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Quantification of crystal morphology
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ELSEVIER
CRYSTAL GROWTH
Journal of Crystal Growth 137 (1994) 1—11
Quantification of crystal morphology M.E. Glicksman
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M.B. Koss, V.E. Fradkov, M.E. Rettenmayr, S.S. Mani
Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, New York /2180, USA
Abstract Morphological measurement constitutes an important experimental subject in crystal growth and materials science, and is currently receiving renewed attention because of the rapid advances occurring in computer technology, coupled with the concomitant sharp reductions in the cost of digital image processing. Image processing applied to the quantification of microstructural images is currently being used in our laboratory to increase the understanding of interfacial dynamics during crystal growth and to analyze the kinetics of microstructural evolution. Quantification of microstructural and crystal growth morphologies, such as the measurement of dendritic tip radii, crystallite size distributions, and crystallite shapes, provides the geometric foundation needed for interpreting interfacial dynamics during crystal growth and an objective description of morphogenesis accompanying solid—liquid and solid—solid phase transformations. Automated methods employed, and, in part, developed by the authors to measure these morphological and kinetic parameters, using advanced statistical and stereological methods, are reviewed in this paper. Some of the techniques disclosed here are currently being refined even further to achieve improved precision in the quantification of crystal growth morphology.
1. Introduction
Most engineering materials are microstructurally heterogeneous and usually evolve toward their final microstructures in distinct stages during the materials processing history. For example, nucleation, crystal growth, deformation, grain growth, phase coarsening, etc., are some of the individual processes that might be involved. Numerous characteristics of the evolving microstructure are measured to predict the behavior of the material in service. The length scale of the primary crystallizing phase, for example, is an important microstructural parameter, usually measured subsequent to the solidification process. _______
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Corresponding author,
Specifically, an important morphological length
scale of practical concern in liquid—solid phase transformations is the tip radius of the dendritic crystals formed initially in a melt and, at a later processing stage, the size of the evolving primary phase crystallites. Another microstructural parameter of significance is that needed to quantify the crystallite shape which is an ill-defined geometric parameter that plays an important role for describing the structure of dense multiphase systems with relatively high volume fractions of primary solid phase. These metrical and non-metneal morphological measurements are typically executed manually by procedures that tend to be tedious and inconsistent; hence, the authors have developed several specialized computer-based imaging systems to perform them as an automated quantitative analysis on images captured
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ME. Glicks,nan et a!. /Journal of Crystal Growth /37 (1994) 1—/l
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either in situ or post mortem to the crystal growth process. These image analysis techniques have been developed specifically to acquire quantitathe representations of interfacial structures produced by various crystal growth processes, and resolved over length scales determined by and appropriate to the type of microscopy employed to produce the original image.
2. Image analysis
to describe complex microstructures, because their value can be deduced independently of the size, shape, or spatial and topological arrangements of the individual phases. This paper will concentrate on two aspects of morphological quantification of microstructures through stereologic measures observed in or after a crystal growth process: the first is measurement of the steady-state tip radius in an in situ thermal dendnitic crystal, which is formed initially in the freezing process; the second is the measurement of crystallite domain size, shape, and distribution
Image analysis concerns quantification and classification of images and objects of interest, in
this case the structure and form of crystalline phases as determined by their interfacial configuration. Here, we will deal explicitly with manipulation, processing, and analysis of optical images, all performed objectively and automatically by a computer. Recent interest in this subject by crystal growers and other materials scientists has
as observed in a quenched two-phase microstructure as the crystal—melt system evolves isothermally over time.
3. Measurement of in situ crystal morphology Dendritic growth is perhaps the most common
been motivated in part by the impressive and
form of crystallization, usually occurring wher-
rapid advances in digital computer technology,
ever metals and alloys solidify under relatively
The rapidly falling costs of digital imaging hardware and image analysis software have now firmly placed such options within economic reach, and finally make practical truly automated, customizable solutions to quantitative morphology problems occurring over a spectrum of crystal growth and materials science applications, The major concern to the image analyst is retention of maximum information from an accurately recorded original image, and minimization of the influence of various unavoidable extrinsic interferences during the image processing and analysis. The automated procedures developed in our laboratory to analyze images perform sophisticated corrections for extrinsic image interferences, such as non-uniform object lighting, or variable photographic exposure and development, Measurements performed to extract so-called global stereological parameters can he easily performed on the images processed. Stereological measures, which are important in microstructural analysis, are used to describe the three-dimensional geometric and topologic features of an object or collection of objects which comprise a
shallow temperature gradients, such as in most casting and welding processes. The dendritic morphology affects a material’s properties and its subsequent deformation processing responses. Understanding dendritic crystal growth is considered the crucial first step in controlling these properties and responses. Essential to the understanding of dendritic crystal growth is obtaining accurate, quantitative experimental data on the steady-state aspects of dendnitic growth kinetics and morphology. The overall experimental task of measuring the size and shape of steady-state dendritic crystals, for either comparison with other experiments or to verify theory, is, as will be shown, not a trivial one. This is evidenced by noting the typical experimental uncertainties encountered in such measurements, which are typically ca. ±10% in the measured dendrite tip radii, ~ or ca. ±5%in the measurement of dendrite velocities, V [1—5]. Also, one should note that dendritic
microstructure. Such measures are used routinely
scaling theory usually requires evaluation of these
variables combined as the nearly constant product VR~0,which would typically exhibit a cornhined uncertainty of ±25%.Our research on the
ME. Glicksman et al. /Journal of Crystal Growth 137 (1994) 1—I /
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crystal growth of high-purity (6—9’s) succinontrile (SCN) crystals indicates that these measurement uncertainties can be reduced by as much as a factor of one-half to one-third through appropriate applications of digital image processing. The following is a brief discourse and perspective commentary (to be published elsewhere in full detail), on how to measure the steady-state tip radius of a dendritic crystal by the application of modern digital image processing, Initially, even prior to producing an image, there arises the experimental question whether the dendritic crystal chosen for imaging is sufficiently isothermal and well isolated from the per-
One of the first quantitative techniques used to accomplish profiling analysis of in situ dendritic growth images was the matching parabola method [1,2,41, in which a family of confocal parabolas of different radii of curvature is projected against the magnified image of the selected dendritic tip, taking into account the experimental image magnification and the scaling of the parabolas. The radius of the best matching parabola is chosen to obtain the measured dendrite tip radius. A second useful approach is to locate the position of the tip, move along the growth axis of the dendrite an appropriate meremental distance, L1, back from the tip, and then
turbing influences caused by other actively growing interfaces. Indeed, measuring displacements of the dendrite tip at sequential times during crystal growth would yield suitable kinetic data (the growth speed) only growth. if the dendritic crystalaxial is exhibiting steady-state Also, to insure that a dendrite is in fact sufficiently isolated from its neighbors to avoid transient growth, the distance, 1i1, from a point near the tip to the closest other active solid—liquid interface must be greater than the estimated thermal or solute diffusion length. The next step needed in measuring the curvature of an SCN dendrite tip is to apply an empirically confirmed, yet theoretically based, geometnc assumption: i.e., the tip profile projected on a (100) plane in the (100) direction is, by virtue of the bce symmetry, a parabola. Thus, near the tip, the dendrite is a three-dimensional crystal growing close to the ideal form of a paraboloid of revolution. Consequently, the approaches used in making dendritic tip radii measurements have been based on analyses that compare the observed image of the dendrite tip to the “best” fitting parabola. Once this parabola is determined, it is straightforward to calculate the corresponding tip radius of curvature as the unique, linear metrical parameter that sets all the length scales of the developing dendritic crystal. There are several alternative procedures to select the best paraboloid, and thereby, to characterize the dendritic shape from a photographic or video
