ESTIMATION OF FRACTAL DIMENSION ON INNER STRUCTURE OF LEAF SAMPLES BY USING THE BOX COUNTING METHOD

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ESTIMATION OF FRACTAL DIMENSION ON INNER STRUCTURE OF LEAF SAMPLES BY USING THE BOX COUNTING METHOD

Elio Conte^1,2 & Maria Pieralice³

1I.S.P A., CNR- Bari, Italy

2School of International Advanced Studies on Applied Theoretical and non Linear Methodologies of Physics, Bari, Italy

3Department of Biomedical Sciences and Oncology, University of Bari “Aldo Moro”, Italy Corresponding Author: [email protected]

ABSTRACT

Organisms support continual exchange with the environment so that they maintain in a state far from their thermodynamic equilibrium. The plants maintain themselves under low entropy conditions, a necessary prerequisite to life. The concept of fractal dimension to describe structures, which look the same at all length scales, was first proposed by Mandelbrot Objects are usually referred to as self-similar to indicate their scale-invariant structure. The common characteristic of such fractal objects is that their length depends on the length scale used to measure it, and the fractal dimension tells us the precise nature of this dependence. Estimation of fractal dimension of leaf shape was recently performed form various authors. We estimated Fractal Dimension of different kinds of leaves looking at their inner structure until to the cellular nucleus. The results of the applied methodology resulted rather satisfactory so that in following papers we will apply it to investigation of plant structures under different experimental conditions as plant stress and per oxidation.

1. INTRODUCTION

Plants as all the other living beings are open systems supporting constantly material and energetic exchanges with the environment. Under such conditions of continual exchange with the environment the organisms can be maintained in a state far from their thermodynamic equilibrium. The plants maintain themselves under low entropy conditions, a necessary prerequisite to life. Let us give a rapid look to the basic structures. The stomata are the main path for higher plants to exchange matter and energy with environment. Stomata operate responding homogeneously for heterogeneously to environmental conditions. By varying the width of the stomata pores plants are able to control the flux of CO₂ and H₂O between the leaf and the surround air in response to changes in environmental factors. However, not all leaf stomata are open at the same time at a particular moment. This behaviour is named patchy stomata closure, in which stomata of different locations on the leaf show different responses to similar perturbations. According to Cordon et al. [1], during strong oscillatory behaviour or even at a steady-state stomata conductance, a different number of behaviours in the dynamics of the photosynthetic activity may be found.

The cuticle is the protective layer covering the outer cell layer of the green, aerial parts of land plants. Cuticles protect plants against desiccation, UV radiation and various kinds of physical, chemical and (micro) biological agents. Biological systems from the molecular to the ecological level are often controlled by non-linear feedback mechanisms. The non linearity is often considered the antechamber of possible chaotic dynamics [2]. Some authors have observed chaotic dynamics in plants in different physiological processes. The possible existence of chaotic dynamics in endogenous rhythms in certain CAM species was identified [3]. Other authors [4] found that varying light intensities caused changes in the patterns of photosynthetic temporal behaviours, ranging from periodic to chaotic behaviour. In other studies [5] it was observed period-doubling and period-tripling in higher plants, which suggested two different routes to chaos in leaf temperature responses of maize, tomato, and brewed plants.

Although the stomata response to the environment involves complex mechanisms influenced by many factors, stomata movements appear to be governed by two principal control circuits, one involving gradients of CO2 and the other involving gradients of H2O vapour. Patch wise stomata closure is especially pronounced under stress situations, such as water deficit [6].In addition to the use of methodologies for analysis of non linear , possibly chaotic, regimes, it has became of valuable interest in the last decade the use of fractal analysis with particular interest in physiology [2]

The concept of fractal dimension to describe structures, which look the same at all length scales, was first proposed by Mandelbrot [7]. Such objects are usually referred to as self-similar to indicate their scale-invariant structure. In simple terms, the common characteristic of such fractal objects is that their length depends on the length scale used to measure it, and the fractal dimension tells us the precise nature of this dependence. Estimation of fractal dimension of leaf shape was recently performed form various authors [8]. The aim of the presentstudy is to examine and to determine the morphology of the system (the leaf) under consideration.

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While so many studies have been devoted to the analysis of the rough profile of leaf, other research has started to be performed, devoted to the analysis of Fractal Dimension to the inner structure of leaf. These last studies are promising since may give important information about the status of the plant, and its condition as example under stress or per oxidation. In order to advance in this field it is required first of all to attempt to establish standard methodologies of analysis. This is the initial perspective that we attempt to start to delineate by the present paper.

Consequently, the aim of the present study was to estimate the possible fractal dimension of a sample of leaf in order to begin to rationalize how to standardize the methodology for subsequent studies on plant physiology.

2. MATERIALS AND METHODS

Samples (1 cm x1cm) of leaf of Pelargonium peltatum, of Pelargonium senile, of Saintpaula reticule and of Alliums cape L were collected, cleaned from duct and wiped and thus submitted to direct observation by digital microscope at 100x .

