Performance Assessment of Methods for Estimation of Fractal DimensionFrom Scanning Electron Microscope Images

DUBRAVKO RISOVIĆ1AND ŽIVKO PAVLOVIĆ2

1Molecular Physics Laboratory, Rudjer Bošković Institute, Zagreb, Croatia2Faculty of Technical Sciences, Graphic Engineering and Design, Novi Sad, Serbia

Summary: Processing of gray scale images in order todetermine the corresponding fractal dimension is veryimportant due to widespread use of imaging technologiesand application of fractal analysis in many areas ofscience, technology, and medicine. To this end, manymethods for estimation of fractal dimension from grayscale images have been developed and routinely used.Unfortunately different methods (dimension estimators)often yield significantly different results in a manner thatmakes interpretation difficult. Here, we report results ofcomparative assessment of performance of several mostfrequently used algorithms/methods for estimation offractal dimension. To that purpose, we have usedscanning electron microscope images of aluminum oxidesurfaces with different fractal dimensions. The perfor-mance of algorithms/methods was evaluated using thestatistical Z‐score approach. The differences betweenperformances of six various methods are discussed andfurther compared with results obtained by electrochemi-cal impedance spectroscopy on the same samples. Theanalysis of results shows that the performance ofinvestigated algorithms varies considerably and thatsystematically erroneous fractal dimensions could beestimated using certain methods. The differential cubecounting, triangulation, and box counting algorithmsshowed satisfactory performance in the whole investi-gated range of fractal dimensions. Difference statistic isproved to be less reliable generating 4% of unsatisfactoryresults. The performances of the Power spectrum,Partitioning and EIS were unsatisfactory in 29%, 38%,and 75% of estimations, respectively. The results of thisstudy should be useful and provide guidelines toresearchers using/attempting fractal analysis of images

obtained by scanning microscopy or atomic forcemicroscopy. SCANNING 9999:XX–XX, 2013. ©2013 Wiley Periodicals, Inc.

Key words: fractal analysis, fractal dimension, grayscale image, SEM, TEM, AFM, EIS

Introduction

Processing of gray scale images in order to recognizefractal patterns and determine the corresponding fractaldimension is very important due to widespread useof imaging technologies and application of fractalanalysis in many areas of science, technology, andmedicine. In particular, fractal analysis based on imageprocessing is nowadays commonly used and widespreadin areas such as materials science and technology andmedicine.

Considering for example the functional properties ofmaterials one finds that they are often determined bystructure of their surface and its characteristics. Thetopography of a surface can be determined using imagingmethods such as scanning electron microscopy (SEM),scanning tunneling microscopy (STM), and the atomicforce microscopy (AFM), and various contact and non‐contact profilometric methods. The results of suchanalysis are usually presented as a set of differenttopography (roughness) parameters characterizing theconsidered surface (Stout and Blunt, 2000;ASME, 2009). In that context, recently it has beenshown that the fractal dimension is one of most relevantparameters in characterization of surface topography(Bigerelle et al., 2003). Fractal dimension is correlatedwith various surface roughness parameters (Russ, 2001;Chappard et al., 2003), related to the basic materialsproperties, and can provide information on subtlestructural changes of surfaces or even mechanismsleading to its formation (Risović et al., 2002, 2003;McRae et al., 2002).

The image‐based fractal approach to surface charac-terization is based on self‐similarity (of patterns) at

Address for reprints: Dubravko Risović, Molecular Physics Laboratory,Rudjer Bošković Institute, POB 180, HR‐10002, Zagreb, CroatiaE‐mail: drisovic@irb.hr

Received 4 October 2012; Accepted with revision 10 January 2013

DOI 10.1002/sca.21081Published online XX XX 2013 in Wiley Online Library (wileyonlinelibrary.com).

SCANNING VOL. 9999, 1–10 (2013)© Wiley Periodicals, Inc.

different scales. Its advantage is that it is insensitive tostructural details and provides a single descriptor of thesurface topography: the fractal dimension D compre-hending the information on the degree of surfacecomplexity (Mitchel and Bonnell, 1990; Liet al., 2003). The fractal dimension lies within the range2 � D � 3, where a smooth surface has a value ofD ¼ 2, and an increased value of D represents a surfacewith increased complexity. Its application is convenientsince fractal models comprise topography parameterswhich are independent of the resolution of theinstrument.

Another approach for determination of fractaldimension of porous/rough surfaces is provided by theelectrochemical impedance spectroscopy (EIS) (Nyikosand Pajkossy, 1985; Lasia, 1999). EIS studies theresponse of an electrolyte/surface interface system to theapplication of a periodic small amplitude ac signal. Thesemeasurements are carried out at different ac frequenciesand analysis of the system response contains informationabout the interface, its structure, and reactions takingplace there (ibid.).

