Grain Reconstruction from Voxel Data

A three-dimensional EBSD measurement arrives as voxels, whether it comes from serial sectioning or from a diffraction technique that probes the volume. Each voxel carries a position, a phase and an orientation. A grain is a phase-homogeneous, spatially connected region of such voxels. This page asks how the segmentation criterion changes the grains we measure. Unlike the simulated multiphase volume in the overview, this example uses the DREAM.3D IN100 data so that one phase can be coloured by orientation.

plottingConvention.default('y↑→x');
how2plot = plottingConvention.default3D;

Load the voxel data

EBSD3.load reads the cell data of a DREAM.3D file into an EBSD3square, the voxel counterpart of a square grid EBSD map. The voxels form a three-dimensional array, so ebsd(i,j,k) addresses one of them. Every further scalar array of the file becomes a property.

fname = fullfile(mtexDataPath,'EBSD','SmallIN100_MeshStats.dream3d');
ebsdImported = EBSD3.load(fname);
ebsd = ebsdImported

ebsd = EBSD3square (y↑→x)
 
 Phase    Orientations  Mineral         Color  Symmetry  Crystal reference frame
     1  1000000 (100%)  unknown  LightSkyBlue       432                         
 
 Properties: ConfidenceIndex, grainId, FeatureReferenceMisorientations, Fit, GBManhattanDistances, ImageQuality, KernelAverageMisorientations, ParentIds, QPManhattanDistances, SEMSignal, TJManhattanDistances
 Scan unit : um
 X × Y × Z : [-37 -12] × [10 35] × [-29 -4]
 Grid size (square): 100 × 100 × 100

Reconstruct the grains

calcGrains works as its two-dimensional counterpart: two neighbouring voxels belong to the same grain when their misorientation angle stays below the threshold. Voxels are 6-connected. The boundary between two grains consists of the voxel faces they share, each stored as two triangles, so every grain is a closed surface.

[grains,ebsd] = calcGrains(ebsd,'angle',5*degree)

grains = grain3d (y↑→x)
 
 Phase  Grains  Volume  Mineral  Symmetry  Crystal reference frame
     1     812   15625  unknown       432                         
 
 boundary faces: 757562
 
 Properties: meanRotation, GOS
 
 
ebsd = EBSD3square (y↑→x)
 
 Phase    Orientations  Mineral         Color  Symmetry  Crystal reference frame
     1  1000000 (100%)  unknown  LightSkyBlue       432                         
 
 Properties: ConfidenceIndex, grainId, FeatureReferenceMisorientations, Fit, GBManhattanDistances, ImageQuality, KernelAverageMisorientations, ParentIds, QPManhattanDistances, SEMSignal, TJManhattanDistances
 Scan unit : um
 X × Y × Z : [-37 -12] × [10 35] × [-29 -4]
 Grid size (square): 100 × 100 × 100

The summary lists the grains with their volume and mean orientation. The second output is the same voxel data with a grainId per voxel. Plotting the mean orientation colours every face on the outside of the volume.

plot(grains,grains.meanOrientation,'edgeAlpha',0.1,'micronbar','off')
setCamera(how2plot)

The colour is constant over each grain and changes sharply at the grain boundaries. The staircase texture of the outer faces is the voxel grid itself; the boundary follows the voxel faces exactly.

Volumes

Every grain is a closed surface, so its volume follows from the divergence theorem and equals the number of its voxels times the voxel volume. The volumes of all grains add up to the measured box.

[sum(grains.volume), prod(size(ebsd)) * ebsd.dx * ebsd.dy * ebsd.dz]

ans =
       15625       15625

Most grains are small. The largest grain is worth a look on its own.

histogram(grains.volume)
xlabel('grain volume (length units^3)')
ylabel('number of grains')

[~,id] = max(grains.volume);
plot(grains(id),'micronbar','off','edgeAlpha',0.2)
setCamera(how2plot)

Grains stored in the file

DREAM.3D and GrainMapper3D files carry the grain id the vendor software assigned to every voxel, which the importer stores as ebsd.grainId. The flag 'grainId' builds the grains from these ids instead of the orientations, so the stored segmentation becomes a grain3d with the same closed surfaces, volumes and boundary as a reconstruction of our own.

grainsStored = calcGrains(ebsdImported,'grainId')

grainsStored = grain3d (y↑→x)
 
 Phase  Grains  Volume  Mineral  Symmetry  Crystal reference frame
     1     813   15625  unknown       432                         
 
 boundary faces: 757564
 
 Properties: meanRotation, GOS

The second output of calcGrains replaces ebsd.grainId, so the stored segmentation must be read from ebsdImported. Grain labels are arbitrary: subtracting the two ID arrays would not measure agreement. First compare a physically meaningful quantity, the equivalent-sphere diameter, \(d_V=(6V/\pi)^{1/3}\), for indexed grains in each reconstruction.

gMTEX = grains('indexed');
gStored = grainsStored('indexed');
dMTEX = (6*gMTEX.volume/pi).^(1/3);
dStored = (6*gStored.volume/pi).^(1/3);
diameterEdges = linspace(0,max([dMTEX;dStored]),25);

histogram(dMTEX,diameterEdges,'DisplayName','MTEX, 5 degrees')
hold on
histogram(dStored,diameterEdges,'DisplayName','stored segmentation')
hold off
legend('Location','best')
xlabel('equivalent-sphere diameter (length units)')
ylabel('number of grains')

Similar distributions do not establish voxel-by-voxel agreement. They do show whether the two segmentations give similar grain sizes. To locate discrepancies, compare their grainId properties on the same slice.

Small grains

A handful of voxels with a stray orientation form a grain of their own. The option minPixel removes indexed grains below a number of voxels. Their voxels are marked notIndexed and form notIndexed grains, exactly as in two dimensions.

[grainsClean,ebsdClean] = calcGrains(ebsdImported,'angle',5*degree,'minPixel',10);
[length(grains('indexed')), length(grainsClean('indexed'))]

ans =
   812   749

A cutoff is a choice of minimum resolved volume: ten voxels correspond to the volume below. It can suppress isolated indexing errors, but also remove real small grains. Check the affected regions before interpreting a loss of fine grains as a material feature.

minimumVolume = 10 * ebsd.dx * ebsd.dy * ebsd.dz
removedFraction = nnz(ebsdImported.isIndexed & ~ebsdClean.isIndexed) / ...
  nnz(ebsdImported.isIndexed)

minimumVolume =
    0.1562
removedFraction =
   2.5200e-04

The grain boundary

grains.boundary is a grain3Boundary. Each face records the two voxels it separates, ebsdId, the two grains, grainId, and the misorientation between the two voxels. Faces on the outside of the volume have a zero in the second column of grainId. The misorientation angle across the inner faces shows where the low angle boundaries are.

gB = grains.boundary;
isInner = all(gB.grainId > 0,2) & all(gB.isIndexed,2);
histogram(gB.misorientation(isInner).angle ./ degree)
xlabel('local misorientation angle (degrees)')
ylabel('number of mesh faces')

Next

Smoothing removes voxel steps before surface measurements. Properties measures the reconstructed grains, and Operations relates sections to their parent grains. Boundary Network measures internal interfaces. The Three-Dimensional Grains page reads the surface mesh DREAM.3D stored alongside the voxels.