Chaos, Solitons & Fractals 69 (2014) 179–187

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Chaos, Solitons & FractalsNonlinear Science, and Nonequilibrium and Complex Phenomena

journal homepage: www.elsevier .com/locate /chaos

On 2D generalization of Higuchi’s fractal dimension

http://dx.doi.org/10.1016/j.chaos.2014.09.0150960-0779/� 2014 Elsevier Ltd. All rights reserved.

E-mail addresses: sladjana@imsi.rs, sladjana@imsi.bg.ac.rs, sladjana@ibiss.bg.ac.rs

Sladjana SpasicUniversity of Belgrade, Department of Life Sciences, Institute for Multidisciplinary Research, Kneza Viseslava 1, 11030 Belgrade, Serbia

a r t i c l e i n f o

Article history:Received 10 July 2013Accepted 25 September 2014

a b s t r a c t

We propose a new numerical method for calculating 2D fractal dimension (DF) of a surface.This method represents a generalization of Higuchi’s method for calculating fractal dimen-sion of a planar curve. Using a family of Weierstrass–Mandelbrot functions, we constructWeierstrass–Mandelbrot surfaces in order to test exactness of our new numerical method.The 2D fractal analysis method was applied to the set of histological images collected dur-ing direct shoot organogenesis from leaf explants. The efficiency of the proposed method indifferentiating phases of organogenesis is proved.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Fractal dimension [1], as a spatial metric, is broadlyapplied in many applications for signals and images analy-sis. Many methods have been developed for estimatingfractal dimension of curves [2–4] or images [5–7]. Man-delbrot [8] proposed the term ‘‘fractal dimension’’ to qual-ify objects with the same structure at all scales. Althoughthere are plenty of definitions, we used the term fractaldimension because it refers to a fractional (non-integer)number. Generally, fractal dimension, as a measure, couldbe related to quantifying the geometry of complex or noisyshapes and objects [9,10].

Over the last 15 years, fractal and multifractal geome-tries were applied extensively in many medical signaland image analysis applications like pattern recognition,texture analysis and segmentation. Many algorithms forthe calculation of fractal dimension such as the isarithm,variogram, probability, box-counting, and triangular prismmethods have been proposed and tested their benefits andtheir limits [11–16]. Although different fractal calculationalgorithms sometimes give different results, they arewidely applied [17]. For complex surfaces, it was foundthat the modified triangular prism method, proposed byClarke [5] and modified by Jaggi [13], was the most reliable

estimator when compared with the isarithm and vario-gram methods [18,19]. Higuchi proposed the method forcalculating fractal dimension of curve in plane [2]. Higu-chi’s method is frequently used for analysis of rat brainactivity in time domain after injuries [20,21] and to com-paring the brain complexity at the distinct brain levels dur-ing stable anesthesia induced by different anesthetics [22],also in monitoring the depth of anesthesia and sedation inhuman [23,24]. The changes of complex bursting neuronalactivity in garden snail in altered physiological states havebeen detected by Higuchi’s fractal dimension [25]. Thismethod has been confirmed as very accurate, simple andfast to assess changes in the EEG, ECoG and neuronal activ-ity in different experimental conditions. This motivated usto introduce a simple and fast new method to calculate 2Dfractal dimension of surface in the space, as a generaliza-tion of Higuchi’s method. We aimed to analyze histologicalpreparations in neuroscience, botany, etc.

In practice, different fractal methods such as so-calleddilation method of measuring the capacity fractaldimension, mass-radius and the box-counting methodshave been successfully applied in neuroscience, for neuro-nal cell image analysis [26,27]. Thus, mentioned methodsallows morphological characterization of retinal ganglioncells into different groups [26] as well as quantitativemorphological discrimination between proliferating andnon-proliferating microglia cells [27] and analysis ofdendritic tree images of cerebellar Purkinje neurons during

180 S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187

ontogeny and phylogeny [28]. Hausdorff’s dimension, alsoknown as the similarity dimension has been successfullyapplied for plant identification by analyzing leaf complex-ity [29] while Minkowski fractal dimension was used forleaf shape analysis [30]. Box-counting fractal dimensionenables discrimination between individuals of plantspecies Anthyllis cytisoides under different slope exposureby comparing branch architecture [31].

