Measuring Surface Topography by Scanning ElectronMicroscopy. II. Analysis of Three Estimators of SurfaceRoughness in Second Dimension and Third Dimension
Rita Dominga Bonetto,1* Juan Luis Ladaga,2 and Ezequiel Ponz1
1Centro de Investigación y Desarrollo en Ciencias Aplicadas Dr. Jorge J. Ronco (CINDECA) CONICET—UNLP,47 No. 257-CC 59, 1900 La Plata, Argentina
2Facultad de Ingeniería de la Universidad Nacional de Buenos Aires, Departamento de Física—Laboratorio de Láser,Paseo Colón 850, Ciudad Autónoma de Buenos Aires, Argentina
Abstract: Scanning electron microscopy ~SEM! is widely used in surface studies and continuous efforts arecarried out in the search of estimators of different surface characteristics. By using the variogram, we developedtwo of these estimators that were used to characterize the surface roughness from the SEM image texture. Oneof the estimators is related to the crossover between fractal region at low scale and the periodic region at highscale, whereas the other estimator characterizes the periodic region. In this work, a full study of these estimatorsand the fractal dimension in two dimensions ~2D! and three dimensions ~3D! was carried out for emery papers.We show that the obtained fractal dimension with only one image is good enough to characterize the roughnesssurface because its behavior is similar to those obtained with 3D height data. We show also that the estimatorthat indicates the crossover is related to the minimum cell size in 2D and to the average particle size in 3D. Theother estimator has different values for the three studied emery papers in 2D but it does not have a clearmeaning, and these values are similar for those studied samples in 3D. Nevertheless, it indicates the formationtendency of compound cells. The fractal dimension values from the variogram and from an area versus steplog–log graph were studied with 3D data. Both methods yield different values corresponding to differentinformation from the samples.
Key words: dynamic programming, fractal dimension, surface roughness, SEM, stereomicroscopy
INTRODUCTION
Many questions arise when the roughness of materials isstudied in several scientific and technological areas. In thisarea of study, scanning electron microscopy ~SEM! is veryversatile and its strong use is due to the large depth of field.Different roughness estimators are used in order to analyzethe SEM images. Bonetto and Ladaga ~1998! suggested twoestimators ~dper and dmin! to characterize the SEM imagetexture together with the fractal dimension D ~Bonettoet al., 2002; Ladaga & Bonetto, 2002!. The FERImage pro-gram ~Bianchi & Bonetto, 2001! allows the calculation ofthese estimators by means of the variogram, that is, with alog–log graph of variance versus step. The variogram is oneof the methods used for surface-roughness studies of self-affine samples. Although other methods exist for these typesof samples, such as the Fourier method or the slit-islandmethod ~Mandelbrot et al., 1984!, we use the variogrammethod because we are interested in samples with a Brown-
ian fractal behavior at low scale and a quasiperiodic orperiodic behavior at high scale. The variogram is veryefficient in detecting this periodicity whose estimator is thedper parameter. However, in the analysis of the texture of anSEM image, the variogram is also efficient in detecting thecross-over between the region where the fractal is evidencedand the periodic region ~dmin!.
It is necessary to emphasize the difference in meaningbetween the dmin parameter and other discussed crossovers,for example, in Feder ~1988! or the break point found byKaye ~1989!. These parameters correspond to the slopechange as a consequence of the presence of two differentfractals at two different scales. The dmin parameter is validonly for the variogram when periodic or quasi-periodicimage textures are studied, and this parameter correspondsto the end of the periodic region. Russ ~1994! studiedexamples of some crossovers or break points related to thestraight-line intercept of the fractal graphs that is used inthe lacunarity or topothesy equations. The lacunarity is aparameter introduced by Mandelbrot ~1982! that describesthe texture of a fractal. Different fractal images can beconstructed with the same fractal dimension but with dif-ferent texture because they have different lacunarity. This
Received November 8, 2004; accepted May 23, 2005.*Corresponding author. E-mail: [email protected]
Microsc. Microanal. 12, 178–186, 2006DOI: 10.1017/S143192760606003X Microscopy AND
parameter is widely used in studies of fractal and multifrac-tal surfaces. There are applications of lacunarity in differentfields such as ecology, image processing, medicine, etc. ~fordetails about the lacunarity, cf. Mandelbrot ~1982!, Feder~1988!, Barnsley et al. ~1988!, Russ ~1994!!. We know that itis necessary to apply the multifractal concept for a wholestudy. However, we will show that the study with an SEMstereo pair complements the characterization with the studyof an only SEM image with the variogram, allowing a gooddescription of the studied sample surface.
