- Email: [email protected]
Two stereological correction methods: Allocation method and kernel transformation method
MineralsEngineering,Vol. 11, No. 8, pp. 707-715. 1998 © 1998ElsevierScienceLtd All fightsreserved 0892-6875/98/$ - see frontmatter 0892-6875(98)00057-0
Pergamon
TWO STEREOLOGICAL CORRECTION METHODS: ALLOCATION METHOD AND KERNEL TRANSFORMATION METHOD
R.G. FANDRICH ~, C.L. SCHNEIDER t and S.L. GAY* § Centre for Mining Technology and Equipment/University of Queensland, PO Box 883, Kenmore, Qld. 4069, Australia. E-mail: [email protected] i" Utah Comminution Center, University of Utah, 135 S 1460 E Rm 306, Salt Lake City, LIT 84112-0114, USA :~ Julius Kruttschnitt Mineral Research Centre, Isles Road, Indooroopilly, Qld. 4068, Australia
(Received 7 April 1998; accepted 21 May 1998)
ABSTRACT
Image analysis of polished particle sections is a common technique for measuring the liberation spectrum of mineral particle populations. The liberation distributions measured from particle sections differ from the actual distribution by what is known as the stereological error. This error always overestimates the degree of liberation by amounts dependent on factors such as the type of measurement (eg, linear or areal), particle composition and texture. For example, a particle may be composite, but the section may reveal only one phase making it apparently liberated Numerous methods have been proposed for correcting this error. Some of them are not appropriate for real mineralogical textures and are therefore practically useless. This paper focuses on two existing stereological correction methods for binary mineralogical systems: the Gay allocation method and the transformation kernel method. The transformation kernel correction procedure was found to provide somewhat more accurate and detailed results for the iron oxide ore investigated © 1998 Elsevier Science Ltd. All rights reserved. Keywords Liberation analysis; iron ores
INTRODUCTION Over the last decade an increasingly accepted technique for the gathering mineral liberation data has become that of image analysis of polished particle sections. It has now almost become routine for mineral processing plants to measure liberation spectra at crucial points within their circuits using image analysis. In the past, one aspect of particle section liberation measurement that has been inappropriately dealt with, and sometime ignored, is the stereological error associated with linear and areal measurements. The stereological error arising from areal measurements is illustrated in Figure 1 where a sectioned binary particle is shown. If the darker phase is the phase of interest then the particle has an approximate volumetric grade of 0.4. The three areal sections through the particle each show different grades. Sections 1 and 3 are liberated while area section 2 has an areal grade approximately that of the particle volumetric grade. This example shows how large the error can sometimes be. The Figure also demonstrates how the error always occurs in one
707
708
R.G. Fandfich et
al.
large the error can sometimes be. The Figure also demonstrates how the error always occurs in one direction. That is, the measured areal (or linear) liberation distribution will always be more liberated than the actual volumetric distribution. This comes from the fact that liberated sections can be produced from composite particles but composite sections cannot be produced from liberated particles.
/
/I
S
/ / / /
Fig. 1 Sectioned binary particle illustrating stereological error. The purpose of this work is to determine the stereological correcting characteristics of the transformation kernel and the allocation methods when applied to a particular binary iron oxide ore system. The work was performed by the principal author as part of his PhD candidature.
ALLOCATION METHOD The allocation method was developed by Gay [1,2] and is a theoretical general correction based on geometric probability equations. The approach uses the fact that when particles of a particular grade are sectioned there are nine geometric probability equations which relate the resulting sections with the original grade of the particles. The relationships involve parameters that are not conventionally used in image analysis. For example, for a section, one key parameter is the average distance between pixels of the same mineral. Consequently, unlike most other stereological corrections, Gay's method requires digitised images of the particle sections. The allocation method can be applied to linear intercept data.
Two stereological correction methods
709
Using the known geometric relationships, the allocation method uses a sorting algorithm to allocate sections into grade bins. The approach is to allocate the sections into the bins so that for each bin, each of the nine geometric probability equations are satisfied within statistical accuracy. As there are ten composite bins, there are then 90 geometric probability equations being used.