In geology and planetary science, fractal method relat-ing concentrations and distances allows separation of geo-chemical anomalies from background [32]. Morency andChapleau [33] applied the mass-radius and box-countingmethods to the characterization of urban-related statessuch as population dispersion or transit share examinedas a function of distance to central business district. As itcan be seen different fractal methods has been developedand adapted to cope with whole series of problems inmany applied scientific areas. Study by Jelinek and Fernan-dez [26] has confirmed that calculated fractal dimensionvalues of the same cell images could be different with dif-ferent applied fractal methods, even method that measuresame type of fractal dimension showed statistical differ-ences between fractal dimension values for identical cellgroup. So, in different field of science and especially in biol-ogy where the variability is rule more than a coincidence itis necessary to apply different fractal methods to obtaincomplete picture of structures, processes, etc. Higuchi frac-tal dimension has been already successfully used for bio-logical signal analysis (EEG, MEG, EMG) but practicallythere are no studies about application of Higuchi’s meth-ods for morphological analysis of cells and tissue especiallyof plant histological images. This point certainly encour-ages us to seek for an application of Higuchi fractal dimen-sion as a new method for study of plant cell images.

This paper is organized as follows: Section 2 introducesthe method for calculating fractal dimension and estimateserrors of the proposed method, based on its application onWeierstrass–Mandelbrot functions. In Section 3, wedescribe and discuss the application of 2D Higuchi’s fractaldimension to biological images. We discuss all results ofthis paper and give related conclusions in Section 4.

2. 2-D generalization of Higuchi’s fractal dimension

2.1. A method for calculating 2D Higuchi’s fractal dimension

Our idea on generalization is based on Higuchi’smethod for estimating fractal dimension values of a planarcurve [2]. All details concerning the originally proposedHiguchi’s method are described in Appendix 1. The gener-alization is related to the analysis of surface in space.

Let f represent a surface in 3-dimensional Euclideanspace. Graph of the function f, G(f) is the set of points(i, j,xij) R3, i = 1,. . .,M, j = 1,. . .,N,where xij = f(i,j).

Let X be a (M � N)-dimensional matrix where xij are itselements and i = 1,. . .,M, j = 1,. . .,N. Then the matrix X canbe seen as a discrete representation of the function f.

We link into triangles, three by three, the points of thegraph G(f) corresponding to the coordinates (i, j) on the xygrid, like in Fig. 1A. We chose the step equal to one,

meaning that we take all the points (i, j,xij) into account.The set of the triangles formed in that way approximatethe surface f. We can therefore approximate the area ofthe surface f by the sum of the areas of the resulting trian-gles. We then increase the step size and repeat the process.For the step size equal to two we choose every secondpoint from the starting matrix X along rows and columnsforming new matrix, etc.

To put it formally, let kmax, be a free parameter repre-senting the maximum step size. For k = 1,. . .,kmax, let usconstruct k new matrices Xm

k as:

Xmk ¼

xm;m xmþk;m xmþ2k;m ::: xmþpk;m

xm;mþk xmþk;mþk xmþ2k;mþk ::: xmþpk;mþk

::: ::: ::: ::: :::

xm;mþsk xmþk;mþsk xmþ2k;mþsk ::: xmþpk;mþsk

26666664

37777775ð1Þ

where m = 1, 2,. . .,k,p = int[(M-m)/k], s = int[(N-m)/k], andint(r) is integer part of the real number r. In that way, fork = 2 we formed new matrix Xm

2 , for k = 3 we formed matrixXm

3 ,... and so on up to the k = kmax.Then, we can approximate the surface f by triangles

whose vertexes are given by the matrices Xmk . We illus-

trated on Fig. 1 top view of such surfaces.The sum of the areas of the triangles, Am(k), was com-

puted for each of the k matrix Xmk representing surfaces:

AmðkÞ ¼ CX½M�m

k �

i¼1

X½N�mk �

j¼1jxmþði�1Þk;mþjk � xmþði�1Þk;mþðj�1Þkj�

�jxmþik;mþjk � xmþði�1Þk;mþjkjþjxmþik;mþjk � xmþik;mþðj�1Þkj�jxmþik;mþðj�1Þk � xmþði�1Þk;mþðj�1Þkj

�ð2Þ

where C ¼ 12k4

ðN�1Þ½N�m

k �ðM�1Þ½M�m

k �.