record the width of the dendrite, 14’, the distance between the opposing optical edges of the profile. The locally estimated tip radius, R., corresponding to the width measured at the distance is 2/8L~,which is a geometric formula L,that R, Wç can be derived easily from the equation of a parabola [2]. The dendritic tip radius for, ~ is then calculated from the average of the sampled
image. The most useful of these analysis proce-
dures are discussed next,
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local estimates, viz., R~1~(1/n)~’1R1. The last approach described in this paper for analyzing in situ dendrite images to obtain steady-state tip radii is based on obtaining a least-squares regression curve that fits the sampled optical edge to a parabolic equation [2,5]. One of the free fitting coefficients of this equation is mathematically related to the radius of the regressed parabola, and therefore, to that of the dendrite. Generally, the exact mathematical form of the equation, and the choices of what constitutes the independent (versus the dependent) variable are not significant. However, the final calculated crystal radius is dependent on the number of sampling points used to define the crystal—melt optical edge, the scatter in those points, and the distribution of those points over the whole curve. Thus, for a small dendrite tip, with a relatively large (with respect to the tip radius) scatter in the coordinates of the points defining the optical edge, the radius results would be based on just a few edge points, predominately clustered near the tip, and would therefore depend sensitively on the exact mathematical form chosen to represent the parabolic and on the particular choice of independent variable. =
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ME. Glicksman eta!. /Journal of Crystal G,oi’th /37 (1994) 1—1/
Curve fitting proved to be the best method for
analyzing the authors’ recent experiments on dendritic crystal growth in SCN [6]. The radii determination based on measuring W2/8L is strongly dependent on the choice made for the location of the tip. In addition, this method weights all the r• measurements equally in the calculation of an average tip radius. Of course, these problems are not insurmountable; but by the time one responds carefully to the two criti-
cisms mentioned above, one has de facto approached the image analysis based on the curve fitting method anyway. The matching parabola method is clearly the weakest approach. This method relies critically on deft hand—eye judg-
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ments, and, thus, is most susceptible to operatordependent errors or subjective judgments. It is also the least quantitative approach with respect to assessing the error estimation and in explicitly
exposing the heuristic assumptions employed. Not incidentally, this last paragraph also moves the discussion from the most computationally intensive method, especially when using a non-linear form (non-linear in the coefficients, not the vanables) for the parabolic equation, to the least computationally intensive method.
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Common to this discussion of profiling methods is the assumption that the dendrite profile is perfectly aligned along some arbitrary (vertical) y-axis. This assumption is not critical, because we have shown that a small angle of rotation (<4
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Fig. I. Computer enhanced gray-scale image of a dendritic tip and the located edge used in the regression procedure to calculate the tip radius.
to the dendritic edge data scarcely affects the calculated radius results when using these methods. Furthermore, an additional fitting parameten, the rotation, 6, can he added to these methods. Experience with this type of image analysis shows that it is best to determine U independently from the tip radius, or from the series of
tip images, and then account for any rotation from the y-axis so that one avoids introducing an additional freely varying parameter in the tip radius measurements or in the curve fitting procedures. The authors have incorporated many of these considerations into a convenient semi-automated image analysis system titled computer aided dendnite analysis program (CADAP). Fig. 1 shows a typical computer enhanced gray scale image of an
SCN dendrite photograph used with CADAP, and the computed crystal—melt interface edge on which a CADAP curve fitting routine will operate [6]. The first step in applying the CADAP procedure is to capture the image. The primary consid-
eration is to capture the image at a large magnification, and obtain a high spatial resolution in
terms of the number of computer lines, or pixels, per micron, and, therefore, a high precision in the dendrite edge loci. A larger image magnification also increases the maximum number of edge points used in the analysis. However, the enhanced resolution must be balanced against a decreased field of view. The field of view must
M.E. Glicksman eta!. /Journal of Crystal Growth 137 (1994) 1—11
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extend at least several tip radii behind the den-
tized and processed with this assumption in mind.
drite tip in order for the data to fit a parabola. The compromise in applying CADAP was to Select the largest magnification that still allows measurements along the axis of dendrite for a minimum distance of 4—5 tip radii, After the image is captured, several subsequent processing operations were performed. First, any irregularities in the image capture procedures were corrected. In this case, the image correction was obtained by removing the background non-uniformities in the lighting used to digitize the original 35 mm photographic negative. The non-linear effects introduced by the film developing process can also be corrected from photographs taken of the incident light transmitted through a series of neutral density filters. These densitometric films were developed by cxactly the same processing as were the dendrite photographs. The image is then inverted to ohtam a positive digitized image that duplicates the
Values other than 50% transmission have been used in the literature [3]. For the images used in this study, widths of the edge function are on the order of 10—20 ~m (defined as 10% to 90% of the background illumination surrounding the
actual light intensity profile at the dendrite before illumination of the undeveloped film. In addition, several standard image processing transforms were tested, such as a median transform to eliminate single pixel noise, and an averaging transform that tends to smooth out or defocus sharp edges [71.The use of these transforms, both singly and in combination, had little effect on the tip radius calculations. The explanation for this was most likely that the curve fitting routines employed in the CADAP procedure mathematically accomplish what the digital transforms do by image processing. Part of the success of our analysis, without recourse to a lot of digital image processing, may be based on the fact that the optical system that records the dendritic growth is shadowgraphic, and, therefore, provides some real time analog image processing. Thus, image processing transforms may still be useful for other dendritic growth recording methods or tip radius calculation methods, The final step prior to the radius calculation step is to create a list of ordered pairs that define the measured dendrite edge. In applying CADAP one assumes that the crystal—melt interface occurs at 50% transmission of the average background of the processed image. Images were digi-
edge), so that the choice of where to locate the
edge is quite important. We are currently expenimenting with other choices and with gradient edge detection methods. However, if the edge function width is small enough, these considerations become far less critical. Images with narrower edge functions are well worth the effort invested in the optical system to create them. To convert the located edge to a tip radius
measurement, we self-consistently fit the edge loci ordered pairs to an equation for a parabola, linear in it fitted coefficients, y a + bx + cx2, where the tip radius, ~ equals l/2c. The data range was restricted on the y-coordinate to values between 0 and 2R 11~, and we iterated the curve fitting and data limiting until the final tip radius converges. The particular choice of how many tip radii are used in the self-consistent iterative process (1-, 3-, or 4R1~,instead of 2R1~) is important in as much that it changes the value of the final calculated tip radius. Furthermore, the choice of 2r is not universal [3]. This raises the issue of whether the region around the dendnite tip is truly a parabola. Currently we are using a numerical modeling study to determine (in the same manner that we analyze dendrite edges) how the CADAP system analyses true parabolas in the presence of Gaussian noise that stochastically alters the “edge”. Although unanswered questions remain, our current analysis indicates that, after digital image processing, a least-squares regression of the edge points defining the interfacial profile provides a valid approach to quantifying the morphology of dendrites. Clearly the use of a powerful computer is imperative to accomplishing both the image processing and the regression analysis. However, the use of a powerful computer, per Se, is not a panacea. One must employ the best optical techniques to insure that high-quality images are mitially captured, that steady-state and controlled, isolated growth conditions are achieved experi=
0
ME. Glick.s,nan eta!.