TRANSFORMATION K E R N E L M E T H O D The transformation kernel stereological correction method for particle sections involves finding solutions
~gvlD) to the transformation equation 1
F(&ID)=f
(1)
o
where F(gilD) is the cumulative distribution of linear (i=l) or areal (i=a) grades and K(gilgvD) is the cumulative distribution of linear or areal grades generated by particles that have volumetric grade gv and size D i e . the transformation kernel. Equation 1 was first proposed as a method of stereological correction in 1982 [3] and was derived from well-established principles of geometric probability. Determination of the transformation kernel K(gilgv,D) can only be performed experimentally through linear or areal distribution measurements of particles of known composition. A parametric model for the transformation kernel based on these measured distributions was proposed by Schneider [4]. Schneider's model parameterises the transformation kernel using incomplete beta functions by implementing separate functional forms to characterise the liberated and the unliberated portions of the linear or areal distributions as functions of volumetric grade. Two functional forms are possible: one for a symmetrical kernel and another for a more parameter intensive unsymmetrical kernel. Solving the transformation equation once the kernel is known presents various problems due to the ill conditioned nature of K(gilgv,D). As functional forms for arbitrary grade distributions do not exist, the discrete form of the transformation equation is used: N
(2)
e,'=E n=l
where
Kmn is the N x N discrete version of the kernel. As Kran is not exact and Fi m is not error free, the
use of direct inversion of
Kmn to solve for fly is not feasible. Equation 1 is a Fredholm integral equation
of the first kind and hence techniques for solving this form of ill-posed problem are required. Several regularisation and optimisation procedures are available to solve the transformation equation. One method is to use Tikhonov regularisation and constrained Rosenbrock Hillclimb optimisation [4]. The method employed in this work was an entropy regularisation method based on Hansen's [5] general solution to discrete ill-posed problems. The objective function for this method is 12
(3)
IIIO0(KL - F,)II÷>, n=l
12
and is subject to the constraints
1 and n=!
for n = 1, 2 .... 12 and where X is the regularisation
710
R.G. Fandrich et
al.
parameter. A detailed description of the implementation of this method to mineral liberation problems is given by King and Schneider [6].
EXPERIMENTAL The initial experimental work that made this particular study possible was conducted by Stewart [7] in 1975 and involved the fractionation of iron oxide ore particles into 21 grade fractions through heavy liquid and fluidised bed separation techniques. The iron oxide phase of the ore was an approximate 50:50 hematite/magnetite mix whilst the gangue consisted of a silicate phase of predominantly quartz. This ore has been the subject of previous liberation studies [8] due to its recognised binary nature. Five of the narrow grade fractions were initially used to construct the transformation kernel. These fractions underwent QEM*SEM areal section liberation measurements. The cumulative areal liberation distributions obtained describe the number of particle sections within the standard 12 grade classes; the 2 liberated grade classes and the 10 composite classes each of grade range 0.1. The five narrow grade fractions that were used to construct the transformation kernel by the method briefly described above were also measured and corrected with both methods. With the transformation kernel now constructed other known distributions were required to test the two methods. Thus, three different grade distribution were manufactured from six further composite narrow grade fractions: 1.
2. 3.
A "uniform" grade distribution consisting of equal volumetric proportions of particles from each narrow grade fraction. A "high" grade distribution consisting of a greater volumetric proportion of particles from higher grade fractions. A "low" grade distribution consisting of a greater volumetric proportion of particles from lower grade fractions.
It is recognised that these distributions are not entirely typical of what might be found in a real industrial circuit and that is not the aim of this work. The aim of the study is to test the stereological correction procedures on "known" distributions and the use of the above manufactured distributions was the best way the achieve that. Great care was taken during the manufacturing of these grade distributions to ensure that the mounted samples for QEM*SEM liberation analysis would provide the most representative polished section possible. Both correction methods were then applied to the measured areal distributions and the results compared. The areal distributions from the five narrow grade fractions used in the construction of the kernel were also corrected using both methods.
RESULTS For the purposes of comparing the measured areal, corrected volumetric and actual volumetric distributions, the cumulative form of the liberation distributions are used. These distributions are shown in Figures 2 to 5. Two forms of the actual volumetric distribution are represented in each Figure. Firstly there is the actual volumetric curve which is generated from the upper and lower densities of the each of the six component grade fractions and the assumption that the grade distribution within those limits is linear. The adjusted actual volumetric curve is a representation of the actual volumetric curve in the 0.1 grade interval domain. The reason for this second "adjusted" representation is that it provides and indication of the "target" distribution for the correction methods with the same curve resolution that the corrections provide. The resulting volumetric liberation spectra from the five narrow grade fractions are considered first. Figure 2 shows the volumetric distributions resulting from both correction methods as well as the measured areal, actual volumetric (thin solid line) and adjusted volumetric (thick solid line) distributions for the 0.47 grade fraction. The effect of stereological error can clearly be seen by comparing the actual volumetric with the measured areal distributions. The measured areal spectra has the same average grade or first moment but
Two stemologicalcorrectionmethods
711
has a much greater variance than the actual distribution. This is the overestimation of liberation mentioned earlier. Comparison of the two correction methods reveals a "sharper" volumetric approximation produced by the transformation kernel method. The allocation method generally produced volumetric distributions of greater variance. That is, particles were allocated to a wider range of grade bins on either side of the actual grade. This result was typical for the other narrow grade fractions except for the 0.11 grade fraction. Here both correction methods produced practically the same sharp volumetric distribution. The better performance of the transformation kernel method is not surprising as the transformation kernel was constructed using this very data and the correction is, in a sense, just reproducing the data. The quality of this reproduction is a function of the quality of the kernel parameterisation fit to the measured liberation data. If the models used to characterise the kernel fit poorly to the measured data then the reproduction of the narrow grade fractions will also be correspondingly poor.