Am(k) was averaged for all m, forming the mean value ofthe surface area A(k) for each k = 1,...,kmax as

AðkÞ ¼Pk

m¼1AmðkÞk

ð3Þ

An array of mean values A(k) was obtained and S is theslope of the least squares linear best fit from the plot ofln(A(k)) versus ln(1/k2):

S ¼ lnðAðkÞÞ= lnð1=k2Þ: ð4Þ

The fractal dimension (DF) was estimated as

DF ¼ Sþ 1: ð5Þ

To underline again, N, M and kmax are the free parame-ters chosen by analyst. The value N �M represents thenumber of matrix elements of X.

2.2. Weierstrass–Mandelbrot Functions

In order to explore exactness of our new numericalmethod, we applied previously described approach toWeierstrass–Mandelbrot (WM) functions. These functionsare suitable [3,34–36] because their fractal dimension (D)is given theoretically. In what follows we will distinguishtheoretical fractal dimension, D, from the numericallyestimated one, DF.

Fig. 1. A graphical illustration of the method for construction k new matrices Xmk and computation the area Am(k) for each of the k matrix representing

surfaces, Xmk (top view of surfaces).

S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187 181

Let us observe a family of WM functions as cosine func-tions of t, having two parameters, c and H:

WcHðtÞ ¼

X1l¼1

c�lH cosð2pcltÞ ð6Þ

where 0 < H < 1, c > 1, and l 2 N The fractal (box-counting)dimension of the graph of the WM function defined with(6), is given by D = 2 � H [34–36]. Moreover, it is widelybelieved that the Hausdorff dimension of the graph ofWM function is also equal 2 � H, at least for most valuesof c , but this has not been proved rigorously [36]. Besico-vich-Ursell theorem in [35] gives the equality between2 � H and Hausdorff dimension of the graph of WM func-tion under certain conditions.

Let us construct a finite-sum approximation to a WMfunction Wc

HðtÞ defined with (6), for l = 1,. . .,200. Size ofanalyzed area is t [0, 1]. We generated Wc

HðtÞ family forthe following discrete values of parameters: ci = 1.3,1.5,2.0,3.0,4.0,5.0; i = 1, . . .,6 and Hj = 0.9,0.8,. . .,0.1; j = 1,. . .,9;and t [0,1]. We denote by Nc and NH number of differentc and H values respectively. Hence, Nc = 6 and NH = 9. So,we used the equation (6) and these parameters to form

(nx1)-dimensional vectors WcH . In order to generate a sur-

face from WM function (Eq.(6)), we formed (n � n)-dimen-sional matrices Mc

H given by:

McH ¼Wc

HðWcHÞ

T ð7Þ

In case of surface given by matrix McH , the fractal dimen-

sion of the contours along x and y - axes direction areequal, thus, the surface is isotropic. In the Section concern-ing ‘‘3-D fractals and texture characterization of spatialsurfaces’’ Klonowski [37] and Pentland in [38] wrote thatsurface’s fractal dimension is one plus the contours’dimension, even for anisotropic surfaces. Hence, we gener-ated a new family of WM functions as cosine functions of t,in matrix representation as Mc

Hwith given theoretical frac-tal dimension by D = 3 � H.

