Weighted Burgers Vector Analysis

The "Weighted Burgers Vector" is a method to analyze geometrically necessary dislocations from EBSD maps, complementary (or alternatively) to GND analysis. The reference description is in Wheeler et. al., The weighted Burgers vector: a new quantity for constraining dislocation densities and types using electron backscatter diffraction on 2D sections through crystalline materials

The "Weighted Burgers Vector" is a moving window technique interpreting the orientation gradient in a given neighborhood and at a given distance to be caused by dislocation lines which crosscut the observation surface. It is weighted/biased by the cosine of the angle between the map-normal the dislocation line direction.

In MTEX the weighted Burgers vector is computed by the function ebsd.weightedBurgersVec, which offers two methods. This page is in two parts, one for each.

The integral method is the default. It derives the WBV from the orientation gradient in x- and y- direction, obtained using a convolution kernel similar to a Prewitt operator, except that the corners are weighted by 0.5. This is comparable to what is termed the "integral method" by other implementations. The size of the kernel can be adjusted in order to cope with noisy data.

The gradient method derives the WBV from the last column of the dislocation density tensor. It resolves finer detail, but it differentiates the data directly and so amplifies whatever noise is in it - high angular resolution EBSD data, or denoising prior to the analysis, is required. Which neighbours enter that derivative can be chosen, and that choice is what the second half of this page is about.

We start by importing the same data as for the GND example.

% import the EBSD data
mtexdata single

ebsd = EBSDsquare (y↓→x, row↓→col)
 
 Phase  Orientations  Mineral         Color  Symmetry  Crystal reference frame
     1  10201 (100%)       Al  LightSkyBlue      m-3m                         
 
 Properties: oldId
 Scan unit : um
 X x Y x Z : [6 → 16] x [4 → 14] x [0 → 0]
 Normal vector: (0,0,1)
 Square grid  :101 x 101

We reconstruct grains because later on, we do not want to compute the WBV across grain boundaries

[grains,ebsd] = calcGrains(ebsd,'angle',2.5*degree,'minPixel',6);

% we will use the noisy data later on
ebsdN = ebsd;

% denoise the data
F = halfQuadraticFilter;
ebsd = smooth(ebsd,F,'fill',grains);

Part 1: the integral method

The default integral method is a moving window over the map, so it needs the data in matrix form. Our map already is - that is how it was imported, see Square and Hex Grids - and for a map that is not, gridify is called internally. The gradient method in part 2 has no such requirement, see there.

wbv = weightedBurgersVec(ebsd)

wbv = vector3d (y↓→x)
 size: 101 x 101

The WBV is returned in specimen coordinates as a list of vector3d. We can inspect its magnitude (in 1/scanunit) and direction.

plot(ebsd,wbv.norm,'refFrame','on')
mtexColorbar
mtexTitle('WBV magnitude')

Visualizing the WBV

In order to visualize the direction of the WBV in specimen coordinates, we can use a directional color key.

cK = HSVDirectionKey(wbv);
plot(ebsd,cK.direction2color(wbv),'FaceAlpha',wbv.norm/0.22)
mtexTitle('WBV in specimen coordinates')

nextAxis
plot(cK,'figSize','tiny')
mtexTitle('directional color key')

We could also display the WBV as small arrows. If we allow for any magnitude, the plot would become quite cluttered. Hence, we will only display those vectors which have a reasonably large magnitude. Next to it we plot the distributions of WBV in a spherical projection.

cond = wbv.norm > quantile(wbv.norm,0.85);

plot(ebsd,cK.direction2color(wbv),'FaceAlpha',wbv.norm/0.22)
hold on
quiver(ebsd(cond),wbv(cond),'color','k','autoScaleFactor', 2, 'antipodal','linewidth',0.5);
hold off
mtexTitle('WBV in specimen coordinates')

nextAxis
plot(wbv,'weights',wbv.norm,'contourf')
mtexTitle('WBV density distribution')

The WBV in crystal coordinates

In order to inspect the WBV in crystal coordinates we transform them using the orientations and choose a fitting directional color key

% transform to crystal reference frame
wbvC = inv(ebsd.orientations) .* wbv;

% define a directional color key respecting crystal symmetry
cKC = HSVDirectionKey(wbvC);

% plot the data
plot(ebsd,cKC.direction2color(wbvC),'FaceAlpha',wbv.norm/0.22)
mtexTitle('WBV in crystal coordinates')

% plot the color key
nextAxis
plot(cKC)
mtexTitle('directional color key')

% overlaid with the contour lines of its density distribution
hold on
plot(wbvC,'weights',wbvC.norm,'contour', ...
    'contours',0.2:0.1:2,'linecolor','k','ShowText','on', ...
    'linewidth',2)
hold off

Effect of windowSize

In case there are reasons why the EBSD data cannot be denoised, the WBV can also be computed with respect to a larger neighborhood. The integer specified with with 'windowSize' gives a 2*n+1 square across which the WBV is computed. The default is a 3-by-3 box.

