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Textural terms are difficult to quantify, as they are often textural analysis is to cut thin sections in a rock sampleand to study the outlines of the grains in this plane underan optical microscope. X-ray tomography is a promising

Journal of Volcanology and Geothermal Rquite subjective, particularly for igneous rocks. Quan-1. Introduction

The scientific description of rocks, or petrography,involves observations and measurements at variousscales and various levels of details. A generaldescription, including the mode and the texture, isusually enough for the geologist to understand in broadterms the nature and origin of a rock. However, thecomplexity of the processes involved in the crystalliza-tion and textural evolution of rocks requires more andmore quantification in order to test the available models.

tification therefore requires objective accurate measure-ments and appropriate statistical analysis. In thiscontribution a complete method of optical texturedetermination, from the thin section to the finaldiagrams, is presented.

Textural analysis generally consists in measuring thesize, the shape, the orientation and the position of thegrains in a rock. Dihedral angle measurements can alsoprovide valuable information (Holness et al., 2005), ascan c-axis orientations (e.g., Heilbronner and Pauli,1993). The simplest and cheapest way to performmanagement and visualisation of data. Automatic detection of grain edges is performed by watershed segmentation on digitalpictures of the thin section. After vectorization of the segmented picture, the resulting map of grain boundaries is edited andcorrected manually. GIS software allows the user to associate each grain with attributes such as phase name, position, size, aspectratio, orientation and convexity. The grains can then be classified according to one or several attributes. The spatial distribution ofthe different classes of grains can be visualised with colour-coded maps or quantified by cluster analysis. A weakly foliatedquartzite and an igneous cumulate are taken as examples to show how invisible patterns are made evident with these kinds of maps.The interpretation of these spatial distributions remains problematic, but future development of 2D or 3D numerical models oftextural evolution makes this technique promising. 2006 Elsevier B.V. All rights reserved.

Keywords: watershed segmentation; GIS software; textural analysis; crystal size distribution; pattern; foliationTextural analysis of thin sections of rocks can be performedAbstract

with a Geographic Information System (GIS) program to improveThe use of watershed segmtextural analysi

Joseph

Department of Earth Sciences, University of Cambridg

Received 1 July 2004;Available onlin Tel.: +44 1223 333433; fax: +44 1223 333450.E-mail address: jbar02@esc.cam.ac.uk.

0377-0273/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.jvolgeores.2005.09.017ation and GIS software forf thin sections

raud

wning Street, Cambridge, CB2 3EQ, United Kingdom

ed 30 September 2005February 2006

esearch 154 (2006) 1733www.elsevier.com/locate/jvolgeoresand impressive technique for direct 3D analysis(Philpotts et al., 1999), but the boundary between two

nd Getouching grains of the same mineral cannot be resolved.X-ray tomography is then ideal for imaging clasts inlavas or porphyroblasts in metamorphic rocks (Carlsonet al., 1999). More generally, when the imagingtechnique is based on a chemical property of theminerals (like atomic number for backscattered electronimaging, BSE) and when a major part of the rock iscomposed of the same mineral (like plagioclase in agabbro) then only a continuous touching frameworkappears on the images. Thus, the size, shape andorientation of the constituting crystals cannot bemeasured.

Consequently, grain-edge outlining is still themandatory first step of most textural analysis methods.This is generally done either by hand directly on thin-section photographs or scans (Higgins, 2000; Bergerand Roselle, 2001; Boorman et al., 2004), or on screenwith a drawing package (Jerram et al., 2003). Manyauthors have proposed automatic methods of grainboundary detection (Launeau et al., 1994; Goodchildand Fueten, 1998; Bartozzi et al., 2000; Heilbronner,2000; McEwan et al., 2000; Thompson et al., 2001; vanden Berg et al., 2002; Tarquini and Armienti, 2002;Perring et al., 2004). None of these methods has beenproved to be completely satisfactory, especially forcomplex polymineralic plutonic rocks. Indeed, most ofthese methods are based on the colour contrast of acrystal with its neighbours. The grain outline is thenextracted by grey-scale thresholding or by gradientfiltering. This may work well with BSE images ofvolcanic rocks because crystals are typically embeddedin a vitreous matrix without touching each other.However, for holocrystalline rocks, a set of pictureshas to be taken at many different orientations of crossedpolars, at the expense of simplicity (Goodchild andFueten, 1998; Heilbronner, 2000; Perring et al., 2004).Another possibility is to use orientation contrast (OC)images obtained by scanning electron microscopy(SEM, Bartozzi et al., 2000). This method has theadvantage of exploiting directly the changes in crystallattice orientation, allowing grains and sub-grains to beidentified objectively. However, it is not obvious thatthis method can be easily used for polymineralic rocks.The method described here makes use of simple opticalmicroscopy and of a set of pictures taken at threecrossed-polars orientations only. A powerful segmenta-tion algorithm normally used for medical imaging hasbeen employed and the results of this so-calledwatershed segmentation are particularly good withholocrystalline rocks.

This method of textural analysis also makes extensive

18 J. Barraud / Journal of Volcanology ause of Geographic Information System (GIS) software.GIS software has revolutionised geography and geolog-ical mapping because it offers a unique way to visualise,manipulate and analyse spatial data. Petrographic dataalso have spatial components and the spatial distributionof minerals and of their attributes should providesignificant information about the processes that occurredduring a rock's history. However, most techniques oftextural quantification lose track of the spatial informa-tion by averaging a parameter (e.g., size, orientation) orby calculating a unique quantity that describes thespatial distribution pattern (SDP, Jerram et al., 1996,2003). While these quantities are useful to comparedifferent samples, a better way to visualise a spatialdistribution is to display directly the quantity on a map.A GIS program allows the user to visually identifydifferent populations of crystals by overlaying severalparameters. Quantification can be subsequently per-formed on each individual population. This articleprovides two examples of applications for metamorphicand igneous rocks.

2. Methodology

The method comprises seven stages, from acquisitionof the images to statistical analysis and display of theresults (Fig. 1).