100
i:
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80
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°°oOm
..I
A.,,,,~,
/~r,, ' I
70
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l/
6o
I ,'&/ j
o Actual volumetric
),~ E
11
30
I f
20
10 00.0
"-=- Measured areal
.~ s t ~"
" =" Allocation volumetric
o• ' "= ¢
¢"
¢.•
0.1
0.2
0.3
/ , / ~
"-~" Transformation volumetric
/1._ 0.4
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
1.0
grade
Fig.2 Actual, measured and corrected liberation distributions for the 47% grade fraction.
'°° /
9° T 80 t
"~
40-1-
x ~. - ~ e " " / " = ~ / " / I .~." ,,~¢" f f ~ d ¢ ~
3,, 1 8
u/
2Ol-
10 31(,~
. , ' ~ "~. " ' ~ ' Y
"-=- Measured areal
/" i . . ~ ~
...............
~
0-
0.0
• Actual volumetric • = ~ s t e d actual volumetric
0.1
0.2
:
:
:
I
I
0.3
0.4
i
:
( tr,c
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
1.0
grade
Fig.3 Actual, measured and corrected liberation distributions for the uniform grade distribution.
712
R.G. Fandrich et
al.
A more valid test of each method is that provided by the three manufactured grade distributions. The uniform grade distribution contained equal amounts of each fraction by volume whilst the high and low grade distributions were volumetrically weighted to their respective ends of the grade spectrum. Figures 3, 4 and 5 show the distributions for the uniform, low and high grade distributions, respectively. Again, here the actual volumetric distributions are constructed as mentioned earlier. Each linear upward slope represents the grade range of one of the six narrow grade fractions used to manufacture the distribution. The adjusted actual volumetric curve shows the cumulative distribution in the 0.1 interval grade domain. As before the stereological error can be seen in each Figure by the overestimation of liberation by the areal distribution.
100
=,
__
,
- -
90
E 80 g
7o
•-
50
/,~ ~'
. Actual volumetric
--AOiu. ~
2o
[
...
--"
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r,o
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~
'= I 0.1
0
0.0
I
I
I
0.2
0.3
0.4
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
1.0
grade
Fig.4 Actual, measured and corrected liberation distributions for the low grade distribution.
100
[/
90
/
80
I -'-'Adjusted actual volumetric
70
I I "-=- Measured I
60
-/'/~'o Actual volumetric
.//'V
areal
/ ~j'/] ,4" . f ! - =
/''"
Allocation volumetric
I --x.
Transformation
,
volumetric
~" I
=7,~
x
-
~ j l ~ /
30
10
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o-
0.0
"-
;0.1
~=" ° ~, = I " .,.. x " ~ , ~ = = = ~ -= 0.2
/ J
~
=
=
t
t
I
t
0.3
0.4
0.5
0.6
0.7
0.8
0.9
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grade
Fig.5 Actual, measured and corrected liberation distributions for the high grade distribution.
Two stereologicalcorrection methods
713
Uniform Grade Distribution The first observation to make about the volumetric spectra produced by both methods for the uniform grade distribution, shown in Figure 3, is regarding the prediction of liberated particles. The allocation method predicts some liberated particles of both phases whereas the transformation kernel method correctly predicts no liberated particles. The second observation concerns the general shape of the cumulative liberation distribution curves. The transformation kernel volumetric distribution has features that reflect the step-like nature of the actual volumetric distribution. It has steeper and flatter sections whereas the volumetric distribution resulting form the allocation method is smoother, running straight through the steps of the actual volumetric curve. The allocation volumetric curve reflects more the behaviour of the adjusted actual volumetric curve. The features of the transformation kernel volumetric distribution are however, not always in the correct grade domain position. For example, the steep increase corresponding to the grade fraction at about g = 0.42 in the actual volumetric curve appears slightly to the left, or at a lower grade, in the transformation kernel volumetric curve.
Low Grade Distribution The curves for the low grade distribution, shown in Figure 4, behave similarly to the uniform grade distribution. The allocation method correction predicted some liberated iron oxide where none was present and produced a smooth curve. The transformation kernel correction correctly predicted no liberated particles and again has more features. The feature "shift" mentioned in the uniform grade distribution case for the g = 0.42 fraction also appears here. This results in the large gap between the actual and the transformation volumetric curves at about g = 0.35.
