Shape Analysis of Particles by an Image Scanner and a Microcomputer: Application to Agglomerated Aerosol Particles

Algorithms are presented for feature extraction of the geometrical shape of parti cles. The system consists of a microcomputer and an image scanner which scans the micrograph of particles and transmits the compressed image data. Methods for shape analysis are: 1) calculation of fundamental particle shape parameters ( Feret diameter, area, perimeter, first moment, second moment), 2) fractal analysis, 3) opening method and 4) separation of circular primary particles from an agglomerate. The connectivity of particle boundary is recognized for five cases (continuation, termination, creation, split and merge), and the fundamental shape features are calculated according to it. The validity and accuracy of these methods are examined by comparing the calcu lated and theoretical shape parameters for standard figures. A new index is proposed for the description of the structuring elements number of a two-dimensional particle shape in opening analysis. These methods are applied to the shape analysis of agglomerated aerosol particles generated from an electric furnance and by the CVD method, and it is shown that the fractal dimension as a particle distribution is useful for the quantitative description of shape.


Introduction
Agglomerated aerosol particles are found in various fields, such as coal combustion, diesel exhaust gases and nuclear reactor safety. They have complicated shapes like chain or cluster aggregates. Investigation of the dynamic behavior of these agglomerated particles is required for evaluating the measurements on these aerosols or their effects on human health. The relationship between the geometrical shape and the dynamics of a non-spherical particle has not yet been studied sufficiently, except for particles having simple configurations. It is necessary to find a shape parameter (or parameters) that can clearly explain the relation be- t This report was originally printed in J. Aerosol Research, Japan, 2, 117-127 (1987) in Japanese, before being translated into English with the permission of the editorial committee of the Japan Association of Aerosol Science and Technology, Japan. 2 tween the geometrical shape and the dynamic behavior of irregular-shaped particles. Quantification of the geometrical shape of aerosol particles is performed mainly for two-dimensional microscopic images. Computer image processing is indispensable for the morphological analysis of a particle having a complicated shape.
Recently, image processing technology has significantly advanced so that even bad quality image can be processed at a high-speed using image signal processors. These commercially available image processing instruments are very expensive and cannot be used easily, while the image scanner, which is used for inputting images, such as drawings into a microcomputer, is relatively inexpensive. Moreover, it has the advantages of low image distortion and high resolution due to the use of Charge Coupled Devices (CCD). Therefore, the scanner is effective for processing micrograph images of particles with a microcomputer. This paper discusses the methods for calculating various shape parameters from micrographs using an image scanner and a microcomputer. The methods are then applied to the shape analysis of agglomerated aerosol particles.
2. Image Processing Using Image Scanner 2. 1 Calculation of fundamental particle shape parameters The calculation of particle shape parameters is performed for a binary coded image that is converted from the gray picture of an original micrograph. The threshold value for binary coding is set at an appropriate level by changing the preset level of the image scanner (PC- INSO 1,NEC), and the binary coded images are displayed on a CRT by the microcomputer (PC-980 1, NEC). The scanner reads the image data of a particle micrograph in the horizontal direction using CCD and transmits the compressed binary image data to the microcomputer by sliding the micrograph in the vertical direction with a roller. We have adopted sequential method for processing an image because the scanner transmits data, scanning the field of view in a raster scan mode, and the method also saves data storage area.
The following conditions should be satisfied for the 8-connected continuation of one pair of points (start point Sk (i) and end point Ek (i) of the k-th edge points) detected on the scan line i, and the h-th edge points on the scan line (i-I): The connectivity at each edge point is recognized as one of five cases shown in Fig. 1.
Creation Termination Continuation Merge Fig. 1 Illustration of the connectivity of particle boundary at the scan line i.
Two or more pairs of edge points on the scan line (i-I) are continuous to the single pair of edge points on line i, and the same label is reassigned to the corresponding objects.
Two or more pairs of edge points on the scan line i are continuous to the single pair of edge points on line (i-I), and the label of the object on line (i-I) are assigned to the ones on line i. Based on the recognition described above, we can calculate the fundamental particle shape parameters, such as horizontal or vertical Feret diameter, area, perimeter, first moment and second moment.
(a) Horizontal Feret diameter: where iS and iE are the edge points designated label j.
(b) Vertical Feret diameter: where Yc and Yr are the scan line numbers ( Y coordinate) when conditions ( 1) and (2) are satisfied, respectively.
(c) Area: A= ~{E(i)-S(i)} (d) Perimeter: (6) The correction method proposed by Taniguchi I) is adopted to reduce errors caused by digitization of the image in the diagonal direction. (e) First moment: (11) Using these fundamental shape parameters, we can calculate diameter of the circle of equal projected area, diameter of the circle of equitl perimeter, center of gravity, radius of gyration, direction of major axis, circularity, anisometry and bulkiness. In cases of Merge and Split, the basic parameters are corrected. The number of particles equals the difference between the total event number of Creation and that of Merge for particle boundaries having different labels.