close all
newMtexFigure('layout',[2,4])

% first we plot again the WBV form the denoised dataset
wbv = weightedBurgersVec(ebsd);
nextAxis(1,1)
plot(ebsd,wbv.norm); hold on
mtexTitle('WBV norm (denoised) / box = 3')

nextAxis(2,1)
plot(wbv,'weights',wbv.norm,'contourf','antipodal')
mtexTitle('density distribution')

% next we plot the WBV form the noisy dataset

for ws = [1 2 3]

  wbv = weightedBurgersVec(ebsdN,'windowSize',ws);
  nextAxis(1,ws+1)
  plot(ebsdN,wbv.norm); hold on
  mtexTitle(['WBV norm / box =' num2str(2*ws+1)])

  nextAxis(2,ws+1)
  plot(wbv,'weights',wbv.norm,'contourf','antipodal')
  mtexTitle('density distribution')

end
mtexColorbar

Here we see that there is some difference between the noisy and the denoised data in a 3-by-3 and larger neighborhoods. For larger window sizes, we see that there is of course a loss of detail (and empty spaces appear around unfilled points) and a decrease in the norm of WBV, since high orientation gradients are spread by the larger window size. At the other hand, the distribution of the WBV in the pole figures becomes sharper, because there is less scatter in the WBV over larger areas.

Part 2: the gradient method

The second method takes the WBV from the last column of the dislocation density tensor, i.e. from the curvature tensor and ultimately from ebsd.gradient. Unlike the integral method it is computed on the virtual lattice and needs no grid at all - it works on a plain EBSD, on a phase subset, on rotated or sheared grids, and on maps that do not lie in the xy plane. ebsd happens to be gridified here, which changes nothing.

We use the denoised map throughout this part. Differentiating a noisy map point by point is exactly what this method must not be given.

wbv = weightedBurgersVec(ebsd,'gradient');

close all
cK = HSVDirectionKey(wbv);
plot(ebsd,cK.direction2color(wbv),'FaceAlpha',wbv.norm/0.2)
mtexTitle('WBV in specimen coordinates' )

nextAxis(1,2)

plot(wbv,'weights',wbv.norm,'antipodal','contourf')
mtexTitle('density distribution')

Comparing this with the map from part 1 we observe that for denoised data the gradient based method results in a more detailed map.

Choosing the stencil

Which neighbours enter the derivative is selected with the 'stencil' option, passed straight through to ebsd.gradient. There are three choices, and they trade detail against robustness in the same way 'windowSize' does for the integral method - only at the scale of a single pixel.

The default is 'oneSided' so that existing numbers do not move. Let us compute all three on the denoised map.

stencils = {'oneSided','1hop','full'};

close all
newMtexFigure('layout',[2,3])

for k = 1:3

  wbv = weightedBurgersVec(ebsd,'gradient','stencil',stencils{k});

  nextAxis(1,k)
  plot(ebsd,wbv.norm,'micronbar','off')
  mtexTitle(['WBV norm / ' stencils{k}])

  nextAxis(2,k)
  plot(wbv,'weights',wbv.norm,'contourf','antipodal')
  mtexTitle('density distribution')

end
mtexColorbar

The three maps show the same structures, but progressively smoothed. The effect is easiest to read off the extremes rather than the mean, since widening the stencil averages the sharpest gradients away first.

for k = 1:3
  wbv = weightedBurgersVec(ebsd,'gradient','stencil',stencils{k});
  notNan = ~isnan(wbv);
  fprintf('%-9s  defined at %d of %d pixels,  mean |W| = %.4f,  max |W| = %.4f\n',...
    stencils{k}, nnz(notNan), numel(wbv), mean(wbv(notNan).norm), max(wbv(notNan).norm));
end

oneSided   defined at 10198 of 10201 pixels,  mean |W| = 0.0288,  max |W| = 0.4340
1hop       defined at 10200 of 10201 pixels,  mean |W| = 0.0277,  max |W| = 0.2562
full       defined at 10200 of 10201 pixels,  mean |W| = 0.0271,  max |W| = 0.2169

The mean barely moves - 0.029 to 0.027 - while the maximum drops by half, which says the stencil is acting on the sharpest gradients and leaving the bulk of the map alone. Whether that is noise being removed or structure being lost is a question about the data, not about the method.

So the choice is the familiar one between locality and noise. 'oneSided' keeps the sharpest features and the largest magnitudes, but a forward difference over one pixel is the least robust estimator of a derivative there is. 'full' is the most stable, at the price of smoothing genuine short range structure. '1hop' sits between the two and is symmetric, which 'oneSided' is not.

Note that this is a different knob from 'windowSize' in part 1, even though both trade detail for stability. 'windowSize' enlarges the integration loop and can be pushed to 5-by-5, 7-by-7 and beyond to cope with genuinely noisy data; the stencil only ever reaches the immediate neighbours of a pixel. If your data is noisy enough that 'full' is not sufficient, denoise it or use the integral method, rather than expecting the stencil to make up the difference.