High Grade Distribution The same observations previously made can be seen in the high grade distribution results shown in Figure 5. Liberated iron oxide particles are predicted by the allocation method and not by the transformation kernel method. The feature shift cited above for the g = 0.42 grade fraction again appears here, causing a large deviation of the transformation kernel curve from the actual volumetric curve in the grade range g = 0.4 to 0.7. The allocation volumetric curve also produces a large deviation from the actual volumetric curve in this range. The fact that both corrections incorrectly predicted here indicates that the actual volumetric distribution, derived from narrow grade densities, was not correctly represented by the image analysis performed. No conclusion will be drawn based on these results.
DISCUSSION The two stereological correction methods considered are quite different in their approach; one being exclusively based on the theoretical geometrical equations and the other relying on direct texture measurement and characterisation along with good regularisation and optimisation. Implementation of the transformation kernel method requires an appropriate kernel for texture under investigation. This is no trivial matter. The experimental liberation data required to construct the transformation kernel for a specific mineralogical system is difficult and even sometimes not possible to obtain, thus making the method less practical. Determination of a transformation kernel of a real mineralogical binary texture requires liberation measurements of narrow grade fractions of the particular texture. This has so far only be done for two real mineralogical systems. Many kernels have been calculated from simulated particles such as those generated by capped sphere models and PARGEN. These kernels only simulated very simple textures and have been found to be inappropriate for real mineralogical textures. An example of this is the comparison of stereological methods performed by Lin et al. [9] where a capped sphere kernel was inappropriately applied to a mineralogical texture and thus, naturally produced poor results. A response to this work was published by King and Schneider [10].
714
R.G. Fandrichet al.
A procedure for determining the most appropriate kernel for a particular texture has been proposed by King and Schneider [6] and is simply based on L-curve characteristics associated with the discrete ill-posed problem solution developed by Hansen [5]. As more and more kernels are calculated the probability of finding an appropriate one for the application under consideration, via the L-curve technique, will increase. This makes the transformation kernel method more accessible for variety of a mineralogical systems. Both methods have only been successfully applied to two-phase or binary systems at this stage. Efforts are currently underway to expand both methods to multi-phase systems. The first attempts to produce and run a tertiary kernel have already been made. Some mention of the difficulties associated with this work and the corresponding errors associated with this image analysis should be made. The most difficult task to perform in this study was to ensure that the particle sections being presented for image analysis were as representative of the desired volumetric grade distribution as possible. Meticulous sampling was required firstly, to manufacture the desired grade distributions from the narrow grade fractions and secondly, to present a representative polished section for QEM*SEM liberation analysis. If the particle sections presented for image analysis are not representative of the true particle population then no stereological method has any chance of correcting accurately. The image analysis results obtained for the high grade distribution indicate that such a sampling problem occurred for this case. How accurately the transformation kernel reflects the texture of the mineralogical system has a large bearing on the transformation kernel method's performance. The feature shift mentioned earlier is attributable to the closeness of the parametric kernel model fit to the measured data. The correlation between the parametric model fit and the measured data for the unliberated sections is at its poorest in the volumetric grade range about the grade g = 0.4. Thus, certain errors in this region of the spectrum are to be expected. This illustrates the usefulness of having an unsymmetrical kernel modelling scheme available to cope with textures that produce unsymmetrical or irregular liberation data.
CONCLUSIONS A comparison of the two stereological correction procedures when applied to this iron oxide ore reveals that the transformation kernel method generally provides a more accurate and detailed solution. The transformation kernel correction generally reflected the shape of the actual volumetric cumulative liberation spectra measured. The allocation method did not capture certain features of the actual volumetric distributions and occasionally incorrectly predicted liberated particles when none existed. The corrections resulting from the allocation method reflected the behaviour of the adjusted actual volumetric curves. Reliable binary stereological correction methods for real mineralogical ore systems are available to provide acceptably accurate volumetric liberation data.
ACKNOWLEDGMENTS
The authors would like to thank Dr P.B.S Stewart for the use of his fractionated iron oxide ore particles for this study.
REFERENCES 1.
2. 3.
Gay, S.L., Liberation modelling using particle sections, PhD Dissertation, University of Queensland, 1994. Gay, S.L., Stereological equations for phases within particles, Journal of Microscopy, 179, Pt. 3, 1995, pp. 297-305. King. R.P., The prediction of mineral liberation from mineralogical texture. Proc. XIV Int. Mineral Processing Congress, Toronto, 1982, pp. 11-17.
Two stereologicalcorrectionmethods ,
5.
,
7. 8. 9.
10.
