An Effective Sem-Based Image Analysis System for Quantitative Mineralogy t

R.P. King and C.L. Schneider Comminution Center, University of Utah* An effective image-analysis system for use in the study of particulate mineralogical material is described. The available commercial image-processing and image-analysis software systems do not usually include adequate algorithms for the effective analysis of multiphase mineralogical textures. Filtering algorithms are usually inadequate for the accurate removal of image noise without compromising the integrity of phase edges. Although most systems offer good algorithms for the analysis of binary images, algorithms for ternary and higher order images are almost non-existent and these are essential for the analysis of mineralogical material in the particulate state. Existing systems make no provision for effective stereological analysis of image data nor for the interpretation of microstructure using models based on stochastic and integral geometry. This paper describes the development of an image analysis system based on an SEM equipped with secondary electron and back-scattered electron detectors, an image memory for the storage of digital images captured at slow scan speeds and a SUN workstation for image processing and image analysis. The algorithms and procedures described include a full range of linear-intercept analysis routines used in quantitative mineralogy.


Introduction
Mineralogy has always been related to image analysis to a great extent.Luster, color and macroscopic texture are fundamental characteristics of visual nature that were probably used at the very birth of mineralogy as a science.The establishment of scientific mineralogy was possible due to the development of optical microscopy.The very first image-processing procedures in this field can be considered to be the use of filters and polarized light in order to modify the appearance of the individual minerals being viewed.Since then, image analysis and processing have evolved enormously.Technological progress in related fields like mineral processing and extractive metallurgy have, at the same time, drastically increased the demand for quantitative information from mineralogical specimens.This quantitative information must be accurate enough for the measured values to be used as parameters for simulation models, process design procedures and control strategies.Some examples of particle population properties that can be measured by image analysis are particle size distribution, particle composition and particle composition distribution (the liberation spectrum), surface area per unit volume and interphase area per unit volume of phase, the last being a measure of mineralogical texture.Measurements must be made on cross sections of carefully mounted particle specimens because mineralogical materials are opaque.By cross-sectioning, the particles' internal microstructure is exposed for the so important textural characterization and liberation spectra measurement.Furthermore, the spatial interpretation of the one-or two-dimensional information extracted from such cross-sections can be accomplished by means of a variety of stereological procedures that have been developed in recent years.
Several stereological relationships are used in quantitative mineralogy.Those based on linear-intercept analysis are the most accurate and convenient and are very well understood theoretically.The measurement of the one-dimensional linear-intercept distribution is accomplished by superimposing a frame of linear probes on the image containing the features.If this procedure is repeated for several images from the same particle population, the distribution of linear-intercept lengths becomes smooth enough to be used as an unambiguous measure of that particular feature.Then the stereological relationships are used to convert it to the necessary three-dimensional information.This procedure is quite straightforward if perfect phase-discriminated images are available.In order to obtain such images, an effective image analysis system coupled with a set of image-processing routines is required.
A comprehensive set of image-processing routines for digitized images that can be used for quantitative mineralogy is presen!ed for the first time in this paper.A detailed description of the stages involved between image acquisition and stereological transformation for real mineral particles from test ores is presented.

Image analysis system
The generation of a good image is an essential precursor to any serious image-analysis work.When analyzing mineralogical structures, it is essential to start with an image that resolves all of the features of interest and provides sufficient contrast to distinguish separate phases that must be analyzed.Particular attention must be paid to feature edges and phase boundaries.These play an important role in the generation of accurate statistical analysis of multiphase materials.
There are essentially three methods that are commonly used for the generation of images: optical microscopy, scanning-electron microscopy, and SEM coupled with x-ray microanalysis.The imaging system of choice will depend on the problems and materials that need to be studied.Probably the most commonly used system is the SEM equipped with both backscattered and secondary electron detectors.The back-scattered electron detector is good for mineral phase identification because the intensity of the output is proportional to the average atomic mass in the region of the specimen under the electron beam.However, closely related minerals cannot be readily distinguished, and for complex mineralogies, x-ray microanalysis is essential for routine use.