Consequently, DF range (2 < DF < 3) was evenly occu-pied: Dj = 2.1,2.2,. . .,2.9. Number of generated functions(matrices) was Nc NH = 54. We analyzed these functionsat different resolutions in order to examine how the reso-lution affects the error value of the calculated DF. For theresolution, we chose r = 100, 200, 400 points per linearinterval [0,1] so that dimensions of matrices Mc

H were

Fig. 2. Graphics of Weierstrass–Mandelbrot functions (A-I), for two parameters, c = 4 and Hj = 0.90,0.80,. . .,0.10, characterized by theoretical fractaldimension D (Dj = 2.1,2.2,. . .,2.9), an example of plane (J), a surface generated by random generated numbers (K), and surface representing Gaussian noise(L) with corresponding DF, calculated by our method and algorithm.

182 S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187

100 � 100, 200 � 200 or 400 � 400 respectively. Therefore,the number of analyzed WM functions (their surfaces)given by Mc

H was 3 � Nc NH = 162. For each of the 162 WMfunctions we estimated DF, so that DF values were calcu-lated for each square window (window sizeM � N = 100 � 100 points), without overlap, using kmax = 8.Let us remind that the value N x M represents the number

of windows elements (or whole matrix X), and kmax is thefree parameter representing the maximum step size for k(see Section 2.1). The parameter kmax determines the rangeof values for k. Thus, each (nxn)-dimensional matrix Mc

H

was divided into smaller (M � N)-dimensional matrices(square windows). For example, (400 � 400)-dimensionalmatrix Mc

H was divided into (100 � 100)-dimensional

Fig. 3. Square of Residuals (SR, panels A, B, C) and Mean Square Errors (MSE, panel D), obtained by applying our new method to a family of Weierstrass–Mandelbrot functions, by varying resolution (100,200,400 points per linear unit), with predefined values of parameter ci = 1.3,1.5,2.0,3.0, 4.0,5.0 andfractal dimension Dj = 2.1,2.2, 2.3,. . .,2.9. Averaged Squares of Residuals (AVG) are presented by bold black line on panels A, B, and C.

S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187 183

windows. Individual DF values from all squares windowswere averaged, in order to obtain a final DF value for a par-ticular matrix.

As an example, values of estimated DF and theoreticallygiven D for the family of WM functions, characterized byci = 4, and their graphs are presented on Fig. 2. Similarresults were obtained for other values of ci.

As illustration of the method exactness, we also exam-ined fractal dimension values of plane surface (Fig. 2J), sur-faces generated by random numbers (Fig. 2K), and surfacerepresenting Gaussian noise (Fig. 2L). Estimated (DF) andtheoretically given (D) fractal dimension values are pre-sented on Fig. 2J, K, L. In order to facilitate the interpreta-tion of results: the D value of plane or smooth surfacewas estimated to be �2, and the D of more complex sur-faces notably surfaces generated by random white noisewas estimated to be �3.

All the calculations described in Sections 2.1 and 2.2were performed in MatLab R2010a.

2.3. Error estimation of the proposed method – based on itsapplication on Weierstrass–Mandelbrot functions

In order to estimate the error of this method, weapplied the method from Sec. 2.1 to the same family of

Weierstrass–Mandelbrot functions. For easier interpreta-tion, we denote by Mi

j Weierstrass–Mandelbrot functioncharacterized by Dj and ci, with DFðM

ijÞ its fractal dimension

computed by the described procedure. We estimated threetypes of errors:

- Square of Residuals (SR)

SRðci;DjÞ ¼ DFðMijÞ � Dj

� �2ð8Þ

- Mean Square Error (MSE)

MSEðciÞ ¼1

NH

Xj

SRðci;DjÞ ð9Þ

- Averaged Squares of Residuals (AVG)

AVGðDjÞ ¼1

Nc

Xi

SRðci;DjÞ ð10Þ

We presented these errors on Fig. 3 by varying resolu-tion (100,200,400 points per linear unit), with predefinedvalues of parameter ci = 1.3,1.5,2.0,3.0,4.0,5.0 and fractaldimension Dj = 2.1,2.2,2.3,. . .,2.9.

By comparing Fig. 3A, B, and C for fixed value of c, it canbe noticed that SR values decrease with increase in resolu-tion. AVG defined by (10) have been monotonouslydecreasing with respect to the values of fractal dimension,

184 S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187

D, and sufficiently small (<0.004) for medium and highervalues (D > 2.4) of theoretical fractal dimension (seeFig. 3A,B,C).