The FERImage program also allows the attainment ofthe fractal dimension from Fourier power spectrum. Thislast option was implemented in order to obtain fractaldimensions of those samples that presented high-frequencynoise. Nevertheless, this option can also be used in the studyof any type of surfaces, particularly anisotropic surfacesfrom the Fourier transform and the use of the topothesyrelated to the straight-line intercept in the Fourier powerspectrum. However, the Fourier method is not as efficient asthe variogram method for the dmin calculation. A morecomplete study that shows the relationship between themaxima of the Fourier power spectrum and the minima ofthe variogram can be seen in Ladaga and Bonetto ~2002!.The FERImage program also allows us to obtain polargraphs of the slope value ~related to the fractal dimension!or the intercept value ~related to the topothesy L! of thestraight line in the variogram to study anisotropic images.The topothesy turned out to be more sensitive to anisotro-pies than the fractal dimension ~Sayles & Thomas, 1978;Russ, 1994; Thomas et al., 1999; Bonetto et al., 2002!.
Russ and Russ ~1987! have obtained an empiric corre-lation between the original, surface-fractal dimension and atextural parameter that was obtained from the brightnessdifferences of an SEM image. Russ ~1994! carried out acomplete documented investigation about different aspectsof fractal surfaces and especially in the relationship betweenthe fractal dimension of the original surface and the oneobtained from image texture. Skands ~1996! found a linearcorrelation between the surface topographic properties andthe secondary electrons emitted from them by using SEM.Russ ~2002! presented a whole study on the image process-ing and particularly about new approaches related to thetopographic analysis and fractal dimensions.
An important goal is to verify the validity of the hypoth-esis that the textural parameters D, dper , and dmin have aphysical meaning related to the surface roughness by usingthe same tool: SEM in this case. For that purpose, an SEMimage stereo pair and a robust algorithm is required toobtain reliable height data. In the recently developed EZE-Image program ~Ponz et al., 2006!, fast cross correlation andtwo stages of dynamic programming are used to obtain adense disparity map based on Sun’s algorithm ~Sun, 2002!.The program also allows us to obtain a dense height map byusing suitable equations. In addition, it allows us to obtain aset of parameters and three functional indices ~Dong et al.,1994! that characterize the third-dimension ~3D! surface
topography. The EZEImage program also allows us to calcu-late the fractal dimension from the variogram and from alog–log graph of areas versus steps. In the first case, step isthe length corresponding to two different positions in thedense height map, and in the second case, step is theelemental square size of projection.
In this work, a study was carried out with three samplesof FEPA emery papers to verify the correspondence amongthe three estimators: D, dper , and dmin in the second dimen-sion ~2D! and 3D. The height images or dense height mapsobtained from the EZEImage program was used as theinput image in the FERImage program to obtain D, dper ,and dmin in 2D and 3D. In this article, a coincidence be-tween the fractal dimension values obtained in 2D and 3Dwith the variogram will be shown. It will be an importantverification since it is easier to work with only one SEMimage. On the other hand, it will be shown that the fractaldimension values obtained from the areas versus step willbe related fundamentally to roughness characteristics of theparticles. Regarding the dmin parameter, it will be observedthat in 2D, it represents the minimum cell with enoughstatistical weight to produce periods, whereas in 3D, itrepresents the average particle sizes of the emery papers.The dper parameter in 2D is a measurement of the averagedistance among the particles in the SEM images, and it hasdifferent values for the three studied, emery-paper samples.It will be shown that the dper parameter does not have aclear meaning in 3D, and its values are similar for the threestudied samples.
