Automatic Classification of Graphite in Cast Iron - CiteSeerX

Centro de Tecnologia Mineral Ministério da Ciência e Tecnologia Coordenação de Análises Minerais – COAM Setor de Caracterização Tecnológica - SCT

Automatic Classification of Graphite in Cast Iron

Otávio da Fonseca Martins Gomes (CETEM) Sidnei Paciornik (DCMM/PUC-Rio)

Rio de Janeiro Julho/2005 CT2005-062-00 – Microscopy and Microanalysis 11 (4): 363371, 2005.

www.journals.cambridge.org/jid_MAM

Microscopy Microanalysis

Microsc. Microanal. 11, 363–371, 2005 DOI: 10.1017/S1431927605050415

AND

© MICROSCOPY SOCIETY OF AMERICA 2005

Automatic Classification of Graphite in Cast Iron Otávio da F.M. Gomes 1,2 and Sidnei Paciornik 1, * 1 Department of Materials Science and Metallurgy, Catholic University of Rio de Janeiro, Rua Marquês de S. Vicente, 225, sala 501L, Gávea, Rio de Janeiro, RJ, 22453-900, Brazil 2 CETEM—Centre for Mineral Technology, Av Ipê, 900, Ilha do Fundão, Rio de Janeiro, RJ, 21941-590, Brazil

Abstract: A method for automatic classification of the shape of graphite particles in cast iron is proposed. In a typical supervised classification procedure, the standard charts from the ISO-945 standard are used as a training and validation population. Several shape and size parameters are described and used as discriminants. A new parameter, the average internal angle, is proposed and is shown to be relevant for accurate classification. The ideal parameter sets are determined, leading to validation success rates above 90%. The classifier is then applied to real cast iron samples and provides results that are consistent with visual examination. The method provides classification results per particle, different from the traditional per field chart comparison methods. The full procedure can run automatically without user interference. Key words: cast iron, automatic classification, ISO-945, shape factor, image analysis

I NTR ODUCTION Cast iron can be defined as a Fe–C alloy with carbon content above approximately 2 wt%, often resulting in free carbon in the form of graphite particles ~Callister, 2003!. It is, therefore, a material composed of graphite particles dispersed in a metallic matrix. Cast iron presents graphite particles of different shapes that directly affect its thermo-mechanical properties. Fracture toughness and ductility depend strongly on the shape of the graphite particles. Particles with nodular shapes improve these properties whereas more elongated particles or ones with irregular contours are detrimental due to stress concentration points ~Van Vlack, 1964!. Thus, cast iron is classified according to the shapes of its graphite particles. The ISO-945 standard ~ISO, 1975! presents six classes to characterize the different shapes of graphite particles. Figure 1 shows the six reference images that represent these classes. Class I is called lamellar because of the characteristic shape of the particles observed in plane sections. Actually, the lamellae are interconnected, forming colonies that can only be perceived in three dimensions. Graphite particles of Class I are typical of gray cast iron. Classes IV, V, and VI are called, respectively, irregular nodular, indistinct nodular, and regular nodular or spheroidal and are the classes that correspond to malleable ~IV and V! and nodular ~V and VI! cast iron. Indeed, nodular cast Received June 21, 2004; accepted February 18, 2005. *Corresponding author. E-mail: [email protected]

iron is generally characterized by the presence of at least 80% graphite particles in classes V and VI and by the absence of particles from classes I and II ~Fargues & Stucky, 1994!. Class III is comprised of vermicular cast iron ~also known as compacted graphite iron!, thus called when it contains at least 80% of particles in this class, and also particles of classes V and VI ~Stefanescu et al., 1988!. Class III represents graphite particles with a shape intermediate between classes I and VI and that can also form colonies. Similarly, vermicular cast iron is intermediate between gray and nodular cast iron, presenting thermal properties similar to gray cast iron and mechanical properties closer to nodular cast iron ~Stefanescu et al., 1988!. Class II, called crab or spiky due to its shape with sharp branches that resemble crab legs, represents a shape for graphite particles that does not correspond to any specific kind of cast iron. The particles in this class appear due to the degeneration of class VI particles in the production of cast irons when there are impurities or excessive nodulizing components ~Hecht, 1995!. Particles in this class also appear in rapidly cooled hypereutectic gray irons ~Hecht, 1995!. Traditionally, the classification of cast iron is made by visual comparison of optical micrographs with the reference charts from ISO-945. Obviously, a simple visual comparison is subjective and cannot be automated in the industrial environment. Thus, it is of great interest to have a reliable automated approach for the classification of existing classes of graphite shapes. A computer-aided system can automate this process using image analysis ~IA!, which brings new possibilities to microstructure characterization. Image analysis provides