Finally, Mean Square Errors defined by (9) mainlydecrease for higher resolutions, except in case of ci = 4.0,5.0. However, it results MSE < 0.018 in all cases (Fig. 3D).

3. Procedure for the application of 2D Higuchi’s fractaldimension to biological images

The suggested procedure for application of 2D fractalanalysis to biological images consists of two steps: imagepreprocessing (Fig. 4A) and estimating DF by presentedmethod with statistical analysis (Fig. 4B).

In this work, 2D fractal analysis method was applied tothe collection of microscopical images (Leitz DMRB photo-microscope Leica, Wetzlar, Germany, 10 � objective).Images for the analysis (Fig. 4A) have been obtained fromhistological preparations of samples collected during60 days of Tacitus bellus direct shoot organogenesis fromleaf explants in vitro [39]. Histological examination byexperts [39] revealed three successive stages of directshoot organogenesis: stage of initiation of shoot organo-genesis (1st stage), stage of meristemoids formation (2ndstage) and stage of shoot buds formation (3rd stage). Ouraim was to discover whether the DF could be useful inquantitative characterization of different stages oforganogenesis.

At the first step in preprocessing, the original histolog-ical images were converted to gray scale and auto contrastcorrection was performed using Adobe Photoshop 7.0.From each selected histological image the ‘‘informativepart’’ was cropped that represents a corresponding stageof shoot bud formation with adequately small part of sur-rounding tissue of primary leaf explant. The last step inpreprocessing to analysis (Fig. 4A) was resizing images toresolutions of 94 pixels/inch. Size of these new images,prepared for the analysis, was 14 � 8 inches (1316 � 752pixels or 989632 points). In the first step of images pro-cessing (Fig. 4B), gray scale images were converted intothe matrices with elements in R[0,1]. Conversion has per-formed with MatLab function im2double that converts the

Fig. 4. Block diagram of image pro

intensity image to double precision numeric field, rescalingthe data to numeric value in real interval [0,1] so that:black was converted to 0 and white to 1. Finally, for eachimage the resulting matrix with elements in R[0,1] isobtained. Matrix size was 1316 � 752 elements (points).

The resulting matrices were subjected to fractal analy-sis (Fig. 4B). We choose the parameters N = 100, M = 100and kmax = 8 on the basis of our preliminary analysis andexperiences with optimum choice of kmax value. Thus, eachmatrix was divided into smaller (M � N)-dimensionalmatrices (square windows). DF values were calculated foreach square window (window size M � N = 100 � 100points), without overlap, using kmax = 8. Individual DF val-ues from all squares windows were averaged, in order toobtain a final DF value for a particular image. It is possibleto calculate DF for whole images. However, this is not rec-ommended if the image is extremely inhomogeneous tex-ture. In such case DF value does not represent true measureand division into square windows is advisable.

We tested differences between DF values of histologicalimages in three phases of organogenesis at 94 pixel/inchresolution. We used One-way ANOVA with phase of organ-ogenesis as a factor with three levels (1st, 2nd and 3rdstage). The dependent variable in our analysis was the DFvalue of histological image. Therefore, we analyzed 18images: 6 images per each phase. Statistical analysis wasperformed by the use of SPSS 13.0 for Windows. All detailsabout the used statistical methods are shown in theAppendix 2.

On the Fig. 5, we presented one sample for each stage ofshoot organogenesis. Mean of fractal dimension values forthe 1st stage was 2.29 ± 0.03, for the 2nd stage 2.24 ± 0.05,and for the 3rd stage 2.19 ± 0.01. We note the decline of DFvalues during shoot meristem differentiation. One-wayANOVA test showed significant differences in DF valuesbetween three morphogenic stages (F = 16.552, p < 10�4)and Post-Hoc Duncan test differentiated phases into threeseparate subsets at the level of significance of 0.05(Table 1). We therefore proved that three successive stagesof direct shoot organogenesis are characterized by differ-ent DF values. Thus, each stage is different from the othertwo in terms of the DF values.

cessing techniques applied.