2.1. Image acquisition

In order to achieve the best results, it is important tostart with a good-quality digital image of the thinsection. The resolution has to be high enough to allowthe smallest grains to be imaged correctly. The image istypically taken with a digital camera attached to amicroscope. Film scanners have been used by severalauthors and provide images that may be suitable forCSD studies (Boorman et al., 2004). However, theirresolution may not be high enough for shapedescription. The images shown in this article havebeen taken with a Nikon Coolpix 5000 that delivers25601920 pixels images. The images are taken withthe 2 or 4 lens of the microscope, depending on theaverage grain size. A mosaic of 33 images coversabout half a standard thin section when the 2 lens isused. The final image contains around 45 millionpixels and the resolution is about 400 pixels permillimetre. This high-resolution image will be usedlater in the workflow for the manual correction of themap of grain boundaries. However, the file has a verylarge size, resulting in a dramatic increase incalculation time in the next step of segmentation. The

othermal Research 154 (2006) 1733image is therefore downsampled with Photoshop to a

nd GeJ. Barraud / Journal of Volcanology asmaller size (about 30 million pixels). This value is acompromise that has been found to preserve the qualityof the segmentation and to offer a reasonable executiontime (see below).

2.2. Noise reduction and watershed segmentation

One of the novelties of the method is that automaticgrain edge detection is performed by watershed

Fig. 1. Workflow of the method.segmentation. A state-of-the-art image-processing codecalled ITK (for Image Toolkit, http://www.itk.org) isused for this purpose. ITK offers almost all the filtersand segmentation algorithms currently available formedical imaging. ITK is an open-source softwarepackage implemented in C+, and is cross-platform(Unix, Windows and MacOS X). The toolkit is acollection of libraries and the user writes applicationscombining the several filters that will be applied to theinput image in a sequence. The method presented makesuse of an application that is furnished with ITK andsimply called WatershedSegmentation1.

The strategy of watershed segmentation is to treat animage as a height function, i.e., the grey level of a pixelcorresponds to the altitude of a point on a surface (Fig.2a). The image of a thin section is treated like a DigitalElevation Model where grains with uniform colourdefine regions at different altitudes. A gradient filter isthen applied so that grain edges can be seen as ridgesand grains as valleys. If this virtual topography is nowflooded, the water will collect in the basins. The sizeof these basins will grow with increasing level of waterwhen they merge with adjacent basins. The remainingridges of the flooded landscape are the grain boundaries.

The implementation of the watershed segmentationalgorithm in ITK involves an initial smoothing of theraw image by the so-called anisotropic diffusion filter(Perona and Malik, 1990). This filter removes a largeamount of noise while preserving the sharpness of grainedges. Anisotropic diffusion includes a variable con-ductance term that, in turn, depends on the differentialstructure of the image. Thus, the smoothing effect willbe less pronounced at edges, as measured by highgradient magnitude.

All the filters are applied on the three channels ofRGB colour pictures simultaneously by convertingscalar-valued RGB images into a vector-based repre-sentation in the 3D RGB space. After anisotropicdiffusion, the height function is calculated with agradient operator on the pixel magnitude. Finally, thewatershed segmentation is performed. The output is acolour image in which each grain has a different colour(Fig. 2).

Four user-defined parameters control the entireprocess: conductance and iteration for the smoothingfilter, threshold and flood level for the segmentationfilter. A suitable combination of parameters has to befound by trial and error to avoid over- or under-segmentation (Fig. 2cd). The conductance controls thesensitivity of the diffusion to the presence of high-contrast edges. The number of iteration controls the time

19othermal Research 154 (2006) 1733during which diffusion occurs, accentuating the blurring

nd Ge20 J. Barraud / Journal of Volcanology aeffect. Typical values for conductance and iteration arein the ranges 28 and 26, respectively.

Threshold and level are set as a fraction (valuebetween 0 and 1) of the maximum depth of the inputimage. The threshold corresponds to the backgroundnoise, which is removed (flattened) before segmenta-tion. As most of the noise was removed already bydiffusion, this parameter is fixed for this study at a lowvalue of 0.01. The level value allows the user tominimize over-segmentation by establishing a minimumwatershed depth. Adjacent regions are merged if theircombined depth falls below the minimum. A high valuetherefore results in a lower segmentation. The segment-ed images in this article were produced with a level inthe range 0.160.30.

During the process of finding the correct parameters,the treated images are visually compared with theoriginal. Nevertheless, due to complexity of naturaltextures, it is necessary to make manual corrections atthe next step. A slightly oversegmented result is

Fig. 2. Watershed segmentation of a thin section. (a) Principles of the methograin. The grain boundaries are defined by the steep gradients on both sides offlooded by water. The basins are filled up to a level called the watershed deptha troctolite (crossed polars and lambda plate, bottom length 6.5 mm). (c) Over(e) Correct segmentation (level=0.3).othermal Research 154 (2006) 1733therefore preferable because merging two objects toremove a false boundary is easier than drawing amissing boundary manually.

The smoothing filter is by far the slowest step of theprocess. It takes 18 min to filter a 5000-by-3698 imagewith 2 iterations on a PC equipped with a Pentium 4 2.6GHz processor. It takes 40 min for the same image with6 iterations.

2.3. Vectorization and editing

The segmented image is a raster (pixel-based) imagethat has to be converted to a vector format beforeediting. The built-in raster-to-vector conversion utilityof ArcGIS 8.2 was used in this study. Each colour patchof the segmented image is converted into a polygon. Thevector format has many advantages: (1) grain bound-aries consist of nodes and lines, allowing easy editing,(2) overlaps and gaps do not exist because twoneighbouring grains share a single boundary, (3) there

d: the curve on the left shows an imaginary intensity profile through athe intensity high. The gradient image defines a topography that can be, which controls the final amount of segmentation. (b) Original image ofsegmented result (level=0.16). (d) Undersegmented result (level=0.4).

nd Geis no loss of quality at high magnification, (4) each grainis identified as a single object with various attributes likename, area, perimeter, length, etc.

Curved boundaries are approximated by straightlines, so the distance between two nodes on the outlineshould be as small as possible. The accuracy of thevectorization is controlled by a parameter called clustertolerance in ArcGIS, which for polygons corresponds tothe minimum distance between two nodes. Theoretical-ly, the best result would be achieved if this distance wereequal to the size of the original pixels. However, thiswould increase excessively both file size and displaytime. The correct value depends on the complexity ofthe texture, the grain size, and the capacity of thecomputer. For this study, acceptable accuracy wasachieved with a minimum distance of 10 m (3 to 5pixels).