A BRIEF REVISION OF THE DEFINITIONOF THE THREE TEXTURAL ESTIMATORS~D, dper, AND dmin!
The variance of the brightness distribution in SEM imagescorresponding to a Brownian fractal surface ~Mandelbrot,1982; Feder, 1988; Van Put, 1991! is
V~s! � ^Dz 2 &� s 2H, ~1!
where Dzi is the gray-level difference between two-differentpositions in the SEM digital image for a step i of length s, inpixels or microns. The parameter H is related to the fractaldimension D � 3 � H, and H � 1/2 is the value for theordinary Brownian case ~Einstein, 1905!.
The slope of the linear region in the log–log graph ofthe variance versus step ~variogram! characterizes the frac-tal behavior of the image texture. Generally, on engineeredsurfaces there is a linear region at low scale and a periodicregion at high scale ~e.g., a variogram with a linear andperiodic region can be seen in Fig. 2!. This region can bedescribed with a subset of the spatial frequency spectrarepresented by “k wave vectors” on the orthogonal Carte-sian system. This subset contains only the maximums of the
Three Estimators of Surface Roughness in 2D and 3D 179
Fourier spectrum ~minimums in the variogram! and byprojecting these k wave vectors on the pair of orthogonalcoordinates axes the following is obtained:
^k 2 & � ^kx2 &� ^ky
2 &, ~2!
where
^kx2 & � Svxi kxi
2
and
^ky2 &� Svyi kyi
2 ~kxi � 2p/sxi and kyi � 2p/syi !
with sxi ~syi ! corresponding to the minimums on the vario-gram by each axis and vxi ~vyi ! are the statistical probabili-ties of kxi
2 and kyi2 , respectively:
vxi � ~DVxi /VMxi !/Sj DVxj /VMxj ;
vyi � ~DVyi /VMyi !/Sj DVyj /VMyj ,
where VMxi and VMyi are the values of the left-lateral maxi-mums of the variance corresponding to the minimums sxi
and syi , respectively; DVxi and DVyi are the differences be-tween those maximum and minimum values. As ^k2& is arotational invariant so is its inverse. Bonetto and Ladaga~1998! defined a rotational invariant parameter dper , whichhas length dimension and characterizes the periodic regionof the image as follows:
dper �2p
M^kx2 &� ^ky
2 &. ~3!
The dmin parameter corresponds to the inferior end ofthe periodic region and is related to the smallest particle~cell! size with enough statistic weight to produce periods inthe variogram ~Bonetto & Ladaga, 1998!. This parameter isgenerally easily visible from the graph but when this doesnot occur, the dmin parameter can be obtained as the inter-section between the straight line that fits the low scaleregion where the fractal is measured, and the straight linethat better fits the maximums of the periodic region.
3D SUR FACE TOPOGRAPHY FROM SEMSTEREO PAIR
In stereomicroscopy, the apparent separation between twopoints seen from different angles are directly related to theparallax or disparity by providing information in the 3D.Therefore, to obtain information of the 3D from a stereopair of SEM images, it is necessary to take two images byrotating the specimen at a small angle.
When operating SEM in the usual way, that is, at angleshigher than 08 and at any M magnification, the heightdifference z~i, j ! is not directly proportional to the disparityand it has a more complicated expression ~Lane, 1972!:
where W and M are the working distance and the magnifi-cation, respectively; f1 and f2 are the tilt angles correspond-ing to the left image ~lower angle! and the right image~higher angle!, respectively; x1 is the pixel position ~i, j !whose height value needs to be known on the left image,and x2 is the same for the right image. These pixel positionsare measured on the epipolar line, that is, perpendicular atthe tilt axis and by taking the image center as the coordinateorigin. These height values are measured with respect to aplane that contains the tilt axis and that forms an angle of~90 � f1! with the optic axis. The x1 and x2 parameters aremeasured in the same units as W. The EZEImage programuses this expression to obtain the dense height map fromwhich it is possible to obtain all the topographic parametersthat are necessary to characterize the surface.