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Figure 1. Reference images for the six classes of graphite particles according to standard ISO-945 ~ISO, 1975!.

efficient ways of measuring graphite shapes with greater speed and statistical quality. Moreover, digital methods allow the application of pattern recognition and artificial intelligence techniques to automate the classification procedure. Once a set of reference images of the different graphite types is provided to the measurement system, the recognition of unknown samples can be made automatically in a typical supervised classification procedure ~Duda & Hart, 1973; Gonzalez & Woods, 1992!. There remains, however, the problem of how to characterize the graphite shape. The present article proposes an automatic method, based on IA, to classify graphite particles in cast irons according to the ISO-945 standard. A supervised classification routine is employed, in which the reference images from the ISO-945 are used as a training set. To represent the graphite shapes, several shape descriptors are tested.

S UPERVISED C LASSIFICATION A supervised classification procedure comprises three stages: training, validation, and classification. In the training stage,

a known set of objects, the training set, is used to build a knowledge base. Several parameters—the parameter set— are measured for each object, establishing a parameter space, where each object is represented as a parameter vector, a dot in this space. Thus, the defined classes can be understood as regions in the parameter space. The validation or test stage estimates the performance of the classification system, also considering its generalizing capability. Several validation methods have been described in the literature. One of the simplest is the resubstitution estimate ~Toussaint, 1974!. This method consists of the classification of the training set objects and the evaluation of the success rate. However, it is an overoptimistic estimate that does not consider the generalizing capability of the system. An improved validation method is the holdout estimate ~Toussaint, 1974!. The available known objects are split into two complementary sets: the training set and the validation set. After the training stage, the validation set objects are classified and the success rate is determined. However, if the total number of objects is small, the results are sensitive to the specific choice of objects for training/ validation. This is indeed the case when using the ISO-945

Automatic Classification of Graphite in Cast Iron

reference images, as they contain a relatively small number of objects. Thus, different partitions with fixed size are taken and the average success rate is obtained ~Duda & Hart, 1973; Toussaint, 1974!. If the validation stage indicates that the training is successful, classification is then possible. First, a new training set is built with all available known objects and the training stage is carried out again. Unknown objects can then be recognized through the measurement of the same parameters and the use of a decision method—the classifier—that will attribute each unknown object to one of the previously defined classes. In the cast iron problem, the number of classes is defined by the ISO-945 standard. The training set is comprised of the reference charts. Each chart contains a certain number of graphite particles that represent each of the six classes. The challenge is to choose the parameter set that will group particles of the same class, accepting their intrinsic variability and, at the same time, provide maximum discrimination between the six classes. In practice, a single parameter is seldom enough to distinguish between two or more classes and the parameter space is multidimensional. On the other hand, adding more parameters to the set does not necessarily imply an improvement in classification ~Chen, 1973!. The increase in dimensionality of the parameter space requires an increase in the size of the training set to assure good statistics, and this is not always possible. Moreover, a large parameter set may reduce the generalizing capability of the classification system. Thus, one should select the smallest set among the available parameters. If time is not a critical issue, the choice of parameters for an optimal set may be made through an exhaustive search of the set that reaches the best success rate in the validation stage.