Although the fractal approach is very useful,calculating quantitative properties like fractal dimen-sion from an image is still elusive and challenging.Different algorithms have been proposed and tested(Pentland, 1984; Clarke, 1986; Chen et al., 1989;Molteno, 1993; Sarkar and Chaudhuri, 1994; Van Putet al., 1994; Liu and Chang, 1997; Zahn and Zösch,1997; Chen et al., 2003; Losa et al., 1994–2005; Sunet al., 2006; Lopes and Betrouni, 2009; Li et al., 2009)and many studies have been developed to investigatethe reliability of fractal dimension estimation withdifferent algorithms applied to different fractal func-tions. Several studies have reported that differentmethods (dimension estimators) often yield significant-ly different results in a manner that makes interpreta-tion difficult (Dubuc et al., 1989; Douketis et al., 1995;Vazquez et al., 1995; Chen et al., 2003; Brewer andDi Girolamo, 2006; Oczeretko et al., 2008; Liuet al., 2010).

However, despite considerable efforts to explain andsolve it, the problem of inconsistent results derivedfrom various fractal calculation algorithms persists, andthe need for objective assessment of relevant algo-rithms/methods by introduction of some quantitativemeasure for the performance quality is still unsatisfied.Our aim was to satisfy that need, at least to a degree.To that purpose, in this comparative study, weextended the number of comparisons of fractaldimension estimation algorithms to more often usedtypes suitable for analysis of gray scale images. Inparticular, we applied six different algorithms/methodsfor determination of fractal dimension to SEM grayscale surface images. The values of fractal dimensioninferred from analysis of SEM images were furthercompared with results obtained by EIS on the same

samples. We assessed the performance of each algo-rithm/method using as a figure of merit its Z‐scoredetermined applying statistical methods compliant withrecommendation of the (ISO 13528:2005) standard(ISO 13528:2005). This standard is regularly used invarious intercomparisons for assessment of results oftests on unknown samples. The results of this studyshould be useful and may provide guidelines toresearchers using/attempting fractal analysis of imagesobtained by SEM, STM, AFM, or of various kinds ofmedical images, as well as those interested in generalimage processing and analysis.

Materials, Theory, and Methods

Materials

In this study, we used gray scale SEM images ofaluminum oxide films (non‐imaging areas of litho-graphic printing plates) obtained by electrochemicalroughening and anodizing of 0.3 mm thick AA1050aluminum foils. We used several types of lithographicprinting plates of different vendors (further on denotedA–D) with different surface topography and rough-ness of the aluminum oxide areas. The different surfacestructure and roughness of investigated samples stemfrom different manufacturing methods which includechemical and electrolytic graining processes asexplained in the literature (Lin et al., 2001). Thisselection was motivated by two reasons: firstly, asthe size and quality of the grained surface micro-structure influence the printing performance anddurability of the printing plates they are manufacturedaccording to stringent, standardized procedures(ISO 12218:1997, ISO 12647‐2:2004) resulting insurfaces of controlled and reproducible roughnesssuitable for the purpose of this study, and secondly,for the reasons of technological importance and need ofoxide films characterization.

In this investigation 24 SEM images of aluminumoxide film samples with different surface topographies(i.e., fractal dimension) were used. SEM images ofaluminum oxide films of thermal printing plates AgfaThermostar P970 CtP and Fuji Brillia LH‐PIE CtP, weredenoted A and B respectively. Samples from groups Care D were aluminum oxide films of conventionalprinting plates Cinkarna Kemolit PO7 and AgfaMeridian P51, respectively.

Scanning Electron Microscopy

SEM micrographs of the investigated samples wereobtained with JEOL JMS T300 scanning electronmicroscope. To assure the uniform electrical propertiesand to avoid the charging/discharging of aluminum oxide

2 SCANNING VOL. 9999, 9999 (2013)

surfaces, the aluminum plate samples (5 mm � 5 mm)were gold coated (� 30 nm thick) by magnetron sputterSC7620 Quorum Technologies, Polaron. The micro-graphs of all samples were recorded at magnification1,500�. In this way we ensured that the image coverssurface area large enough to provide a satisfactoryscaling range for estimation of fractal dimension andavoidance of possible influence of localized topographyeffects. The selection of particular magnification has noinfluence on estimation of fractal dimension as we haveshown previously (Mahović‐Poljaček et al. 2008). Thedimension of micrographs was 2,135 � 1,592 pixels(width � height).

Determination of Fractal Dimension From Gray‐ScaleImages

The image‐based technique is amethod that enables thequantified data on a fractal object to be obtained from itsdigital image. The images may be digital or have to bedigitalized and sometimes need to be processed so that thefeatures of fractal objects canbe extracted.There aremanymethods for extraction of fractal dimension from an imageall including numerous variations (Pentland, 1984;Clarke, 1986; Dubuc et al., 1989; Theiler, 1990; MillerandReifenberger, 1992;Chen et al., 1993;Molteno, 1993;Sarkar and Chaudhuri, 1994; Van Put et al., 1994;Douketis et al., 1995; Vazquez et al., 1995; Zahn andZösch, 1997;Brewer andDiGirolamo,2006;Oczeretkoetal., 2008; Li et al., 2009; Liu et al., 2010).