The polygons can then be edited in order to correctthe unavoidable artefacts of automatic segmentation.There are two types of artefacts: missing boundaries andfalse boundaries (oversegmentation). Missing bound-aries occur when the colour gradient between two grainsis not high enough to be detected by the previoussegmenting process. Oversegmentation is the reverse:the presence of fractures, cleavages and plagioclasetwinning causes irrelevant subgrains to be created.Olivine grains are typically impossible to automaticallycontour correctly because of multiple fractures andalteration products. They tend to form a mosaic of 5 to20 pieces. However, it is relatively easy to reform thecorrect outline by merging the subgrains.

The editing step is the longest of the method. Animage of the thin section underlies the map of polygons,so that missing or false boundaries can be identified andcorrected. The microscope is necessary for complexsituations, e.g., very small grains or altered areas.Ultimately, a precision of 515 m on the position of aboundary can be achieved. The smallest grain diameterthat can be measured is about 10 m. However, theprecision depends on the starting picture and can beincreased if a higher magnification is used during thefirst stage.

Finally, this method of grain boundary detectionmay be called semi-automatic in the sense thatsegmentation artefacts are manually corrected. Thissort of interactive method was tested by Heilbronner(2000) and proved to be both accurate and fastcompared with the fully manual procedure. I did nottry to make a similar comparison for the present methodbecause it seems evident in both cases that it is still atime-consuming and painstaking task. Nevertheless, for

J. Barraud / Journal of Volcanology asimple textures (one or two phases, few fractures), thistechnique is undeniably fast and accurate (see examplesbelow).

2.4. Analysis

2.4.1. Measure of lengthThere are many ways to measure the length and the

width of an object (Fig. 3a). Higgins (2000) reviewedvarious methods and favoured the longest distancebetween two points on the contour. However, thisobjectively defined length is generally not the one thepetrographer would choose subjectively as it is notalways in the perceived main orientation of the object.As the value of the aspect ratio is commonly associatedwith the orientation of the shape, the length used in thisstudy is equal to the projection of the object in thedirection parallel to its major orientation or direction ofelongation (Fig. 3a). This is sometimes called the boxlength. The width will be the projection in a perpendic-ular direction. The major orientation is determined bycalculating the moments of inertia of the shape aroundthe centroid (e.g., Mulchrone and Choudhury, 2004).The nodes around the outline of the object are used forthe calculation. The principal directions are given by theeigenvectors of the matrix of the moments. Oneadvantage of this method is to measure both lengthand orientation at the same time.

Another commonly-used measure of length is thelong axis of the best ellipse fitting the object. However,different methods exist to fit an ellipse to a shape (e.g.,Mulchrone and Choudhury, 2004; Launeau, 2004).Moments of inertia are very popular but again, differentresults will be obtained depending on the way themoments are calculated (boundary or region-basedmethods). Moreover, the best-fitting ellipse can becalculated with a minimisation algorithm, or with theeigenvalues of the tensor. Ultimately, the size of theellipse can be adjusted, so as to give it the same area asthat of the object.

In order to illustrate this variety of methods, sixdifferent measures of length were made on a set ofconvex quadrangles (Fig. 3b). Three programs wereused: the MATLAB toolbox PolyLX of Lexa (2001), thepopular image-processing software ImageJ (Rasband,2005), and the recent program SPO2003 by Launeau(2004). PolyLX allows the user to compute the momentsof inertia of arcGIS polygons (boundary-based method)and to determine the box length in the calculateddirection as described previously. For this purpose, theroutine aorten was modified and now gives both boxlength and length of the major axis of the fitted ellipse

21othermal Research 154 (2006) 1733(aorten Major=22e1, where e1 is the biggest

nd Ge22 J. Barraud / Journal of Volcanology aeigenvalue). The longest length was also computed withPolyLX.

ImageJ gives the major axis of the best-fitting ellipsecalculated with a pixel-based method. The differencebetween this and aorten is also that the best-fittingellipse is computed by equalising the second momentsof the shape and the ellipse. Moreover, the area of thisellipse is normalised to the area of the object. SPO2003provides an additional measurement of the major axis ofthe best-fitting ellipse (pixel-based method, ellipse axesgiven by eigenvalues, not area-normalised) and of thebox length.

The results are summarised on a plot showing thelength calculated with these methods vs. the box lengthcalculated with PolyLX (Fig. 3c). It can be seen that thelongest length is sometimes exactly equal to the boxlength. This occurs when the shape is axisymmetric. TheSPO2003 box length is very close to the PolyLX boxlength, but systematically slightly smaller. The differ-ence may be due to the difference in image format,vector-based for PolyLX and pixel-based for SPO2003.

Fig. 3. Length and shape descriptors. (a) Definitions used in this study. (b)obtained by the method of the moments (aorten routine of the PolyLX toolbeach object. (c) Compilation of different methods of length measurement appThe range of convexity values obtained for various regular and irregular shaothermal Research 154 (2006) 1733The major axes of ellipses given by ImageJ andSPO2003 are very similar, the small difference beingdue to the normalisation. These two major axes arealmost always smaller than the box length, while aortenMajor is consistently longer than the box length.However, it can be shown that aorten Major isproportional to the other two major axes, as they are allbased on moments of inertia.

Finally, the choice of a type of measurement isdifficult and may depend on the application. The boxlength was chosen in this study because it is easy toimplement and similar to the actual dimensions of sub-rectangular crystals like plagioclase. In addition, thecomparison with other methods shows that it givesintermediate values, far from extremes.

2.4.2. Shape descriptorsNumerous shape descriptors are available. Shape

description has many applications in different domainsand this has resulted in a rather confused literature. Adescriptor may have different definitions depending on

Quadrangles with a range of orientation and shape. A fitting-ellipseox) is shown, together with the major semi-axis drawn at the center oflied on the previous quadrangles. The straight line has a slope of 1. (d)pes.

nd Gethe author. For example, compactness is often defined asthe ratio of squared perimeter to the area of an object. Inthis case, it has a minimum at 4 for a circle andapproaches infinity for complex irregular objects.However, Iivarinen et al. (1997) define compactnessas the ratio of the perimeter of a circle with equal areato that of the original object to the original perimeter. Inthis case, the parameter ranges between 0 and 1.