mum value corresponding to perfect geometric shapes and the minimum corresponding to irregular shapes. Both the standard shapes and their theoretical models can vary widely and there are several parameters described in the literature. Table 1 lists many of these factors, showing their definitions and the characteristics to which they are more sensitive. It is worth mentioning that the names and definitions of these parameters vary in the literature ~Russ, 1995!. The graphite shapes presented in the ISO-945 standard vary from irregular, elongated, and branched shapes to circles, when going from Class I to Class VI. Thus, in the attempt to characterize these shapes, dimensionless factors that quantify the deviation from a circle have been proposed. The first parameter in Table 1, the circular shape factor ~CSF!, is 1 for a perfect circle and decreases when the perimeter increases against the area. However, if the object contour is irregular, the perimeter can increase even if the overall shape is still close to a circle. Thus, the CSF is more sensitive to contour irregularities. The second parameter in Table 1, the roundness, depends on the maximum caliper ~maximum feret @Fmax #! of the object. Again, it assumes a value of 1 for a perfect circle and decreases when Fmax increases against the area. Thus, it is more sensitive to the elongation of the object. Grum and Sturm ~1995! proposed an intermediate parameter between the circular shape factor and roundness. This parameter models the circle based on its area, perimeter, and maximum feret, being thus sensitive both to elongation and contour irregularities. This parameter is defined as

f⫽

Dimensionless Shape Factors These factors are dimensionless parameters derived from the basic geometrical measurements ~area, perimeter, calipers, etc.!. They generally vary between 0 and 1, the maxi-

16{A 2 . p{P{~Fmax ! 3

~1!

Hecht ~1995! proposed the use of two parameters:

C HARACTERIZING G RAPHITE S HAPE There is no universal definition for the shape of an object. Intuitively, the shape of an object is described by comparison with another one or through the characteristics of its contour. Thus, in image analysis, shape is characterized by quantifying some contour property or the difference between a given object and a reference shape. These shape description parameters should, in principle, be independent of size, position, and rotation of objects in an image. There are two main types of parameters: dimensionless shape factors and contour measurements ~Gomes, 2001!.

365

P2 4{p{A

~2!

3{~Fmax ! 2 . 4{A

~3!

a⫽ and

b⫽

These factors are, in fact, variations of the CSF and roundness. Together, they are sensitive to elongation and contour irregularities. Wojnar ~2000! states that a single parameter is not enough to characterize graphite shapes. Nevertheless, he proposed a parameter, based on the ratio between the diameters of the largest inscribed and smallest circumscribed circles, that is simultaneously sensitive to branching, elongation, and contour irregularities.

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Table 1.

Shape Factors

Name Circular shape factor ~Russ, 1995! Roundness ~Russ, 1995! Modified CSF ~Grum & Sturm, 1995! Grum CSF ~Grum & Sturm, 1995! Aspect ratio ~Russ, 1995! Modification ratio ~Wojnar, 2000! Branching factor ~Wojnar, 2000! Convexity ~Russ, 1995! Solidity ~Russ, 1995!

Definition 4{p{A CSF ⫽ P2 4{A Round ⫽ 2 p{Fmax 4{A CSFm ⫽ P{Fmax 16{A2 CSFg ⫽ 3 p{P{Fmax Fmin AR ⫽ Fmax W MR ⫽ Fmax W BF ⫽ Fmin Pc Conv ⫽ P A Sol ⫽ Ac

Sensitivity Circular shape and contour irregularities Circular shape and elongation Circular shape, elongation and contour irregularities Circular shape, elongation and contour irregularities Elongation Elongation and branching Branching and bending Convex shape and contour irregularities Convex shape, thin and long ramifications

Notation: A: surface area, A c : convex area, P: perimeter, Pc : convex perimeter, Fmin : minimum feret, Fmax : maximum feret, W: largest inscribed circle diameter.

Contour Measurements and the Average Internal Angle Contour measurements are functions computed from the relative position of contour pixels and between these pixels and the center of gravity of the object, and basic geometric measurements. Based on the radii of the contour pixels ~their distance to the center of gravity!, parameters like the average radius, the variance of the radii, and their dispersion about the average can be defined ~Weeks, 1996!. More complex contour measurements can be defined from the so-called contour signature analysis. Based on the radii, angle, and sequential position of pixels in the contour, a characteristic signature function can be obtained. Many parameters can be extracted from this signature, such as the Fourier descriptors ~Gonzalez & Woods, 1992!. Due to their nature, contour measurements are generally more sensitive to the irregularities of the contour ~Gomes, 2001!. However, they are computationally more demanding than the dimensionless shape factors. The present article proposes an approximate, albeit simple, method to compute some contour measurements that will be useful for shape classification. Based on a binary image ~Fig. 2a! containing the objects to be analyzed, the following sequence is followed: 1. The pixels of the objects are painted with gray level k 2 ~Fig. 2b!;

