Materials Characterization 46 (2001) 317 – 327
Microcrack identification on particulate-reinforced metal matrix composite Sylvie Yottea,*, Denys Breyssea, Joe¨lle Rissa, Somnath Ghoshb a
Centre de De´veloppement des Ge´osciences Applique´es, Baˆtiment de ge´ologie, Avenue des Faculte´s, F33405 Talence cedex, France b Department of Aerospace Engineering/Applied Mechanics/Aviation, The Ohio State University, 327 Boyd Laboratory, 155 W. Woodruff Avenue, Columbus, OH 43210, USA Received 28 February 2000; accepted 9 December 2000
Abstract This is a study of particle damage based on image analysis of light optical micrographs obtained by serial sectioning of an Al – Si – Mg alloy reinforced with Si particles. To quantify damage in the material, it is necessary to determine the cracked surface area. Unfortunately, there is a deviation in the gray level distribution on the images, so that a simple thresholding does not work. In this study, four methods for allowing the identification of crack pixels on an image are investigated. Three of the methods do not identify the crack pixels to within an acceptable error. The fourth method, which is based on the isolation of each particle and the definition of crack pixels through a gray level step over the particle pixels, is found acceptable and is, hence, chosen. An application is done on three images, and the damage areas in the three sections are compared. D 2001 Elsevier Science Inc. All rights reserved. Keywords: Image analysis; Segmentation; Metal matrix composite; Crack
1. Introduction Particle metal matrix composites are now increasingly used for automotive and aerospace structures because of developments in both composite design and fabrication techniques. Thus, the necessity of knowing better the material becomes important in order to simulate the behavior of the structure for composite design and dimensioning. With this aim, a multiscale simulation of composite damage could be developed. Such a program has to be fed with material data. The objective of the present study is * Corresponding author. Tel.: +33-5-56-84-65-88; fax: +33-5-56-80-71-88. E-mail address:
[email protected] (S. Yotte).
to develop a material analysis capability necessary for such a numerical simulation. The material studied here is a sintered Al – Si – Mg alloy containing 0.4% Mg and 20.4% Si particles. A part of the Si is associated with Mg in the form of Mg2Si, another part remains in the solid solution of Al – Si alloy and the third part is a precipitate of Si particles. The detailed processing has been described by Li et al. [1,2]. Specimens are prepared and loaded in tension parallel to the extrusion direction at various levels of deformation. For each level, serial sectioning is performed, and micrographs (light optical microscopy) are taken for each section at various magnifications in order to have a representation of the damaged volume (see Refs. [1,2]). The damage observed on the micrographs comes from particle
1044-5803/01/$ – see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S 1 0 4 4 - 5 8 0 3 ( 0 0 ) 0 0 11 4 - 5
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Fig. 1. Flowchart of the whole study: the paper is devoted to the crack analysis part.
microcracking. Fig. 1 shows a flowchart of the whole process. A study of the 2D micrographs (particle and crack analysis) has been already done by Li et al. [2,3], as well as the 3D reconstruction from serial sectioning images. However, the method used for crack identification lacks precision and misses some cracks. Thus, the aim of this paper is to improve crack identification. In the following, four different methods for crack identification are proposed. To process for analysis, the images are scanned with a Sony CCD-IRIS camera. The analysis is done using a commercial image analysis software package, MISIS. Three images at 500 magnification are studied (1 pixel = 0.125 mm2). They will be called Images 1, 3 and 5. These come from a micrograph series of 5% strained composite. They are gray level images: pixels are colored from black (gray level = 0) to white (gray level = 255). A Look-Up Table transformation (LUT) can be used, which sets in white most of the matrix pixels and allows the description of particles using the whole gray level range (0 – 254, with the 255 gray level representing the matrix pixels). This improves the visual comparison between the images obtained after processing.