RESULTS AND DISCUSSION
Figure 1 shows SEM images of 444 � 444 pixels of emerypapers FEPA #1000, #800, and #500 at 100� magnification~Fig. 1a, b, and c, respectively!. The images were obtainedwith a scanning electron microscope Philips SEM 505. Italso is possible to see the dense height maps correspondingto the previous SEM images ~Fig. 1d, e, and f, respectively!.The dense height maps were obtained with two SEM imagestaken at 10 and 20 deg, respectively and at an optic workingdistance of 22 mm.
In Tables 1 and 2, the obtained values using the vario-gram are shown for the D fractal dimension estimators andthe dmin and dper parameters for the SEM images, and forthe dense height maps corresponding to Figure 1. Becauseof the definition of the dper parameter, it is indispensable totake different angles. This parameter was obtained from theaverage of the ^k2&measured at 0, 15, 30, 45, 60, and 75 deg.Due to digitization effects, we have observed an increase ofthe fractal dimension when the obtained average value istaken at different angles in the image. In this study, thefractal dimension was obtained by scanning the image inhorizontal and vertical directions and then taking the aver-age, because for these samples, the fractal dimensions areapproximately equal in all directions. Although the digitiza-tion effect in the dmin parameter exists, it is better tocalculate the average of different angles to take into accountthe influence of asymmetries in the cells or particles. In all
180 Rita Dominga Bonetto, Juan Luis Ladaga, and Ezequiel Ponz
Three Estimators of Surface Roughness in 2D and 3D 181
cases, the dmin parameter was obtained by calculating theaverage of the obtained values at 0, 15, 30, 45, 60, and75 deg.
Using these tables, it is possible to conclude thefollowing:
Fractal dimension estimator. Both the SEM image textureand the manifested roughness in the height data show asimilar behavior. The observed fluctuations are due to thefact that the obtained fractal dimension, correspondingto the whole emery paper, is the result of at least threeeffects: ~1! the effect corresponding to the edge of theparticles ~“chessboard effect”! with a fractal dimension;2.5, ~2! the own roughness of each particle, and ~3! theeffect of the adhesive ~glue or resin! with a fractal dimen-sion bigger than 2.5 by the noise in the image due to thecontained little texture in this image region. The particlefractal dimensions are surely smaller than 2.5 as it can beseen in the tendency to smaller values of the fractaldimensions for emery papers with increasing particle size~decreasing particle densities!, where the influence of thechessboard effect is less predominant.
Parameter dmin. With respect to height image, the minimumcell with enough statistical weight to produce valleys inthe variogram is related to the average size of the distri-bution of particles and not to the minimum particle sizeas happened with SEM images. The gray levels on anSEM image correspond to a discontinuous function be-cause they are related to the slope and not to the height.But the height data that were obtained from the stereopair correspond to a continuous function, which in manypoints could be nonderivative. To understand the differ-ent meanings of the dmin parameter, it is necessary to takeinto account that the curve gets far from the fractalregime in the variogram of the emery papers when thestep size is bigger than the particle size, with enoughstatistical weight to produce periods ~and whose edge isidentified by the sharp slope change!. But in the heightdata, the change of the Brownian fractal to a periodicregime is almost exclusively due to the average size distri-bution and equal to ^S&, that is, the periods can onlyappear when exceeding the size of the most probableparticles. This fact is due to the continuity in the heightson these engineered surfaces that forces them to staywithin a minimum range. The meaning of dmin for an-other type of samples surely will not be the same. As withany other crossover or break point, the meaning of thedmin must be carefully interpreted in each studied example.