Image collection and digitalization
Digital image storage is essential for subsequent image analysis.Not all scanning-electron microscopes are equipped with digital storage and some digital interface may be required.Variable scan rates and digital frame averaging are useful facilities to incorporate in a digital interface.
The digital image should be capable of being displayed in real time so that the operator can adjust focus, magnification, background intensity and contrast.The intensity and contrast in the captured image are probably the most important parameters that can ensure effective phase discrimination.Fig. 1 shows a back-scattered image of cross-sectioned sphaleritedolomite particles.This image is a weighted average of eight individually digitalized frames scanned at slow scanning speeds.Intensity and contrast have been adjusted to produce very good phase discrimination.
The peaks in the image intensity spectra should be well separated, and the required portion of the spectrum should fill the whole of the available digital intensity range.This should be done by adjustment of the microscope controls.Various digital histogramstretching algorithms that are often used to correct a poorly adjusted image are at best useless and more often introduce unwanted artifacts into the image and should be avoided.Fig. 2 shows an example of a good histogram for a raw image containing three phases.
Background intensity variations should be corrected before any further image processing.Variations in background intensity are observed in most images.In the optical microscope these result from variations in illumination intensity and in the SEM from variations in detector collection efficiencies at different points on the sample surface.Various interactive algorithms are available to correct images for background intensity variations, but we have found that each detector configuration should be accurately mapped for background intensity variations.Additive correction is more effective than multiplicative correction.The background should always be adjusted to the same level in all images that will be processed automatically so that bias due to overall variations in image intensity can be avoided.

Phase discrimination
Mineralogical analysis usually relies on accurate definition of mineral phases, and some preprocessing of the image is necessary to ensure proper phase discrimination.The elimination of the halo that will always surround the phases with higher grey values whenever more than two phases are discriminated is probably the most important image preprocessing operation that is required.This is called delineation filtering, and it converts the steep grey-level ramps that define the phase boundaries in a grey-level image into sharp steps so that the transition from one phase to the next is accomplished in a single pixel step.The effect of delineation is illustrated in Fig. 3  and 4. The insets show magnified portions of the main image so that single pixels are visible.The effect of the delineation filter is clearly visible.Delineation filtering will generally improve phase separation.The grey-level histogram after delineation is shown in Fig. 5, which can be compared with the histogram of the original image in Fig. 2. The significant improvement in phase separation is clearly evident.The removal of the halo effect is particularly important for mineral liberation studies.A halo will always register the feature as unliberated and will grossly distort any estimation of surface exposure of the mineral.
The delineation algorithm used is defined by (1) where gmax and gmin are the largest and smallest grey levels in a roughly circular neighborhood of pixel i.
In order to make the algorithm efficient, only those The halo at the interface between the bright phase and the dark background is visible.The inset is a magnification of the region inside the small square.
Fig. 4 Image that has been discriminated after delineation filtering.The halo has been eliminated.The removal of the halo is particularly important for mineral liberation studies.pixels i are examined that have a two-dimensional gradient greater than a predetermined level.The choice of neighborhood size and gradient threshold should be established for each mineral and particle type.Once determined, these should be fixed for all subsequent analysis of the same material.
Correct phase discrimination usually cannot be achieved by grey-level thresholding alone.The fluctuations in grey level within a phase are usually too great and there will always be some pixels in each phase that are too high or too low so that they are allocated to the wrong phase by thresholding.Some additional information is usually required that can Grey level Fig. 5 Intensity histogram of an image showing good phase discrimination after delineation filtering.
characterize each phase unambiguously.The best characterization is undoubtedly an elemental composition at each pixel as can be determined by x-ray microanalysis.This is a comparatively expensive approach, and we have found other characterizations that work well.In a three-phase image, the bright phase can be characterized by thresholding the image at a level past the phase peak in the grey-value histogram.This produces a highly fragmented image of the bright phase as illustrated in Fig. 6.Any spurious regions that appear in the original discriminated image are very unlikely to produce any highlevel fragments so that selection of only those features in the discriminated image that are touched by at least one high-level fragment produces a very good binary image of the required phase.This procedure is significantly better than the use of any low-pass or median filter for the elimination of small spurious regions.Both of these filters make significant changes to the feature outlines, which in tum introduce unacceptable errors in the measured distribution of linear intercepts.

