- Email: [email protected]
Chapter 15 The Use of Fractal Dimension to Characterize Individual Airborne Particles
E.J. Karjalainen (Editor), Scientific Computing and Automation (Europe) 1990 0 1990 Elsevier Science Publishers B.V., Amsterdam
173
CHAPTER 15
The Use of Fractal Dimension to Characterize Individual Airborne Particles P. K. Hopkel, G. S. Casuccio2, W. J. Mershon2, and R. J. Lee2 1 Department of Chemistry, Clarkson University, Potsdam, Ny 13699-5810, USA and 2R.J. Lee Group, 350 Hochberg Road, Monroeville, PA 15146, USA
Abstract The computer-controlled scanning electron microscope has the ability to characterize individual particles through the fluoresced x-ray spectrum for chemical composition data and image analysis to provide size and shape information. This capability has recently been enhanced by the addition of the capacity to capture individual particle images for subsequent digital processing. Visual texture in an image is often an important clue to the experienced microscopist as to the nature or origin of the particle being studied. The problem is then to provide a quantitative measure of the observable texture. The use of fractal dimension has been investigated to provide a single number that is directly related to observed texture. The fractal dimension of the object can easily be calculated by determining cumulative image properties of the particle such as perimeter or area as a function of magnification. Alternative methods that may be more computationally simpler have also been explored. The fractal dimension for a variety of particles of varying surface characteristics have been determined using these different computational methods. The results of these determinations and the implications for the use of fractal dimension in airborne particle characterization will be presented.
1. Introduction Computer-Controlled Scanning Electron Microscopy (CCSEM) with its associated x-ray fluorescence analysis system is capable of sophisticated characterization of a statistically significant number of individual particles from a collected particulate matter sample. This characterization includes elemental analysis from carbon to uranium and along with scanning and image processing can analyze an individual particle in less than 2 seconds although longer analysis times are often needed for more complete particle analysis. This
174
automated capability greatly enhances the information that can be obtained on the physical and chemical characteristics of ambient or source-emitted particles. RecenUy improved data analysis methods have been developed to make use of the elemental composition data obtained by CCSEM [l, 21. Howevcr, we are currently not taking full advantage of the information available from the instrument. In addition to the x-ray fluorescence specUum and the size and shape data available from the on-line imagc analysis systcm, it is now possible to store the particle image directly. Major improvements in automated particle imaging by the RJ Lee Group now allows the automatic capturc of single particle imagcs. Thus, a 256 gray level, 256 pixel by 256 pixel images of a large number of particles can be easily obtained for more detailed off-line analyses of shape and particle texture. These analyses may provide much clearer indications of thc particle’s origin and what has happened to it in the atmosphere. In this report, one approach, fractal analysis, will be applied to a series of images of particles with known chemical compositions in order to explore the utility of the fractal dimension for characterizing particle texture.