2. An average filter with kernel size k ⫻ k is applied to the modified image ~Fig. 2c!; 3. The edges of the original binary image are detected and multiplied by the filtered image ~Fig. 2d!. The image ~Fig. 2d! resulting from these few operations contains the edges of the particles in which each pixel has a gray level ~n! that is an integer value correlated to the internal angle ~u! of the object at that particular position. Actually, the internal angle ~u! can be roughly calculated in degrees as shown below:

冋 冉 冊册

u ⫽ 180 ⫹ n ⫺

k ⫹1 360 . {k { 2 2 k ⫺1

~4!

To clarify the meaning of u, and the use of equation 4, a magnified test particle was prepared as shown in Figure 3a. In this figure, the gray squares represent pixels of the object and the dark gray squares represent edge pixels. The parameter k was chosen at its smallest value, 3, and a corresponding 3 ⫻ 3 pixel neighborhood is shown superimposed on the top edge of the object. Following the procedure outlined above, every object pixel is painted with gray level k 2 ⫽ 9. The resulting pixel values in the neighborhood are shown in Figure 3b. The application of a k ⫻ k average filter to the modified image

Automatic Classification of Graphite in Cast Iron

367

Figure 2. Image processing sequence for obtaining the local internal angle: binary image ~a!; object with gray level k 2 ~b!; image blurred with a k ⫻ k average filter ~c!; resulting image in which each edge pixel has an intensity proportional to local internal angle ~d!.

Figure 3. Obtaining the AIA: test object with superimposed analysis neighborhood ~a!; painting object pixels with gray level k 2 ~b!; obtaining the average value n ~c!; the internal angle at this position ~d!.

leads to a value n that corresponds to the number of object pixels in the neighborhood. In this case, n⫽

5{k 2 ⫽ 5. k2

~5!

The central pixel of the neighborhood then receives the value n ~Fig. 3c!. The internal angle at this position, 1358 in

this case ~Fig. 3d!, can then be calculated from equation 4 as follows:

冋 冉 冊册

u ⫽ 180 ⫹ 5 ⫺

3 ⫹1 360 ⫽ 135. {3 { 2 2 3 ⫺1

~6!

Based on the image in Figure 2d it is easy to obtain several parameters such as, for instance, the average internal

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Otávio da F. M. Gomes and Sidnei Paciornik

angle ~AIA! for an object. As described in the following sections, this parameter is useful for classifying graphite shapes in cast iron. Variable k parameterizes this measurement. It defines the kernel size ~k! for the average filter and the gray level ~k 2 ! that paints the objects in the binary image. Choosing a small value for k leads to great sensitivity to small irregularities in the contour. However, these small irregularities derive much more from the discrete pixelization than from the real shape of objects. In the limit, with k ⫽ 3, there are only seven possible values for n ~2, 3, 4, 5, 6, 7, and 8! that correspond respectively to internal angles of 08, 458, 908, 1358, 1808, 2258, and 2708. On the other hand, a large value of k increases this measurement resolution but allows adjacent objects to interfere in the measurement, biasing the result. Moreover, AIA is inversely proportional to k and directly proportional to object size. Thus, an optimum value for k must be chosen. Even with these approximations, the experiments described in the following indicate the validity of the use of AIA in the classification of graphite shapes in cast iron.

Size-Related Parameters As stated previously, the shape description parameters should be independent of size. However, size and shape are strongly correlated properties ~Wojnar, 2000!. The discrete pixelization of an object image hampers its contour representation. This effect is inversely proportional to object size. The larger an object is in an image, the more pixels it contains, leading to a more accurate representation. On the other hand, in smaller objects, formed by few pixels, the representation is worse, causing a significant deterioration in shape measurements ~Livens, 1998!. Thus, the addition of size-related parameters to the parameter set introduces size discrimination in parameter space that may be helpful to compensate for the size–shape correlation. In fact, experiments described in the following sections confirm this assumption. Table 2 lists three size-related parameters used in the present work, with their definitions.