2 Fractal dimension
The shape parameters described in Section 2. 1 cannot completely reflect the two structural characteristics of an agglomerated particle profile, that is, the boundary consists of fine primary particles and macro-scale irregular form. Kaye 2 ) applied the concept of "fractal" proposed by Mandelbrodt3) to the shape analysis of powders and analyzed the structures using the structured walk method. To apply that concept to this image processing system, a new calculation method has been examined.
If the length of a particle boundary is measured with scale An and n steps are required for 4 measurement, the length Ln of the boundary is given by (12) When a straight line is measured for several values of An, a linear relationship holds between Ln and An. If the length of complicated boundary, such as a coastline, is measured in the same manner, the length increases with the shortening of An because the fine structure of the boundary appears. This relationship can be expressed as, (13) for fractal boundaries, where a is constant, and ~ is negative value. With Eq. (12) = Eq. (13 ), the following equation is obtained: A log-log plot of n versus An generates a linear line. The fractal dimension d represented below can be obtained from the slope of the line: The fractal dimension is calculated by the following procedures which are different from the sequential method in Section 2. 1. a) First, a particle binary image is stored in the VRAM (resolution: 640x400) of the microcomputer, and the outline is extracted. b) The profile can be extracted by detecting the edge points on a specific line and carrying out a logical operation between the image data on the specific scan line and on the upper one, as well as on the lower one. c) Next, grid length An is chosen as a basic value for scale conversion. d) The number N of the intersection points of the particle boundary and grid lines (X and Y axes) is counted, as shown in Fig. 2, for a series of lengths An. e) The fractal dimension d is obtained as the absolute value of the slope of the line in the log-log graph of N versus An. However, N should be corrected to unity within specific grid interval if the boundary intersects two or more times.
A one-dimensional measure is used to determine the fractal dimension of a particle boundary, as shown in Fig. 2. If open space, such as a netlike agglomerate, exists in the particle, the following procedure can be applied in order to obtain the porosity, that is, the fractal dimension for a two-dimensional measure.
The gravity center of a particle image is picked, and then a series of nested squares of different sizes are placed around it. The number of particle pixels in each square is counted 4 ).

3 Opening method
Fractal dimension is effective for quantitatively describing the properties of the complicated boundary of an aerosol particle. However, the dimension cannot express the isotropic properties of a two-dimensional shape or the elemental particle numbers of agglomerated particle. To measure particle size distribution, Matheron has mathematically developed the concept "opening" (refer to reference 5 for details). Domon 6 ) has applied this for quantitatively describing the morphology of mouse parotid glands.
Since a square lattice is used in this system, n/2 operations of the erosion-dilation processing to a binary coded image on square grids can be considered as the opening process of size n.
If the total pixel number of binary images is P(n), the pixel number of particles for the opening size n is given by D(n) = P(n-1)-P(n ). A two-dimensional image can be expressed using the number distribution of particles, that is, the distribution of D(n).
KONA No. 6 (1988) For instance, the distribution D(n) of a figure consisting of only one structuring element (a square in this system) has only one peek at n which corresponds to its side. The distribution has some peaks if the figure consists of some elements having different sizes, like chain agglomerates.
In practical processing, binary image data is stored in the VRAM of the microcomputer, and the number of pixels are counted after activating erosion-dilation operations n times.