715
Schneider, C.L., Measurement and calculation of liberation in continuous milling circuits, PhD Dissertation, University of Utah, 1995. Hansen, P.C., Regularisation Tools, a MATLAB package for analysis and solution of discrete illposed problems, Version 2.0 for MATLAB 4.0, Report UNIC-92-03, 1993. Available in postscript from Netlib ([email protected]) from the library NUMERALGO. King, R.P. & Schneider, C.L., Stereological correction of linear grade distributions for mineral liberation, accepted for publication in Powder Technology, 1997. Stewart, P.B.S., Composite particles, Masters Thesis, Imperial College, London 1975. Sutherland, D., Gottlieb, P., Jackson, R., Wilke, G. & Stewart, P.S.B., Measurement in section of particles of known composition, Minerals Engineering, 1,(4), 1988, pp. 317-326. Lin, D., Gomez, C.O. & Finch, J.A., Comparison of stereological correction procedures for liberation measurements, Transactions, Mineral processing and Extractive metallurgy, 104, 1995, C155-61. King, R.P. & Schneider C.L., Comparison of stereological correction procedures for liberation measurements, Transactions, Mineral processing and Extractive metallurgy, 106, 1997, C51-4.
C o r r e s p o n d e n c e on papers published in Minerals Engineering is invited, preferably by email to [email protected], or by Fax to +44-(0)1326-318352
Pergamon
TWO STEREOLOGICAL CORRECTION METHODS: ALLOCATION METHOD AND KERNEL TRANSFORMATION METHOD
R.G. FANDRICH ~, C.L. SCHNEIDER t and S.L. GAY* § Centre for Mining Technology and Equipment/University of Queensland, PO Box 883, Kenmore, Qld. 4069, Australia. E-mail: [email protected] i" Utah Comminution Center, University of Utah, 135 S 1460 E Rm 306, Salt Lake City, LIT 84112-0114, USA :~ Julius Kruttschnitt Mineral Research Centre, Isles Road, Indooroopilly, Qld. 4068, Australia
(Received 7 April 1998; accepted 21 May 1998)
ABSTRACT
Image analysis of polished particle sections is a common technique for measuring the liberation spectrum of mineral particle populations. The liberation distributions measured from particle sections differ from the actual distribution by what is known as the stereological error. This error always overestimates the degree of liberation by amounts dependent on factors such as the type of measurement (eg, linear or areal), particle composition and texture. For example, a particle may be composite, but the section may reveal only one phase making it apparently liberated Numerous methods have been proposed for correcting this error. Some of them are not appropriate for real mineralogical textures and are therefore practically useless. This paper focuses on two existing stereological correction methods for binary mineralogical systems: the Gay allocation method and the transformation kernel method. The transformation kernel correction procedure was found to provide somewhat more accurate and detailed results for the iron oxide ore investigated © 1998 Elsevier Science Ltd. All rights reserved. Keywords Liberation analysis; iron ores
INTRODUCTION Over the last decade an increasingly accepted technique for the gathering mineral liberation data has become that of image analysis of polished particle sections. It has now almost become routine for mineral processing plants to measure liberation spectra at crucial points within their circuits using image analysis. In the past, one aspect of particle section liberation measurement that has been inappropriately dealt with, and sometime ignored, is the stereological error associated with linear and areal measurements. The stereological error arising from areal measurements is illustrated in Figure 1 where a sectioned binary particle is shown. If the darker phase is the phase of interest then the particle has an approximate volumetric grade of 0.4. The three areal sections through the particle each show different grades. Sections 1 and 3 are liberated while area section 2 has an areal grade approximately that of the particle volumetric grade. This example shows how large the error can sometimes be. The Figure also demonstrates how the error always occurs in one
707
708
R.G. Fandfich et
al.
large the error can sometimes be. The Figure also demonstrates how the error always occurs in one direction. That is, the measured areal (or linear) liberation distribution will always be more liberated than the actual volumetric distribution. This comes from the fact that liberated sections can be produced from composite particles but composite sections cannot be produced from liberated particles.
/
/I
S
/ / / /
Fig. 1 Sectioned binary particle illustrating stereological error. The purpose of this work is to determine the stereological correcting characteristics of the transformation kernel and the allocation methods when applied to a particular binary iron oxide ore system. The work was performed by the principal author as part of his PhD candidature.
ALLOCATION METHOD The allocation method was developed by Gay [1,2] and is a theoretical general correction based on geometric probability equations. The approach uses the fact that when particles of a particular grade are sectioned there are nine geometric probability equations which relate the resulting sections with the original grade of the particles. The relationships involve parameters that are not conventionally used in image analysis. For example, for a section, one key parameter is the average distance between pixels of the same mineral. Consequently, unlike most other stereological corrections, Gay's method requires digitised images of the particle sections. The allocation method can be applied to linear intercept data.
Two stereological correction methods
709
Using the known geometric relationships, the allocation method uses a sorting algorithm to allocate sections into grade bins. The approach is to allocate the sections into the bins so that for each bin, each of the nine geometric probability equations are satisfied within statistical accuracy. As there are ten composite bins, there are then 90 geometric probability equations being used.
