Simulating Pole Figure data edit page

Simulating pole figure data from a given ODF is useful to investigate pole figure to ODF reconstruction routines. Let us start with a model ODF given as the superposition of 6 components.

cs = crystalSymmetry('orthorhombic');
mod1 = orientation.byAxisAngle(xvector,45*degree,cs);
mod2 = orientation.byAxisAngle(yvector,65*degree,cs);
model_odf = 0.5*uniformODF(cs) + ...
  0.05*fibreODF(Miller(1,0,0,cs),xvector,'halfwidth',10*degree) + ...
  0.05*fibreODF(Miller(0,1,0,cs),yvector,'halfwidth',10*degree) + ...
  0.05*fibreODF(Miller(0,0,1,cs),zvector,'halfwidth',10*degree) + ...
  0.05*unimodalODF(mod1,'halfwidth',15*degree) + ...
  0.3*unimodalODF(mod2,'halfwidth',25*degree);
plot(model_odf,'sections',6,'silent','sigma')

In order to simulate pole figure data, the following parameters have to be specified

  1. an arbitrary ODF
  2. a list of Miller indece
  3. a grid of specimen directions
  4. superposition coefficients (optional)
  5. the magnitude of error (optional)

The list of Miller indece

h = [Miller(1,1,1,cs),Miller(1,1,0,cs),Miller(1,0,1,cs),Miller(0,1,1,cs),...
  Miller(1,0,0,cs),Miller(0,1,0,cs),Miller(0,0,1,cs)];

The grid of specimen directions

r = regularS2Grid('resolution',5*degree);

Now the pole figures can be simulated using the command calcPoleFigure.

pf = calcPoleFigure(model_odf,h,r)
pf = PoleFigure
  crystal symmetry : mmm
  specimen symmetry: 1
 
  h = (111), r = 72 x 37 points
  h = (110), r = 72 x 37 points
  h = (101), r = 72 x 37 points
  h = (011), r = 72 x 37 points
  h = (100), r = 72 x 37 points
  h = (010), r = 72 x 37 points
  h = (001), r = 72 x 37 points

Add some noise to the data. Here we assume that the mean intensity is 1000.

pf = noisepf(pf,1000);

Plot the simulated pole figures.

plot(pf)

ODF Estimation from Pole Figure Data

From these simulated pole figures we can now estimate an ODF,

odf = calcODF(pf)
odf = ODF (mmm → xyz)
 
  Uniform portion:
    weight: 0.46042
 
  Radially symmetric portion:
    kernel: de la Vallee Poussin, halfwidth 5°
    center: 29758 orientations, resolution: 5°
    weight: 0.53958

which can be plotted,

plot(odf,'sections',6,'silent','sigma')

and compared to the original model ODF.

calcError(odf,model_odf,'resolution',5*degree)
ans =
    0.0814

Exploration of the relationship between estimation error and number of pole figures

For a more systematic analysis of the estimation error, we vary the number of pole figures used for ODF estimation from 1 to 7 and calculate for any number of pole figures the approximation error. Furthermore, we also apply ghost correction and compare the approximation error to the previous reconstructions.

e = [];
for i = 1:pf.numPF

  odf = calcODF(pf({1:i}),'silent','NoGhostCorrection');
  e(i,1) = calcError(odf,model_odf,'resolution',2.5*degree);
  odf = calcODF(pf({1:i}),'silent');
  e(i,2) = calcError(odf,model_odf,'resolution',2.5*degree);

end

Plot the error in dependency of the number of single orientations.

close all;
plot(1:pf.numPF,e,'LineWidth',2)
ylim([0.07 0.32])
xlabel('Number of Pole Figures');
ylabel('Reconstruction Error');
legend({'Without Ghost Correction','With Ghost Correction'});