4 Primary particle separation from an agglomerated particle
The method of erosion-dilation processing described in Section 2. 3 can be applied to estimate original primary particles from the two-dimensional image of an agglomerated aerosol particle which consists of fine spherical particles if the degree of particle overlapping is slight. When the degree is significant, the separation is impossible using this method. The iterative method 7 ) or the method for feature extraction and hierarchical decomposition for a closed curve using averaging operations 8 ) has been applied to separate the above with satisfactory results 9 ). If these methods were to be applied to our system, all calculations would have to be processed by the software of the microcomputer, resulting in an overloading. Then, we have examined a new processing method.
When primary particles are assumed to be spherical, the coordinates of centers and radii must be determined. To obtain these, a part of their arcs should be estimated from the agglomerated particle. First, an outline of the twodimensional image is extracted by the operation described in Section 2. 2. The curvature of the boundary changes abruptly on the intersection points where different circles meet. This can be expressed by the tangent angle becoming negative at the points. Therefore, we calculate the tangent angles at each discrete outline point with the boundary following, and select the candidate points for calculating parameters of circles during positive tangent angles.
To reduce the effects of digitization errors, the original image is subjected to one dilationerosion process, and the coordinates of the boundary points are smoothed by the interpolation using the gravity center of three points. Next, three parameters (the coordinates of the center and the radius of a circle) are estimated by the non-linear least squares method. The initial value of each parameter is obtained as follows: the center is the intersection point of the perpendicular bisectors of two chords (one is formed by candidate points having an initial number and an intermediate number; the other is formed by candidate points having a final number and an intermediate number), the radius is the distance between the center and the candidate point having intermediate number.
In case that the parameters obtained by the above-mentioned steps should indicate a circle, the program may sometimes recognize them as indicating different circles. We define the distance function P, which depends on the distance between centers of each circle t::,.r, written in A threshold value of P is determined by the discriminant analysis method to) in order to judge that some pairs of circles indicate an identical circle. If the pair of circles indicate an identical circle, the parameters are recalculated by reorganizing the candidate points on the boundary. In our calculation dth is assumed to be 10. If a large circle should include a smaller one, another judgement is done in order to reorganize the parameters.
The programs described in Section 2. 1 are written in the assembly language when the programs (for controlling input/output to or from the image scanner and VRAM) require high-speed processing, and the others are in the C language. These programs are modularized.

1 Fundamental shape parameters
The measurement accuracy depends on two kinds of errors: one is caused by the algorithm itself, and the other is due to binary coding. The latter is attributed to the photographic contrast or the characteristics of the image scanner.
A study on the accuracy of the algorithm was carried out by Kuga 11 ) and the authors 12 ) using model images generated by a computer. They have reported that the largest error occurs in perimeter measurement. Since the threshold level of the image scanner can be only changed into 8 levels, we used photographs having higher contrast. In the following, we have discussed the errors caused by the fluctuation of light source and electrical circuits.
As shown in Fig. 3, the standard figures have been created and the shape parameters of these figures have been calculated. The sizes of figures were changed into 4 or 5 levels, and the average values of 20 measurements were obtained. The figures, except for a circle, are input by rotating them to some degree. Tables 1 and 2 list the mean ratio of the measured to theoretical shape parameter and the coefficients of variation. The perimeter and the area represent the diameter equivalent to the circle perimeter and the area, respectively. The sizes shown in Tables 1 and 2 represent the pixel numbers equivalent to: the circle diameter; the side of a square and triangle; the diameter of the structuring element circle for a doublet and triplet (same size); and the smaller circle diameter for a doublet (different size). In this case, the diameter of the larger circle is twice that of the smaller one. Table 1 shows that accuracy increases with the increase in the figure size. When the number of pixels exceeds 70, the difference be- tween the measured and theoretical shape parameters (excluding the radius of gyration of triangles) are lower than 1%, and the coefficients of variation are stable. The error increases for pixel numbers below 20. Especially for a triangle, the rotation of the figure causes significant digitization errors in the diagonal direction, and the relative error of the area reaches approximately 10% for 10 pixels. However, as real particles seldom have such shapes, the relative errors of measured shape parameters are estimated to be about 5% or less for 20 pixels picture. Table 2 shows the accuracy for simple agglomerated models. In this case, the error between the measured and theoretical shape parameters of the area is larger than that of the perimeter. For the agglomerate whose elemental particles have the same size (pixels = 1 0 ), the relative errors of the measured shape parameters are smaller than that of the elemental circle, and those errors become at the same level as the pixel number increases. This indicates that the errors caused by drawing the figure cannot be negligible, and the agglomerate is considered to be a circle whose diameter is bigger than that of the elemental circle due to the larger degree of fusion of the elemental circles. On the other hand, the relative error in the doublet consisting of different sized circles depends mainly on the larger circle.