2. Fractal analysis The texture of a surface is produced by its material componenls and the process by which it is formed. A technique callcd fractal analysis pcrmits detailed quantification of surface tcxlurc comparable to the detail perceived by the human eye. Although the fractal technique is mathcmatically complex, it is simpler than a Fourier analysis and it is wcll-suited for CCSEM. The fractal dimension of the surface is characteristic of the fundamental nature of the surface being measured, and thus of the physics and chemistry of that formation process. It is the objective of this feasibility study to determine whcthcr or not this concept is applicable to the characterization of individual particlcs and to heir classification into groups that can further be examined. Fractal analysis is based on the fact that as a surface is examined on a finer and fincr scale, more fcaturcs bccome apparent. In the same way, if an aggregated propcrty of a surface such as its surface area is measured, a given value will be obtained at a given lcvcl of detail. If the magnification is increased, additional surface elements can be vicwcd, adding to the total surface area measured. Mandelbrot [3] found, however, that the property can be related to the mcasurement element as follows
where P(E) is thc value of a measured property such as length, arca, volume, etc., E is thc fundamental mcasurcment element dimension, k is a proportionality constant, and D is the fractal dimension. This fractal dimension is then characteristic of the fundamcnlal
175
nature of the surface being measured, and thus of the physics and chemistry of that formation process. A classic example of understanding the fractal dimension is that of determining the length of a boundary using a map. If one starts with a large scale map and measures, for instance, the length of England’s coastline, a number is obtained. If a map of finer scale is measured, a larger number will be determined. Plotting the log of the length against the log of the scale yields a straight line. Mandelbrot has demonstrated how one can use the idea of fractal dimensionality in generating functions that can produce extremely realistic looking “landscapes” using high resolution computer graphics. The reason these pictures look real is due to their inclusion of this fractal dimensionality, which provides a realistic texture. Thus, it can be reasonably expected that from appropriate image data incorporating the visual texture, a fractal number can be derived that characterizes that particular textural pattern [4]. The fractal dimension of the surface can be determined in a variety of ways. For example, the length of the perimeter can be determined for each particle by summing the number of pixels in the edge of the particle at each magnification. The fractal dimension can then be obtained as the slope of a linear least-squares fit of the log(perimeter) to log (magnification).Alternatively, the fractal dimension of each image could be calculated by dctcrmining the surface area in pixels at each magnification and examining the log(area) against the log(magnificati0n). In addition, the fractal dimension can be determined from the distributions of the intensities of pixels a given distance away from each given pixel as described by Pentland [41. Another approach for the surface fractal dimension determination has been suggested by Clarke [ 5 ] . Clarke’s method is to measure the surface area of a rectangular portion of an object by approximating the surface as a series of rectangular pyramids of increasing base size. First, the pyramids have a base of one pixel by one pixel with the height being the measured electron intensity. The sum of the triangular sides of the pyramid is the surface area for that pixel. Thcn a two pixel by two pixel pyramid is used followed by a four by four pixel pyramid until the largest 2” size square is inscribed in the image. The slope of the plot of the log of the surface area versus the log of the area of the square is 2 - D where the slope will be negative. Clark‘s program was written in the C computer language. It has been translated into FORTRAN and tested successfully on the data sets provided in Clarke’s paper. This approach appears to be simple to use and computationally efficient and will be the primary method used for this study.
3. Results and discussion To test the ability of fractal dimension to distinguish among several different particle compositions, sccondary elcctron images were obtained for several particles each of sodium chloride, sodium sulfate, and ammonium sulfate. In addition, images of several
176
particles whose composition was not provided to the data analyst were also taken. The particles analyzed are listed in Table 1. To illustrate the Clarke method, Figure 1 shows a 16 level contour plot of the secondary electron intensities from an image of a ammonium bisulfate particle observed at a magnification of 180. The rectangle inscribed in the particle is the arca over which the fractal dimension was determined. The log(surface area) is plotted against the log(area of the unit pyramid) for the four particle types in Figures 2-5. The results of the fractal dimension analysis are also given in Table 1. The “unknown” compound was (NH4)HSOd. It can be seen that there is good agreement of the fractal dimensions determined for a TABLE 1 Particles analyzed for fractal dimension. No.