Generally, the existing algorithms for computing Dare either based on a geometrical or a stochasticapproach. They correspond to alternative definitions ofthe dimension of a set (Soille and Rivest, 1996).Geometrical algorithms consider the graph of an imageas a tridimensional object. Stochastical algorithmsassume that the image can be considered as a fractalBrownian function, a particular class of fractal with welldefined statistical properties (Pentland, 1984). In thiscontext triangulation, box/cube counting, coveringblanket represent Geometrical algorithms, while vario-gram/variance method and power spectrum are examplesof stochastical algorithms. Although, the methods and/oralgorithms differ, they obey to the same protocolsummarized by the three steps:

– Measure the quantities of the object using variousstep sizes.

– Plot log (measured quantities) versus log (stepsizes) and fit a least‐squares regression linethrough the data points.

– Estimate D as the slope of the regression line.

In this study, we restricted our selection to thefollowing; most frequently used methods, generallyavailable through open source software:

Difference statistics (dynamic scaling) method(Pentland, 1984)

The method, suitable for gray scale images, relies onestablishing the difference function Df(e) as a function ofspatial scale:

Df ðeÞ ¼ h f ðx; yÞ � f ðx0; y0Þ½ �2i1=2 ð1Þwhere f(x0, y0) is the value of the considered function(representing height or intensity) at reference point and f(x,y) is its value at a point at distance e from the referencepoint.Thebrackets denoteaveragingof thedifferenceoverspatial extent of the considered surface/image. If theconsidered function is the height h then the differencefunction Dh(e) is also known as height–height correlationfunction.

Now, for e � j (Dubuc et al., 1989; Falconer, 1990;Li et al., 2003):

Df ðeÞ ¼ eH ð2ÞHere, j is lateral correlation length and H ¼ d � D isHurst or Hölder exponent related to the embedding andfractal dimensions d and D, respectively.

Hence, taking the logarithm of both sides ofEquation (6) and letting e to go to zero, the fractaldimension is obtained as (Falconer, 1990):

D ¼ lime��!0 d � logDf ðeÞloge

� �ð3Þ

Box counting (Voss, 1988) and differential cube countingmethods (Sarkar and Chaudhuri, 1992, 1994)

Methods are derived directly from the definition ofbox‐counting fractal dimension. Both algorithms arebased on the following steps: a cubic lattice with latticeconstant l is superimposed on the z‐expanded image.Initially l is set at X/2 (where X is the length of edge of thesurface), resulting in a lattice of 2 � 2 � 2 ¼ 8 cubes.The lattice constant l is then reduced stepwise by a factorof 2 and the process repeated until l equals to the distancebetween two adjacent pixels. The box counting methodcounts the number of boxes N(l) which contains at leastone pixel of the image. The slope of a plot of log N(l)versus log 1/l gives the fractal dimension Df directly.

In the differential cube counting method N(l) iscounted in a different manner. For each cube column, thecubes are numbered bottom up starting from one. Foreach column: the number of the upper most boxcontaining some pixel of the surface is denoted j andthe lowest box containing any surface pixels are denotedi. For each cube size l, N(l) is computed as the sum ofj � i þ 1 over all columns. Df is estimated from theslope of a plot of log N(l) versus log 1/l.

Triangulation method (Clarke, 1986)Method is very similar to cube counting method and is

also based directly on the definition of the box‐countingfractal dimension. The method works as follows: a grid

D. Risović and Ž. Pavlović: Performance assessment in estimation of fractal dimension 3

of unit dimension l is placed on the image. This definesthe location of the vertices of a number of triangles.When, e.g. l ¼ X/4, the image surface is covered by 32triangles of different areas inclined at various angles withrespect to the x–y plane. The areas of all triangles arecalculated and summed to obtain an approximation of thesurface area S(l) corresponding to l. The grid size is thendecreased by successive factor of 2, as before, and theprocess continues until l corresponds to distance betweentwo adjacent pixel points. The slope of a plot of S(l)versus log 1/l then corresponds to Df � 2.

Power spectrum method (Saupe, 1988; Zahn andZösch, 1997)

Method is based on the power spectrum dependenceof fractional Brownian motion

PðvÞ � v�ðg�2DÞ ð4ÞIn the power spectrum method, every line height

profiles that forms the image is Fourier transformed andthe power spectrum evaluated and then all these powerspectra are averaged. Fractal dimension is evaluatedfrom the slope b of a least‐square regression line fit to thedata points in a log–log plot of power spectrum as

Df ¼ 8

2� b

2ð5Þ

Partitioning (Van Put et al., 1994; Li et al., 2009;Podsiadlo and Stachowiak, 2000)

A variation of the box counting method involving arepeated application of a simple partitioning procedure tothe image, eventually dividing it into a large number ofsubsets, each covered by a small box. Initially a box (in ad‐dimensional space) containing all the points in the set(pixels in the image) is divided evenly into 2d boxes ofequal size each containing a set of points. This process isthen applied recursively; each of the 2d boxes issubdivided as the original box, creating another 22d

boxes. The empty boxes are not partitioned any further.As this process continues m times, it will give N(e) (thenumber of boxes of size e needed to cover the consideredset) for logarithmically spaced values of e, [N(2�1), N(2�2), …, N(2�m)], as required by box counting methodto estimate D in the limit e ! 0.

