How to index grains and access shape properties.

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Accessing individual grains |

Indexing by a Condition |

Indexing by orientation or position |

Grain-size Analysis |

Spatial Dependencies |

Grains have several intrinsic properties, which can be used for statistical, shape as well as for spatial analysis

Let us first import some EBSD data and reconstruct grains

plotx2east mtexdata forsterite ebsd = ebsd(inpolygon(ebsd,[5 2 10 5]*10^3)); grains = calcGrains(ebsd) plot(ebsd) hold on plot(grains.boundary,'linewidth',2) hold off

grains = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 0 1139 4052 notIndexed 1 242 14093 Forsterite mmm 2 177 1397 Enstatite mmm 3 104 759 Diopside 12/m1 X||a*, Y||b*, Z||c boundary segments: 10420 triple points: 903 Properties: GOS, meanRotation

The variable `grains` is essentially a large vector of grains. Thus when applying a function like area to this variable we obtain a vector of the same lenght with numbers representing the area of each grain

grain_area = grains.area;

As a first rather simple application we could colorize the grains according to their area, i.e., according to the numbers
stored in `grain_area`

plot(grains,grain_area)

As a second application, we can ask for the largest grain within our data set. The maximum value and its position within a
vector are found by the Matlab command `max`.

[max_area,max_id] = max(grain_area)

max_area = 3.8945e+06 max_id = 1617

The number `max_id` is the position of the grain with a maximum area within the variable `grains`. We can access this specific grain by direct indexing

grains(max_id)

ans = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 1 1548 Forsterite mmm boundary segments: 424 triple points: 70 Id Phase Pixels GOS phi1 Phi phi2 1617 1 1548 0.0129383 167 81 251

and so we can plot it

hold on plot(grains(max_id).boundary,'linecolor','red','linewidth',1.5) hold off

Note that this way of addressing individual grains can be generalized to many grains. E.g. assume we are interested in the
largest 5 grains. Then we can sort the vector `grain_area` and take the indices of the 5 largest grains.

[sorted_area,sorted_id] = sort(grain_area,'descend'); large_grain_id = sorted_id(1:5); hold on plot(grains(large_grain_id).boundary,'linecolor','green','linewidth',1.5) hold off

By the same syntax as above we can also single out grains that satisfy a certain condition. I.e., to access are grains that are at least half as large as the largest grain we can do

condition = grain_area > max_area/2; hold on plot(grains(condition).boundary,'linecolor','red','linewidth',1.5) hold off

This is a very powerful way of accessing grains as the condition can be build up using any grain property. As an example let us consider the phase. The phase of the first five grains we get by

grains(1:5).phase

ans = 0 0 0 0 2

Now we can access or grains of the first phase Forsterite by the condition

condition = grains.phase == 1; plot(grains(condition))

To make the above more directly you can use the mineral name for indexing

`grains('forsterite')`

ans = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 242 14093 Forsterite mmm boundary segments: 7670 triple points: 795 Properties: GOS, meanRotation

Logical indexing allows also for more complex queries, e.g. selecting all grains perimeter larger than 6000 and at least 600 measurements within

condition = grains.perimeter>6000 & grains.grainSize >= 600; selected_grains = grains(condition) plot(selected_grains)

selected_grains = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 5 5784 Forsterite mmm boundary segments: 1893 triple points: 244 Id Phase Pixels GOS phi1 Phi phi2 798 1 1447 0.013232 166 127 259 876 1 1075 0.00805971 153 68 237 999 1 1044 0.00753116 89 99 224 1617 1 1548 0.0129383 167 81 251 1620 1 670 0.0870135 143 78 274

One can also select a grain by its spatial coordinates using the syntax `grains(x,y)`

x = 12000; y = 4000; plot(grains); hold on plot(grains(x,y).boundary,'linewidth',2,'linecolor','r') plot(x,y,'marker','s','markerfacecolor','k',... 'markersize',10,'markeredgecolor','w') hold off