/Journa!
of Crystal Growth /37 (/994) 1—/1
mentally, and that only highly reproducible den-
affine volume correlation, which shows that in
dnitic crystals are grown and analyzed properly.
dense microstructures each particle or crystal domain develops its local diffusion field within a
4. Crystallite size and shape measurements After crystallites form from a melt, the microstrueture gradually evolves toward thermodynamic equilibrium. Typically, the initially formed dendrite branches increase in average size as they interact via diffusion to reduce the total interfacial energy of the system. This process, known as late-stage phase coarsening, or Ostwald ripening, was investigated theoretically by Lifshitz, Slyozov, and Wagner (LSW) [8,9], who considered the crystal dynamics in an infinitesimally dilute, twophase system. To expand the scope and applicahility of the classical LSW approach [8,9], and to understand the dynamics of Ostwald ripening in densely crystallized systems, microstructural evolution studies of dense two-phase systems are underway at our laboratory. In addition, the interesting departure of real multiphase systems from the limiting assumptions used in LSW theory are currently under active theoretical development. Specifically, departures from LSW behavior are usually introduced theoretically by introduction of a second transport-related length scale. The arguments justifying a second length scale are often heuristic, and this length scale is introduced through a variety of models sometimes as a screening distance for interparticle diffusion, or as a “cut-off radius” defining the average “environment”, or even as the outcome of statistical correlations occurring among the crystallites. The current theoretical uncertainty regarding the dynamics of phase coarsening again underscores the importance of performing careful —
experiments to measure length scale evolution
kinetics in crystallizing systems. The quantification of morphology through stereology and image processing techniques will now he discussed from this standpoint, Recently Marsh [10] has developed a nonheuristic, physically-based solution to the prob1cm of selecting a second length scale in partially crystallized solid—liquid microstructures. This development introduces the concept of a linear
restricted volume of matrix, or melt phase, that is itself proportional to the particle’s own volume. The correlation suggests that the shape and size of the crystallites determine the average growth rate. A quantitative metallographic study of the evolving particles adds to the fundamental understanding of how solid—liquid microstructures evolve geometrically and kinetically during isothermal processing. In order to automate the metallographie quantification for application to high solid fraction systems for comparison to theones like that of Marsh [101,the analyzed image must contain distinct particles, each independent and distinguishable. This makes the image processing of a three-dimensional, densely interacting collection of particles a far more challenging problem in morphological quantification than that discussed above for a single dendrite.
The standard [11—13]stereological procedures used in quantitative characterization of crystallites usually involve counting of linear intercepts and points measured on random planar sections, followed by conversion of the intercept counts to crystallite sizes, shape factors, volume fraction,
and other stereologically significant parameters. Global morphological and distributional parameters are then obtained through conversion of the stereological data obtained from two-dimensional sections to information about the morphology in three-dimensional space. One then obtains either three- or two-dimensional crystallite population descriptions, and the statistical properties of the structure is usually described by distribution functions. A digitized gray-scale crystallization microstructure is shown in Fig. 2, which has been prepared for metallographic recording by suitably polishing and etching. It can be clearly seen that the high volume fraction of primary (dark gray) solid phase leads to a large number of contacts between neighboring crystallites. These connections among the particles make it difficult for to identify the shapes and positions of the individual crystallites comprising the microstructure. This is particularly troublesome when attempting to ap-
M.E. Glicksman et a!. /Journal of Crystal Growth /37 (1994) 1—11
7
tallites are interconnected to each other through
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Fig. 2. A typical digitiied gras-scale image ‘.sith the darker region being the solid phase.
ply computer assisted techniques for making automated stereological measurements. In fact, most automated image analyzing systems assume some specific, idealized, shape for the crystallites, typically a sphere or ellipsoid, which simplifies the subsequent image analysis algorithms, but then usually results in some loss of valuable shape information. Automated commercial systems like Optirnas®, suitable for DOS-based personal cornputers and work stations, incorporate proprietary procedures to separate touching crystallites. Such image processors are often limited by the morphological configuration cases that the software allows, such as the maximum number of interpartide contacts and the topological features of the interconnected crystallites. We have developed a more general algorithm, implemented on a Macintosh based system, which is broadly applicable to metallographic samples where individual crys-
(a)
(b)
multiple local contacts. Even the complex situation shown in Fig. 2, where the crystallites are involved in numerous and multiple contacts, can now be handled virtually fully automatically by this algorithm which allows all the distinguishable cases of particle morphologies and topological connections. The basic principle behind image morphology analysis is to employ various shape structuring elements in order to implement the required analysis tasks such as edge detection, segmentation, and feature detection [14]. Morphological tools offer a unified and powerful approach to a number of image description problems, such as the one shown here for identifying each individual crystallite in a dense collection of particles. Complex, convoluted and ramified interfacial shapes that are commonly encountered during unstable crystal growth pose similar structure analysis problems. These image measurement tools are particularly useful in image analysis applications dealing with morphological features that are highly irregular in form and often interconnected, as observed, for example, in metallographic samples containing a large volume fraction of the dispersed phase. The most important attribute in the development of this general procedure is that it does not require one to make any simplifying assumptions regarding the crystallite geometry or arrangements, and it defines the structuring element precisely so that the loss of quantifiable detail on the morphology is minimized during image processing.
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Fig. 3. (a)—(c) Optimizing threshold value to convert a gray-scale image to a binary by minimizing granularity. (d) Filtering the binary image to remove small particles.