Fig. 5. Images obtained from histological preparations of samples collected during T. bellus direct shoot organogenesis from leaf explants in vitro (on the leftpanels – A) and their corresponding numerical matrix representation as surfaces (on the right panels – B) are presented. DF values of histological images arecalculated using our new method, using parameters N = 100 and kmax = 8. Mean DF and standard deviation for each stage of organogenesis are presented (B).

Table 1Differences in mean FD values for three morphogenic stages were tested.Results of Post-Hoc Duncan Test are displayed. a = 0.05.

Stage n Subset

1 2 3

3rd stage 6 2.1867 2.29331.0002nd stage 6 2.2400

1st stage 6

Sig. 1.000 1.000

S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187 185

4. Discussion and conclusion

In this paper we suggest a new method to compute 2Dfractal dimension measure as a generalization of Higuchi’sfractal dimension originally proposed by Higuchi [2].

We used a family of Weierstrass–Mandelbrot functionsand generated surfaces which approximate their graphswith planar triangles. Theoretically known fractal dimen-sion values of the WM functions (D) were used in orderto prove the method. Our new method is more accuratefor higher fractal dimension values as demonstrated byAveraged Squares of Residuals computation.

Our method is similar to triangular prism method [5],but it is much simpler. Both of them are geometrical meth-ods as are the covering blanket [40], the flat structuring ele-ment [41,42], box dimension and lacunarity [43,44]. Asmany others, our method can be used for computing fractaldimension on greytone images. Rivest, Soille and colleague[45–47] showed that robustness of image measurementsare obtained if the two following conditions are satisfied:invariance to translations and rotations, and dimensionalbehavior. In [47], the authors analyzed some methods forthe fractal dimension of grayscale images and their

186 S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187

dimensional behavior. Some of them like triangular prismmethod, covering blanket, and box dimension are notdimensional. They claim that ‘‘the surface area measure-ment of the graph of a greytone image is not dimensional:

S KtKxGð f Þ½ �–kktt kkx

x S½Gð f Þ� ð11Þ

where G(f) denote graph of the function f, S surface, Kx ascaling of factor kx of the image plane, and Kt a lineartransformation of the intensity values of gain kt and kx, kt

are the real numbers.In this paper, our goal was to generalize Higuchi’s

method through the simple and fast new method of calcu-lating 2D fractal dimension. Despite its great importance,we have not investigated the robustness of Higuchi’s frac-tal dimension. Our preliminary research indicates that it isinvariant with respect to the rotations and translations, butit is not expected to be dimensional, based on statement(11) from [47]. This is left for the future work.

To illustrate the usefulness of our suggested method inthe image analysis, we discriminated different morpogenicstages of shoot organogenesis by estimating DF values. Thisis our preliminary study and we plan to extend ourresearch to different growth and developmental processesin plants and animals. Among other things, we expect touse presented method for the quantification of synchroni-zation during selected processes. Higuchi’s method ismuch simpler than other similar methods, it allows fastcomputational analysis of images and it can be used aloneor in combination with other methods. Application of Hig-uchi’s method has advantage over other fractal methodswhen it comes to speed, accuracy, simplicity and time nec-essary for analysis.

Acknowledgments

This work was supported by project OI 173045,financed by the Serbian Ministry of Education, Scienceand Technological Development.

I am very grateful to Dr. Aleksandra Mitrovic for hisinvaluable collaboration in analysis of possible biologicalapplication of suggested method and Ivana Petrovic forcritical reading of this manuscript. I am also very gratefulto Dr. Dušica Janoševic, Dr. Snezana Budimir, Dr. JelenaBogdanovic Pristov, co-authors in Mitrovic et al., 2012 forproviding the collection of images obtained from histolog-ical preparations.