Edge tracing and feature identification
Most image-analysis operations are performed on binary or ternary images.A wide range of logical and morphological operations is useful to analyze various aspects of the image.A few of the standard operations are essential for mineral liberation work.Since liberation is essentially a property of the particulate state, identification of individual features in the image that arise from separate particles is of crucial importance.The basis for feature identification is edge tracing.
If the edge of a feature is traced, the feature is identified simply by filling all pixels within the edge, which for particles, will always be a closed region.An edge pixel of a feature must satisfy two requirements: the pixel must belong to the feature and must be connected to the background in at least one of the 168 Fig. 6 A "fingerprint" image of the bright phase in a 3-phase image.This is useful for the elimination of artifacts.
four orthogonal directions (4-contiguity).This definition requires a definition of the background.A 4-path in the image is any sequence of pixels in which each pixel is connected to its leader and to its follower in one of the four orthogonal directions.An 8-path in the image is any sequence of pixels in which each pixel is connected to its leader and to its follower in one of eight directions.The background of an image is the collection of all pixels that are connected to the image frame by at least one 4-path.The use of the 4-path rather than the 8-path prevents the background from penetrating the features across very narrow diagonal openings at the feature edges.These normally arise from slight imperfections in the greylevel thresholding at the feature edges and do not usually signify an actual small concavity on the particle surface.Several efficient edge-tracing algorithms are available, and these are easy to incorporate into an image-analysis package.

Quantitative measurements
Image analysis involves the measurement of geometrical features in the image.These include area, perimeter, orientation, location and various derived geometrical properties.

Measurement of linear features and area
The simplest geometrical measurements to make on a digitized image are linear lengths and areas.The area of a feature is simply equal to the total number of pixels that are required to represent the feature multiplied by the area occupied by each pixel.The error introduced, because it is not possible to represent the curved boundary of a feature exactly by means of rectangular pixels, is small provided that the features are significantly larger than the pixel size.The length of a linear element is calculated from the coordinates of the pixels at each end of the segment using Because these measurements of area and length are not significantly affected by the digitalization of the image, they are preferred for quantitative work.However geometrical properties that cannot be measured entirely using linear segments and area are also often required and then the digitalization can have very significant consequences.

Quantitative measurement of perimeters -
the fractal problem The measurement of perimeter length of any feature in an image can often provide useful information.For example it may be used in equation ( 12) below to calculate the surface area per unit volume of a particular mineral or phase and, by restricting the measurements to perimeters that separate two specific phases, the interphase area between them in the three-dimensional structure can be obtained.
The measurement of perimeter is unfortunately subject to a great deal of uncertainty because it is dependent on the digital pixel size and the resolution at which the original image is generated.A feature edge is defined through a number of steps that must be taken to follow the feature boundary.Steps may be horizontal, vertical or diagonal.Each step progresses from one edge-pixel to a neighboring edge-pixel.The perimeter length of a feature in a digitized image is calculated as perimeter length (3) where Nh and Nv and Nct are the number of steps in the horizontal, vertical and diagonal directions respectively, and Ch and Cv are the calibration factors in the horizontal and vertical directions respectively.
The ratio of diagonal to linear steps in the edge tends to increase as the pixel size decreases and the total number of pixels increases more than proportionately as the image is enlarged relative to the pixel size.This is the fractal problem, and unfortunately, most surfaces in nature do not approach a limiting perimeter length as the image resolution increases indefinitely.As a result, it is not possible to make measurements of perimeter length by image analysis in any absolute sense.It is possible to make only comparative measurements of the perimeter of different features when they are imaged at the same magnification and resolution.An approximate procedure that can be used to correct the measured perimeter for the fractal effect has been developed and is quite successful when applied to the kind of images that are generated when mineralogical particles are studied.The procedure corrects the measured perimeter to a standard magnification (and therefore resolution) that is convenient for the job on hand.We usually refer back to the resolution at the smallest magnification used.
The empirical correction is developed as follows.A series of monosize particle samples are prepared from a population that shows no variation in distribution of particle shape with particle size.Such populations of particles are difficult to prepare, but a sample of material that has been prepared by crushing without undue handling and screening of the products is usually adequate.The mono size fractions are separated by careful screening between closely sized screens.The separate size fractions are mounted, sectioned and polished, and images are generated at a magnification that is proportional to the representative size of the sample.The average perimeter per unit are a (BA) of each sample is plotted on log-log coordinates against the ratio of particle size of the sample ( d p) to the reference particle size ( d P *).Let M represent the magnification used to generate the image of particle size d P and M* the magnification used to generate the image for particle size dP *.Let Crr(M) represent the fractal correction factor at magnification M defined so that the measured value of BA corrected for the fractal effect varies exactly as dp• 1 .From equations ( 4) and ( 5) Since the calibration experiment is performed with the magnification proportional to dP- Measured values of perimeter per unit area at any magnification can be corrected using the connection factor in equation ( 8) to reduce all measurements to the reference magnification.