reflection of the light from one crack border to the other and no light is reflected from inside the crack to the microscope lenses. Thus, the darker parts of the image are considered as cracks. These cracks can be detected by the human eye and provide the reference for result comparison. Image 1 (Fig. 2) shows a left-to-right deviation in the gray level distribution. There is a gradual darkening of the image from one edge to the other (Fig. 3); thus, a constant gray level does not always represent cracks. This prohibits the use of a global thresholding method, which would give the cracked areas directly. A small threshold value would identify correctly a crack in some part of the image but a whole particle as a crack in another part, since a high value would
2. Problem linked with crack identification In actuality, cracks are holes in the specimen surface. For the observation through optical microscopy, light falls on the specimen surface. There is
Fig. 2. Image 1: pixels show lower (darker) and lower gray levels from A to B.
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Fig. 3. Gray level deviation with the pixel location on an image along the AB line, examples of gray level distribution are taken here on five particles along the AB line.
neglect the crack in the former part and identify correctly the cracks of the latter. The segmentation (extraction of objects from an image) algorithms address this problem. Several methods exist, which may be classified in two main families. The first class of algorithms uses shape as major information leading to the extraction. For example, Einstein et al. [4] analyze cell nuclei by assuming that they have an ellipsoid shape. Four points of the nucleus are selected (by the user) to define the extreme left, right, bottom and top of the object. Then, they are linked by four quarters of an ellipse, which defines the nucleus contour. The second family uses histograms of gray level. The processing is based on global or local criteria. Methods based on global criteria can be used first, and among them, the simple thresholding. The gray level determines which pixels belong to an object and which ones do not belong. Therefore, a selection of pixels on the basis of their gray level is performed. From this basic idea, more sophisticated methods are developed. Ammouche [5] uses a combination of thresholding and shape identification for microcrack identification in concrete. A threshold analysis is done, which reveals all the defects in the concrete. Cracks are then identified by testing their shape: a comparison between the object area and the area of the circumscribed circle gives the objects whose aspect ratio is low and, thus, will be considered as microcracks. Macquaire and Grillon [6] have solved a problem similar to the present one for the case of glass epoxy composites. A threshold is achieved for each gray level between 0 and 255. An opening (see Ref. [7] for a definition) is applied on each resulting binary image with a structuring element perpendicular to the cracks and greater than the already known crack width. Thus, all the cracks disappear from this image, and the remaining objects are assumed not to be cracks. Then, the difference between the initial binary image and the transformed one gives some of
the cracks and eliminates the other objects. The union of all these binary resulting images gives an image of all the cracks. The thresholding could also be done after transformation of the gray level histogram. Breen and Williams [8] have achieved a transformation in order to use the same automatic threshold for an image series from a film. They have first defined one of the images as a reference (Fig. 4). The function transforms each gray level of an image i into the gray level of the reference one, for which the value of the distribution function is the same. The pixels whose gray level was gini get a new gray level gnew. Their frequency remains the same as previously, but some initially adjacent gray level classes could be bulked together. Thus, they minimize the contrast difference between the images and allow a similar analysis for all the images. Another way consists of using local gray level information. Such a method has been used by Darwin et al. [9] to perform automatic crack identification on cement paste. Three different parts are identified on the image (Fig. 5). The C zones are the noncracked regions, the B zones are the transition region and the A zone is the crack itself. The image is read in the four directions of the square grid (horizontal, vertical and two diagonals), with each direction being analyzed independently line after line, pixel after pixel. For each line, each pixel gray level is compared with the mean gray levels of the crack pixels on the same line. If the difference Idif between the current gray level and crack mean gray level exceeds one of two defined ranges dB or dC, the pixel is identified as belonging to the transition Region B (dB < Idif < dC) or to the noncracked Region C (Idif > dC). A pixel identified as a crack pixel by scanning one of the four directions is said to belong to the crack. Thus, the beginning or end of the crack contour can be detected. In the same vein, Barba et al. [10] have used the concept of entropy, defined for each pixel on the gray level distribution on a local neighborhood. The
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Fig. 4. An illustration of the principle of histogram matching [8]. Ref is the gray level repartition related to the first image of the series and taken as the reference. Thus, for all the images, whose gray level repartition is f(u), the cumulative function fcum(u) is determined. This allows the inverse function refcum 1 to be applied on fcum(u) and, thus, gives the normalized function fnorm.