There are also descriptors with different names andalmost the same definition (e.g., ellipticity, elongationand Grain Shape Index all refer to the same ratiobetween area and length). A limited set of independentshape descriptors is preferable because they allow anefficient classification when used in combinations(Iivarinen et al., 1997). In this study only two shapedescriptors are used: aspect ratio and convexity.

The most commonly used shape descriptor ingeology is the so-called aspect ratio (AR) betweenlength and width of an object. The aspect ratio has, forexample, various applications in structural geology as itcan be related to finite strain under certain assumptions(Mulchrone, 2003). Moreover, knowing the averageaxial ratio allows a better conversion of 2D measure-ments into true 3D Crystal Size Distributions (CSD,Higgins, 2000). The average axial ratio can also becompared with the bulk alignment factor (definedbelow) to estimate the extent of recrystallisation ofigneous cumulates under compaction (Meurer andBoudreau, 1998; Boorman et al., 2004).

The second shape descriptor used in this study is theconvexity. The convexity is a measure of the irregularityof the shape. There are several ways to define theconvexity and a definition based on the boundary ratherthan the area has been chosen because it is moresensitive to small defects (Zunic and Rosin, 2004). Theconvexity C is then:

C PconvP

;

where Pconv is the perimeter of the convex hull of theshape and P is the perimeter of the shape. The convexhull is the smallest convex set that includes the shape(Fig. 3a). The convexity varies between 0 and 1 andequals 1 for a convex shape. Fig. 3d shows typicalconvexity values computed for simple geometricshapes. A euhedral crystal is generally convex outwards.Reactive dissolution, pressure solution, embayment, andbending are processes that may alter the shape andreduce the convexity of a crystal (see discussion).

Area, perimeter, and convexity were computedwithin ArcGIS. The length, width and orientation were

J. Barraud / Journal of Volcanology acalculated with PolyLX (Lexa, 2001).2.4.3. Strength of foliation and alignment factorThe alignment factor is calculated from the bulk

orientation tensor. In order to take into account theelongation of each crystal, the technique has beenmodified from Meurer and Boudreau (1998) andWheeler et al. (2003) as follows: the orientation n(measured clockwise from an axis parallel to the sideedge of the image) and length Ln of crystal n (n from 1to N) are used first to calculate an individual tensorT (n):

T n Ln cos2an cosan sinan

cosan sinan sin2an

:

The bulk orientation tensor is then:

Dij 1N XNn1

T nij :

The eigenvalues e1 and e2 (with e1Ne2) of this matrixare then used to calculate a normalized alignment factorAF:

AF 100 e1e2e1

:

This number describes the degree of coherence ofgrain orientations and ranges from 0 (no significantalignment) to 100 (all the grains are aligned). Moreover,the eigenvectors of matrix D give the principaldirections of the population.

In order to estimate the influence of the crystalelongation on the AF parameter, two idealisedpopulations were studied (Fig. 4): a single populationcomprising two identical rectangles separated by anangle 2, and a mixed population comprising twodifferent rectangles (AR=3 and AR=1.1). A com-parison is also made with the alignment factor AFmbused by Meurer and Boudreau (1998), and Boormanet al. (2004). This parameter does not take intoaccount the shape of the crystals and is based on thenormalised orientation tensor (Harvey and Laxton,1980):

M 1N cos

2an cosan sinancosan sinan sin2an

:

If 1 is the highest eigenvalue of this matrix, then

AFmb 2 k150:The comparison shows that AFmb does not depend on

the axial ratio of the crystals (Fig. 4c), which may be a

23othermal Research 154 (2006) 1733problem if passive deformation like compaction is to be

nd Ge24 J. Barraud / Journal of Volcanology adescribed (Launeau, 2004). To avoid this problem,Boorman et al. (2004) used only the 40 longest grains.One may alternatively use the proposed AF that givesmore importance to the direction of elongated crystalsand therefore provides high values for all in the case ofa mixed population. However, one should keep in mindthat this parameter is still not unique and does not allowone to return to the original population. A rose diagrammust be provided to correctly estimate the range andspread of crystal orientations.

Another independent estimate of the Shape Pre-ferred Orientation (SPO) is performed with theintercept method (Launeau and Robin, 1996): inter-cepts are counted along a set of lines, which scan theimage in every direction (0 to 180), at regularintervals, each time the line crosses a grain boundary.This method actually computes the direction of thepreferred grain-boundary orientations. The calculationwas performed with the program SPO2003 of Launeau(2004). Interestingly, the great sensitivity of thismethod may provide important secondary directions

Fig. 4. Variations of the alignment factors (AF and AFmb) for a singlepopulation of two rectangles of identical axial ratio (a, AR=3) and fora mixed population of two rectangles with different ARs (b). The graph(c) shows that the two populations give the same result with the use ofAFmb, whereas a distinction can be made with the proposed AF.that can subsequently be used to classify the grainsinto different populations. Spatial statistics then givesignificant information on the texture, as explainedbelow.

2.4.4. Spatial statisticsOne of the main advantages of using GIS software

over conventional drawing packages is the possibility ofassigning attributes to each polygon. The attributes ofthe polygons are recorded in a table associated with thevector file. Commonly used attributes are: phase, areaand perimeter, orientation, as well as various shapedescriptors, as described above. The results aredisplayed directly on the digitised thin section in theform of a map. Various kinds of maps can be produced,showing the spatial distribution of a parameter with acolour code. More sophisticated interpolation maps canalso be computed: the value of a parameter is assigned tothe centre of each grain and the values between thesepoints are interpolated according to the Inverse DistanceWeighting algorithm (IDW). IDW assumes that eachmeasured point has a local influence that diminisheswith distance. It weights the points closer to theprediction location greater than those farther away. Inthis study, weights are simply proportional to the inversedistance. This creates a surface that smoothes theprevious patchwork maps and makes it easier to detect,for example, clusters of small grains or zones of highlyconvex grains.