All these algorithms measure space‐filling propertiesexcept the power spectrum method, which measuresharmonic components pertaining to the magnitude ofoscillations at different scales. These algorithms can beused with any type of gray scale image regardless of itssource. However, care must be taken that the dimensionof considered image provides at least the minimalmagnitude scaling region required for reliable estimationof fractal dimension, which is estimated to be (1,024)2

pixels (Mitchel and Bonnell, 1990). To satisfy thisrequirement in this study we used SEM images withdimensions 2,135 � 1,592 (width � height) pixels—

well above the required minimal size of the scalingregion.

For estimation of fractal dimension of a gray scaleimage applying different algorithms three open sourcesoftware: Gwyddion (2011), Fractal 3e (2011), andFraclab (2011) were used.

Determination of Fractal Dimension From EIS

EIS measurement on samples from which SEMimages were taken were performed by the immersion ofthe samples in unstirred, aerated 3.5% (w/w) K2SO4

solution at 25 � 1°C. The area of the samples exposed tothe solution was 1 cm2. The electrochemical cellconsisted of reference electrode Ag, AgCl, platinumcounter electrode and working electrode (anodizedaluminum sheet). Electronic equipment comprehendedan EG&G Lock‐in‐amplifier Type 5210 in combinationwith an EG&G Potentiostat/Galvanostat Type 263 A atan open circuit potential. EIS spectrum was measuredover the frequency range from 10 mHz to 64 kHz undercontrolled potentionstatic conditions. Capacitance andresistance data were obtained by software PowerSine EISSoftware Equivalent Circuit (Boukamp, 1989).

The interpretation (analysis) of experimental dataobtained by EIS on rough/porous electrodes reveals thepresence of non‐trivially frequency dependant elementsin the measured spectrum. In many cases this dispersivebehavior can be excellently described over a wide rangeof frequency v by an empirical relationship (“constantphase element”—CPE):

Y ¼ 1

Z¼ TðivÞa ð6Þ

Here, Y and Z denote admittance and impedance,respectively. The values of exponent a are between 0.5and 1. In their seminal paper Nyikos and Pajkossy (1985)related this CPE exponent to the fractal dimension D:

a ¼ 1

D� 1or D ¼ aþ 1

að7Þ

The value of frequency dispersion parameter a can beinferred from analysis of EIS data using the proceduredescribed previously (Risović et al., 2008), or from anadequate equivalent circuit modeling (Mahović‐Poljačeket al., 2012). Then, Equation (7) yields the correspondingfractal dimension.

Description of the Statistical Tools Used for theEvaluation of Results

The normality of data were checked by the Shapiro–Wilk normality test (an improvement on themore generalKolmogorov–Smirnov test) (Shapiro and Wilk, 1965).

4 SCANNING VOL. 9999, 9999 (2013)

The statistical evaluation of the data obtained byapplication of different algorithms for estimation of Dwas conducted following recommendations of the(ISO 13528:2005) standard.

The performance of the algorithms in estimation of Dwas evaluated with the assignment of a figure of merit,the Z‐score according to the following formula (§ 7.4,ISO 13528:2005):

Z ¼ DA � DRef

sRefð8Þ

where Z is the Z‐score value; DA ¼ average result ofthe algorithm on a given sample;DRef ¼ assigned value;and sRef ¼ deviation unit.

The Z‐score, as the figure‐of‐merit, classifies theperformance of a particular algorithm as follows:

jZj � 2—satisfactory; 2 < jZj � 3—doubtful;

jZj > 3—unsatisfactoryð9Þ

The assigned value, DRef, for the evaluation of thealgorithm performance was defined as the median of allthe results obtained by multiple application of all

considered method on a given sample (§5.6,ISO 13528:2005). The deviation unit (sRef) wasdetermined to be 2% according to the “fit‐for‐purpose”criteria (§6.3, ISO 13528:2005(E)), based on the case‐bycase consideration of the measurand (D), and its targetvalue, including repeatability of the applied method.

Grubbs’s test (1969) was applied for detection ofpossible outliers (results which are sufficiently differentfrom all other results to warrant further investigation).

Results and Discussion

Representative SEM images of samples from eachgroup are shown in Figure 1. Depending on the appliedmethod the values of estimated fractal dimensions ofconsidered samples spanned the range 2.1 � D � 2.75.The median values of fractal dimensions Dmedian variedbetween the sample groups from2.50 for groupD samplesto2.57forBgroup.Themedianvaluesoffractaldimensionfor groups A and C were 2.54 and 2.52, respectively.

The estimated fractal dimensions of aluminum oxidefilm samples from groupsA andBobtainedwith differentmethods are depicted in Figures 2 and 3, respectively.