1 1 1
2 2 3 3
4 5 5 6 6
7 7
8 8
9 9 10 10 11 11 12 12
Particle Type Na2S04 Na2S04 Na2S04 Na2S04 Na2S04 Na2S04 Na2S04 (NH4hSO4 (NH4hS04 (NH4)2so4 (NH4hS04 (NH4hS04 NaCl NaCl NaCl NaCl NaCl NaCl Unknown 1 Unknown 1 Unknown 2 Unknown 2 Unknown 3 Unknown 3
Magnification
180 1,000 5,000 250 1,000 100 1,000 150 330 3,500 270 1,000 330 1,000 330 2.500 350 3,000 275 8.500 1,600 7,000 2.000 7,000
Fract a1 Dimension
Uncertainty
2.41 2.44 2.33 2.42 2.42 2.36 2.43 2.28 2.23 2.19 2.26 2.20 2.27 2.27 2.29 2.13 2.27 2.15 2.36 2.25 2.38 2.30 2.37 2.29
0.01 0.01 0.02 0.01 0.01 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.02 0.01 0.02 0.02 0.04 0.03 0.03 0.03
f
*
f f f f f f f f f f f f f f f f f f f f f f
a) Excludes Particle 1 at 5,000~and Particle 3 at 1OOx. b) Excludes Particle 5 at 3,500~and Particle 6 at 1,000~. c) Excludes Particle 8 at 2.500~and Particle 9 at 3,000~. d) Excludes Particle 10 at 8 , 5 0 0 ~and Particles 11 and 12 at 7.000~.
Average Value
2.412a
2.233b
2.275c
2.370d
177
Figure 1. Image of an NH4HS04 particle (left) showing inscribed rectangle within which the surface area is determined (right).
series of similar composition particles with some notable exceptions. Most of the high magnification views show dimensions significantly lower than that obtained at lower magnification.It is not yet clear why this result has been obtained, but there appears to be a potential systematic variation related to image magnification. However, for the set of images taken at a magnification such that they fill approximately 60% of the screen area, there is a consistent pattern of fractal dimension for each chemical compound. There is a sufficiently small spread in fractal dimension values that might be useful in conjunction
1-33Ox 1-1ooox v 2-33Ox
1-275x 1-850Ox v 2-1600x v 2-7OOOx
0
0
h
v
2-2500r
rd
0
3-35Ox 3-3000r
k
6 -
o
,. 0
, I
Log ( U n i t Pyramid)
Figure 2. Plot of log(surface area) against log(unit pyramid) for the three N a ~ S 0 4particles at each magnification.
I
I
3-2OOOx
I
I
I
2
3
4
5
Log (Unit Pyramid)
Figure 3. Plot of log(surface area) against log(unit pyramid) for the three (NH&S04 particles at each magnification.
178 I
I
I
v
1
7 ,
I 0
0
1-15Ox 2-33Ox
--
v
2-35OOx
1-180W
1~1OOOX 1-5OOOx 2-250x 2-1000W
3-IOOx
3p1000x
to
I
0
1
2
3
4
‘
5
Log ( U n i t P y r a m i d )
Figure 4. Plot of log(surface area) against log(tnit pyramid) for the three NaCl particlcs at each magnification.
30
I 1
2
3
4
5
Log ( U n i t Pyramid)
Figure 5. Plot of log(surface area) againqt log(unit pyramid) for the three (NH4);?S04 particles at each magnification.
with the fluoresced x-ray intensities in classifying the particles into types for use in a particle class balance [2] where additional resolution is necded beyond that provided by the clcments alone. These results certainly provide a stimulus for further study. It is clcar there is a rclationsliip between the measured fractal dimension and the texture observed in the imagcs. Further work is needed to dcterinine a method for routinely obtaining reliable fraclal dimcnsions from secondary electron images.
Acknowledgements The work at Clarkson University was supported in part by the National Science Foundation under Grant AThl89-96203.
References 1.
2. 3. 4. 5.
Kim DS, Hopke PK. The classification of individual particles based on computer-controllcd scanning electron microscopy data. Aerosol Sci Techno1 1988; 9: 133-151. Kim DS, Hopke PK. Source apportionment of the El Paso acrosol by particlc class balance analysis. Aerosol Sci ‘IkchnoZl988;9: 221-235. Mandelbrot R. The Fractal Geometry of Nature. San Francisco, CA: W.H. Freeman and Co., 1982. Pcntland AP. Fractal-based description of nuturd scenes. SRI Technical Note No. 280, SRI Intcmational, Mcnlo Park, CA, 1984. Clarke KC. Computation of thc fractal dinicnsion of topographic surfaccs using the triangular prism surface area method. Computers & Ceosci 1986; 12: 713-722.