Maps are a convenient way to visualize the data.However, one may prefer to summarize the informationinto a single number to compare different samples. Theaverage is commonly used but, unfortunately, it may notreflect the actual heterogeneity of a population.Classification into bins allows the identification ofdifferent populations within the sample. One can nextcombine several parameters in order to bring to lightsome significant correlation in the data. For example, apopulation may be defined as the grains whose aspectratio is higher than 1.5 and length less than 1 mm.However, a bias may be introduced if the classificationis done arbitrarily. In order to choose class limitsobjectively, natural breaks (also called smart quantiles)are used in this study: the features are divided intoclasses whose boundaries are set where there arerelatively big jumps in the data values. This algorithmis a one-dimensional example of the K-means clusteringmethod (e.g., Weisstein, 2005). Given a desired numberof classes, K, the natural breaks method partitions thedata into K subsets that minimize the sum of thespreads within each subset. The spread is measured as

othermal Research 154 (2006) 1733the sum of squares of the residuals, which are the

nd GeJ. Barraud / Journal of Volcanology adifferences between a value and the average value of thesubset. An iterative procedure is used to find theminimum. The resulting classes do not have the samesize, nor the same number of values. This is acompromise method between Equal Interval (equal-sized classes, different numbers of values in each class)and Quantile (unequal-sized bins, same number ofvalues).

The different populations are then displayed individ-ually, showing their spatial distribution. The resultingpattern can then be quantified by calculating the ratio R(Jerram et al., 1996), which is the ratio between themean nearest neighbour distance (NND) of objects inthe sample and the mean NND expected for a randomdistribution of points with the same population densityas the sample. The mean NND for the sample, rA, isdefined as

rA P

rN

Fig. 5. Quartzite. (a) RGB stack of grey-scale images taken at three different oboundary detection. Truncated grains at the edges of the image were remove25othermal Research 154 (2006) 1733where r is the NND, and N is the number of individualsmeasured. The mean NND for a random distribution ofpoints, rE, is defined as

rE 12 qp

where is the density of the observed distribution.Finally, R is defined as

R rArE

:

The value of R is combined with the porosity (in thiscase, the proportion of grains that are not in the studiedpopulation) to decide if a distribution of points israndom, clustered, or ordered. The next section willshow that SDP analysis and statistical analysis can becoupled in a relevant manner.

rientations of crossed polars (0, 30, and 60). (b) Result of automaticd. Polygons are coloured according to their length.

3. Examples

3.1. Quartzite

A fine-grained quartzite from the Isle of Islay(Scotland) was selected as a first test of the method.The purpose is to show the possibilities of GIS softwarein petrography, not to address fully a geological issue.The sample comes from the Harker collection at theUniversity of Cambridge (sample no. 41685) andconsists of almost pure quartz with b1% of muscovite(Fig. 5a). The rock presents a weak foliation defined bythe Shape Preferred Orientation (SPO) of quartz and bythe muscovite flakes.

Automatic segmentation was applied on a stack ofthree grey-scale pictures taken at different orientations

of crossed polars (the red, green and blue channelscorrespond to an angle of 0, 30 and 60, respectively,Fig. 5a). The simplicity of the thin section makes itpossible to analyse directly the entire segmented image,without any further manual correction (Fig. 5b).However, the presence of areas with very small grainsresulted in a slightly oversegmented image because ofthe occurrence of muscovite and some alteration in theseareas. In addition, some crystals may present subgrains,which have been considered as distinct grains.

The length ranges between 0.002 and 0.92 mm (Fig.5b) and the width between 0.002 and 0.6 mm. A CSDplot was calculated with the width measurements withthe program CSDCorrections 1.36 of Michael Higgins(Higgins, 2000) on 4500 grains representing the upperhalf of the image. The result shows a gently concave-

plot0.5, toystals

26 J. Barraud / Journal of Volcanology and Geothermal Research 154 (2006) 1733Fig. 6. Crystal size distribution and classification by orientation. (a) CSDat 0.5, half way between a cube and an ellipsoid. A foliation intensity ofweak fabric. (b) Rose diagram of grain orientation. Only the largest cr

directions calculated by the intercept method with the program SPO2003 afteMaps showing the spatial distribution of three classes of grains defined fromfor the grains of the upper half of the picture. The shape number was setgether with an aspect ratio 1 :1 :1.2, was chosen to take into account the(lengthN0.0795 mm) are plotted. (c) Rose diagram of grain boundary

r grain identification. The thick line at 120 is the main direction. (df)the secondary directions at 100, 125 and 167.

different domains of homogeneous SPO can be clearlydefined (Passchier and Trouw, 1996).

This example illustrates how the combination in aPetrographic Information System of several quantitiesmeasured with different methods can be used todiscriminate distinct populations and reveal a distribu-tion pattern. This rock is indeed weakly foliated along ageneral direction around 130 but this description masksthe presence of two scattered populations of crystals thatare orientated differently. An EBSD analysis wouldprovide complementary data on the crystallographicorientations of the crystals and on the grain boundarymisorientations, which can be used to distinguishsubgrains from grains. However, the complete investi-gation and interpretation of this sample is beyond thescope of this article.

27nd Geothermal Research 154 (2006) 1733upwards curve for the intermediate and large grain sizes(Fig. 5c). The plot shows also a minor excess of smallgrains (widthb0.044 mm), which is related to thealready-mentioned oversegmentation in that part of thespectrum. Consequently, it is difficult to tell more aboutthis population of small crystals, so the next investiga-tion must be restricted to the larger grains. The grainswere therefore classified with the natural-breaks methodinto 3 classes (Fig. 5b): small (lengthb0.0795 mm),medium (0.0795b lengthb0.1834) and big (lengthN0.1834 mm). The larger grains consist of medium andbig grains.

As the foliation is defined by the larger grains, thedata can still be used to address the following question:what is the SPO of this weakly foliated rock? The AFvalue of 43 indicates a moderate alignment around thecalculated principal direction at 124. This preferentialorientation can also be seen on a rose diagram (Fig. 6a).However, only a very broad peak can be defined around110150 (the standard deviation is 40). A moresignificant estimate of the SPO is performed with theintercept method. The result shows a principal orienta-tion around 120 and five secondary directions: 20,50, 100, 125 and 167 (Fig. 6b). The regularorganization of these directions every 30 or so suggestsa control by the quartz crystallography (hexagonalsymmetry) during deformation. This feature, togetherwith the presence of subgrains and very small grainsmay be the sign that rotation recrystallisation occurred.This process is defined by Jessell et al. (2004) as theformation of sub-grains by the relative rotation of part ofa crystal lattice with respect to its neighbour, as a resultof the progressive addition of dislocations of the samesign to a sub-grain wall.