2 Fractal dimension
In the same manner as in Section 3. 1, standard figures are created, and image data is input by the image scanner to obtain the fractal dimension of the boundary. Figure 4  The Koch curve has an infinite self-similarity, however, a lower limit exists for drawing the figure or resolution of the computer display. Therefore, the fractal dimension is defined for the range that is larger than the lower limit mentioned above. The dimension is unity for the range below the lower limit. It should be noted that the fractal dimension is defined within a specific finite range because a figure has a finite scale. Figure 5 shows the results of a circle, square and rectangle with an aspect ratio of 2, obtained by the opening method. The abscissa indicates the size of opening normalized by the side length (square root of the area) of the square that is equivalent to the area of the tar-  By comparing the data of squares which are smoothed and not smoothed, shown in Fig. 5 (the square includes some ruggedness that cannot be shown in the figure), a peak in the number distribution of the smallest size is found in the graph for the unsmoothed square. This indicates that the smoothing operation is effective in reducing the effects of the digitization noise.

3 Opening method
After the opening operation, the peak in the distribution for a square appears at the value of ' A corresponding to the length of its side (A. = 1 ). The peak for a rectangle appears at the size of its shorter side. The number distribution shows a good agreement with the theoretical value. The maximum opening size is the side length of the inscribed square for a circle, and the distribution of squares is determined so as to fill the remaining portion of the circle. Consequently, the squares distribute over a relatively wide range of size that is smaller than its peak position, as shown in To evaluate the effectiveness of the opening method that divides a two-dimensional figure into structuring elements, it was applied to a triadic Koch island, to model figures of agglomerates, and to test powders that Kaye used to calculate fractal dimensions. As shown in Fig. 6,
(d)  agglomerates are divided into their structuring elements. For the strongly fused agglomerate consisting of many structuring elements, a peak appears at a point which corresponds to the largest circle of the fused portion. The results shown in Fig. 7 also indicate that the test powders are satisfactorily divided into their structuring elements according to their shapes. The area-weighted average of opening sizes is considered to represent a "squareness" using a standard figure as the quantitative evaluation of shape. Figure 8 indicates the average, except for the Koch island, shows a good relationship to the circularity that is generally used. By considering the ratio of the total number N of distributed figures to the peak value nP, an index as to the structuring elements of a figure, 10 p =N/np (18) can offer the measure of structuring elements number with the total number N as shown in Table 3. For instance, when the agglomerates (c) and (d) in Fig. 6 are compared, the fractal dimension of agglomerate (d) is larger than that of (c), and p of (c) is larger than that of (d), indicating that figure (c) includes more structuring elements than (d). N of (d) is larger than that of (c). This fact indicates that agglomerate (c) consists of more elements of the same size than (d).

4 Separation of primary particles from an agglomerate
As described in Section 3. 3, the opening method can estimate the size and the number of primary particles, but the method is not effective in estimating primary particles when they are strongly fused. Figure 9 illustrates the estimated results of structuring element circles from agglomerate models according to the method in Section 2. 4. In each model, the estimated results are satisfactory, and the estimation error of the radius is within 2%. However, some circles are not identical, though their arcs are on the same circumference. This was considered to be caused by the digitization errors resulting from the input condition. Therefore, we had to dialogically change the threshold value in the discriminant analysis in Section 2. 4.

1 Particles generated by an electric furnance
A boat containing a granular lead with no silver was fed into an electric furnance. The generated aerosol particles were guided into a coagulation chamber by flowing N 2 gas, thereby obtaining agglomerated aerosol particles 13 ). Since the generated aerosol particles were of a chain structure consisting of spherical primary particles, we applied the separation method described in Section 2. 4.
As shown in Fig. 10, the size distribution of primary particles is log-normal, having a geometric mean diameter of 0.205 ~m and a geometric standard deviation of 1.46. On the other hand, the measurements of about 500 primary particle diameters using calipers show a geometric mean diameter of 0.19 5 pm and a geometric standard deviation of 1.41. This difference suggests that the number of larger diameter particles decreased more than the result obtained by the separation method. This is caused by the following: the parameter estimation was difficult or the estimated diameter KONA No. 6 (1988) was larger than the real diameter because only a small numbers of pixels of an arc were used for estimation, or the change in curvature was too small as shown in Fig. 11. As a result, the discriminant function for reforming the arcs did not work effectively. It is required to en- hance the resolution ability of the image scanner or to manually eliminate excessively large circles to analyze particles that were not separated sufficiently.