What is stereology and why is it needed?
In practice, stereo logy provides a body of methods that can be used to calculate some properties of threedimensional geometric structures using data obtained from two-dimensional sections of these structures.The ability to translate data from two dimensions to three is very important in applied mineralogy because of the frequent use of microscopic observations of minerals in polished or thin section.Some of the stereological procedures give exact results (within the scatter imposed by the need to sample materials for examination), while others are only approximations, but all provide quantitative rather than qualitative or merely descriptive information.It is this quantitative property of stereological methods that is most important in practice.While qualitative descriptions of mineralogical structures are of enormous utility to the mineralogist and have been used for more than a century, they are not entirely satisfactory for generating data that can be used in mineral processing.
The fact that it is always necessary to sample the material under study prior to examination and also because real mineralogical materials show considerable non-uniformity and indeterminacy in texture and structure, it is only possible to make measurements in a statistical sense, and stereological methods are statistical in nature.These methods supply information on the average geometrical properties throughout a structure but yield virtually no information on the geometrical properties of a single element in the structure.The problems that arise because of the indeterminate nature of mineralogical structures continue to attract a great deal of research interest.
In addition to the standard symbols, it will be necessary also to make use of concepts and notations from probability theory.We will use the following symbols.p(x)dx = fraction by number of elements having property x in the range (x, x + dx).f(x)dx = fraction by volume of elements having property x in the range (x, x + dx).This is also called the volume-weighted distribution and is sometimes written as fv(x).
The concept of conditional probability will be denoted by a vertical line so that p (xly)dx = fraction by number of elements having property x in the range (x, x + dx) counted only from among those elements that are characterized by property y.
An important rule converts the conditional probabilities to total joint probabilities p(x, y) = p(xly) p(y) (9) from which follows immediately p(x) = fp(x IY) p (y) dy (10) The integral in equation (10) will always be taken over an appropriate range of the variable y.

Basic stereological relationships that are used in quantitative mineralogy
Perhaps the most well-known and widely used stereological conversion relationships are those that relate point, linear and areal fractions to volume fractions (11) These simple relationships illustrate very nicely the two basic stereological ideas: the conversion of quantitative information from zero to one to two to three dimensions and ability of stereology to yield precise three-dimensional quantities about a threedimensional geometrical structure from measurements made in lower dimensions.The lower dimensional measurements are usually significantly easier to make than the three-dimensional measurements.Often it is impossible, with current measuring instruments, to make the appropriate three-dimensional measurements.In fact, prior to the development of image analyzing computers in the 1970's, it was impossible to make accurate measurements in any dimension other than zero, and point counting was the only viable microscopic measuring technique.
The surface area of irregular 3-D geometrical elements can also be recovered from measurements in lower dimensional spaces using (12) or (13) It must be emphasized at this point that equations (11), ( 12) and ( 13) are valid only when the values are averaged over many individual elements, and it is important to take the ratio of averages not the average of ratios.Thus when evaluating BA for a mineralogical specimens, BA is computed as the total perimeter of a particular phase measured over many typical fields of view divided by the total area of that phase in those same fields of view.
Equations ( 12) and ( 13) may be combined to give All quantities in equation ( 14) are measured in lower dimensional space and equation ( 13) can be used to check the consistency of measurements made during any sampling by image analysis.The last two equalities in equation (11) also provide convenient consistency checks that should always be applied.

KONA No.ll (1993)
A particularly useful relationship exists between I L and the mean intercept length, p., through the phase in question.

(15)
This is particularly useful because p. is so easy to measure by image analysis.In equation ( 15), IL is interpreted as the number of intersections with the surface of the phase per unit length of linear traverse through that phase.Sv is the surface area per unit volume of the phase and B A the perimeter of the phase per unit sectioned area of the phase.
There are a number of other exact stereological formulas, but these are not often used in mineralogy and are not discussed here.Weibel's l) text provides comprehensive discussion of all the classical stereological procedures.