TRANSFORMATION K E R N E L M E T H O D The transformation kernel stereological correction method for particle sections involves finding solutions
~gvlD) to the transformation equation 1
F(&ID)=f
(1)
o
where F(gilD) is the cumulative distribution of linear (i=l) or areal (i=a) grades and K(gilgvD) is the cumulative distribution of linear or areal grades generated by particles that have volumetric grade gv and size D i e . the transformation kernel. Equation 1 was first proposed as a method of stereological correction in 1982 [3] and was derived from well-established principles of geometric probability. Determination of the transformation kernel K(gilgv,D) can only be performed experimentally through linear or areal distribution measurements of particles of known composition. A parametric model for the transformation kernel based on these measured distributions was proposed by Schneider [4]. Schneider's model parameterises the transformation kernel using incomplete beta functions by implementing separate functional forms to characterise the liberated and the unliberated portions of the linear or areal distributions as functions of volumetric grade. Two functional forms are possible: one for a symmetrical kernel and another for a more parameter intensive unsymmetrical kernel. Solving the transformation equation once the kernel is known presents various problems due to the ill conditioned nature of K(gilgv,D). As functional forms for arbitrary grade distributions do not exist, the discrete form of the transformation equation is used: N
(2)
e,'=E n=l
where
Kmn is the N x N discrete version of the kernel. As Kran is not exact and Fi m is not error free, the
use of direct inversion of
Kmn to solve for fly is not feasible. Equation 1 is a Fredholm integral equation
of the first kind and hence techniques for solving this form of ill-posed problem are required. Several regularisation and optimisation procedures are available to solve the transformation equation. One method is to use Tikhonov regularisation and constrained Rosenbrock Hillclimb optimisation [4]. The method employed in this work was an entropy regularisation method based on Hansen's [5] general solution to discrete ill-posed problems. The objective function for this method is 12
(3)
IIIO0(KL - F,)II÷>, n=l
12
and is subject to the constraints
1 and n=!
for n = 1, 2 .... 12 and where X is the regularisation
710
R.G. Fandrich et
al.
parameter. A detailed description of the implementation of this method to mineral liberation problems is given by King and Schneider [6].
EXPERIMENTAL The initial experimental work that made this particular study possible was conducted by Stewart [7] in 1975 and involved the fractionation of iron oxide ore particles into 21 grade fractions through heavy liquid and fluidised bed separation techniques. The iron oxide phase of the ore was an approximate 50:50 hematite/magnetite mix whilst the gangue consisted of a silicate phase of predominantly quartz. This ore has been the subject of previous liberation studies [8] due to its recognised binary nature. Five of the narrow grade fractions were initially used to construct the transformation kernel. These fractions underwent QEM*SEM areal section liberation measurements. The cumulative areal liberation distributions obtained describe the number of particle sections within the standard 12 grade classes; the 2 liberated grade classes and the 10 composite classes each of grade range 0.1. The five narrow grade fractions that were used to construct the transformation kernel by the method briefly described above were also measured and corrected with both methods. With the transformation kernel now constructed other known distributions were required to test the two methods. Thus, three different grade distribution were manufactured from six further composite narrow grade fractions: 1.
2. 3.
A "uniform" grade distribution consisting of equal volumetric proportions of particles from each narrow grade fraction. A "high" grade distribution consisting of a greater volumetric proportion of particles from higher grade fractions. A "low" grade distribution consisting of a greater volumetric proportion of particles from lower grade fractions.
It is recognised that these distributions are not entirely typical of what might be found in a real industrial circuit and that is not the aim of this work. The aim of the study is to test the stereological correction procedures on "known" distributions and the use of the above manufactured distributions was the best way the achieve that. Great care was taken during the manufacturing of these grade distributions to ensure that the mounted samples for QEM*SEM liberation analysis would provide the most representative polished section possible. Both correction methods were then applied to the measured areal distributions and the results compared. The areal distributions from the five narrow grade fractions used in the construction of the kernel were also corrected using both methods.