The SPO can be further investigated by couplingtextural analysis and spatial statistics. The threesecondary directions of the intercept method at 100,125 and 167 suggest that three equal-sized classescentred on these values can be defined. Three maps wereproduced by displaying only the grains whose orienta-tion falls in a given class (Fig. 6ce). These maps showthat each population is fairly scattered, with a fewgroups of touching crystals. In order to quantify thisassumption, further statistical treatment can be appliedto each population individually: the R value equals 1.0for the 3 groups, showing that their spatial distribution isclustered (regardless of the porosity, Jerram et al., 1996).The 116145 population shows this clustering partic-ularly well because it is denser (Fig. 6d). Bands ofsimilarly orientated crystals can be observed, suggestinglocalized accommodation of the deformation. However,

J. Barraud / Journal of Volcanology athis should not be confused with an SC fabric where3.2. Troctolite

For this example, two thin sections of a single sampleof troctolite from the Tertiary layered intrusion of theIsle of Rum, Scotland, were analysed. The sample(RMC022) was collected by Mark Hallworth (Cam-bridge University) from Unit 9, on the northern slopes ofHallival. The reader may refer, for example, to Emeleuset al. (1996) for an introduction to Rum and to Bdard etal. (1988), which is more focussed on Unit 9.

A cursory examination of the hand sample shows thatthe rock has a well-defined magmatic foliation but noobvious lineation. The two thin sections, namedRMC022Vand RMC022H, were cut normal and parallelto the foliation, respectively. The rock is medium grainedand consists of olivine (14.5 vol.%, RMC022H content),plagioclase (82 vol.%), clinopyroxene (2 vol.%), spinelFig. 7. Troctolite from Unit 9 of the Rum layered intrusion (sampleRMC022H).

Fig. 8. Troctolite. (ab) RGB stacks of grey-scale images taken at three different orientations of crossed polars (0, 30, and 60). (a) RMC022V,normal to foliation. The dashed contour shows the analysed area. (b) RMC022H, parallel to foliation. (cd) Maps of grain boundaries after automaticsegmentation and manual correction. (c) RMC022V (675 grains); (d) RMC022H (1662 grains). All the figures are at the same scale.

Fig. 9. Rose diagrams of orientations of plagioclase grains (RMC022H). The long axis is the main direction calculated with the method of Harvey andLaxton (1980). The inner ellipse has an aspect ratio equal to the ratio of eigenvalues (Rf). The data are smoothed by a 15-wide Gaussian curve. (a) Allgrains; (b) big grains only (lengthN0.697 mm); (c) small grains (lengthb0.696 mm). The rose diagrams were obtained with SPO2003 (Launeau,2004).

28 J. Barraud / Journal of Volcanology and Geothermal Research 154 (2006) 1733

pools where olivine crystals are present, or wheresmall crystals are concentrated in clusters of 3 to 20crystals. This strong clustering is reflected in the low

29nd Geothermal Research 154 (2006) 1733(1 vol.%) and some hydrated phases (0.5 vol.%). Eachphase has its own specific size and shape type. Olivineforms big rounded crystals, occasionally showingpointed apophyses (Fig. 7). Plagioclase shows a widerange of grain size and shape, the bigger laths definingthe foliation. Clinopyroxene is interstitial betweenplagioclase or forms thin rinds separating olivine andplagioclase (Fig. 7). Clinopyroxene is believed topseudomorph the final porosity of the solidifying rock(Holness et al., 2005). Finally, spinel forms smallrounded crystals scattered in the thin section.

After segmentation and vectorization of the thinsections, the grain outlines were carefully corrected(Fig. 8). Plagioclase crystals were analysed in term oforientation, size and shape.

The intensities of plagioclase SPOs in the twosections are compared to confirm that no lineation canbe defined. The alignment factor AF is 82 in RMC022V(because of the strong foliation) while it is only 7 inRMC022H (no principal direction can be statisticallydefined). Thus, the rock is well-foliated but not lineated.The absence of lineation is confirmed by the rose ofdirections (Fig. 9a), which shows that the low value forthe in-plane section is due to the presence of crystals ofall orientations. The AF value is also lowered by thepresence of two slightly predominant families of crystals(at 70 and 160) that are roughly perpendicular to eachother.

Another important consequence of this rather strongfoliation is that, in the foliation plane, the biggestcrystals are cut preferentially in their longest dimension.In other words, there is less probability in RMC022H ofintersecting big crystals through a corner and conse-quently there is more chance that a small intersectionbelongs to a small crystal than to a big one. Thisproperty allows one to analyse the crystal sizedistribution without stereological corrections, andtherefore to explore the relationships between positionand crystal size.

The crystal size was classified into 2 categories(small and big) with the natural-breaks algorithm(Fig. 10ab). The modal proportion of small and bigcrystals is 32% and 50%, respectively. Fig. 10bshows that the big crystals form a continuous networkof touching crystals. The alignment factor AF for thispopulation is 10, in accordance with the low valuealready calculated for the entire population. More-over, the rose diagram of directions for big crystalsonly (Fig. 9b) shows a clear organisation into twoperpendicular sets. Hence, the crystal framework iscomposed of a mesh of sub-perpendicular crystal

J. Barraud / Journal of Volcanology achains that covers all the space. The chains defineFig. 10. Maps of RMC022H plagioclases showing: (a) the populationof crystals smaller than 0.696 mm; (b) the population of crystals longer

than 0.697 mm; (c) the population of crystals with an axial ratio lowerthan 1.5.

R-value of the population of small crystals (R=1.0for a content of 30 vol.%).

Clustering is also visible for the population ofcrystals with low aspect ratio, as shown in Fig. 10cwhere only crystals with AR1.5 have been displayed.The R value in that case is equal to 1.0 for a content of27.5 vol.%. A comparison between Fig. 10a and cshows that clusters of small crystals mostly correspondto clusters of low-AR crystals.

The plagioclase crystals can also be classifiedaccording to their convexity. A convexity map and aninterpolation map of the convexity values are shown inFig. 11. The interpolation map shows a geometricpattern that suggests that the variations of convexity arenot random: zones with high-convexity crystals (greenblue zones in Fig. 11b) can be clearly distinguished from

zones of low convexity (redorange zones). Thesezones of high convexity generally correspond to theclusters of small crystals shown previously (highlightedwith boxes no. 1 in Fig. 11b). Low-convexity regionsare related either to large complex-shaped singlecrystals, or to the presence of clinopyroxene (highlight-ed with boxes no. 2 in Fig. 11b).