points where the entropy exceeds a threshold are defined as belonging to the contour of the objects. Arhens et al. [11] used a combination of size 5 5 rank 5 modified median filtering, size 7 Laplace high pass filtering and size 5 5 rank 20 modified median
Fig. 5. Crack identification (after Ref. [9]).
filtering (see Ref. [7] for definition) associated with thresholding for isolating points where the gray level gradient is higher than the gradient of the other neighboring points, and, thus, identified the contour of the objects. Watershed algorithms can be used as the objects and are often locally defined by a gray level maximum. They consist in an analogy with a topographical image: High gray level zones are akin to crests and low gray level zones to catchment basins. Belhomme et al. [12] performed extraction of the DNA contour from breast cancer nuclei with an algorithm based on the watershed method. The contour of an object is defined as the place where the gray level contrast between adjacent pixels reaches a maximum. Therefore, a gradient image obtained from the original one gives an image where boundaries have higher gray level than those of the initial image, the boundary gray level depending of the contrast of the original image. Thus, a watershed operation allowed
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Fig. 6. Application to the example of Fig. 3. The crack pixels identified by an ‘‘expert eye’’ are shown in bold. The minimum for the five particles is identified on the fifth one and equal to 6. The gray levels of each of the four other particles are then transformed so that the minimum is the same for the five parts. Then, a threshold s1 or s2 is applied.
the determination of the local gray level maxima, which represent the DNA contour. In the present case, crack pixels are to be identified from among three possibilities, viz. matrix pixels, particle pixels, crack pixels and crack shapes are not clearly defined. As the cracks are many and small, a method such as the one used by Einstein et al. [4] is not chosen. Four different methods have been developed wherein some ideas of the previous procedures will be used:
use of local minimum to define the pixels belonging to the crack.
Three methods (the first three, see below) are tested on Image 1; the fourth is defined on Image 3 and then applied on Images 1 and 5.
3. Methods of crack identification 3.1. Method 1: deviation correction
definition of the beginning of the crack by a gray level gradient identified on a line; gray level transformation in order to apply a uniform threshold; and
The first concept involves correcting the gray level deviation of the image by multiplying the pixel gray levels with an appropriately chosen function of the
Fig. 7. Image resulting from the crack determination by the first method.
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S. Yotte et al. / Materials Characterization 46 (2001) 317–327 Table 1 Transformation associated with the segmentation by moment conservation Moment 0 1 2 3
Gray level P Pni/n = 1 ng P i i2 Pnigi3 nigi
Binary (n1 + n2)/n = 1 n1z1 + n2z2 n1z12 + n2z22 n1z13 + n2z23
between gmin and gmin + s belong to the crack. Fig. 6 shows this application with the example of Fig. 3:
Fig. 8. Crack characteristics used in the second algorithm.
pixel position. Consequently, the first step is to define the function, which, when multiplied by the pixel gray level, will yield a value lying in a determined gray level interval for all crack pixels. Thus, it is necessary to evaluate this multiplier function, which is expected to sort out a crack pixel from the other pixels. The image is divided into multiple subdomains, such that each domain contains at least one crack. We assume that the subdomains are small enough so that the gray level deviation between two consecutive parts is negligible. Then, the minimum gray level gi is determined for each subdomain i and gmin = min( gi); thus, fi = gmin/gi is the constant value of the correction function on the subdomain i. Thereafter, each minimum will have the same value g min throughout the image. A step value s is chosen such that the crack is determined by thresholding the gray level gmin + s; i.e., all the pixels whose gray level lies
Fig. 9. Application of the second algorithm to the crack determination.