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ME. Glicks,nan eta!. /Journal of Crystal Growth 137 (/994) 1—1/ 1
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The procedure to be described operates on a binary image, and its purpose is to separate and locate each of the initially interconnected crystallites. A binary image is obtained from the original gray-scale image by removing the background density present in the optical unit, The gradient in the background is removed by considering smaller units of the gray-scale image at a given time. The granularity, which is the sum of black particles per black pixel and white particles per white pixel, is measured and minimized to determine the threshold value for the unit in considerFig. 3 shows the importance of minimizing granularity to get a true representation of the crystallites. This step of transforming from a gray image to a binary one is automated to eliminate any hand—eye judgments and to minimize operator subjectivity. The two important fundamental
morphological operations which form the basis for morphological operations to identify and separate individual crystallites are dilation and cr0sion [15]. Erosion is an image processing operation that removes pixels from the edges of crystallites in the binary image. This operation, if applied repeatedly, will eventually separate interconnected crystallites that are touching and remove isolated pixels. Dilation is an image pro-
cessing operation that complements erosion by adding pixels to the edges of objects in a binary image.
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and fillsDilation in holes connects which willdiscontinuous be used laterobjects, in the procedure for separating individual crystallites. The method developed by the authors to identify individual crystallites uses a combination of a modified erode operation to attack the connecting necks selectively, and then applies a complete (uniform) pixel erosion iteratively to find the dentroids of the individual crystallites. Fig. 4b shows an example of the selective erosion operation performed on the binary image in Fig. 4a, where the necks are preferentially attacked. Fig. 4c
shows the image after a complete (uniform) erode
binary
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Fig. 4. (a) A typical digitized gray-scale image converted into aa image. (b) The necks operation. being selective attacked performing the selective erode (c) The image on after complete erode operation has been performed.
ME. Glicksman et al. /Journal of Crystal Growth /37 (1994) 1—Il
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Fig. 5. The image with the crystallite center positions marked. Fig. 6. The image after the skeletonize operation has been performed giving the topology of the system.
has been implemented and where separation between initially interconnected crystallites has been achieved. The combination of these two pixel erosion operations is performed until none of the black pixels is left. This procedure will eventually locate all the positions of the crystallite centers, measured here as the centroid of the particle areas, as demonstrated in Fig. 5. Insofar as crystallites usually have complicated interconnected structures in dense systems, it is also important to determine and retain information on the topology of the original image, which is determined by applying the skeletonize operalion. This image processing operation repeatedly removes pixels from the edges of objects in a binary image until they are reduced to a single pixel-wide curve or skeleton, as shown in Fig. 6. value as compared to the skeleton, The separaThe centers are shown in a different gray-scale tion of the crystallite network is simpler once the skeleton structure is determined. The skeletons are then dilated, i.e., pixel layers are added on either side, marking the layers with respect to sides and skeletons at every stage until all the possible positions Finally, theboundary boundaryend end points are are located. all located, and a straight line “cut” with the shortest possible distance is inserted to separate each of the particles as indicated in Fig. 7. There is some residual error in the separation algorithm which can be reduced by performing a few extra comparison steps and by using some
manual editing. Although these automated procedures are not totally free from error, there are further improvements being considered to get more repeatable, error-free measurements from these analysis tools. There is also a trade off made between selecting the magnification at which the microstructural image is recorded cornpared to the available, and affordable, resolution of the computer hardware and software. There are also some interesting technical questions regarding the gray-scale to binary conversion procedures, where the positions of the crystal—melt interfaces are determined, These image analysis
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ME. G!icksman eta!, /Journal of Crystal Growth 137 (/994) 1—/I
issues are among those being studied currently in our laboratory. Finally, after all the individual crystallites are identified and separated, the stereological measurements of interest can be performed fully automatically to determine the size, shape, and on100
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stereological
tool described here provides affordable access to easily implemented characterization of images for quantitative metallographic
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and morphologic measurements.
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5. Conclusions
-
100 150 200 length range
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The development of computer aided image analysis tools provides a consistent and objective method for morphological quantification. As cxamples we discussed the development of a least-
200
squares regression technique, applied after inclusion of image processing corrections of the crystal—melt interface, and suggested its use for vanous in situ measurements of dendrite tip radii,
150
speeds, and morphologies. This method provides
100
the most accurate, objective approach for quantifying the morphology and kinetics of steady-state crystal growth. The use of a powerful microcomputer or work station is shown to be critical for efficient image processing and regression analysis. The application of an image analysis program
300 ~
entation distributions, see Fig. 8. The shape factor measurement shown is the ratio of the major to minor axis length for the best fitting ellipse. Using the distributions measured for crystallite size, shape, and axis orientation it becomes possible to monitor and describe the microstructural evolution trajectories quantitatively on a purely geometric basis. The measured stereological distributions can the be used as the basis to compare different theoretical predictions. The automated
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Fig. 8. Stereological measurements made on the separated image for (a) size. (h) shape and (c) orientation distributions.
topological assumptions about the shape or eonfigurations of individual erystallites. This analysis technique provides new avenues for providing detailed experimental checks on the limits of theories on phase coarsening in dense microstructures with high volume fractions of a dispersed phase. Finally, this development in the quantification of morphology enables one to perform highly automated measurements of shapes and their distributions in a suitably prepared metallographic sample.
M.E. Glicksman et al. /Journal of Crystal Growth 137 (1994) 1—Il
6. Acknowledgements
[51ER.
This work is being conducted with the support by the National Aeronautics and Space Administration, Space Experiments Division, Microgravity Experiments Branch, under NASA 3-25368, with liaison through the Lewis Research Center and the National Science Foundation under DMR-89-21 35.
161
Rubenstein and M.E. Glicksman. J. Crystal Growth 112(1991) 84.
7. References 111
S.C. Huang and M.E. Glicksman, Acta Met. 701. [21S.H. Tirmizi and W.N. Gill, J. Crystal Growth 488. [3] A. Dougherty and J.P. Gollub, Phys. Rev. A 3043. [4] J.H. Bilgram, M. Firmann and E. Hiirlimann, Growth 96 (1989) 175.
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29 (1981) 85 (1987) 38 (1988) J. Crystal
ME. Glicksman et al., Reprint No. AIAA 93-0260, 31st Aerospace Sciences Meeting & Exhibit, Reno, NV(1993). [7] GA. Baxes, Digital Image Processing (Prentice-Hall. Englewood Cliffs, NJ, 1984). 181 C. Wagner, Z. Electrochem. 65 (1961) 35. [9] tM. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids 10 (1961) 35. [10] S.P. Marsh, PhD Thesis, Rensselaer Polytechnic Institute, Troy, NY (1989). [111 H.E. Exner and H.L. Lukas, Metallography 4 (1971) 325. [12] E.E. Underwood, Quantitative Stereology (Addison-Wesley, Reading, MA, 1970). [13] R.T. DeHoff and F.N. Rhines, Quantitative Microscopy (McGraw-Hill, New York, NY, 1968). 1141 Y. Mahdavieh and R.C. Gonzalez, Advances in Image Analysis (SPIE Optical Engineering Press, The lnternational Society for Optical Engineering, Bellingham, Washington, USA, 1992). 115] M.D. Levine, Vision Man and Machine, McGraw-Hill Series in Electrical Engineering (McGraw-Hill, New York, 1985).