Length measurements and shape descriptors can becombined together in order to further refine theclassification of plagioclase crystals. For example, Fig.12a shows the convexity distribution of the populationof bulky crystals (AR1.5) presented previously. Thiscombination is possible because convexity and axialratio are two independent parameters, as shown by theplot of convexity vs. axial ratio for all the plagioclasecrystals of RMC022H (Fig. 12b). The correlation

interp

30 J. Barraud / Journal of Volcanology and Geothermal Research 154 (2006) 1733Fig. 11. (a) Spatial distribution of convexity in RMC022H. (b) Map of

than plagioclase are filled in white. Boxes numbered 1 show zones of highassociated with clinopyroxene wedges.olated convexity. The grain boundaries are superimposed; phases other

convexity; boxes numbered 2 show regions of low-convexity crystals

Fig. 12. (a) Map of plagioclase crystals with low axial ratio (AR1.5,see Fig. 10c) showing two populations classified according to theconvexity (natural breaks). (b) Plot of convexity vs. axial ratio for allthe plagioclase grains of RMC022H. (c) Plot of convexity vs. lengthfor the same crystals. The contours show density surfaces.

J. Barraud / Journal of Volcanology and Gecoefficient R2 is equal to 0.0032. Conversely, convexityand length seems much more dependent on each other(Fig. 12c), as the correlation coefficient is in that case0.31. Though this relationship is therefore not useful forclassification, it may reflect the occurrence of a texturalprocess that makes small crystals convex and bigcrystals complex-shaped.

4. Discussion and conclusions

Since the pioneering work of Kretz (1969), patterns,arrangements or frameworks of crystals have beenrecently subject to increasing attention (Philpotts et al.,1999; Jerram et al., 2003) and this is partly due toimproved techniques of visualisation. The potentialcontribution of GIS software in this domain can besummarised as follows: (1) the representation of crystalsby polygons extends the possibilities of analysis anddisplay; (2) the improved management of datasets makesit easier to classify the crystals by linking togetherlocation information, phase name and textural para-meters. Thus, one can define a Petrographic Informa-tion System as an integrated method for drawing,measuring, analysing, and displaying data related withthin sections of rocks.

The main advantage of GIS software over otherprograms of image processing and/or textural analysis(e.g., ImageJ, SPO2003, CSDcorrections) is that the usercan still work on the image after the measurements. Mostprograms provide only a table of measurements and thelink with the original geometric database (the grains) islost. Although it is important for textural quantificationto obtain single quantities like main orientation oramount of clustering, a GIS program allows one to returnto the grain topology and identify exactly whichpopulation of grains can be related to a given result.

To conclude, a recapitulation of the methods andresults presented in this article follows:

(1) Watershed segmentation was applied successfullyto the problem of automatic grain boundarydetection.

(2) The different methods of lengthmeasurement werecompared, and the box length was selected for itsconvenience and relevance to plagioclase analysis.

(3) A shape descriptor called convexity was intro-duced. Its independence with another shapedescriptor (axial ratio) was demonstrated with anexample, showing that they can be used incombination for shape classification.

(4) A new alignment factor which takes into account

31othermal Research 154 (2006) 1733the elongation of crystals was proposed.

32 J. Barraud / Journal of Volcanology and Geothermal Research 154 (2006) 1733(5) A statistical algorithm called natural breaks,commonly used for geographic applications, waspresented and its use illustrated with examples.

(6) Examples of maps obtained by crossing severaltextural parameters were shown.

(7) The recognition of different populations ofcrystals and the explanation of the spatialdistributions obtained are two different things.The interpretation of the results requires powerfulcomputer models (like ELLE; Jessell et al., 2001)that can simulate the complex coupling ofprocesses acting on a rock: crystal growth,chemical reactions, solid-state diffusion, grainboundary diffusion, grain boundary migration,and solidliquid interface adjustment. Hence, thesame method of textural analysis can be applied torocks and models in order to compare them in arelevant manner.

(8) As stated in the introduction, textural quantifica-tion requires objective measurements. Objectivitycan be improved by increasing the precision ofmeasurements, which has been done by perfectingthe methodology (use of high resolution pictures,painless correction of errors by the use of a vectorformat). Classification is also performed objec-tively by the appropriate algorithm. However, thepossibility of objective comparison is the key toobjective measurement. This article presents newtextural parameters and their range of variationsfor basic shapes and two natural examples.Comparison with numerical models would be astep further. Thus, a better understanding oftextural processes is essential for a real objectivityof textural quantification.

Acknowledgements

Scottish Natural Heritage granted permission toconduct fieldwork on the Isle of Rum. I gratefullyacknowledge Steve Laurie who gave me access to theHarker collection. Alan Boyle and Alan Boudreau arethanked for insightful reviews that improved andclarified the paper. This work was financially supportedby the European Community's Human PotentialProgramme under contract HPRN-CT-2002-000211(EUROMELT).

References

Bartozzi, M., Boyle, A.P., Prior, D.J., 2000. Automated grain boundarydetection and classification in orientation contrast images. Journal

of Structural Geology 22, 15691579.Bdard, J.H., Sparks, R.S.J., Renner, R., Cheadle, M.J., Hallworth,M.A., 1988. Peridotite sills and metasomatic gabbros in theeastern layered series of the Rhum Complex. Journal of theGeological Society (London) 145, 207224.

Berger, A., Roselle, G., 2001. Crystallization processes in migmatites.American Mineralogist 86 (3), 215224.

Boorman, S., Boudreau, A., Kruger, F.J., 2004. The lower zone-criticalzone transition of the Bushveld Complex: a quantitative texturalstudy. Journal of Petrology 45 (6), 12091235.

Carlson, W.D., Denison, C., Ketcham, R.A., 1999. High-resolution X-ray computed tomography as a tool for visualization andquantitative analysis of igneous textures in three dimensions.Electronic Geosciences 4 (3).

Emeleus, C.H., Cheadle, M.J., Hunter, R.H., Upton, B.G.J., Wads-worth, W.J., 1996. The Rum layered suite. In: Cawthorn, R.G.(Ed.), Layered Intrusions. Elsevier Science, pp. 403439.