Appendix 1:. A method for calculating Higuchi’s fractaldimension of a planar curve

The method for calculating fractal dimension of curve inplane has been proposed by Higuchi 1988. [2]. Higuchi’sfractal dimension is a nonlinear measure of waveformcomplexity in time domain. Discretized functions or sig-nals could be analyzed as time sequences x(1),x(2), . . .,x(N). From the starting time sequence, it was con-structed new self-similar time series Xm

k as:

Xmk : xðmÞ; xðmþ kÞ; xðmþ 2kÞ; :::; xðmþ int ðN � kÞ=k½ �kÞ

for m = 1,2,. . .,k where m is initial time; where k is timeinterval, k = 1,. . .,kmax;kmax is a free parameter, and int(r)

is integer part of the real number r. ‘‘The length’’ of curveLm(k) was computed for each of the k time series or curvesXm

k .

LmðkÞ ¼1k

Xint N�mk½ �

i¼1jxðmþ ikÞ � xðmþ ði� 1ÞkÞj

� �N � 1

int N�mk

� k

" #

where N is the length of the original time series X and(N � 1)/{int[(N �m)/k]k} is a normalization factor. Lm(k)was averaged for all m forming the mean value of the curvelength L(k) for each k = 1,. . .,kmax as

LðkÞ ¼Pk

m¼1LmðkÞk

An array of mean values L(k) was obtained and the DFwas estimated as the slope of least squares linear best fitfrom the plot of ln(L(k)) versus ln(1/k):

DF ¼ lnðLðkÞÞ=lnð1=kÞ:

In practice, the original curve or signal can be dividedinto smaller parts with or without overlapping, called‘‘windows’’. Then, the method for DF values computingshould be applied to each window where N should be seenas the length of the window and DF values are calculatedfor each window, without overlap. Individual DF valuescan be averaged across all windows for the whole curve.In that case, mean DF value should be used as a measureof curve complexity. This procedure is very useful in caseof the non-stationary signal analysis, such as an electroen-cephalographic (EEG) and electrocorticographic (ECoG)signals or many other biomedical signals. Higuchi’smethod is applied using MATLAB software by Spasic thathas been validated in some of our previous publications[20–22,25]. After many tests, our results confirm that theDF is relatively independent of the length of the windowN, but much more dependent of the parameter kmax.

In order to facilitate the interpretation of DF values ofthe signals, the DF value of smooth curve (for example lin-ear, a low frequency sine wave or less complex curve) wasestimated to be �1. The DF of random white noise or morecomplex curve was estimated to be �2. DF values couldbe > 2.0 because this is numerical method for computingfractal dimension. In practice, calculated DF values can beslightly greater than 2.0 because it is a numerical methodfor calculating the fractal dimension.

Appendix 2:. statistical analysis used in section 3

The one-way analysis of variance (ANOVA) is used todetermine whether there are any significant differencesbetween the means of more independent groups. Theone-way ANOVA cannot tell us which specific groups weresignificantly different from each other; it only tells us thatat least two groups were different. Before we use this anal-ysis, we need to check the different assumptions that ourdata must meet in order for the one-way ANOVA to giveus a valid result. However, the ANOVA test is robust to var-iance heterogeneity when sample size are equal or approx-imately equal even sample size is very small andheterogeneity is not very large (Joanne C. Rogan and H. J.Keselman, 1977; Glass & Stanley, 1970). In the ANOVA test,

S. Spasic / Chaos, Solitons & Fractals 69 (2014) 179–187 187

F is the final test statistic and p is its associated level of sig-nificance. The F is obtained by dividing the Mean Squarebetween groups by Mean Square within groups. In thiscase we have F = 16.552. The F is significant at least atthe alpha level less than .0001, which falls well belowthe required .05 alpha level.

In our case, differences between calculated FD valuesfrom 6 histological images per each of three phases duringshoot organogenesis were compared using ANOVA test.Phase of organogenesis was used as independent variable(or factor) with three levels (1st, 2nd, and 3rd phase) andDF value of histological image as dependent variable. Sincewe have three groups in our study design, determiningwhich of these groups differ from each other is importantand we can do this using a Duncan post hoc test to differ-entiate homogeneous subsets at the level of significancea = 0.05. Thereafter, the post hoc Duncan test was applied.All statistical analyses were performed using the SPSS 13.0software.

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