2 Particles generated by the CVD method
We also applied the above described method to the shape analysis of aerosol particles generated by the CVD (Chemical Vapor Deposition) method. The aerosol particles are generated as follows: the Al 2 0 3 aerosol particles were generated using oxidation of aluminium acetylacetonate (AlAA) in C0-0 2 flame. AIAA is introduced into the flame in a finely sprayed ethanol solution by a ultrasonic nebulizer with CO. Clean air flowed from the peripheral portion of the burner to introduce the particles into the plenum chamber for stabilization. The aerosol particles were sampled using thermopositor and subjected to observation using an electron microscope. Figure 12 shows an example of TEM micrographs of the generated particles. The primary particles are approximately spherical in shape, and the size distribution is log-normal, having a geometrical mean diameter of 5 ~ 9 nm and a geometrical standard deviation of 1.2 ~ 1.3.
As shown in Fig. 13, the fractal dimension of the generated aerosol particle changes significantly at the point that corresponds to the size of the primary particles. This indicates that there are two shape structures of the agglomerated particle, i.e. the Euclidean boundary (d= 1) of the spherical primary particle and the complicated boundary (d= 1.34) of the entire agglomerated particle.
If the concentration of the solution and the equivalence ratio are constant, the particle shape shifts from web to cluster aggregates with the increase in the CO flow rate. Circularity can be used as an index for quantifying such difference in shape of agglomerated particles. The circularity, however, is calculated for a single particle and has a specific distribution for each condition of particle generation. Thus, the obtained data was summarized using an arithmetic mean of circularities for 300 ~ 500 particles as shown in Fig. 14. When AlAA concentration is 10 rnol/rn 3 , the circularity is proportional to the CO flow rate, coinciding with the visual recognition. However, for the concentration of 20 rnol/rn 3 , the circularity is not proportional to the CO flow rate. When the number percentage of fine particles is large on a micrograph, significant error in the digital image processing occurs because of poor resolution. As a result, the percentage of particles having a cirGularity of nearly unity increases, hence the mean value of circularities cannot completely express shape irregularity. This leads to possible disparity with the visual recognition result. The size distribution of primary particles is relatively rnonodisperse, so agglomerated particles are considered to be self-similar as a group. We select radius of gyration rg of an agglomerated particle as a basic scale, and perimeter P or area A as a measurement quantity. Then for KONA No.6 (1988) the agglomerated particles, the relationship between r g and P (or A) can be expressed as, P (or A) a: r/ as shown in Fig. 15. Here, d is the fractal dimention for a particle size distribution. This method can improve upon the lack of information, and it can eliminate erroneous information on the shapes of fine particles. The fractal dimensions in Fig. 14 are obtained from the relationship between the perimeter (in this case, diameter of the circle of equal perimeter) and the radius of gyration. The results show that the method can quantitatively represent the characteristics of shape that cannot be described with circularity. The dimension agrees satisfactorily with that obtained from a single agglomerate in Fig. 13, and this indicates the consistency of both methods.

Conclusions
By using an image scanner and a microcomputer, shape analysis of particles was performed relatively inexpensively for the microscopic images. The calculation of fundamental shape parameters from a micrograph containing many particles was sequentially carried out by corn-bining the classification of the connectivity of the particle boundary and the conventional calculation algorithm of shape parameters. The accuracy of this method is of approximately 2% error when the number of pixels is 40 or more. We have developed the fractal and opening analysis method of a single particle, and the separation method of spherical primary particles from an agglomerate particle. Moreover, the particle shape can be described by parameters that relate to the index of structuring elements number proposed in the opening analysis. In the applications of these methods to agglomerated aerosol particles generated by the two methods, we introduced the fractal dimension for particle groups, which is obtained for the entire range of the particle size distribution. The fractal dimension, in this case, is determined from the relationship between the radius of gyration of a single particle as a standard scale and the perimeter as a measurement value. The dimension can quantitatively express the characteristic change of shape caused by the different conditions of generation, while it cannot be described with the mean value of circularities. Nomenclature