for a small step value s1, the algorithm will identify correctly the cracks whose gray level were initially the highest but only a part of the other cracks. for a large step value s2, the algorithm will yield correctly the crack whose initial gray level was about gmin, but it will also select a large part of the particles of high gray level (in the left side of the diagram of Fig. 6). Fig. 7 shows an application to the image using a low step value s. Not only are the cracks only partially recognized but segments of particles are also selected. Thus, even with a continuous transformation, the method cannot be used for identifying cracks on images. 3.2. Method 2: identification through gray level gradient In this method, the idea is that the crack seen on the picture has a contrast with the less gray areas. Thus, the first step is not to define a crack as a gray level, below which everything is cracked, but as a set of pixels whose boundary is defined by a fixed gray level gradient. The crack pixels must fulfill the following three criteria (see Fig. 8).
Fig. 10. Application of the third algorithm to the crack determination.
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Table 2 Application of the third algorithm to ternary repartitions Gray level distribution (%) 150
120
90
Estimated crack pixel percentage (%)
40 35 70 20
40 35 20 70
20 30 10 10
41 48 21 52
(1) The gray level is lower than 255 smin. (2) There is a limited increase in the gray level of adjacent pixels. This increase is proscribed by two limits, viz. s1 and s2. The maximum limit s2 avoids the confusion between the transition from matrix to particle and the transition from particle to crack, which has a lower gradient. The assumption is that the former gray level transition is higher than the latter. (3) There is a quasiconstant gray level (the variation between two consecutive pixels does not exceed a given limit sf). The parameters sf, s1, s2 and smin have to be determined on the gray level image before the cracks are identified. The image is analyzed horizontally line after line from left to right. This direction is chosen because there are very few transitions from matrix to crack, but almost all transitions are from matrix to particle and then particle to crack, which is not true for the vertical direction. For each line, the algorithm consists of recognizing the beginning of a crack one line after the other using the first two conditions. The crack is then described by a limited variation of the gray level. Analyzing the image one line after the other and from left to right, we decide that the third pixel of a set of three belongs to a crack when:
the first one is recognized not to belong to the matrix; and the gray level difference between the third and the first pixels is greater than s1 but smaller than s2. The crack is then assumed to begin from the third pixel; the subsequent gray level pixels are treated as crack pixels and the end of the crack is searched for. If the variation of the gray level between two consecutive pixels stays below sf, then the pixel still belongs to the crack. If not, the last pixel is not registered as a crack pixel. Thus, the crack end is detected and the following pixels are treated with the first condition in order to detect the next crack. The result of this algorithm is shown in Fig. 9. The crack starts are correctly identified, but some
Center of the two binary classes z1
z2
99 96 100 107
145 145 148 140
particle contours are selected, too. The crack end is not identified, which comes from the application of the criterion on two consecutive pixels. A better result could be obtained by a comparison with the mean value of the crack pixels that are already read. 3.3. Method 3: thresholding of each particle Because the cracks are constituted of pixels that are darker than particle pixels, the idea is to select crack pixels by the thresholding of each particle. The method used is that of segmentation by moment conservation. This method is chosen because it is adapted to low contrast images [7]. The distribution between two classes is such that the first four moments of the binary distribution are the same as those of the initial gray scale image. Table 1 depicts the transformation. Here, n1 and n2 are respectively the pixel amount of the two new classes, z1 and z2 are their centers, ni is the pixel with a gray level gi and n is the total number of pixels. The first step in this method is to select one particle after another. The image is thresholded and each particle becomes an object that can be selected. Its pixel coordinates are read, and the corresponding gray level is read on the gray scale image. Thus, a gray level histogram of the particle can be constructed, and with it, the automatic thresholding is performed. Fig. 10 presents the resulting image. The
Fig. 11. Result of the first stage of the procedure (X = 40%), nmin = minimum.
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Fig. 12. Result of the first step of the procedure (X = 33%), nmin = mean of the six minimum gray level values.