ELSEVIER
CRYSTAL GROWTH
Journal of Crystal Growth 137 (1994) 1—11
Quantification of crystal morphology M.E. Glicksman
‘~,
M.B. Koss, V.E. Fradkov, M.E. Rettenmayr, S.S. Mani
Materials Engineering Department, Rensselaer Polytechnic Institute, Troy, New York /2180, USA
Abstract Morphological measurement constitutes an important experimental subject in crystal growth and materials science, and is currently receiving renewed attention because of the rapid advances occurring in computer technology, coupled with the concomitant sharp reductions in the cost of digital image processing. Image processing applied to the quantification of microstructural images is currently being used in our laboratory to increase the understanding of interfacial dynamics during crystal growth and to analyze the kinetics of microstructural evolution. Quantification of microstructural and crystal growth morphologies, such as the measurement of dendritic tip radii, crystallite size distributions, and crystallite shapes, provides the geometric foundation needed for interpreting interfacial dynamics during crystal growth and an objective description of morphogenesis accompanying solid—liquid and solid—solid phase transformations. Automated methods employed, and, in part, developed by the authors to measure these morphological and kinetic parameters, using advanced statistical and stereological methods, are reviewed in this paper. Some of the techniques disclosed here are currently being refined even further to achieve improved precision in the quantification of crystal growth morphology.
1. Introduction
Most engineering materials are microstructurally heterogeneous and usually evolve toward their final microstructures in distinct stages during the materials processing history. For example, nucleation, crystal growth, deformation, grain growth, phase coarsening, etc., are some of the individual processes that might be involved. Numerous characteristics of the evolving microstructure are measured to predict the behavior of the material in service. The length scale of the primary crystallizing phase, for example, is an important microstructural parameter, usually measured subsequent to the solidification process. _______
*
Corresponding author,
Specifically, an important morphological length
scale of practical concern in liquid—solid phase transformations is the tip radius of the dendritic crystals formed initially in a melt and, at a later processing stage, the size of the evolving primary phase crystallites. Another microstructural parameter of significance is that needed to quantify the crystallite shape which is an ill-defined geometric parameter that plays an important role for describing the structure of dense multiphase systems with relatively high volume fractions of primary solid phase. These metrical and non-metneal morphological measurements are typically executed manually by procedures that tend to be tedious and inconsistent; hence, the authors have developed several specialized computer-based imaging systems to perform them as an automated quantitative analysis on images captured
0022-0248/94/$07.OO © 1994 Elsevier Science By. All rights reserved SSDI 0022-0248(93)E0479-Q
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ME. Glicks,nan et a!. /Journal of Crystal Growth /37 (1994) 1—/l
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either in situ or post mortem to the crystal growth process. These image analysis techniques have been developed specifically to acquire quantitathe representations of interfacial structures produced by various crystal growth processes, and resolved over length scales determined by and appropriate to the type of microscopy employed to produce the original image.
2. Image analysis
to describe complex microstructures, because their value can be deduced independently of the size, shape, or spatial and topological arrangements of the individual phases. This paper will concentrate on two aspects of morphological quantification of microstructures through stereologic measures observed in or after a crystal growth process: the first is measurement of the steady-state tip radius in an in situ thermal dendnitic crystal, which is formed initially in the freezing process; the second is the measurement of crystallite domain size, shape, and distribution
Image analysis concerns quantification and classification of images and objects of interest, in
this case the structure and form of crystalline phases as determined by their interfacial configuration. Here, we will deal explicitly with manipulation, processing, and analysis of optical images, all performed objectively and automatically by a computer. Recent interest in this subject by crystal growers and other materials scientists has
as observed in a quenched two-phase microstructure as the crystal—melt system evolves isothermally over time.
3. Measurement of in situ crystal morphology Dendritic growth is perhaps the most common
been motivated in part by the impressive and
form of crystallization, usually occurring wher-
rapid advances in digital computer technology,
ever metals and alloys solidify under relatively
The rapidly falling costs of digital imaging hardware and image analysis software have now firmly placed such options within economic reach, and finally make practical truly automated, customizable solutions to quantitative morphology problems occurring over a spectrum of crystal growth and materials science applications, The major concern to the image analyst is retention of maximum information from an accurately recorded original image, and minimization of the influence of various unavoidable extrinsic interferences during the image processing and analysis. The automated procedures developed in our laboratory to analyze images perform sophisticated corrections for extrinsic image interferences, such as non-uniform object lighting, or variable photographic exposure and development, Measurements performed to extract so-called global stereological parameters can he easily performed on the images processed. Stereological measures, which are important in microstructural analysis, are used to describe the three-dimensional geometric and topologic features of an object or collection of objects which comprise a
shallow temperature gradients, such as in most casting and welding processes. The dendritic morphology affects a material’s properties and its subsequent deformation processing responses. Understanding dendritic crystal growth is considered the crucial first step in controlling these properties and responses. Essential to the understanding of dendritic crystal growth is obtaining accurate, quantitative experimental data on the steady-state aspects of dendnitic growth kinetics and morphology. The overall experimental task of measuring the size and shape of steady-state dendritic crystals, for either comparison with other experiments or to verify theory, is, as will be shown, not a trivial one. This is evidenced by noting the typical experimental uncertainties encountered in such measurements, which are typically ca. ±10% in the measured dendrite tip radii, ~ or ca. ±5%in the measurement of dendrite velocities, V [1—5]. Also, one should note that dendritic
microstructure. Such measures are used routinely
scaling theory usually requires evaluation of these
variables combined as the nearly constant product VR~0,which would typically exhibit a cornhined uncertainty of ±25%.Our research on the
ME. Glicksman et al. /Journal of Crystal Growth 137 (1994) 1—I /
3
crystal growth of high-purity (6—9’s) succinontrile (SCN) crystals indicates that these measurement uncertainties can be reduced by as much as a factor of one-half to one-third through appropriate applications of digital image processing. The following is a brief discourse and perspective commentary (to be published elsewhere in full detail), on how to measure the steady-state tip radius of a dendritic crystal by the application of modern digital image processing, Initially, even prior to producing an image, there arises the experimental question whether the dendritic crystal chosen for imaging is sufficiently isothermal and well isolated from the per-
One of the first quantitative techniques used to accomplish profiling analysis of in situ dendritic growth images was the matching parabola method [1,2,41, in which a family of confocal parabolas of different radii of curvature is projected against the magnified image of the selected dendritic tip, taking into account the experimental image magnification and the scaling of the parabolas. The radius of the best matching parabola is chosen to obtain the measured dendrite tip radius. A second useful approach is to locate the position of the tip, move along the growth axis of the dendrite an appropriate meremental distance, L1, back from the tip, and then
turbing influences caused by other actively growing interfaces. Indeed, measuring displacements of the dendrite tip at sequential times during crystal growth would yield suitable kinetic data (the growth speed) only growth. if the dendritic crystalaxial is exhibiting steady-state Also, to insure that a dendrite is in fact sufficiently isolated from its neighbors to avoid transient growth, the distance, 1i1, from a point near the tip to the closest other active solid—liquid interface must be greater than the estimated thermal or solute diffusion length. The next step needed in measuring the curvature of an SCN dendrite tip is to apply an empirically confirmed, yet theoretically based, geometnc assumption: i.e., the tip profile projected on a (100) plane in the (100) direction is, by virtue of the bce symmetry, a parabola. Thus, near the tip, the dendrite is a three-dimensional crystal growing close to the ideal form of a paraboloid of revolution. Consequently, the approaches used in making dendritic tip radii measurements have been based on analyses that compare the observed image of the dendrite tip to the “best” fitting parabola. Once this parabola is determined, it is straightforward to calculate the corresponding tip radius of curvature as the unique, linear metrical parameter that sets all the length scales of the developing dendritic crystal. There are several alternative procedures to select the best paraboloid, and thereby, to characterize the dendritic shape from a photographic or video