173
CHAPTER 15
The Use of Fractal Dimension to Characterize Individual Airborne Particles P. K. Hopkel, G. S. Casuccio2, W. J. Mershon2, and R. J. Lee2 1 Department of Chemistry, Clarkson University, Potsdam, Ny 13699-5810, USA and 2R.J. Lee Group, 350 Hochberg Road, Monroeville, PA 15146, USA
Abstract The computer-controlled scanning electron microscope has the ability to characterize individual particles through the fluoresced x-ray spectrum for chemical composition data and image analysis to provide size and shape information. This capability has recently been enhanced by the addition of the capacity to capture individual particle images for subsequent digital processing. Visual texture in an image is often an important clue to the experienced microscopist as to the nature or origin of the particle being studied. The problem is then to provide a quantitative measure of the observable texture. The use of fractal dimension has been investigated to provide a single number that is directly related to observed texture. The fractal dimension of the object can easily be calculated by determining cumulative image properties of the particle such as perimeter or area as a function of magnification. Alternative methods that may be more computationally simpler have also been explored. The fractal dimension for a variety of particles of varying surface characteristics have been determined using these different computational methods. The results of these determinations and the implications for the use of fractal dimension in airborne particle characterization will be presented.
1. Introduction Computer-Controlled Scanning Electron Microscopy (CCSEM) with its associated x-ray fluorescence analysis system is capable of sophisticated characterization of a statistically significant number of individual particles from a collected particulate matter sample. This characterization includes elemental analysis from carbon to uranium and along with scanning and image processing can analyze an individual particle in less than 2 seconds although longer analysis times are often needed for more complete particle analysis. This
174
automated capability greatly enhances the information that can be obtained on the physical and chemical characteristics of ambient or source-emitted particles. RecenUy improved data analysis methods have been developed to make use of the elemental composition data obtained by CCSEM [l, 21. Howevcr, we are currently not taking full advantage of the information available from the instrument. In addition to the x-ray fluorescence specUum and the size and shape data available from the on-line imagc analysis systcm, it is now possible to store the particle image directly. Major improvements in automated particle imaging by the RJ Lee Group now allows the automatic capturc of single particle imagcs. Thus, a 256 gray level, 256 pixel by 256 pixel images of a large number of particles can be easily obtained for more detailed off-line analyses of shape and particle texture. These analyses may provide much clearer indications of thc particle’s origin and what has happened to it in the atmosphere. In this report, one approach, fractal analysis, will be applied to a series of images of particles with known chemical compositions in order to explore the utility of the fractal dimension for characterizing particle texture.
2. Fractal analysis The texture of a surface is produced by its material componenls and the process by which it is formed. A technique callcd fractal analysis pcrmits detailed quantification of surface tcxlurc comparable to the detail perceived by the human eye. Although the fractal technique is mathcmatically complex, it is simpler than a Fourier analysis and it is wcll-suited for CCSEM. The fractal dimension of the surface is characteristic of the fundamental nature of the surface being measured, and thus of the physics and chemistry of that formation process. It is the objective of this feasibility study to determine whcthcr or not this concept is applicable to the characterization of individual particlcs and to heir classification into groups that can further be examined. Fractal analysis is based on the fact that as a surface is examined on a finer and fincr scale, more fcaturcs bccome apparent. In the same way, if an aggregated propcrty of a surface such as its surface area is measured, a given value will be obtained at a given lcvcl of detail. If the magnification is increased, additional surface elements can be vicwcd, adding to the total surface area measured. Mandelbrot [3] found, however, that the property can be related to the mcasurement element as follows
where P(E) is thc value of a measured property such as length, arca, volume, etc., E is thc fundamental mcasurcment element dimension, k is a proportionality constant, and D is the fractal dimension. This fractal dimension is then characteristic of the fundamcnlal