Fig. 14. Initial image after the use of a LUT (103 – 155), Image 3.
crack pixels are adequately identified, but many particle pixels are mistaken for crack pixels. Thus, the area of the cracks is highly overestimated. In order to understand better the mechanism of this method, consideration is given to arbitrary simple histograms that are representative of the material. In the case of the previous image, the crack pixel gray levels lie between 80 and 120, depending on the particle location. The particle pixel gray levels are greater than 100. Consequently, the arbitrary histograms are divided into three pixel classes of constant gray level, one representing the crack whose gray level is fixed at 90 and the two others representing the remaining pixels belonging to the uncracked part of the particle whose gray levels are 120 and 150. Since particles may exhibit little or high microcracking levels, different frequency distributions are tested for the three classes and various crack pixel frequencies are tested, from 10% to 30% of the total particle area. As the gray level repartition on the particle is not uniform and may vary from one particle to the next, the distribution in the particle between the 120 and 150 classes is also varied. Four distributions are chosen to represent different possibilities in the pixel distribution on a particle (see Table 2).
Three results are given with the centers of the two binary resulting classes z1 and z2 and the percentage of pixels identified as crack pixels (pixels belonging to the lower class) by the algorithm (Table 2). In all the cases, the crack size is overestimated, as in the previous image. For the last two cases, with the same initial crack pixel frequency (90 gray level class), the recognized crack is twice as small when the contrast with particle is higher. A major influencing factor for a better crack identification seems to be the percentage of pixels in the gray level 150 class. The recognized crack is greater (52%) for the lower pixel percentage (20%) and decreases to 21% as the 150class pixel percentage becomes higher (70%). The center of the crack class always lies between the 90 and the 120 classes. Thus, a separation with this limit would identify the crack correctly, but it would be meaningless. More generally, a method based on the gray level histogram appears unable to give a satisfactory solution to this problem, because the only criterion for crack identification is the gray level and not the number of pixels belonging to the crack or to the particle.
Fig. 13. Initial image, Image 3.
Fig. 15. Image 3 (both cracks and uncracked particles); s = 40, X = 30%.
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Fig. 16. Transformed binary image of Image 3 (cracks only).
Fig. 18. Image 3; s = 30, X = 30%.
3.4. Method 4: identification through gray level gradient on each particle
surrounding pixels. Therefore, the step s does not catch any other crack pixel at all. Thus, instead of defining nmin as the very minimum, it can be defined as the mean of the first six lower values of the particle gray level histogram. Fig. 12 shows how this changes the result. (b) After binarisation of the resulting image (the cracks are then in black), first, a closing of the holes lying inside the black pixels sets is realized. Then, a closing (see Ref. [7]) with a linear three-pixel structuring element, parallel to the crack, is done to remove the parasite pixels not belonging to the crack. Figs. 13 – 19 illustrate the complete procedure. The initial image is shown in Fig. 13. A LUT is applied in order to modify the gray level distribution, which displays the 256 gray levels between the initial levels 103 and 155 (see Fig. 14).
The method consists of the following two stages. (a) Each particle is isolated in the same way as previously done (see Section 3.3), and the minimum gray level nmin is identified on the particle gray level histogram. It is preascertained that the crack is made up of all the pixels whose gray level lie between nmin and nmin + s, where s is a chosen step. In this method, the uncracked particles are also selected. In order to retain only those pixels belonging to a cracked particle, it is decided that for all ‘‘identified cracks’’, if the ratio between crack area and particle area is above a certain value X, the cracks are ignored and the particles are considered as uncracked. The parameters s and X have to be determined as a function of the image. Fig. 11 shows the result of this procedure on the previous image. The recognized uncracked particles are in gray and the crack pixels are in black. On this image, crack pixels are well identified. The circled point is a pixel belonging to a crack. Its gray level is the minimum for the particle, but the local contrast is high and its gray level is much lower than that of the
Fig. 17. Comparison of the identified cracks with the real ones for Image 3; the circles show the unrecognized cracks.