Table 2.

Size-Related Parameters Name

Definition

Area Aspect product Box

A AP ⫽ Fmax{Fmin Box ⫽ Fmax{Fp max

Notation: A: surface area, Fmin : minimum feret, Fmax : maximum feret, Fp max : feret perpendicular to Fmax

were tested: nine dimensionless shape factors ~Table 1!, three size-related parameters ~Table 2!, and the AIA. The software also offers three classifiers: the Euclidean distance, the Mahalanobis distance, and the Bayesian classifier ~Duda & Hart, 1973!, which were tested and compared. In all tests described in the following, the Bayesian classifier performed better and all results shown refer only to this classifier. After training and testing, the determined optimal parameter set and classifier were applied to real cast iron samples of three different types. The samples were prepared for optical microscopy and observed at 100⫻ magnification. Images of 10 fields per sample were acquired with a conventional video camera and digitized with a frame grabber to 640 ⫻ 480 pixels. This relatively low resolution was acceptable in this case because it is similar to the resolution of the reference images. Image processing and analysis followed the standard sequence ~Vieira & Paciornik, 2001; Gomes, 2001! as follows:

• • • • •

pre-processing to correct uneven illumination using a large high-pass filter; automatic segmentation of graphite particles through the Otsu algorithm ~Otsu, 1979!; postprocessing comprising the elimination of very small particles and of particles touching the edges of the field; measurement of the parameters in the optimal set; classification.

R ESULTS

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D ISCUSSION

E XPERIMENTAL M ETHODS Determining the Optimal Parameter Set Digitized images of the ISO-945 charts were obtained at 512 ⫻ 512 pixels. Each of these images contains about 20 particles. All parameters mentioned above were implemented in macro routines of the KS400 software ~Carl Zeiss Vision!. Although the software provides some of the basic dimensionless shape factors and size-related parameters and allows easy definition of additional ones, all contour measurements required separate programming using the internal macro programming language. In total 13 parameters

As stated above, the reference images of the ISO-945 were used for training and validation using both resubstitution and holdout estimates. In resubstitution all objects in the reference images were used for training and validation. Naturally, this option provided the best success rates, some of them close to 100%. These rates were initially used to determine the optimal parameter set and the best value for k that parameterizes the AIA. An exhaustive search procedure was employed, testing a total of 8191 combinations of the 13 parameters for 7 k values ~3, 5, 7, 9, 11, 13, and 15!.

Automatic Classification of Graphite in Cast Iron

Table 3.

Global Success Rate ~%! as a Function of the AIA k Parameter

k

3

5

7

9

11

13

15

Maximum # Sets 99%

97 —

98 —

99 7

99 58

99 10

98 —

98 —

Table 4.

Global Success Rate ~%! as a Function of the Parameters Used

Parameter

A

AP

Box

Size a

CSF

Round

CSFm

CSFg

AR

MR

BF

Conv

Sol

AIA

Max. with Max. w/o

99 98

99 99

99 99

99 98

99 99

99 99

99 99

99 99

99 99

99 99

99 99

99 98

99 99

99 96

a

369

Size refers to size-related parameters. See text for details.

The results are summarized in Tables 3, 4, and 5. In these tables the global success rate for the six ISO-945 classes taken as whole is used as comparison parameter. Table 3 illustrates the influence of k. The first row, entitled Maximum, shows the maximum success rate for the six classes as a function of k. The maximum success rate obtained is 99%, for parameter sets containing AIA with k ⫽ 7, 9, or 11. The second row, entitled # Sets 99%, shows for how many parameter sets the 99% value was obtained. For k ⫽ 9, 58 parameter sets reached the 99% success rate, against just 10 sets for k ⫽ 11 and 7 sets for k ⫽ 7. For other values of k, the 99% rate was not reached. Thus, k ⫽ 9 was chosen as the optimum value and all results shown in the following use this value. Table 4 shows the global success rate for the six classes of the ISO-945 as a function of the several shape and size parameters used. The first row, entitled Max with, shows the maximum success rate obtained with a parameter set that contains the parameter. It can be seen that the maximum success rate obtained is 99%. Moreover, this maximum success rate can be reached with parameter sets containing each tested parameter, meaning that none of the parameters individually caused misclassification. Similarly, the second row, entitled Max w/o, is the maximum success rate reached without the parameter. Thus, this row shows the importance of each parameter. The fourth column ~Size! is useful to analyze the influence of the three size-related parameters ~A, AP, and Box!. In this regard, w/o means without any size-related parameter and with means at least one of them. Without at least one size-related parameter, more specifically the area ~A!, the global success rate does not go above 98%. However, as the last column of the table shows, the AIA is the decisive parameter. If it is not used, the maximum rate is just 96%, against 99% when it is employed. Even