RESULTS For the purposes of comparing the measured areal, corrected volumetric and actual volumetric distributions, the cumulative form of the liberation distributions are used. These distributions are shown in Figures 2 to 5. Two forms of the actual volumetric distribution are represented in each Figure. Firstly there is the actual volumetric curve which is generated from the upper and lower densities of the each of the six component grade fractions and the assumption that the grade distribution within those limits is linear. The adjusted actual volumetric curve is a representation of the actual volumetric curve in the 0.1 grade interval domain. The reason for this second "adjusted" representation is that it provides and indication of the "target" distribution for the correction methods with the same curve resolution that the corrections provide. The resulting volumetric liberation spectra from the five narrow grade fractions are considered first. Figure 2 shows the volumetric distributions resulting from both correction methods as well as the measured areal, actual volumetric (thin solid line) and adjusted volumetric (thick solid line) distributions for the 0.47 grade fraction. The effect of stereological error can clearly be seen by comparing the actual volumetric with the measured areal distributions. The measured areal spectra has the same average grade or first moment but
Two stemologicalcorrectionmethods
711
has a much greater variance than the actual distribution. This is the overestimation of liberation mentioned earlier. Comparison of the two correction methods reveals a "sharper" volumetric approximation produced by the transformation kernel method. The allocation method generally produced volumetric distributions of greater variance. That is, particles were allocated to a wider range of grade bins on either side of the actual grade. This result was typical for the other narrow grade fractions except for the 0.11 grade fraction. Here both correction methods produced practically the same sharp volumetric distribution. The better performance of the transformation kernel method is not surprising as the transformation kernel was constructed using this very data and the correction is, in a sense, just reproducing the data. The quality of this reproduction is a function of the quality of the kernel parameterisation fit to the measured liberation data. If the models used to characterise the kernel fit poorly to the measured data then the reproduction of the narrow grade fractions will also be correspondingly poor.
100
i:
--
//
90-
"
i ,/,, =o
80
" ......
°°oOm
..I
A.,,,,~,
/~r,, ' I
70
~
l/
6o
I ,'&/ j
o Actual volumetric
),~ E
11
30
I f
20
10 00.0
"-=- Measured areal
.~ s t ~"
" =" Allocation volumetric
o• ' "= ¢
¢"
¢.•
0.1
0.2
0.3
/ , / ~
"-~" Transformation volumetric
/1._ 0.4
I
I
I
I
I
0.5
0.6
0.7
0.8
0.9
1.0
grade
Fig.2 Actual, measured and corrected liberation distributions for the 47% grade fraction.
'°° /
9° T 80 t
"~
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Fig.3 Actual, measured and corrected liberation distributions for the uniform grade distribution.
712
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A more valid test of each method is that provided by the three manufactured grade distributions. The uniform grade distribution contained equal amounts of each fraction by volume whilst the high and low grade distributions were volumetrically weighted to their respective ends of the grade spectrum. Figures 3, 4 and 5 show the distributions for the uniform, low and high grade distributions, respectively. Again, here the actual volumetric distributions are constructed as mentioned earlier. Each linear upward slope represents the grade range of one of the six narrow grade fractions used to manufacture the distribution. The adjusted actual volumetric curve shows the cumulative distribution in the 0.1 interval grade domain. As before the stereological error can be seen in each Figure by the overestimation of liberation by the areal distribution.
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Two stereologicalcorrection methods
713
Uniform Grade Distribution The first observation to make about the volumetric spectra produced by both methods for the uniform grade distribution, shown in Figure 3, is regarding the prediction of liberated particles. The allocation method predicts some liberated particles of both phases whereas the transformation kernel method correctly predicts no liberated particles. The second observation concerns the general shape of the cumulative liberation distribution curves. The transformation kernel volumetric distribution has features that reflect the step-like nature of the actual volumetric distribution. It has steeper and flatter sections whereas the volumetric distribution resulting form the allocation method is smoother, running straight through the steps of the actual volumetric curve. The allocation volumetric curve reflects more the behaviour of the adjusted actual volumetric curve. The features of the transformation kernel volumetric distribution are however, not always in the correct grade domain position. For example, the steep increase corresponding to the grade fraction at about g = 0.42 in the actual volumetric curve appears slightly to the left, or at a lower grade, in the transformation kernel volumetric curve.
Low Grade Distribution The curves for the low grade distribution, shown in Figure 4, behave similarly to the uniform grade distribution. The allocation method correction predicted some liberated iron oxide where none was present and produced a smooth curve. The transformation kernel correction correctly predicted no liberated particles and again has more features. The feature "shift" mentioned in the uniform grade distribution case for the g = 0.42 fraction also appears here. This results in the large gap between the actual and the transformation volumetric curves at about g = 0.35.
High Grade Distribution The same observations previously made can be seen in the high grade distribution results shown in Figure 5. Liberated iron oxide particles are predicted by the allocation method and not by the transformation kernel method. The feature shift cited above for the g = 0.42 grade fraction again appears here, causing a large deviation of the transformation kernel curve from the actual volumetric curve in the grade range g = 0.4 to 0.7. The allocation volumetric curve also produces a large deviation from the actual volumetric curve in this range. The fact that both corrections incorrectly predicted here indicates that the actual volumetric distribution, derived from narrow grade densities, was not correctly represented by the image analysis performed. No conclusion will be drawn based on these results.