If 0 g0 103, then g1 = 0. If 103 g0 155, then g1=(255/52) ( g0 103). If 155 g0 255, then g1 = 255. The algorithm is then used with a step s of 40 and an X value of 30% (Fig. 15). The image is finally
Fig. 19. Image 3; s = 30, X = 40%.
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Table 3 Cracked areas obtained from Image 3 at 500 magnification for various values of s and X
Image
s
X (%)
Cracked area (mm2) before binary processing
Trial 1 Trial 2 Trial 3
40 30 30
30 30 40
656 498 538
Crack area (mm2) after binary processing 759 548 645
treated by a threshold in order to highlight the cracks and eliminate the remaining pixels of the cracked particles that have been set to a different gray level by the previous operation (Fig. 16). A comparison is done by adding the crack boundary of the binary image (Fig. 16) to the gray level image (Fig. 13). The result shown in Fig. 17 leads to the conclusion that, in spite of a high step s, the program has not recognized some cracks. In fact, the high step leads to identification of a part of the particle containing a crack as being a crack. The crack ratio X criterion eliminates such a crack because it is too high, and the particle is finally assumed to be uncracked. Lowering the step s is a solution for resolving such cases. The result is shown in Fig. 18 with a step of size of 30. Increasing the percentage X also allows the identification of new cracks that might have been overlooked with the previous percentage. Fig. 19 shows the image obtained with X = 40%. Table 3 summarizes the results of the various trials. The first case (s = 40, X = 30%) leads to an overestimation of the cracked area, but it also underestimates the number of cracked particles. In fact, only cracks in the large particles are recognized as cracks, but they exhibit a higher area than their actual area. Increasing X from 30% to 40% (Trials 2 and 3) leads to a limited increase of the cracked area [8% before application of Stage (b)]. These two results can then be used to define the two limits of the actual cracked area. The closing procedure eliminates the isolated pixels, which do not belong to the crack, but renders connected pixels that
Fig. 20. Cracked particles of Image 5.
belong to the crack. This last procedure has a larger influence than the parasite pixel elimination, as the cracked area is finally increased. The difference between the cracked areas in Trials 2 and 3 is approximately 8% before treatment and increases to 17% after the binary treatment. Table 4 and Figs. 20 and 21 summarize the results obtained when the method was applied to the two other images, Images 1 and 5. In the case of Image 5, the optimal (s, X) set is obtained after visual comparison with what the ‘‘expert eye’’ would consider to be cracks. In Table 4, the result obtained by setting X = 100% shows what the program identifies as cracks if the uncracked particles are not eliminated. The cracked area decreases as soon as they are eliminated. A step of s = 13 is preferred because it appears to identify the cracks better, even when high percentage X values are used. However, an area fraction of X = 40% is too high since noncracked pixels are also included. For this image (Image 5), the optimum value of X appears to lie between 25% and 30% (this is equivalent to a difference of 9% in the cracked area). For Image 1, a good result was obtained by using a treatment involving s = 30 and X = 40%.
Table 4 Cracked area for Images 1 and 5 for the 500 magnification Image
Trial
s
X
Cracked area (mm2)
5 5 5 5 5 1
1 2 3 4 5 1
10 10 13 13 13 30
100 25 25 30 40 40
2200 403 364 397 646 515
Fig. 21. Cracked particles of Image 1.