In order to select all grains with a certain orientation one can do

% restrict first to Forsterite phase grains_fo = grains('fo') % the reference orientation ori = orientation('Euler',350*degree,50*degree,100*degree,grains('fo').CS) % select all grain with misorientation angle to ori less then 20 degree grains_selected = grains_fo(angle(grains_fo.meanOrientation,ori)<20*degree) plot(grains_selected)

grains_fo = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 242 14093 Forsterite mmm boundary segments: 7670 triple points: 795 Properties: GOS, meanRotation ori = orientation size: 1 x 1 crystal symmetry : Forsterite (mmm) specimen symmetry: 1 Bunge Euler angles in degree phi1 Phi phi2 Inv. 350 50 100 0 grains_selected = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 12 2979 Forsterite mmm boundary segments: 1603 triple points: 173 Id Phase Pixels GOS phi1 Phi phi2 359 1 63 0.0129176 177 140 250 389 1 1 0 167 128 260 622 1 352 0.00896432 153 122 252 636 1 305 0.0156453 164 115 268 680 1 331 0.0198713 152 143 247 798 1 1447 0.013232 166 127 259 1297 1 289 0.0315393 166 147 260 1549 1 1 0 1 47 279 1609 1 48 0.0169152 158 139 259 1629 1 10 0.0193004 172 125 261 1660 1 129 0.00814837 1 47 279 1662 1 3 0.00475795 1 47 279

Let's go back to the grain size and analyze its distribution. To this end, we consider the complete data set.

mtexdata forsterite % consider only indexed data for grain segmentation ebsd = ebsd('indexed'); % perform grain segmentation [grains,ebsd.grainId] = calcGrains(ebsd)

grains = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 1080 152345 Forsterite mmm 2 515 26058 Enstatite mmm 3 1496 9064 Diopside 12/m1 X||a*, Y||b*, Z||c boundary segments: 43912 triple points: 3417 Properties: GOS, meanRotation ebsd = EBSD Phase Orientations Mineral Color Symmetry Crystal reference frame 1 152345 (81%) Forsterite light blue mmm 2 26058 (14%) Enstatite light green mmm 3 9064 (4.8%) Diopside light red 12/m1 X||a*, Y||b*, Z||c Properties: bands, bc, bs, error, mad, x, y, grainId Scan unit : um

Then the following command gives you a nice overview over the grain size distributions of the grains

hist(grains)

Sometimes it is desirable to remove all boundary grains as they might distort grain statistics. To do so one should remember
that each grain boundary has a property `grainId` which stores the ids of the neigbouring grains. In the case of an outer grain boundary, one of the neighbouring grains has
the id zero. We can filter out all these boundary segments by

% ids of the outer boundary segment outerBoundary_id = any(grains.boundary.grainId==0,2); plot(grains) hold on plot(grains.boundary(outerBoundary_id),'linecolor','red','linewidth',2) hold off

Now `grains.boundary(outerBoundary_id).grainId` is a list of grain ids where the first column is zero, indicating the outer boundary, and the second column contains the
id of the boundary grain. Hence, it remains to remove all grains with these ids.

% next we compute the corresponding grain_id grain_id = grains.boundary(outerBoundary_id).grainId; % remove all zeros grain_id(grain_id==0) = []; % and plot the boundary grains plot(grains(grain_id))

finally, we can remove the boundary grains by

grains(grain_id) = [] plot(grains)

grains = grain2d Phase Grains Pixels Mineral Symmetry Crystal reference frame 1 968 116882 Forsterite mmm 2 480 23877 Enstatite mmm 3 1467 8871 Diopside 12/m1 X||a*, Y||b*, Z||c boundary segments: 40105 triple points: 3383 Properties: GOS, meanRotation

Beside the area, there are various other geometric properties that can be computed for grains, e.g., the perimeter, the diameter, the equivalentRadius, the equivalentPerimeter, the aspectRatio, and the shapeFactor. The following is a simple scatter plot of shape factor against aspect ratio to check for correlation.

% the size of the dots corresponds to the area of the grains close all scatter(grains.shapeFactor, grains.aspectRatio, 70*grains.area./max(grains.area))

plot(grains,log(grains.aspectRatio))

One interesting question would be, whether a polyphase system has dependence in the spatial arrangement or not, therefore, we can count the transitions to a neighbour grain

`%[J, T, p ] = joinCount(grains,grains.phase)`

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