Parameter dper . In the cases when the density of the elemen-tary cells with enough statistical weight to produce peri-ods is not very high, the dper parameter is a measurementof the average distance among them on the SEM images,that is, in 2D. When the image contains a high density ofsmall cells, the dper parameter does not have the previousmeaning because a new arrangement of elementary cells~compound cells! with enough statistical weight to pro-duce periods could be found. This fact can be observedthrough the dper /dmin relationship: 7.62, 5.19, and 4.92 forthe #1000, #800, and #500 emery papers, respectively. Theinterpretation of the dper parameter is not direct in 3Dsince it was defined for 2D. However, even in this case, itcan be observed that the dper estimator would indicate thetendency to form compound cells by observing the valuesof the dper /dmin: 5.36, 5.09, and 4.05 for the #1000, #800,and #500 emery papers, respectively.
There is another way to obtain fractal dimensions withheight data by means of the log–log graph of area versusstep, where step is the elemental-square size of projection.In Table 3, it is shown the obtained fractal dimensions~Dvariogram and Darea! for the emery papers by the variogramand by the log–log graph of area versus step, respectively.The Dvariogram values were obtained through the FERImageprogram by taking the average of both directions, horizon-tal and vertical, and by beginning from different origins.The Darea value was obtained through the EZEImage pro-gram by taking the average of the calculated areas from topto bottom and from bottom to top for different steps.
^
Figure 1. SEM images of 444 � 444 pixels of emery papers FEPA:~a! #1000, ~b! #800, and ~c! #500. Height images ~dense heightmaps! of 444 � 444 pixels corresponding to the previous SEMimages: ~d! #1000, ~e! #800, and ~f! #500.
Table 1. D, dper , and dmin Estimators of SEM Images ~Fig. 1a, b,and c! Obtained by Means of the FERImage Program
*The ^S& column corresponds to the FEPA average grain size.
182 Rita Dominga Bonetto, Juan Luis Ladaga, and Ezequiel Ponz
At this point, it is important to discuss the differentfractal dimensions obtained from the variogram and fromthe log–log graph of area versus step. The fractal dimensionobtained by the last method can be considered essentially abox dimension ~D � 2 � a, where a is the straight lineslope of the log–log graph of area versus step! and is notappropriate for self-affine specimens. Nevertheless, in thecase of self-affine samples, the box-counting method cor-rectly used gives information of the local fractal dimensionand the global dimension ~Feder, 1988!. The local fractaldimension obtained by this method is equal to 3 � H, suchas the fractal dimension obtained by the variogram. Wehave applied the area versus step log–log graph to Browniansurfaces with different fractal dimensions from 2.1 to 2.8and, by studying a range of at least a decade, we haveobserved that the obtained values ~from 2.04 to 2.67, respec-tively! were smaller but closer than those obtained by thevariogram ~2.12 to 2.76, respectively!. These results allow usto state that the area method can be used to obtain anapproximate estimator of the fractal dimension instead ofthe accurate fractal dimension in the case of self-affinesurfaces. Therefore, by using the area method in sampleswhere mixed fractals are present, fractal-dimension estima-tor variations produced by changes in the surface roughnesswill complement the information obtained with the vario-gram method in the study of different processes related tosuch roughness changes.
To understand the obtained Darea values for the emerypapers ~approximately 2.3!, several theoretic examples wereanalyzed. Fractal dimension values were calculated for abinary image built as an arrangement of squares ~named“chessboard” by us! of approximately 13 pixels long on eachside ~Fig. 2!. The black and white values ~0 and 255, respec-tively! were taken as height values. The chessboard can bethought as a “mixed fractal” of a component of topologicaldimension D � 2 ~in the flat regions! and another ofdimension D � 2.5 corresponding to an ordinary Browniancomponent. We must point out that this example has thepurpose of clarifying what happens with two different ef-fects. However, this sample is not specifically a mixed fractalbecause the flat component has a topological dimension 2and the Brownian component is considered fractal onlywhen the fractal dimension is smaller or greater than 2.5.The value 2.5 corresponds to ordinary Brownian textures. Itshould be mentioned that the fractal dimension 2.5 ob-tained by the variogram method in the case of a chessboard
image always occurs independently from the number andsize of the squares, rectangles, or cells on the image. Thisvalue is a consequence of the sharp jump between colors orheights ~a noncontinuous function! and a proof of why thisoccurs can be seen in the appendix in Bonetto et al. ~2002!.This 2.5 value is also obtained in the simple case of animage with only a sharp jump, where obviously the periodsdo not exist ~Fig. 3!. In the case of Figure 2a, a fractaldimension value of 2.502356 0.00077 is obtained by usingthe variogram due to the chessboard effect, whereas in thecase of the calculation with the area versus step log–loggraph, a fractal dimension value at low scale of 2.0472 60.0054 ~Fig. 2b! is obtained. In this example with mixedcomponents, the area method mainly yields information ofthe planar region ~corresponding to the “particle and sub-strate surfaces”!, ignoring the jump information. It also canbe observed in this example that the crossover occurs in
Table 3. The Dvariogram and Darea for the Emery Papers
Figure 2. Fractal dimension obtained from the graph of: ~a! vari-ance ~V ! versus step and ~b! area versus step, corresponding to theimage with 2D particles that appears inside the graph.