Fig 1. Representative SEMmicrographs of aluminum oxide surfaces of thermal printing plates (A and B) and conventional printing plates(C and D) at magnification 1,500�.

D. Risović and Ž. Pavlović: Performance assessment in estimation of fractal dimension 5

It can be seen that the value of inferred fractaldimension of a particular sample depends on appliedmethod and exhibits considerable differences particular-ly for samples from group A.

For some samples the values of fractal dimensioninferred from EIS measurements were significantly lowerthan the corresponding values inferred from SEM images.

The results obtained for samples from groups C and Dare depicted in Figures 4 and 5, respectively. Againconsiderable differences between methods are notice-able. Between the methods, for all samples, the lowestvalues of fractal dimension are those inferred from EISmeasurements. This is compliant with previouslyreported differences between values of fractal dimension

obtained by difference statistic algorithm applied to SEMimages and EIS measurements on the same samples(Risović et al., 2008). These results, although wellcorrelated show systematically lower values of fractaldimensions inferred from CPE exponent in respect tothose obtained from image analysis (ibid.).

The observed differences between fractal dimensionsinferred from EIS and image analysis could be explainedconsidering that the fractal dimension calculated fromEIS should likely be associated with some average of themetal‐oxide and oxide electrolyte interfaces, whereas theSEM image’s fractal dimension is calculated only withinformation from the outer oxide surface. Moreover, ifthe porosity of the outer oxide surface is extreme

Fig 2. The estimated fractal dimensions for group A samples.The overall median value of fractal dimension D(A) ¼ 2.54.Numerical designation of the methods: (1) cube, (2) triangulation,(3) power, (4) partitioning, (5) difference statistics, (6) boxcounting, and (7) EIS.

Fig 3. The estimated fractal dimensions for group B samples.The overall median value of fractal dimension D(B) ¼ 2.57.Numerical designation of the methods: (1) cube, (2) triangulation,(3) power, (4) partitioning, (5) difference statistics, (6) boxcounting, and (7) EIS.

Fig 4. The estimated fractal dimensions for group C samples.The overall median value of fractal dimension D(C) ¼ 2.52.Designation of methods: (1) cube, (2) triangulation, (3) power,(4) partitioning, (5) difference statistics, (6) box counting, and(7) EIS.

Fig 5. The estimated fractal dimensions for group D samples.The overall median value of fractal dimension D(D) ¼ 2.50.Designation of methods: (1) cube, (2) triangulation, (3) power, (4)partitioning, (5) difference statistics, (6) box counting, and (7) EIS.

6 SCANNING VOL. 9999, 9999 (2013)

allowing filling with electrolyte during the EIS measure-ment, then this portion of the oxide layer will not bereadily distinguishable from bulk solution by impedancemeasurements. In this instance, EIS would measure thedimension of the quasi‐surface of a denser layer belowthe surface measured by SEM (McRae et al., 2002).Altogether these effects could result in an apparentsmoothing of the surface leading to systematically lowerfractal dimension.

Statistical evaluation of the data obtained by applica-tion of different algorithms for estimation of D enabledassignment of Z‐score to each method. These results areshown in Figures 6–9.

Plot of figure of merit for the performance ofeach method in evaluation of fractal dimension, i.e. itis Z‐score, for group A samples is depicted in Figure 6.

By inspecting Figure 6 and applying the performancecriteria Equation (9), we conclude that the performance

of methods (1), (2), and (6) in determination of fractaldimension was satisfactory for all samples of group A.Z‐scores of method (3) are not homogenous and varyfrom satisfactory to unsatisfactory, while methods (4) and(5) showed mostly doubtful results and even unsatisfac-tory performance for some samples. The performance ofEIS (method‐7) was unsatisfactory for most samples.

Considering performances in determination of fractaldimension for samples from group B, depicted inFigure 7, one can see that performance of most methodsis satisfactory. Only the results of algorithms (4) and (5)can be in most cases classified as doubtful. For this groupof samples the Z‐score of all methods is ratherhomogenous—for a given algorithm it does not varysignificantly between the samples. The exception is anextremely high Z‐score of EIS on sample #1 (Z > 9).However, the analysis of corresponding value of thefractal dimension shows that this result is an outlier.

Fig 6. Plot of Z‐score values of considered methods for deter-mination of fractal dimension of group A samples. Identification ofthe Z‐score color code: green ¼ satisfactory (|Z| � 2), yellow ¼doubtful (2 < |Z| � 3), red ¼ unsatisfactory (|Z| > 3).Designationofmethods: (1) cube, (2) triangulation, (3)power, (4)partitioning, (5)difference statistics, (6) box counting, and (7) EIS.

Fig 7. Plot of Z values for group B. Identification of the Z‐scorecolor code: green ¼ satisfactory, yellow ¼ doubtful, red ¼ un-satisfactory. Designation of methods: (1) cube, (2) triangulation,(3) power, (4) partitioning, (5) difference statistics, (6) boxcounting, and (7) EIS.