record the width of the dendrite, 14’, the distance between the opposing optical edges of the profile. The locally estimated tip radius, R., corresponding to the width measured at the distance is 2/8L~,which is a geometric formula L,that R, Wç can be derived easily from the equation of a parabola [2]. The dendritic tip radius for, ~ is then calculated from the average of the sampled
image. The most useful of these analysis proce-
dures are discussed next,
=
local estimates, viz., R~1~(1/n)~’1R1. The last approach described in this paper for analyzing in situ dendrite images to obtain steady-state tip radii is based on obtaining a least-squares regression curve that fits the sampled optical edge to a parabolic equation [2,5]. One of the free fitting coefficients of this equation is mathematically related to the radius of the regressed parabola, and therefore, to that of the dendrite. Generally, the exact mathematical form of the equation, and the choices of what constitutes the independent (versus the dependent) variable are not significant. However, the final calculated crystal radius is dependent on the number of sampling points used to define the crystal—melt optical edge, the scatter in those points, and the distribution of those points over the whole curve. Thus, for a small dendrite tip, with a relatively large (with respect to the tip radius) scatter in the coordinates of the points defining the optical edge, the radius results would be based on just a few edge points, predominately clustered near the tip, and would therefore depend sensitively on the exact mathematical form chosen to represent the parabolic and on the particular choice of independent variable. =
4
ME. Glicksman eta!. /Journal of Crystal G,oi’th /37 (1994) 1—1/
Curve fitting proved to be the best method for
analyzing the authors’ recent experiments on dendritic crystal growth in SCN [6]. The radii determination based on measuring W2/8L is strongly dependent on the choice made for the location of the tip. In addition, this method weights all the r• measurements equally in the calculation of an average tip radius. Of course, these problems are not insurmountable; but by the time one responds carefully to the two criti-
cisms mentioned above, one has de facto approached the image analysis based on the curve fitting method anyway. The matching parabola method is clearly the weakest approach. This method relies critically on deft hand—eye judg-
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ments, and, thus, is most susceptible to operatordependent errors or subjective judgments. It is also the least quantitative approach with respect to assessing the error estimation and in explicitly
exposing the heuristic assumptions employed. Not incidentally, this last paragraph also moves the discussion from the most computationally intensive method, especially when using a non-linear form (non-linear in the coefficients, not the vanables) for the parabolic equation, to the least computationally intensive method.
--
Common to this discussion of profiling methods is the assumption that the dendrite profile is perfectly aligned along some arbitrary (vertical) y-axis. This assumption is not critical, because we have shown that a small angle of rotation (<4
)
Fig. I. Computer enhanced gray-scale image of a dendritic tip and the located edge used in the regression procedure to calculate the tip radius.
to the dendritic edge data scarcely affects the calculated radius results when using these methods. Furthermore, an additional fitting parameten, the rotation, 6, can he added to these methods. Experience with this type of image analysis shows that it is best to determine U independently from the tip radius, or from the series of
tip images, and then account for any rotation from the y-axis so that one avoids introducing an additional freely varying parameter in the tip radius measurements or in the curve fitting procedures. The authors have incorporated many of these considerations into a convenient semi-automated image analysis system titled computer aided dendnite analysis program (CADAP). Fig. 1 shows a typical computer enhanced gray scale image of an
SCN dendrite photograph used with CADAP, and the computed crystal—melt interface edge on which a CADAP curve fitting routine will operate [6]. The first step in applying the CADAP procedure is to capture the image. The primary consid-
eration is to capture the image at a large magnification, and obtain a high spatial resolution in
terms of the number of computer lines, or pixels, per micron, and, therefore, a high precision in the dendrite edge loci. A larger image magnification also increases the maximum number of edge points used in the analysis. However, the enhanced resolution must be balanced against a decreased field of view. The field of view must
M.E. Glicksman eta!. /Journal of Crystal Growth 137 (1994) 1—11
5
extend at least several tip radii behind the den-
tized and processed with this assumption in mind.
drite tip in order for the data to fit a parabola. The compromise in applying CADAP was to Select the largest magnification that still allows measurements along the axis of dendrite for a minimum distance of 4—5 tip radii, After the image is captured, several subsequent processing operations were performed. First, any irregularities in the image capture procedures were corrected. In this case, the image correction was obtained by removing the background non-uniformities in the lighting used to digitize the original 35 mm photographic negative. The non-linear effects introduced by the film developing process can also be corrected from photographs taken of the incident light transmitted through a series of neutral density filters. These densitometric films were developed by cxactly the same processing as were the dendrite photographs. The image is then inverted to ohtam a positive digitized image that duplicates the
Values other than 50% transmission have been used in the literature [3]. For the images used in this study, widths of the edge function are on the order of 10—20 ~m (defined as 10% to 90% of the background illumination surrounding the
actual light intensity profile at the dendrite before illumination of the undeveloped film. In addition, several standard image processing transforms were tested, such as a median transform to eliminate single pixel noise, and an averaging transform that tends to smooth out or defocus sharp edges [71.The use of these transforms, both singly and in combination, had little effect on the tip radius calculations. The explanation for this was most likely that the curve fitting routines employed in the CADAP procedure mathematically accomplish what the digital transforms do by image processing. Part of the success of our analysis, without recourse to a lot of digital image processing, may be based on the fact that the optical system that records the dendritic growth is shadowgraphic, and, therefore, provides some real time analog image processing. Thus, image processing transforms may still be useful for other dendritic growth recording methods or tip radius calculation methods, The final step prior to the radius calculation step is to create a list of ordered pairs that define the measured dendrite edge. In applying CADAP one assumes that the crystal—melt interface occurs at 50% transmission of the average background of the processed image. Images were digi-
edge), so that the choice of where to locate the
edge is quite important. We are currently expenimenting with other choices and with gradient edge detection methods. However, if the edge function width is small enough, these considerations become far less critical. Images with narrower edge functions are well worth the effort invested in the optical system to create them. To convert the located edge to a tip radius
measurement, we self-consistently fit the edge loci ordered pairs to an equation for a parabola, linear in it fitted coefficients, y a + bx + cx2, where the tip radius, ~ equals l/2c. The data range was restricted on the y-coordinate to values between 0 and 2R 11~, and we iterated the curve fitting and data limiting until the final tip radius converges. The particular choice of how many tip radii are used in the self-consistent iterative process (1-, 3-, or 4R1~,instead of 2R1~) is important in as much that it changes the value of the final calculated tip radius. Furthermore, the choice of 2r is not universal [3]. This raises the issue of whether the region around the dendnite tip is truly a parabola. Currently we are using a numerical modeling study to determine (in the same manner that we analyze dendrite edges) how the CADAP system analyses true parabolas in the presence of Gaussian noise that stochastically alters the “edge”. Although unanswered questions remain, our current analysis indicates that, after digital image processing, a least-squares regression of the edge points defining the interfacial profile provides a valid approach to quantifying the morphology of dendrites. Clearly the use of a powerful computer is imperative to accomplishing both the image processing and the regression analysis. However, the use of a powerful computer, per Se, is not a panacea. One must employ the best optical techniques to insure that high-quality images are mitially captured, that steady-state and controlled, isolated growth conditions are achieved experi=
0
ME. Glick.s,nan eta!.