Three Estimators of Surface Roughness in 2D and 3D 183
approximately 13 pixels in both graphs. In the variogrammethod, the data were obtained with the FERImage pro-gram by using logarithmic spaced and a correlation coeffi-cient of 0.999. In the area method, the data were obtainedwith the EZEImage program.
In this work, we also studied the case of a fractal imageobtained by adding a fractal Brownian image of a fractaldimension equal to 2.7 with one image similar to a chess-board of “squares,” averaging approximately 14 pixels longon each side. Figures 4 and 5 show two examples analyzedby the variogram and the area methods. In Figure 4a, thechessboard component is predominant due to the fact thatthe square colors are white and black. Figure 5a is similarbut with the chessboard component less predominant be-cause, instead of using black and white colors for thechessboard, two similar gray levels were used. The values offractal dimension at low scale were 2.5120 6 0.0010 and2.6309 6 0.0022 for Figures 4a and 5a, respectively ~theerrors are the corresponding ones to the regression curve!.In this example, it can be observed that in the case ofFigure 4a, the obtained fractal dimension is near the chess-board fractal dimension since the chessboard effect is pre-dominant. In the case of Figure 5a, where the “chessboard”is less predominant, the fractal dimension is near to thebiggest fractal dimension corresponding to the Browniancomponent in this case. In both cases, the dmin occurs at;14 pixels. In Figures 4b and 5b, the same images can beobserved analyzed by means of the area versus step log–loggraph. In both cases ~such as in the case of Fig. 2b!, thejump information in the fractal dimension values is ig-nored. Nevertheless, in the case of Figure 4b, the fractaldimension has a bigger value than 2 ~D � 2.24876 0.0048!since the Brownian component is present. In the case ofFigure 5b, the Brownian component is predominant and
the fractal dimension obtained is 2.560 6 0.018, againsmaller but close to the value obtained by the variogram~2.6309 6 0.0022!. Again, a regime change occurs in ;14pixels as in the variograms of Figures 4a and 5a.
In Table 3, it can be observed that the fractal dimensionestimator Darea is ;2.3 for the three emery papers. Thechessboard effect due to particle edges ~cells in the denseheight maps of Fig. 1d, e, and f!, disappears in the case ofthe log–log graph of the area versus step, as mentionedabove. This fact would allow us to infer that the fractaldimensions of ;2.3 correspond to the overlapping effect ofthe particle-surface roughness and the perturbation of theadhesive for the three studied emery papers. We can assumethat this low value of fractal dimension corresponds to amaximum value of the fractal dimension of particle sur-faces. This assumption is based on the fact that the effect ofthe adhesive contribution ~with fractal dimension .2.5!must be small regarding the particle contribution becausethe resulting value of both effects is significantly ,2.5. As
Figure 3. Fractal dimension obtained from the graph of variance~V ! versus step, corresponding to the image that appears insidethe graph.
Figure 4. Fractal dimension obtained from the graph of: ~a! vari-ance ~V ! versus step and ~b! area versus step, corresponding to theimage which appears inside the graph.