175
nature of the surface being measured, and thus of the physics and chemistry of that formation process. A classic example of understanding the fractal dimension is that of determining the length of a boundary using a map. If one starts with a large scale map and measures, for instance, the length of England’s coastline, a number is obtained. If a map of finer scale is measured, a larger number will be determined. Plotting the log of the length against the log of the scale yields a straight line. Mandelbrot has demonstrated how one can use the idea of fractal dimensionality in generating functions that can produce extremely realistic looking “landscapes” using high resolution computer graphics. The reason these pictures look real is due to their inclusion of this fractal dimensionality, which provides a realistic texture. Thus, it can be reasonably expected that from appropriate image data incorporating the visual texture, a fractal number can be derived that characterizes that particular textural pattern [4]. The fractal dimension of the surface can be determined in a variety of ways. For example, the length of the perimeter can be determined for each particle by summing the number of pixels in the edge of the particle at each magnification. The fractal dimension can then be obtained as the slope of a linear least-squares fit of the log(perimeter) to log (magnification).Alternatively, the fractal dimension of each image could be calculated by dctcrmining the surface area in pixels at each magnification and examining the log(area) against the log(magnificati0n). In addition, the fractal dimension can be determined from the distributions of the intensities of pixels a given distance away from each given pixel as described by Pentland [41. Another approach for the surface fractal dimension determination has been suggested by Clarke [ 5 ] . Clarke’s method is to measure the surface area of a rectangular portion of an object by approximating the surface as a series of rectangular pyramids of increasing base size. First, the pyramids have a base of one pixel by one pixel with the height being the measured electron intensity. The sum of the triangular sides of the pyramid is the surface area for that pixel. Thcn a two pixel by two pixel pyramid is used followed by a four by four pixel pyramid until the largest 2” size square is inscribed in the image. The slope of the plot of the log of the surface area versus the log of the area of the square is 2 - D where the slope will be negative. Clark‘s program was written in the C computer language. It has been translated into FORTRAN and tested successfully on the data sets provided in Clarke’s paper. This approach appears to be simple to use and computationally efficient and will be the primary method used for this study.
3. Results and discussion To test the ability of fractal dimension to distinguish among several different particle compositions, sccondary elcctron images were obtained for several particles each of sodium chloride, sodium sulfate, and ammonium sulfate. In addition, images of several
176
particles whose composition was not provided to the data analyst were also taken. The particles analyzed are listed in Table 1. To illustrate the Clarke method, Figure 1 shows a 16 level contour plot of the secondary electron intensities from an image of a ammonium bisulfate particle observed at a magnification of 180. The rectangle inscribed in the particle is the arca over which the fractal dimension was determined. The log(surface area) is plotted against the log(area of the unit pyramid) for the four particle types in Figures 2-5. The results of the fractal dimension analysis are also given in Table 1. The “unknown” compound was (NH4)HSOd. It can be seen that there is good agreement of the fractal dimensions determined for a TABLE 1 Particles analyzed for fractal dimension. No.
1 1 1
2 2 3 3
4 5 5 6 6
7 7
8 8
9 9 10 10 11 11 12 12
Particle Type Na2S04 Na2S04 Na2S04 Na2S04 Na2S04 Na2S04 Na2S04 (NH4hSO4 (NH4hS04 (NH4)2so4 (NH4hS04 (NH4hS04 NaCl NaCl NaCl NaCl NaCl NaCl Unknown 1 Unknown 1 Unknown 2 Unknown 2 Unknown 3 Unknown 3
Magnification
180 1,000 5,000 250 1,000 100 1,000 150 330 3,500 270 1,000 330 1,000 330 2.500 350 3,000 275 8.500 1,600 7,000 2.000 7,000
Fract a1 Dimension
Uncertainty
2.41 2.44 2.33 2.42 2.42 2.36 2.43 2.28 2.23 2.19 2.26 2.20 2.27 2.27 2.29 2.13 2.27 2.15 2.36 2.25 2.38 2.30 2.37 2.29
0.01 0.01 0.02 0.01 0.01 0.02 0.01 0.02 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 0.02 0.01 0.02 0.02 0.04 0.03 0.03 0.03
f
*
f f f f f f f f f f f f f f f f f f f f f f
a) Excludes Particle 1 at 5,000~and Particle 3 at 1OOx. b) Excludes Particle 5 at 3,500~and Particle 6 at 1,000~. c) Excludes Particle 8 at 2.500~and Particle 9 at 3,000~. d) Excludes Particle 10 at 8 , 5 0 0 ~and Particles 11 and 12 at 7.000~.