Fig 8. Plot of Z values for group C. Identification of the Z‐scorecolor code: green ¼ satisfactory, yellow ¼ doubtful, red ¼ un-satisfactory. Designation of methods: (1) cube, (2) triangulation,(3) power, (4) partitioning, (5) difference statistics, (6) boxcounting, and (7) EIS.

Fig 9. Plot of Z values for group D. Identification of Z‐scorecolor code: green: satisfactory, yellow: doubtful, red: unsatisfac-tory. Designation ofmethods: (1) cube, (2) triangulation, (3) power,(4) partitioning, (5) difference statistics, (6) box counting, and(7) EIS.

D. Risović and Ž. Pavlović: Performance assessment in estimation of fractal dimension 7

Hence, the performance of EIS in determination offractal dimension of B‐samples can be classified assatisfactory inmost cases. Here, it is worth noting that theB‐samples have the highest average fractal dimension ofall groups. The performance of considered methods indetermination of fractal dimension of group C samples(i.e., the corresponding Z‐scores), depicted in Figure 8, issimilar to the results obtained with group A samples.Again, methods (1), (2), and (6) exhibit satisfactoryperformance while EIS performance is unsatisfactory.The performance of other methods (3)–(5) variesdepending on particular sample, resulting in inhomoge-neous Z values.

The Z‐scores obtained for group D samples varybetween the methods, but are very homogenous—for aparticular method there is practically no sampledependence of its Z‐score.

Satisfactory performance of methods (1), (2), and (6)are contrasted with unsatisfactory performance ofmethods (4) and (7), while results of method (5) shouldbe considered doubtful.

Now, considering all results, it could be concludedthat the algorithms based on differential cube counting,triangulation and box counting showed satisfactory(reliable) performance (Z � 2) in determination offractal dimension from gray scale images.

Difference statistic is somewhat less reliable as itgenerated 4% unsatisfactory results (Z‐score � 3).

The performance of the Power spectrum andPartitioning algorithms was unsatisfactory in 29% and38%, respectively. This is in agreement with previousfinding that the performance of power spectrum isdubious when applied to Takagi and Brownian motioncurves, that is fractals in one‐dimension (Brewer and DiGirolamo, 2006). However, both algorithms performedsignificantly better for samples with high average fractaldimension (group B) than for samples with relativelylower average fractal dimension (group D). Suchbehavior (better performance for a higher D) has beenalso reported previously (Soille and Rivest, 1996).

The Relatively poor performance of Power spectrummethod can be explained considering that methods basedon fractional Brownian movement (fBm) work well onlywith surfaces exhibiting an exponential power spectrum.In general, this restriction imposed on the shape of thepower spectrum is not valid, and this may cause errors inthe calculation of fractal dimension (Osborne andProvenzale, 1989). Also, the fBm methods require astatistical self‐affinity with uniform global scaling(Barnsley et al., 1992). In other words, the FBMmethods only work well with a fractal surface in whichthe concept of self‐similarity is extended to permit adifferent global scaling factor for each coordinate axis.

The difference statistics algorithm tested at syntheticsurface images (Takagi and fractional Brownian surfa-ces) performed significantly better at higherD (�2.6) butgiving systematically somewhat higherD values (Dubuc

et al., 1989; Oczeretko et al., 2008). On the other hand,the opposite behavior (a better performance at low andmedium D values than for high D values) was alsoreported for certain algorithms based on investigation onsynthetic—numerically generated surfaces (Douketiset al., 1995).

Determination of fractal dimension fromCPE inferredfrom EIS measurements on samples was unsatisfactoryin 75% of all estimations.

Conclusions

In this study, we used the statistical Z‐score approachto assess and compare the performance of several mostfrequently used algorithms/methods for estimation offractal dimension from gray scale images.

The results of this study showed that the performanceof investigated algorithms varies considerably and thatsystematically erroneous fractal dimensions could beestimated by certain methods.

Generally, the algorithms based on a geometricalapproach in determination of fractal dimension per-formed better than those based on a stochastic approach.In particular, the algorithms based on differential cubecounting, triangulation and box counting showedsatisfactory and reliable performance in the wholeinvestigated range of fractal dimensions. Differencestatistic is proved to be less reliable generating 4% ofunsatisfactory results.

The performance of the Power spectrum and Partition-ing algorithms was unsatisfactory in 29% and 38%,respectively. Significantly better performance wasachieved for samples with higher fractal dimension(D > 2.5) than for samples with relatively lower fractaldimension.

Determination of fractal dimension inferred from EISmeasurements on samples was unsatisfactory in 75% ofall estimations. The values of D were systematically andsignificantly lower than those inferred from gray scaleimages.

It seems that the fractal dimension inferred from gray‐scale SEM images using an appropriate algorithm ismore reliable and a better descriptor of surfacetopography than the fractal dimension inferred fromCPE exponent of EIS measurements, since the latter maybe influenced by other parameters beside pure geometri-cal surface roughness or porosity.