Measurement of size and size distribution
One of the most fundamental geometrical properties of a geometrical structure is its size or the size of the component parts.However, whenever the geometrical structure is not made up entirely of regular geometrical shapes, it is not possible to give a precise definition of size.In addition, the concept of size is not stereologically invariant, and the size of an element of a structure will usually appear to be smaller the lower dimension of the space in which the observation is made.For example, a sphere that is observed in 3-D will give a size equal to its diameter, when observed in a two-dimensional space, the size will be assessed as the diameter of the circle, which is the intersection of the sphere with the two-dimensional sampling plane.This circle will have a diameter less than or, on rare occasions, equal to the diameter of the sphere.If the observation is made in a onedimensional space, the size will be assessed as the length of intersection of a line with the sphere, which will also be less than the diameter of the sphere except on the very rare occasion that the line probe passes exactly through the center of the sphere.A detailed description of how one and two-dimensional measurements on sections or projected areas of particles relates to their size distribution is given in King 2 l, 1982.
Because of the uncertainty associated with any definition of size when shape is not regular, the stereological conversion of size is shape dependent.For spherical elements, the size is unambiguously defined as the diameter of the sphere, and the average is related to the average size measurement in 2-D and 1-D as follows.When the individual elements are not all the same size..., it is the distribution of sizes that is important, and for spherical particles the appropriate stereological transformations are p(A) When the shape is irregular, a general transformation is used to transform the linear intercept distribution to the particle size.distribution.(24) Equation ( 24) is usually used in the cumulative form 00 where Equation ( 25) is the preferred stereological technique for the measurement of size distribution by image analysis.Although equations ( 21) and ( 25) in reality express precisely the same information, the numerical solution of (25) appears to be slightly better behaved than that of equation ( 21).The transformation equations ( 18), ( 19), ( 21), ( 24) and ( 25) are integral equations which are quite difficult to solve, and their solution has attracted a great deal of attention in the literature.Fortunately a reliable numerical procedure has been developed and solutions can now be generated with a great deal of confidence.Remember it is the left-hand sides of these equations that are measured, and the equations must be solved to obtain the volume-weighted distribution of sizes fv(D) in three-dimensional space.In general p(d) is not meaningful when mineralogical materials are studied (since d is not defined for structures of indeterminate shape), and it is only the distribution of linear intercepts p (£) that has an unambiguous meaning and can be measured.

Measurement of mineral liberation
A particularly important transformation in applied mineralogy is the conversion of apparent linear liberation of particulate mineralogical material to the threedimensional liberation spectrum.This is probably the most important image-analysis task for the application of mineralogy to mineral processing.The only satisfactory stereological transformation is based on the measurement of the apparent linear-liberation spectrum.
In equation ( 27) g 1 represents the linear grade measured on a particle transect and g is the volumetric grade of a particle.The solution of this integral equation is also quite difficult and requires some care.The same numerical procedure used for the particle size distribution inversion has been found to give accurate and quick results.This solution is described in Schneider et al. 3 ), 1991.With a reliable numerical solution in hand, equation ( 27) provides an effective and accurate stereological procedure for the measurement of mineral liberation.An alternative solution that uses just the moments of the distribution and forces the spectrum p(g) to correspond to a beta function is not very satisfactory for practical work and does not offer any advantage in convenience or ease of use.Although we do not yet know a great deal about the function p(g 1 ig), some useful approximations have been developed using computer-simulated two-phase particles.These have been confirmed to give accurate solutions for p(g) from measurements made on a variety of ores under conditions ranging from almost complete to almost no liberation.The function p(g 1 lg) is definitely dependent on the mineralogical texture of the ore, and quantitative textural analysis will be required to discover the dependence on texture.It is the need to know this function that has prompted the study of mathematical models for texture using integral geometry techniques, as described in Barbery 4 ), 1991.This will be a major research area in the future.Accurate measurements of p(g 1 lg) are being made at present using natural minerals after very careful and confirmed fractionation of the particle population using magnetic-fluid techniques.
Equation ( 27) is often used in the length-weighted form To generate equation (29), the assumption that p(£1g) = p(£) (30) has been made.This reflects the assumption that particles of different grades generate similar linearintercept distributions, which is satisfactory, provided that either one of the phases does not exhibit specific shape characteristics when broken.