/Journa!
of Crystal Growth /37 (/994) 1—/1
mentally, and that only highly reproducible den-
affine volume correlation, which shows that in
dnitic crystals are grown and analyzed properly.
dense microstructures each particle or crystal domain develops its local diffusion field within a
4. Crystallite size and shape measurements After crystallites form from a melt, the microstrueture gradually evolves toward thermodynamic equilibrium. Typically, the initially formed dendrite branches increase in average size as they interact via diffusion to reduce the total interfacial energy of the system. This process, known as late-stage phase coarsening, or Ostwald ripening, was investigated theoretically by Lifshitz, Slyozov, and Wagner (LSW) [8,9], who considered the crystal dynamics in an infinitesimally dilute, twophase system. To expand the scope and applicahility of the classical LSW approach [8,9], and to understand the dynamics of Ostwald ripening in densely crystallized systems, microstructural evolution studies of dense two-phase systems are underway at our laboratory. In addition, the interesting departure of real multiphase systems from the limiting assumptions used in LSW theory are currently under active theoretical development. Specifically, departures from LSW behavior are usually introduced theoretically by introduction of a second transport-related length scale. The arguments justifying a second length scale are often heuristic, and this length scale is introduced through a variety of models sometimes as a screening distance for interparticle diffusion, or as a “cut-off radius” defining the average “environment”, or even as the outcome of statistical correlations occurring among the crystallites. The current theoretical uncertainty regarding the dynamics of phase coarsening again underscores the importance of performing careful —
experiments to measure length scale evolution
kinetics in crystallizing systems. The quantification of morphology through stereology and image processing techniques will now he discussed from this standpoint, Recently Marsh [10] has developed a nonheuristic, physically-based solution to the prob1cm of selecting a second length scale in partially crystallized solid—liquid microstructures. This development introduces the concept of a linear
restricted volume of matrix, or melt phase, that is itself proportional to the particle’s own volume. The correlation suggests that the shape and size of the crystallites determine the average growth rate. A quantitative metallographic study of the evolving particles adds to the fundamental understanding of how solid—liquid microstructures evolve geometrically and kinetically during isothermal processing. In order to automate the metallographie quantification for application to high solid fraction systems for comparison to theones like that of Marsh [101,the analyzed image must contain distinct particles, each independent and distinguishable. This makes the image processing of a three-dimensional, densely interacting collection of particles a far more challenging problem in morphological quantification than that discussed above for a single dendrite.
The standard [11—13]stereological procedures used in quantitative characterization of crystallites usually involve counting of linear intercepts and points measured on random planar sections, followed by conversion of the intercept counts to crystallite sizes, shape factors, volume fraction,
and other stereologically significant parameters. Global morphological and distributional parameters are then obtained through conversion of the stereological data obtained from two-dimensional sections to information about the morphology in three-dimensional space. One then obtains either three- or two-dimensional crystallite population descriptions, and the statistical properties of the structure is usually described by distribution functions. A digitized gray-scale crystallization microstructure is shown in Fig. 2, which has been prepared for metallographic recording by suitably polishing and etching. It can be clearly seen that the high volume fraction of primary (dark gray) solid phase leads to a large number of contacts between neighboring crystallites. These connections among the particles make it difficult for to identify the shapes and positions of the individual crystallites comprising the microstructure. This is particularly troublesome when attempting to ap-
M.E. Glicksman et a!. /Journal of Crystal Growth /37 (1994) 1—11
7
tallites are interconnected to each other through
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Fig. 2. A typical digitiied gras-scale image ‘.sith the darker region being the solid phase.
ply computer assisted techniques for making automated stereological measurements. In fact, most automated image analyzing systems assume some specific, idealized, shape for the crystallites, typically a sphere or ellipsoid, which simplifies the subsequent image analysis algorithms, but then usually results in some loss of valuable shape information. Automated commercial systems like Optirnas®, suitable for DOS-based personal cornputers and work stations, incorporate proprietary procedures to separate touching crystallites. Such image processors are often limited by the morphological configuration cases that the software allows, such as the maximum number of interpartide contacts and the topological features of the interconnected crystallites. We have developed a more general algorithm, implemented on a Macintosh based system, which is broadly applicable to metallographic samples where individual crys-
(a)
(b)
multiple local contacts. Even the complex situation shown in Fig. 2, where the crystallites are involved in numerous and multiple contacts, can now be handled virtually fully automatically by this algorithm which allows all the distinguishable cases of particle morphologies and topological connections. The basic principle behind image morphology analysis is to employ various shape structuring elements in order to implement the required analysis tasks such as edge detection, segmentation, and feature detection [14]. Morphological tools offer a unified and powerful approach to a number of image description problems, such as the one shown here for identifying each individual crystallite in a dense collection of particles. Complex, convoluted and ramified interfacial shapes that are commonly encountered during unstable crystal growth pose similar structure analysis problems. These image measurement tools are particularly useful in image analysis applications dealing with morphological features that are highly irregular in form and often interconnected, as observed, for example, in metallographic samples containing a large volume fraction of the dispersed phase. The most important attribute in the development of this general procedure is that it does not require one to make any simplifying assumptions regarding the crystallite geometry or arrangements, and it defines the structuring element precisely so that the loss of quantifiable detail on the morphology is minimized during image processing.
(c)
(d)
Fig. 3. (a)—(c) Optimizing threshold value to convert a gray-scale image to a binary by minimizing granularity. (d) Filtering the binary image to remove small particles.
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The procedure to be described operates on a binary image, and its purpose is to separate and locate each of the initially interconnected crystallites. A binary image is obtained from the original gray-scale image by removing the background density present in the optical unit, The gradient in the background is removed by considering smaller units of the gray-scale image at a given time. The granularity, which is the sum of black particles per black pixel and white particles per white pixel, is measured and minimized to determine the threshold value for the unit in considerFig. 3 shows the importance of minimizing granularity to get a true representation of the crystallites. This step of transforming from a gray image to a binary one is automated to eliminate any hand—eye judgments and to minimize operator subjectivity. The two important fundamental
morphological operations which form the basis for morphological operations to identify and separate individual crystallites are dilation and cr0sion [15]. Erosion is an image processing operation that removes pixels from the edges of crystallites in the binary image. This operation, if applied repeatedly, will eventually separate interconnected crystallites that are touching and remove isolated pixels. Dilation is an image pro-
cessing operation that complements erosion by adding pixels to the edges of objects in a binary image.