Average Value
2.412a
2.233b
2.275c
2.370d
177
Figure 1. Image of an NH4HS04 particle (left) showing inscribed rectangle within which the surface area is determined (right).
series of similar composition particles with some notable exceptions. Most of the high magnification views show dimensions significantly lower than that obtained at lower magnification.It is not yet clear why this result has been obtained, but there appears to be a potential systematic variation related to image magnification. However, for the set of images taken at a magnification such that they fill approximately 60% of the screen area, there is a consistent pattern of fractal dimension for each chemical compound. There is a sufficiently small spread in fractal dimension values that might be useful in conjunction
1-33Ox 1-1ooox v 2-33Ox
1-275x 1-850Ox v 2-1600x v 2-7OOOx
0
0
h
v
2-2500r
rd
0
3-35Ox 3-3000r
k
6 -
o
,. 0
, I
Log ( U n i t Pyramid)
Figure 2. Plot of log(surface area) against log(unit pyramid) for the three N a ~ S 0 4particles at each magnification.
I
I
3-2OOOx
I
I
I
2
3
4
5
Log (Unit Pyramid)
Figure 3. Plot of log(surface area) against log(unit pyramid) for the three (NH&S04 particles at each magnification.
178 I
I
I
v
1
7 ,
I 0
0
1-15Ox 2-33Ox
--
v
2-35OOx
1-180W
1~1OOOX 1-5OOOx 2-250x 2-1000W
3-IOOx
3p1000x
to
I
0
1
2
3
4
‘
5
Log ( U n i t P y r a m i d )
Figure 4. Plot of log(surface area) against log(tnit pyramid) for the three NaCl particlcs at each magnification.
30
I 1
2
3
4
5
Log ( U n i t Pyramid)
Figure 5. Plot of log(surface area) againqt log(unit pyramid) for the three (NH4);?S04 particles at each magnification.
with the fluoresced x-ray intensities in classifying the particles into types for use in a particle class balance [2] where additional resolution is necded beyond that provided by the clcments alone. These results certainly provide a stimulus for further study. It is clcar there is a rclationsliip between the measured fractal dimension and the texture observed in the imagcs. Further work is needed to dcterinine a method for routinely obtaining reliable fraclal dimcnsions from secondary electron images.
Acknowledgements The work at Clarkson University was supported in part by the National Science Foundation under Grant AThl89-96203.
References 1.
2. 3. 4. 5.
Kim DS, Hopke PK. The classification of individual particles based on computer-controllcd scanning electron microscopy data. Aerosol Sci Techno1 1988; 9: 133-151. Kim DS, Hopke PK. Source apportionment of the El Paso acrosol by particlc class balance analysis. Aerosol Sci ‘IkchnoZl988;9: 221-235. Mandelbrot R. The Fractal Geometry of Nature. San Francisco, CA: W.H. Freeman and Co., 1982. Pcntland AP. Fractal-based description of nuturd scenes. SRI Technical Note No. 280, SRI Intcmational, Mcnlo Park, CA, 1984. Clarke KC. Computation of thc fractal dinicnsion of topographic surfaccs using the triangular prism surface area method. Computers & Ceosci 1986; 12: 713-722.