Measurement of linear-intercept distribution
Equations ( 21), ( 25) and (29) provide the preferred stereological transformations for the measurement of particle size distribution and mineral-liberation spectrum.The equations are all based on the measurement of the appropriate linear-intercept distributions.Similar equations based on measured area distributions can easily be derived.Although several researchers favor the use of the area distributions, we have a strong preference for the linear-intercept distributions for a number of reasons: (1) The linear intercepts can be measured easily, quickly, accurately, and unambiguously using an image analyzer.(2) A very strong body of theory is available to relate linear-intercept distributions to mineralogical texture.(3) The linearintercept distributions in the phases of the unbroken ore are well defined and easily measured.The concept of the area distribution is not defined for most real mineralogical textures.(4) Areal analysis can be very seriously biased by features in the image that touch even if the contact is through a single pixel contact.Although several feature separation algorithms are available in the literature, none has been found to be reliable enough for automatic image analysis.(5) The correction for the image frame can be implemented with good precision for linear-intercept analysis but not for areal analysis.

Frame correction for linear-intercept analysis
The frame correction for linear-intercept analysis is implemented by defining a guard zone on the righthand side of the image as shown in Fig. 8.Only linear intercepts that start within the active zone to the left of the guard frame are recorded, and all intercepts that are not completed before striking the edge of the frame are rejected.This is illustrated in Fig. 8.The rejection of long intercepts will bias the data against longer intercepts and this bias must be corrected.The correction can be accurately made by A guard frame can be used to eliminate the frame error when doing linear-intercept analysis.Linear intercepts shown as thin lines are rejected.
nothing that an intercept of length e will be unfairly rejected whenever it starts within a zone of width f+M-F.On the other hand, no intercepts of length less than F -M are rejected since these must start within the guard zone in order to strike the image frame.The number of intercepts of length greater than F -M must be corrected upwards by the ratio M/(F-f).If p' CO is the measured linear-intercept distribution, the corrected distribution is given by grade distribution directly and no ancillary estimation of the liberated extremes is required.
The technique used for the inversion of equation ( 29) is based on a finite difference approximation for measurements made on monosize particles 12 PLi = E hijPvJ• fori= 1, 2, ... 12 (32)  distribution using the inversion procedure are shown in Fig. 11.The excellent detail in the calculated distributions is clearly visible, and the very significant differences between the linear and volumetric-grade distributions are clearly evident.

Acknowledgement
The preparation of this paper and the development of several of the techniques described was supported by the Department of the Interior's Mineral Research Institute Program administered by the Bureau of Mines through the Generic Mineral Technology Center for Comminution under grant number G1115149.This paper is based on notes prepared for a workshop on image analysis given at the International Congress on Applied Mineralogy, ICAM'93, Fremantle, June 1993.

Notation and Symbols
It is convenient to use a symbolic notation for the many quantities that are measured directly or can be calculated or estimated from measurements on sections.The most important symbols that are in common use are given in Table 1.Whenever a symbol refers to a particular phase it is represented by means of a lower-case letter.

Fig. 1
Fig. 1 A typical image of multiphase particulate material.

Fig. 3
Fig. 3 Image discriminated without prior delineation filtering.The halo at the interface between the bright phase and the dark background is visible.The inset is a magnification of the region inside the small square.
Fig. 7Method for correcting measured perimeters for the fractal effect.' with i = 1.15.
the diameter of the sphere, d 3 and d 2 the average cubed and squared measured diameters of the circles observed in 2-D and e 2 the average of the squared chord length measured in 1-D.
is the average intercept length that would be observed in a sample having particles of only one size D and £ is the mean of the measured liner-intercept distribution.In equation (21) p(£) and p(£1 D) are distributions by number or numberweighted distributions.In practice, the length-weighted distributions are usually preferred and using the rela-= f PL C£1D) fv (D) dD.

F
Fig.8A guard frame can be used to eliminate the frame error when doing linear-intercept analysis.Linear intercepts shown as thin lines are rejected.
Fig.9 Experimental data illustrating the conversion of linergrade distribution to volumetric-grade distribution.
Fig. 10 Measured linear-grade distributions for progeny particles of various sizes generated by comminution of particles in the size range 710 to lOOOI'm and in the narrow density range 3.5 to 3. 7 glee.

Fig. 11
Fig. 11 The volumetric-grade distribution corresponding to the data shown in Fig. 10.

Table 1 .
List of stereological symbols