The image sensor was 174.5 inch colour CMOS (310 k effective pixels). The colour was 24 bit RGB. As computer interface an USB 2.0 interface was used. Finally a 100x magnification was used for 15” screen.

The results are given respectively in Figures 1,2,3,4.,………

Figure 1 Pelargonium peltatum

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Figure 2 Pelargonium zonale

Figure 3 Saintpaulia rupicola

Figure 4 Allium cepa L

Figure 5 Allium cepa L

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Figure 6 Allium cepa L

Figure 7 Cell of Allium cepa L

Figure 8 Cell of Allium cepa L

Figure 9 Nucleus of Cell of Allium cepa L

The images were analysed using program Haifa 5.1 provided by O. Mescal, based on an improved box counting method where binary images are covered with different grids (box length ε), and the number of boxes N(ε) required to cover the structures of the images is recorded

If an object is fractal, N(ε) increases according to the relation N(

 )  C 

where D is fractal dimension and C is a constant. The fractal dimension can be obtained as

)) log(

/ ) ( log ( lim

0

 



N

D  

 )

The Haifa code is based on counting of squares (black, white, and partially black) from a squared network behind the fractal figure. The difference between calculated and exact values of fractal dimensions obtained using Haifa is very small. The use of the Haifa software for estimation of Fractal Dimension of the digitalized image is particularly advantageous in the sense that the fractal functions implemented in Haifa 5.1 software (for reference on the author and the software see directly the site http://www.fch.vutbr.cz/lectures/imagesci/). Includes three categories of the necessary boxes for the box-counting method: N

W, which contains only the white background; N_B which covers only the black segmented object, N_BW which covers the border of the black object. According to such methodology we obtain three fractal dimension estimations FD_BW. , FD_B+BW and FD_W+BW

In the present paper we use the box counting method as arranged in Haifa. In the subsequent papers we will experience also the method that was introduced from one of the authors [9] previously in analysis of skin pigmented lesions.

3. RESULTS

Let us give the results that we obtained. Each time we give estimation of the box counting

method and fractal dimension estimation also evaluated as function of the intensity

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Pelargonium peltatum

Pelargonium zonale

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Saintpaulia rupicola

Allium cepa L-4

54 Allium cepa L-5

55 Allium cepa L-6

Cell of Allium cepa L-7

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Cell of Allium cepa L-8

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Nucleus of Cell of Allium cepa L-9

The results are summarized in Table 1 .

Table 1: Estimation of Fractal Dimension

BW B+BW W+BW F.D.

Pelargonium peltatum

0.6283  1.9998 ± 0.0000 0.7088 ± 0.0276 0.6283 

Pelargonium zonale

1.3069 ± 0.0465 1.9886 ± 0.0017 1.4124 ± 0.0258 1.3069 ± 0.0465

0.7423 ± 0.0319 1.9996 ± 0.0001 0.8215 ± 0.0291 0.7423 ± 0.0319

1.4333 ± 0.0891 1.9976 ± 0.0010 1.3824 ± 0.0762 1.4333 ± 0.0891

1.6867 ± 0.0707 1.9682 ± 0.0083 1.7425 ± 0.0354 1.6867 ± 0.0707

1.6884 ± 0.0775 1.9233 ± 0.0141 1.8379 ± 0.0239 1.6884 ± 0.0775 Cell of Allium cepa L 7 1.8514 ± 0.0656 1.8651 ± 0.0377 1.9478 ± 0.0194 1.8514 ± 0.0656 Cell of Allium cepa L 8 1.6900 ± 0.0709 1.916 ± 0.0150 1.8471 ± 0.0212 1.6900 ± 0.0709 Nucleus of Cell of

Allium cepa L 9

1.8514 ± 0.0656 1.8651 ± 0.0377 1.9478 ± 0.0194 1.8514 ± 0.0656

58 4. CONCLUSIONS

In the present paper we used the Haifa software from the Institute of Physical and Applied Chemistry, Brno University of Technology, Czech Republic. We obtained various indexes of fractal dimension which characterise the structure of the examined object as the black plane, the black-white border of black object, and the properties of white background. According to the authors the fractal dimension is the slope of the straight line BW.

This method has given satisfactory results that of course in the subsequent papers we will compare with the other different method that we mentioned previously [2].. The obtained results evidence that the fractal dimension increases from 0.6283

 0 . 0243

Such results evidence the importance to apply fractal analysis in these studies since they give direct and valuable information on the level of complexity of the examined structure and maygive important information on the status of the structure in relation to different experimental condition as example the condition of stress of the plant, or its condition of per oxidation. The estimation of the complexity that we reach by using fractal dimension also helps us in understanding basic mechanisms as the necessity of light penetration through the plant. From Table 1 we deduce that some samples gave reduced values of FD respect to the other ones. This result could be indication of the existing relation between structure of the leaf and need of light and temperature requirement. We suggest that some plants need a higher complexity to develop by comparison with the other species.

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