DISCUSSION The two stereological correction methods considered are quite different in their approach; one being exclusively based on the theoretical geometrical equations and the other relying on direct texture measurement and characterisation along with good regularisation and optimisation. Implementation of the transformation kernel method requires an appropriate kernel for texture under investigation. This is no trivial matter. The experimental liberation data required to construct the transformation kernel for a specific mineralogical system is difficult and even sometimes not possible to obtain, thus making the method less practical. Determination of a transformation kernel of a real mineralogical binary texture requires liberation measurements of narrow grade fractions of the particular texture. This has so far only be done for two real mineralogical systems. Many kernels have been calculated from simulated particles such as those generated by capped sphere models and PARGEN. These kernels only simulated very simple textures and have been found to be inappropriate for real mineralogical textures. An example of this is the comparison of stereological methods performed by Lin et al. [9] where a capped sphere kernel was inappropriately applied to a mineralogical texture and thus, naturally produced poor results. A response to this work was published by King and Schneider [10].
714
R.G. Fandrichet al.
A procedure for determining the most appropriate kernel for a particular texture has been proposed by King and Schneider [6] and is simply based on L-curve characteristics associated with the discrete ill-posed problem solution developed by Hansen [5]. As more and more kernels are calculated the probability of finding an appropriate one for the application under consideration, via the L-curve technique, will increase. This makes the transformation kernel method more accessible for variety of a mineralogical systems. Both methods have only been successfully applied to two-phase or binary systems at this stage. Efforts are currently underway to expand both methods to multi-phase systems. The first attempts to produce and run a tertiary kernel have already been made. Some mention of the difficulties associated with this work and the corresponding errors associated with this image analysis should be made. The most difficult task to perform in this study was to ensure that the particle sections being presented for image analysis were as representative of the desired volumetric grade distribution as possible. Meticulous sampling was required firstly, to manufacture the desired grade distributions from the narrow grade fractions and secondly, to present a representative polished section for QEM*SEM liberation analysis. If the particle sections presented for image analysis are not representative of the true particle population then no stereological method has any chance of correcting accurately. The image analysis results obtained for the high grade distribution indicate that such a sampling problem occurred for this case. How accurately the transformation kernel reflects the texture of the mineralogical system has a large bearing on the transformation kernel method's performance. The feature shift mentioned earlier is attributable to the closeness of the parametric kernel model fit to the measured data. The correlation between the parametric model fit and the measured data for the unliberated sections is at its poorest in the volumetric grade range about the grade g = 0.4. Thus, certain errors in this region of the spectrum are to be expected. This illustrates the usefulness of having an unsymmetrical kernel modelling scheme available to cope with textures that produce unsymmetrical or irregular liberation data.
CONCLUSIONS A comparison of the two stereological correction procedures when applied to this iron oxide ore reveals that the transformation kernel method generally provides a more accurate and detailed solution. The transformation kernel correction generally reflected the shape of the actual volumetric cumulative liberation spectra measured. The allocation method did not capture certain features of the actual volumetric distributions and occasionally incorrectly predicted liberated particles when none existed. The corrections resulting from the allocation method reflected the behaviour of the adjusted actual volumetric curves. Reliable binary stereological correction methods for real mineralogical ore systems are available to provide acceptably accurate volumetric liberation data.
ACKNOWLEDGMENTS
The authors would like to thank Dr P.B.S Stewart for the use of his fractionated iron oxide ore particles for this study.
REFERENCES 1.
2. 3.
Gay, S.L., Liberation modelling using particle sections, PhD Dissertation, University of Queensland, 1994. Gay, S.L., Stereological equations for phases within particles, Journal of Microscopy, 179, Pt. 3, 1995, pp. 297-305. King. R.P., The prediction of mineral liberation from mineralogical texture. Proc. XIV Int. Mineral Processing Congress, Toronto, 1982, pp. 11-17.
Two stereologicalcorrectionmethods ,
5.
,
7. 8. 9.
10.
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Schneider, C.L., Measurement and calculation of liberation in continuous milling circuits, PhD Dissertation, University of Utah, 1995. Hansen, P.C., Regularisation Tools, a MATLAB package for analysis and solution of discrete illposed problems, Version 2.0 for MATLAB 4.0, Report UNIC-92-03, 1993. Available in postscript from Netlib ([email protected]) from the library NUMERALGO. King, R.P. & Schneider, C.L., Stereological correction of linear grade distributions for mineral liberation, accepted for publication in Powder Technology, 1997. Stewart, P.B.S., Composite particles, Masters Thesis, Imperial College, London 1975. Sutherland, D., Gottlieb, P., Jackson, R., Wilke, G. & Stewart, P.S.B., Measurement in section of particles of known composition, Minerals Engineering, 1,(4), 1988, pp. 317-326. Lin, D., Gomez, C.O. & Finch, J.A., Comparison of stereological correction procedures for liberation measurements, Transactions, Mineral processing and Extractive metallurgy, 104, 1995, C155-61. King, R.P. & Schneider C.L., Comparison of stereological correction procedures for liberation measurements, Transactions, Mineral processing and Extractive metallurgy, 106, 1997, C51-4.
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