184 Rita Dominga Bonetto, Juan Luis Ladaga, and Ezequiel Ponz
an extra corroboration, we calculated fractal dimensionvalues with the variogram and with the area methods takinga ~f2 � f1! value of 20 deg instead of 10 deg ~by taking theleft image at 10 deg and the right image at 30 deg!. This~f2 � f1! value is too big for the calculation of particleheights, but increases the information corresponding to thecharacteristic roughness of particles ~much smaller! and tothe substrate perturbed by the adhesive. This case would besimilar to the example shown in Figure 5 where the chess-board component is less predominant than the correspond-ing component to the particles plus the adhesive. The fractaldimension estimators obtained by the area methods wereclose to 2.3 in the three emery papers whereas the valuesobtained by the variogram method were close to 2.37.This fact allows us to state that the area method showsthe characteristic roughness of particles perturbed by theadhesive.
It is important to emphasize that the methods thatwork with the height data by means of scanning stylusinstruments or quantitative stereoscopy yield reliable dataof simply-connected surfaces ~i.e., no bridges!. Therefore, inthe case of more complex samples, the obtained fractaldimension values will only be an estimator of the same.Conversely, the fractal dimension values obtained by anonly SEM image yield extra information about the wholeroughness of the sample because this image allows us toshow, for example, bridges. Nevertheless, by an only image,the obtained fractal dimension values may not be accuratevalues because variations in the image brightness for identi-cal slopes can be produced by the influence of differenteffects such as enhanced emission from edges and ridges,sample contamination, big atomic number differences, andso on. That is why we consider the fractal dimension valuesobtained from an only SEM image and those obtained froma height dense map are only estimators of the fractal dimen-sion. However, it is not our objective to get accurate valuesof the fractal dimension with SEM images because its esti-mators are also appropriate to characterize the samples as itwas previously shown.
CONCLUSION
An advantage of fractal analysis from the variogram methodis that it distinguishes surface-roughness characteristicsthrough a parameter that does not depend on the scale ~thefractal dimension! and through parameters that depend onscale ~dper and dmin!. In this work, we show that the fractaldimension estimators obtained by only one SEM image areenough to characterize the surface fractal nature of theemery papers because their behavior is totally similar tothose obtained with 3D height data. The dmin and dper
parameters in 2D and 3D have different meanings. In theemery paper samples studied here in 2D, the dmin parameteris related to the smallest particle size with enough statisticweight to produce valleys in the variogram, whereas in 3D itis related to the average particle size. The dper parameter in2D is a measurement of the average distance among theparticles in the SEM images and has different values for thethree studied emery paper samples. In the 3D case, althoughthe obtained dper parameter values are similar to the threestudied emery-paper samples, they could indicate the forma-tion of compound cells by studying the dper /dmin relation-ship. The possibility of having 3D data allows a moredetailed study of the fractal dimension estimator by usingboth methods: the variogram and the log–log graph of areaversus step. The first method takes into account the fractaldimension of the whole emery paper, which at the least, isthe result of three effects: ~1! the effect corresponding to theedge of the particles, ~2! the roughness of each particle, and~3! the effect of the adhesive ~glue or resin!. The secondmethod takes into account the fractal dimension correspond-
Figure 5. Fractal dimension obtained from the graph of: ~a! vari-ance ~V ! versus step and ~b! area versus step, corresponding to theimage which appears inside the graph.
Three Estimators of Surface Roughness in 2D and 3D 185
ing to the particles and the adhesive where the effect of theparticle edges ~chessboard! is minimized. Although a multi-fractal study should be necessary to characterize completelythese samples, in this work with the three-dimensionalstudy, it was possible to show the influence of at least twoeffects.
ACKNOWLEDGMENTS
This work was supported by Consejo Nacional de Investiga-ciones Científicas y Técnicas ~CONICET! of Argentina. Theauthors thank Dr. Elena Forlerer for supplying of the emerypapers and Dr. Mario Sánchez and Dr. María ElenaCanafoglia for their assistance in the attainment of the SEMimages.
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