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and fillsDilation in holes connects which willdiscontinuous be used laterobjects, in the procedure for separating individual crystallites. The method developed by the authors to identify individual crystallites uses a combination of a modified erode operation to attack the connecting necks selectively, and then applies a complete (uniform) pixel erosion iteratively to find the dentroids of the individual crystallites. Fig. 4b shows an example of the selective erosion operation performed on the binary image in Fig. 4a, where the necks are preferentially attacked. Fig. 4c
shows the image after a complete (uniform) erode
binary
(c)
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Fig. 4. (a) A typical digitized gray-scale image converted into aa image. (b) The necks operation. being selective attacked performing the selective erode (c) The image on after complete erode operation has been performed.
ME. Glicksman et al. /Journal of Crystal Growth /37 (1994) 1—Il
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Fig. 5. The image with the crystallite center positions marked. Fig. 6. The image after the skeletonize operation has been performed giving the topology of the system.
has been implemented and where separation between initially interconnected crystallites has been achieved. The combination of these two pixel erosion operations is performed until none of the black pixels is left. This procedure will eventually locate all the positions of the crystallite centers, measured here as the centroid of the particle areas, as demonstrated in Fig. 5. Insofar as crystallites usually have complicated interconnected structures in dense systems, it is also important to determine and retain information on the topology of the original image, which is determined by applying the skeletonize operalion. This image processing operation repeatedly removes pixels from the edges of objects in a binary image until they are reduced to a single pixel-wide curve or skeleton, as shown in Fig. 6. value as compared to the skeleton, The separaThe centers are shown in a different gray-scale tion of the crystallite network is simpler once the skeleton structure is determined. The skeletons are then dilated, i.e., pixel layers are added on either side, marking the layers with respect to sides and skeletons at every stage until all the possible positions Finally, theboundary boundaryend end points are are located. all located, and a straight line “cut” with the shortest possible distance is inserted to separate each of the particles as indicated in Fig. 7. There is some residual error in the separation algorithm which can be reduced by performing a few extra comparison steps and by using some
manual editing. Although these automated procedures are not totally free from error, there are further improvements being considered to get more repeatable, error-free measurements from these analysis tools. There is also a trade off made between selecting the magnification at which the microstructural image is recorded cornpared to the available, and affordable, resolution of the computer hardware and software. There are also some interesting technical questions regarding the gray-scale to binary conversion procedures, where the positions of the crystal—melt interfaces are determined, These image analysis
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ME. G!icksman eta!, /Journal of Crystal Growth 137 (/994) 1—/I
issues are among those being studied currently in our laboratory. Finally, after all the individual crystallites are identified and separated, the stereological measurements of interest can be performed fully automatically to determine the size, shape, and on100
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______
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______
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stereological
tool described here provides affordable access to easily implemented characterization of images for quantitative metallographic
60
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and morphologic measurements.
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5. Conclusions
-
100 150 200 length range
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The development of computer aided image analysis tools provides a consistent and objective method for morphological quantification. As cxamples we discussed the development of a least-
200
squares regression technique, applied after inclusion of image processing corrections of the crystal—melt interface, and suggested its use for vanous in situ measurements of dendrite tip radii,
150
speeds, and morphologies. This method provides
100
the most accurate, objective approach for quantifying the morphology and kinetics of steady-state crystal growth. The use of a powerful microcomputer or work station is shown to be critical for efficient image processing and regression analysis. The application of an image analysis program
300 ~
entation distributions, see Fig. 8. The shape factor measurement shown is the ratio of the major to minor axis length for the best fitting ellipse. Using the distributions measured for crystallite size, shape, and axis orientation it becomes possible to monitor and describe the microstructural evolution trajectories quantitatively on a purely geometric basis. The measured stereological distributions can the be used as the basis to compare different theoretical predictions. The automated
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was described for measuring the kinetics of evolving crystallites, by determining stereologically the size and shape distributions in three dimensions without recourse to simplifying geometrical or
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Fig. 8. Stereological measurements made on the separated image for (a) size. (h) shape and (c) orientation distributions.
topological assumptions about the shape or eonfigurations of individual erystallites. This analysis technique provides new avenues for providing detailed experimental checks on the limits of theories on phase coarsening in dense microstructures with high volume fractions of a dispersed phase. Finally, this development in the quantification of morphology enables one to perform highly automated measurements of shapes and their distributions in a suitably prepared metallographic sample.
M.E. Glicksman et al. /Journal of Crystal Growth 137 (1994) 1—Il
6. Acknowledgements
[51ER.
This work is being conducted with the support by the National Aeronautics and Space Administration, Space Experiments Division, Microgravity Experiments Branch, under NASA 3-25368, with liaison through the Lewis Research Center and the National Science Foundation under DMR-89-21 35.
161
Rubenstein and M.E. Glicksman. J. Crystal Growth 112(1991) 84.
7. References 111
S.C. Huang and M.E. Glicksman, Acta Met. 701. [21S.H. Tirmizi and W.N. Gill, J. Crystal Growth 488. [3] A. Dougherty and J.P. Gollub, Phys. Rev. A 3043. [4] J.H. Bilgram, M. Firmann and E. Hiirlimann, Growth 96 (1989) 175.
11
29 (1981) 85 (1987) 38 (1988) J. Crystal
ME. Glicksman et al., Reprint No. AIAA 93-0260, 31st Aerospace Sciences Meeting & Exhibit, Reno, NV(1993). [7] GA. Baxes, Digital Image Processing (Prentice-Hall. Englewood Cliffs, NJ, 1984). 181 C. Wagner, Z. Electrochem. 65 (1961) 35. [9] tM. Lifshitz and V.V. Slyozov, J. Phys. Chem. Solids 10 (1961) 35. [10] S.P. Marsh, PhD Thesis, Rensselaer Polytechnic Institute, Troy, NY (1989). [111 H.E. Exner and H.L. Lukas, Metallography 4 (1971) 325. [12] E.E. Underwood, Quantitative Stereology (Addison-Wesley, Reading, MA, 1970). [13] R.T. DeHoff and F.N. Rhines, Quantitative Microscopy (McGraw-Hill, New York, NY, 1968). 1141 Y. Mahdavieh and R.C. Gonzalez, Advances in Image Analysis (SPIE Optical Engineering Press, The lnternational Society for Optical Engineering, Bellingham, Washington, USA, 1992). 115] M.D. Levine, Vision Man and Machine, McGraw-Hill Series in Electrical Engineering (McGraw-Hill, New York, 1985).