Open Matlab File in the Editor MTEX

Ghost Effect Analysis

Explains the ghost effect to ODF reconstruction and the MTEX option ghostcorrection.

On this page ...
Introduction
Construct Model ODF
Simulate pole figures
ODF Estimation
Compare RP Errors
Compare Reconstruction Errors
Plot the ODFs
Calculate Fourier coefficients
Calculate Reconstruction Errors from Fourier Coefficients
Plot Fourier Coefficients

Introduction

A general problem in estimating an ODF from pole figure data is the fact that the odd order Fourier coefficients of the ODF are not present anymore in the pole figure data and therefore it is difficult to estimate them. Artefacts in the estimated ODF that are due to underestimated odd order Fourier coefficients are called ghost effects. It is known that for sharp textures the ghost effect is relatively small due to the strict non-negativity condition. For weak textures, however, the ghost effect might be remarkable. For those cases MTEX provides the option ghost_correction which tries to determine the uniform portion of the unknown ODF and to transform the unknown weak ODF into a sharp ODF by substracting this uniform portion. This is almost the approach Matthies proposed in his book (He called the uniform portion phon). In this section we are going to demonstrate the power of ghost correction at a simple, synthetic example.

Construct Model ODF

A unimodal ODF with a high uniform portion.

cs = crystalSymmetry('222');
mod1 = orientation('Euler',0,0,0,cs);
odf = 0.9*uniformODF(cs) + ...
  0.1*unimodalODF(mod1,'halfwidth',10*degree)
 
odf = ODF  
  crystal symmetry : 222
  specimen symmetry: 1
 
  Uniform portion:
    weight: 0.9
 
  Radially symmetric portion:
    kernel: de la Vallee Poussin, halfwidth 10°
    center: (0°,0°,0°)
    weight: 0.1
 

Simulate pole figures

% specimen directions
r = equispacedS2Grid('resolution',5*degree,'antipodal');

% crystal directions
h = [Miller(1,0,0,cs),Miller(0,1,0,cs),Miller(0,0,1,cs)];

% compute pole figures
pf = calcPoleFigure(odf,h,r);

plot(pf)

ODF Estimation

without ghost correction:

rec = calcODF(pf,'noGhostCorrection','silent');

with ghost correction:

rec_cor = calcODF(pf,'silent');

Compare RP Errors

without ghost correction:

calcError(pf,rec,'RP')
ans =
    0.0081    0.0440    0.0599

with ghost correction:

calcError(pf,rec_cor,'RP')
ans =
    0.0142    0.0267    0.0282

Compare Reconstruction Errors

without ghost correction:

calcError(rec,odf)
ans =
    0.1024

with ghost correction:

calcError(rec_cor,odf)
ans =
    0.0054

Plot the ODFs

without ghost correction:

plotODF(rec,'sections',9,'silent','sigma')
Plotting ODF as sigma sections, range: 0° - 160°

with ghost correction:

plotODF(rec_cor,'sections',9,'silent','sigma')
Plotting ODF as sigma sections, range: 0° - 160°

radial plot of the true ODF

close all
plotFibre(odf,Miller(0,1,0,cs),yvector,'linewidth',2);
hold all

radial plot without ghost correction:

plotFibre(rec,Miller(0,1,0,cs),yvector,'linewidth',2);

radial plot with ghost correction:

plotFibre(rec_cor,Miller(0,1,0,cs),yvector,'linestyle','--','linewidth',2);
hold off
legend({'true ODF','without ghost correction','with ghost correction'})

Calculate Fourier coefficients

Next we want to analyze the fit of the Fourier coefficients of the reconstructed ODFs. To this end we first compute Fourier representations for each ODF

odf = FourierODF(odf,25)
rec = FourierODF(rec,25)
rec_cor = FourierODF(rec_cor,25)
 
odf = ODF  
  crystal symmetry : 222
  specimen symmetry: 1
 
  Portion specified by Fourier coefficients:
    degree: 25
    weight: 1
 
 
rec = ODF  
  crystal symmetry : 222
  specimen symmetry: 1
 
  Portion specified by Fourier coefficients:
    degree: 25
    weight: 1
 
 
rec_cor = ODF  
  crystal symmetry : 222
  specimen symmetry: 1
 
  Portion specified by Fourier coefficients:
    degree: 25
    weight: 1
 

Calculate Reconstruction Errors from Fourier Coefficients

without ghost correction:

calcError(rec,odf,'L2')
ans =
    0.3391

with ghost correction:

calcError(rec_cor,odf,'L2')
ans =
    0.0294

Plot Fourier Coefficients

Plotting the Fourier coefficients of the recalculated ODFs show that the Fourier coefficients with ghost correction oscillates much more the Fourier coefficients with ghost correction

true ODF

close all;
plotFourier(odf,'linewidth',2)

keep plotting windows and add next plots

hold all

Without ghost correction:

plotFourier(rec,'linewidth',2)

with ghost correction

plotFourier(rec_cor,'linewidth',2)
legend({'true ODF','without ghost correction','with ghost correction'})
% next plot command overwrites plot window
hold off