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4. Conclusions
References
Four crack identification methods have been tested. One of these methods, based on identification of cracks in particles through a gray level gradient, allows a good evaluation of the cracked area. However, the values of two parameters have to be determined by a human expert for each image and, thus, the method cannot be considered as totally automatic. Visual comparison between the real image and the final result given by the algorithm is required. Once the values of the parameters have been chosen, the pattern of cracks is automatically selected and crack measurements can be quantified. This method is based on the assumption that the particles are small enough not to be affected by global deviation in the gray level. For evaluating the method at the same magnification, three images have been analyzed, and in the three cases, the cracks have been fairly well identified. Thus, an evaluation of the damage on each section is possible. For example, Images 1 and 3 lead to comparable crack areas (about 515 mm2 for Image 1 and between 548 and 645 mm2 for Image 3), whereas the crack area estimated through Image 5 is about 30% smaller (397 mm2). The visual comparison remains necessary to optimize the parameter values. However, the crack identification is thereafter straightforward, and the whole pattern of cracking can be highlighted. The main interest is that more sophisticated measurements can be performed on these patterns (distance between cracks, crack size, etc). For higher magnification, if the particle size is not small when compared to the image size, the method will require modification, for example, by working the same way but on parts of the particle. The method yields crack patterns that can be used as input data in numerical computations or used for validating such computations (see, for example, Ref. [13] or Ref. [14]). Moreover, the results can be analyzed to determine the representative volume element size in the material microstructure. The previous results, showing significant differences between the crack areas of three images of the same material, mean that the corresponding images are probably too small for consideration as a representative volume linked to this parameter as soon as damage is considered. This method was used in this study for some images showing a deviation in the gray level distribution. It can obviously be used for crack identification on images having a similar deviation, but other ways for crack determination could be based upon its concept.
[1] Li M, Ghosh S, Rouns TN, Weiland H, Richmond O, Hunt W. Serial sectioning method in the construction of 3-D microstructures for particle-reinforced MMCs. Mater Charact 1998;41(2 – 3):81 – 95. [2] Li M, Ghosh S, Richmond O, Weiland H, Rouns TN. Three dimensional characterization and modeling of particle reinforced metal matrix composites: part I. Quantitative description of microstructural morphology. Mater Sci Eng, A 1999;A265:153 – 73. [3] Li M, Ghosh S, Richmond O, Weiland H, Rouns TN. Three dimensional characterization and modeling of particle reinforced metal matrix composites: part II. Damage characterization. Mater Sci Eng, A 1999; A266:221 – 40. [4] Einstein AJ, Gil J, Wallenstein S, Bodian CA, Sanchez M, Burstein DE, Wu HS, Liu Z. Reproductibility of accuracy of interactive segmentation procedures for image analysis in cytology. J Microsc 1997;188(2): 136 – 48. [5] Ammouche A. Identification et quantification automatiques de la microfissuration des be´tons par analyse d’images. Internal report (in French), Universite´ Laval, Faculte´ des sciences et de ge´nie, Canada, 1997. [6] Macquaire B, Grillon F. De´tection automatique de fissures dans les mate´riaux composites. Rev Metall 1994 February:317 – 22. [7] Coster M, Chermant JL. Pre´cis d’analyse d’image (CNRS e´ditions) Paris: Presse du CNRS. [8] Breen EJ, Williams KL. Contrast matching in a sequence of images by histogram normalization. J Microsc 1990;159(2):223 – 7. [9] Darwin D, Nagib Abou-Zeid M, Ketcham KW. Automated crack identification for cement paste. Cem Concr Res 1995;25(3):605 – 16. [10] Barba J, Jeanty H, Fenster P, Gil J. The use of local entropy measures in edge detection for cytological image analysis. J Microsc 1989;156(1):125 – 34. [11] Arhens P, Schleicher A, Zilles K, Werner L. Image analysis of Nissl-stained neuronal perikarya in the primary visual cortex of the rat: automatic detection and segmentation of neuronal profiles with nuclei and nucleoli. J Microsc 1990;157(3):349 – 65. [12] Belhomme P, Elmoataz A, Herlin P, Bloyet D. Generalized region growing operator with optimal scanning application to segmentation of breast cancer images. J Microsc 1997;186(1):41 – 50. [13] Susagna F. Caracte´risation de la microstructure d’un composite Al – Si – Mg par analyse d’images. MSc report (in French), CDGA, Universite´ Bordeaux 1, France, 1999. [14] Yotte S, Breysse D, Riss J, Ghosh S. Image analysis of particle metal matrix composite for damage modelization. Presented at EMD 99, 13th Engineering Mechanics Conference, The Johns Hopkins University, Baltimore, MD, June 13 – 15, 1999.
