 137-147. doi: 10.1016/j.envexpbot.2012.10.003
49. Reeve, R. (1992). A warning about standard errors when estimating the fractal dimension. Comput. Geosci. 18, 89-91. doi: 10.1016/0098-3004(92)90061-U
50. Ruszczynski, A. (2006). Nonlinear Optimization. Princeton, NJ: Princeton University Press.
51. Subramanian, S., Han, L., Dutilleul, P., and Smith, D. L. (2015). Computed tomography scanning can monitor the effects of soil medium on root system development: an example of salt stress in corn. Front. Plant Sci. 6:256. doi: 10.3389/fpls.2015.00256
52. Szustalewicz, A. (2007). "Minimal coverage of investigated object when seeking for its fractal dimension," in Advances in Information Processing and Protection, International Advanced Computer Systems Conference, eds J. Pejas and K. Saeed (Miedzyzdroje), 117-128. doi: 10.1007/978-0-387-73137-7-10
53. Tatsumi, J., Yamauchi, A., and Kono, Y. (1989). Fractal analysis of plant-root systems.Ann. Bot. 64, 499-503.
54. Valdés-Rodríguez, O. A., Sánchez-Sánchez, O., Pérez-Vázquez, A., Caplan, J. S., and Danjon, F. (2013). Jatropha curcas L. root structure and growth in diverse soils. Sci. World J. 2013:827295. doi: 10.1155/2013/827295
55. van Noordwijk, M., Spek, L. Y., and de Willigen, P. (1994). Proximal root diameter as predictor of total root size for fractal branchng models. Plant Soil 164, 107-117. doi: 10.1007/BF00010116
56. Walk, T., Van Erp, E., and Lynch, J. P. (2004). Modelling applicability of fractal analysis to efficiency of soil exploration by roots. Ann. Bot. 94, 119-128. doi: 10.1093/aob/mch116
57. Wang, H., Siopongco, J., Wade, L. J., and Yamauchi, A. (2009). Fractal analysis on root systems of rice plants in response to drought stress. Environ. Exp. Bot. 65, 338-344. doi: 10.1016/j.envexpbot.2008.10.002
58. Yang, Y., Tang, M., Sulpice, R., Chen, H., Tian, S., and Ban, Y. (2014). Arbuscular mycorrhizal fungi alter fractal dimension characteristics of Robinia pseudoacacia L. seedlings through regulating plant growth, leaf water status, photosynthesis, and nutrient concentration under drought stress. J. Plant Growth Regul. 33, 612-625. doi: 10.1007/s00344-013-9410-0
59. Zeide, B., and Pfeifer, P. (1991). A method for estimation of fractal dimension of tree crowns. For. Sci. 37, 1253-1265.