Table 5. Global Success Rate ~%! as a Function of Parameter Set for the Validation Stage Parameter set a

Resubstitution

$A, MR, Conv, Sol, AIA% $A, CSFm, AR, Conv, AIA% $Box, CSF, CSFm, BF, AIA% $Box, AR, Sol, AIA% $A, CSF, CSFg, MR, BF, AIA% $A, Box, MR, Conv, Sol, AIA% $A, Round, MR, Conv, Sol, AIA% $A, CSF, CSFm, MR AIA% $A, CSFm, CSFg, MR, Conv, Sol, AIA% $A, Box, CSF, CSFg, MR, Conv, Sol, AIA%

96 98 97 95 96 96 98 96 98 98

a

Holdout average 92 91 91 91 91 91 91 91 91 91

Only the 10 sets with the highest holdout estimates are shown.

though this 3% difference seems small, it is the largest change caused by a single parameter. For the holdout estimates the object population was partitioned in two complementary sets with the same number of objects ~61!, one serving for training and the other for validation. However, as the total number of objects is small, the results are sensitive to the specific choice of population for training/validation. Thus, 10 different partitions were used and the average success rate of the 10 possibilities was obtained. Average holdout estimates were obtained only for the 2012 parameter sets for which the resubstitution success rates were above 95%. The 10 sets with the highest holdout estimates are shown in Table 5 along with their resubstitution rates. As expected, holdout estimates are lower than the resubstitution results. In Table 5 all parameter sets employ the

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Figure 4. Classification of real samples. Numerical results refer to 10 fields per sample. See text for further details.

AIA, as it was considered a critical parameter. Holdout estimates for sets without the AIA were always at least 3% lower.

Classification of Real Samples Figure 4 depicts typical fields of the real cast iron samples analyzed. The used parameter set was $A, MR, Conv, Sol, AIA%, which provided the best holdout success rate ~Table 5!. In this case, each graphite particle was automatically assigned to one of the six ISO-945 classes and the number of particles in each class, as well as their percentage against the total number of particles, was obtained. The table in Figure 4 shows the classification results for 10 fields per sample. The results are clearly consistent with visual examination of the images. Moreover, the proposed method classifies each individual graphite particle and provides much more detailed information than the traditional chart comparison method, which provides only an overall classification for a given field. The classification of all particles in a given field takes 2 or 3 s on a regular PC.

C ONCLUSIONS An automatic method for the classification of graphite particles in cast iron, based on the ISO-945 standard, was developed. It uses size and shape parameters to build parameter sets that allow discrimination between particles pertaining to the six different classes. A new parameter, the average

internal angle, was proposed and proved to be very relevant in the classification. The training and validation stage of the procedure used the images from the ISO-945 standard as references. As these images are drawn renditions of the graphite particles and contain a small number of particles, they do not constitute a perfect training/validation set. However, as these images are the de facto reference for the traditional chart comparison, it is not unreasonable to use them in the automatic classification procedure. Ideally, images of carefully prepared cast iron samples, each pertaining to a single class, should be used, but no such set of samples/images exists. In principle, training and validation, together with the determination of the optimal parameter set, need not be repeated for the cast iron system. Thus, the classification of real samples can be done quickly and automatically by just applying the developed classifier to real images. The same methodology can be applied to other materials where some kind of classification is currently done visually. If a training set can be obtained and parameter sets can be defined, the automatic method can be faster and more accurate than the traditional method.

A CKNOWLEDGMENTS The authors acknowledge the support of CNPq and FAPERJ, Brazilian funding agencies.

Automatic Classification of Graphite in Cast Iron

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