The results of this study clearly showed that thepopular fractal dimension estimation algorithms pre-sented here should be used cautiously as some mayproduce systematically erroneous fractal dimensionswhen applied to gray scale images.

A solid theoretical foundation that can explainor predict the observed performances of fractaldimension estimators applied to various data sets is stillrequired.

8 SCANNING VOL. 9999, 9999 (2013)

References

ASME. 2009. American National Standard ASME B46. 1‐2009surface texture (surface roughness, waviness, and lay) NewYork: The American Society of Mechanical Engineers.

Barnsley MF, Devaney RL, Mandelbrot BB, et al., editors. 1992.The science of fractal images. New York: Springer‐Verlag.

Bigerelle M, Najjar D, Iost A. 2003. Relevance of roughnessparameters for describing and modeling machined surfaces. JMater Sci 38: 2525–2536.

Boukamp A. 1989. Equivalent circuit, users manual, secondrevised edition. Enschede, The Netherlands: University ofTwente Enschede.

Brewer J, Di Girolamo L. 2006. Limitations of fractal dimensionestimation algorithms with implications for cloud studies.Atmos Res 82: 433–454.

Chappard D, Degasne I, Huré G, et al. 2003. Image analysismeasurements of roughness by texture and fractal analysiscorrelate with contact profilometry. Biomaterials 24: 1399–1407.

Chen CC, Daponte JS, Fox MD. 1989. Fractal feature analysis andclassification in medical imaging. IEEE Trans Med Imaging 8:133–142.

Chen WS, Yuan SY, Hsieh CM. 2003. Two algorithms to estimatefractal dimension of gray‐level images. Opt Eng 42: 2452–2464.

Chen SS, Keller JM, Crownover RM. 1993. On the calculations offractal features from images. IEEE Trans Pattern Anal MachIntell 15: 1087–1090.

Clarke KC. 1986. Computation of the fractal dimension oftopographic surfaces using the triangular prism surface areamethod. Comp Geosci 12: 713–722.

Douketis C, Wang Z, Haslett TL, Moskovits M. 1995. Fractalcharacter of cold‐deposited silver films determined by lowtemperature scanning tunneling microscopy. Phys Rev B 51:11022–11031.

Dubuc B, Zucker SW, Tricot C, Quiniou JF, Wehbi D. 1989.Evaluating the fractal dimension of surfaces. Proc R Soc LondA425: 113–127.

Falconer K. 1990. Fractal geometry, mathematical foundations andapplications Chichester: John Wiley & Sons, Inc.

Fraclab. 2011. Fractal analysis toolbox for signal and imageprocessing‐ Fraclab 2.0 software download and documenta-tion. Available online at: http://fraclab.saclay.inria.fr.

Fractal3e. 2011. Fractal analysis system for Windows –softwaredownload and documentation. Available online at: http://cse.naro.affrc.go.jp/sasaki/fractal/fractal‐e.html.

Grubbs FE. 1969. Procedures for detecting outlying observationsin samples. Technometrics 11: 1–23.

Gwyddion. 2011. Modular program for SPM (scanning probemicroscopy)data visualizationand analysis‐ softwaredownloadand documentation. Available online at: http://gwyddion.net.

ISO 12218:1997, Graphic technology—Process control—Offsetplate making, ISO 1997.

ISO 12647‐2:2004, Graphic technology—Process control forthe production of half‐tone color separations, proof andproduction prints—Part 2: offset lithographic processes.ISO, 2004.

ISO 13528:2005 Statistical methods for use in proficiency testingby interlaboratory comparisons, ISO, 2005.

Lasia A. 1999. Electrochemical impedance spectroscopy and itsapplications. In: Conway BE, Bockris J, White RE, editors.Modern aspects of electrochemistry Vol. 32. New York:Kluwer Academic/Plenum Publishers. pages 143–248.

Li JM, Li Lu, Lai MO, Ralph B. 2003. Image‐based fractaldescription of microstructures. Boston: Kluwer AcademicPublishers.

Li J, Du Q, Sun C. 2009. An improved box‐counting method forimage fractal dimension estimation. Pattern Recognit 42:2460–2469.

Lin CS, Chang C, Fu HM. 2001. AC electrograining of aluminumplate in hydrochloric acid. Mater Chem Phys 68: 217–224.

Liu J, Jiang X, Huang X, Wu S. 2010. Morphologicalcharacterization of super fine pulverized coal particle. Part2. AFM investigation of single coal particle. Fuel 89: 3884–3891.

Liu S, Chang S. 1997. Dimension estimation of discrete‐timefractional Brownian motion with applications to imagetexture classification. IEEE Trans Image Process 6: 1176–1184.

Lopes R, Betrouni N. 2009. Fractal and multifractal analysis: areview. Med Image Anal 13: 634–649.

Losa GA, Merlini D, Nonnenmacher TF, Weibel ER, editors.1994–2005. Fractals in biology andmedicine, Vols. I (1994), II(1998), III (2002), IV (2005). Basel: Birkhäuser Press.