Goodchild, J.S., Fueten, F., 1998. Edge detection in petrographicimages using the rotating polarizer stage. Computers & Geos-ciences 24 (8), 745751.

Harvey, P.K., Laxton, R.R., 1980. The estimation of finite strainfrom the orientation distribution of passively deformed linearmarkers: eigenvalue relationships. Tectonophysics 70 (34),285307.

Heilbronner, R., 2000. Automatic grain boundary detection and grainsize analysis using polarization micrographs or orientation images.Journal of Structural Geology 22, 969981.

Heilbronner, R.P., Pauli, C., 1993. Integrated spatial and orientationanalysis of quartz c-axes by computer aided microscopy. Journal ofStructural Geology 15, 369382.

Higgins, M.D., 2000. Measurement of crystal size distributions.American Mineralogist 85 (9), 11051116.

Holness, M.B, Cheadle, M.J., McKenzie, D., 2005. On the use ofchanges in dihedral angle to decode late-stage textural evolution incumulates. Journal of Petrology 46, 15651583.

Iivarinen, J., Peura, M., Srel, J., Visa, A., 1997. Comparison ofcombined shape descriptors for irregular objects. In: Clark, A.F.(Ed.), 8th British Machine Vision Conference, BMVC'97, Essex,Great Britain, pp. 430439.

Jessell, M., Bons, P., Evans, L., Barr, T., Stwe, K., 2001. Elle: thenumerical simulation of metamorphic and deformation micro-structures. Computers & Geosciences 27, 1730.

Jessell, M., Bons, P., Urai, J., 2004. The Online Microstructure Course.http://www.microstructure.uni-tuebingen.de/moodle/.

Jerram, D.A., Cheadle, M.J., Hunter, R.H., Elliott, M.T., 1996. Thespatial distribution of grains and crystals in rocks. Contributions toMineralogy and Petrology 125 (1), 6074.

Jerram, D.A., Cheadle, M.J., Philpotts, A.R., 2003. Quantifying thebuilding blocks of igneous rocks: are clustered crystal frameworksthe foundation? Journal of Petrology 44 (11), 20332051.

Kretz, R., 1969. On the spatial distribution of crystals in rocks. Lithos2, 3966.

Launeau, P., Cruden, A.R., Bouchez, J.L., 1994. Mineral recognitionin digital images of rocksa new approach using multichannelclassification. Canadian Mineralogist 32, 919933.

Launeau, P., Robin, P.-Y.F., 1996. Fabric analysis using the interceptmethod. Tectonophysics 267, 91119.

Launeau, P., 2004. Evidence of magmatic flow by 2-D image analysisof 3-D shape preferred orientation distributions. Bulletin de laSocit Gologique de France 175 (4), 331350.

Lexa, O., 2001. PolyLXthe MATLAB toolbox for quantitativeanalysis of microstructures. Deformation, Rheology and Tectonics

Conference, Noordwijkerhout, Netherlands.

McEwan, I.K., Sheen, T.M., Cunningham, G.J., Allen, A.R., 2000.Estimating the size composition of sediment surfaces throughimage analysis. Proceedings of the Institution of Civil Engineers.Water, Maritime and Energy 142 (4), 189195.

Meurer, W.P., Boudreau, A.E., 1998. Compaction of igneouscumulates: Part II. Compaction and the development of igneousfoliations. Journal of Geology 106, 293304.

Mulchrone, K.F., 2003. Application of Delaunay triangulation to thenearest neighbour method of strain analysis. Journal of StructuralGeology 25 (5), 689702.

Mulchrone, K.F., Choudhury, K.R., 2004. Fitting an ellipse to anarbitrary shape: implications for strain analysis. Journal ofStructural Geology 26 (1), 143153.

Passchier, C.W., Trouw, R.A.J., 1996. Microtectonics. SpringerVerlag. 325 pp.

Perona, P., Malik, J., 1990. Scale-space and edge detection usinganisotropic diffusion. IEEE Transactions on Pattern Analysis andMachine Intelligence 12, 629639.

Perring, C.S., Barnes, S.J., Verrall, M., Hill, R.E.T., 2004. Usingautomated digital image analysis to provide quantitative petro-graphic data on olivine-phyric basalts. Computers & Geosciences30 (2), 183195.

Philpotts, A.R., Brustman, C.M., Shi, J.Y., Carlson, W.D., Denison, C.,1999. Plagioclase-chain networks in slowly cooled basalticmagma. American Mineralogist 84 (1112), 18191829.

Rasband, W.S., 2005. ImageJ, U. S. National Institutes of Health,Bethesda, Maryland, USA, http://rsb.info.nih.gov/ij/.

Tarquini, S., Armienti, P., 2002. Quick determination of crystal sizedistributions of rocks by means of a color scanner. Image Analysisand Stereology 22, 2734.

Thompson, S., Fueten, F., Bockus, D., 2001. Mineral identificationusing artificial neural networks and the rotating polarizer stage.Computers & Geosciences 27 (9), 10811089.

van den Berg, E.H., Meesters, A.G.C.A., Kenter, J.A.M., Schlager, W.,2002. Automated separation of touching grains in digital images ofthin sections. Computers & Geosciences 28 (2), 179190.

Weisstein, E.W., 2005. K-means clustering algorithm. From Math-Worlda Wolfram Web Resource. http://mathworld.wolfram.com/K-MeansClusteringAlgorithm.html.

Wheeler, J., Jiang, Z., Prior, D.J., Tullis, J., Drury, M.R., Trimby, P.W.,2003. From geometry to dynamics of microstructure: usingboundary lengths to quantify boundary misorientations andanisotropy. Tectonophysics 376 (12), 1935.

Zunic, J., Rosin, P.L., 2004. A new convexity measure for polygons.IEEE Transactions on Pattern Analysis and Machine Intelligence26 (7), 923934.

33J. Barraud / Journal of Volcanology and Geothermal Research 154 (2006) 1733

The use of watershed segmentation and GIS software for textural analysis of thin sectionsIntroductionMethodologyImage acquisitionNoise reduction and watershed segmentationVectorization and editingAnalysisMeasure of lengthShape descriptorsStrength of foliation and alignment factorSpatial statistics

ExamplesQuartziteTroctolite

Discussion and conclusionsAcknowledgementsReferences