Mahović Poljaček S, Risović D, Furić K, Gojo M. 2008.Comparison of fractal and profilometric methods forsurface topography characterization. Appl Surf Sci 254:3449–3458.

Mahović‐Poljaček S, Risović D, Cigula T, Gojo M. 2012.Application of electrochemical impedance spectroscopy incharacterization of structural changes of printing plates. J SolidState Electrochem 16: 1077–1089.

McRae GA, Maguire MA, Jeffrey CA, Guzonas DA, Brown CA.2002. A comparison of fractal dimensions determined fromatomic force microscopy and impedance spectroscopy ofanodic oxides on Zr‐2. Nb Appl Surf Sci 191: 94–105.

Miller S, Reifenberger RJ. 1992. Improved method for fractalanalysis using scanning probe microscopy. J Vacuum SciTechnol B10: 1203–1207.

Mitchel MW, Bonnell DA. 1990. Quantitative topographicanalysis of fractal surfaces by scanning tunneling microscopy.J Mater Res 5: 2244–2255.

Molteno TCA. 1993. Fast O(1V) box‐counting algorithm forestimating dimensions. Phys Rev E 48: R3263–R3266.

Nyikos L, Pajkossy T. 1985. Fractal dimension and fractionalpower frequency dependent impedance of blocking electrodes.Electrochim Acta 30: 1533–1540.

Oczeretko E, Borowska M, Kitlas A, Borusiewicz A, Sobolewska‐Siemieniuk M. 2008. Fractal analysis of medical images in theirregular regions of interest. 8th IEEE International Confer-ence on BioInformatics and BioEngineering, 2008. BIBE2008, pp. 1–6. Athens 2008 DOI: 10.1109/BIBE.2008.4696830

Osborne AR, Provenzale A. 1989. Finite correlation dimension forstochastic systems with power‐law spectra. Physica D 35:357–381.

Pentland AP. 1984. Fractal based description of natural scenes.IEEE Trans Pattern Anal Mach Intell 6: 661–674.

Podsiadlo P, Stachowiak GW. 2000. Scale‐invariant analysis ofwear particle surface morphology II. Fractal dimension. Wear242: 180–188.

Risović D, Gašparović B, Ćosović B. 2002. The impact of fractalgeometry on permittivity and related quantities. J Phys ChemB 106: 9810–9814.

Risović D, Gašparović B, Ćosović B. 2003. Hydrodynamicinfluence on fractal morphology of adsorbed layer of linoleicacid at mercury/electrolyte interface. Colloid Surface A 223:145–156.

Risović D, Mahović Poljaček S, Furić K, Gojo M. 2008. Inferringfractal dimension of rough/porous surfaces—a comparison ofSEM image analysis and electrochemical impedance spec-troscopy methods. Appl Surface Sci 255: 3063–3070.

Russ CJ. 2001. Fractal geometry in engineering metrology. In:Mainsah E, Greenwood JA, ChetwyndDG, editors. Metrologyand properties of engineering surfaces. Boston: KluwerAcademic Publishers.

Sarkar N, Chaudhuri BB. 1992. An efficient approach to estimatefractal dimension of textural images. Pattern Recognit 25:1035–1041.

Sarkar N, Chaudhuri BB. 1994. An efficient differential boxcounting approach to compute fractal dimension of image.IEEE Trans Syst Man Cybernet 24: 115–120.

D. Risović and Ž. Pavlović: Performance assessment in estimation of fractal dimension 9

Saupe D. 1988. Algorithms for random fractals. In: Peitgen H,Saupe D, editors. The science of fractal images. New York:Springer‐Verlag.

Shapiro S, Wilk MB. 1965. An analysis of variance test fornormality (complete samples). Biometrika 52: 591–611.

Soille P, Rivest JF. 1996. On the validity of fractal dimensionmeasurements in image analysis. J Vis Commun Image R 7:217–229.

Stout KJ, Blunt L. 2000. Three dimensional surface topography2nd ed., London: Penton Press.

Sun W, Xu G, Gong P, Liang S. 2006. Fractal analysis of remotelysensed images: a review of methods and applications. Int JRemote Sensing 27: 4963–4990.

Theiler J. 1990. Estimating fractal dimension. J Opt Soc Am A 7:1055–1073.

Van Put A, Vertes A, Wegrzynek D, Treiger B, Van Grieken R.1994. Quantitative characterization of individualparticle surfaces by fractal analysis of scanning electronmicroscope images. Fresenius J Analen Chem 350: 440–447.

Vazquez L, Vara JM, Herrasti P, et al. 1995. Methods of fractalanalysis applied to STM imaging. Chaos Solitons Fract 6:569–573.

Voss R. 1988. Fractals in nature: from characterization tosimulation. In: Peitgen HO, Saupe D, editors. The scienceof fractal images. New York: Springer Verlag. pages 21–70.

Zahn W, Zösch A. 1997. Fractal dimension of thin‐film surfacesobtained by Fourier spectral analysis. Surface Interface Anal25: 488–491.

10 SCANNING VOL. 9999, 9999 (2013)