PoleFigure2ODFGhostCorrection.m

%% Ghost Effect Analysis
%
%%
% 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. Artifacts 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.byEuler(0,0,0,cs);
odf = 0.9*uniformODF(cs) + ...
  0.1*unimodalODF(mod1,'halfwidth',10*degree)

%% Simulate pole figures
%

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

% crystal directions
h = Miller({1,0,0},{0,1,0},{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')

%%
% with ghost correction:
calcError(pf,rec_cor,'RP')

%% Compare Reconstruction Errors

%%
% without ghost correction:
calcError(rec,odf)

%%
% with ghost correction:
calcError(rec_cor,odf)


%% Plot the ODFs

%%
% without ghost correction:
plot(rec,'sections',9,'silent','sigma')

%%
% with ghost correction:
plot(rec_cor,'sections',9,'silent','sigma')

%%
% radial plot of the true ODF
close all
f = fibre(Miller(0,1,0,cs),yvector);
plot(odf,f,'linewidth',2);
hold all

%%
% radial plot without ghost correction:
plot(rec,f,'linewidth',2);

%%
% radial plot with ghost correction:
plot(rec_cor,f,'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)

%% Calculate Reconstruction Errors from Fourier Coefficients

%%
% without ghost correction:
%
calcError(rec,odf,'L2')

%%
% with ghost correction:
calcError(rec_cor,odf,'L2')


%% Plot Fourier Coefficients
%
% Plotting the Fourier coefficients of the recalculated ODFs shows that the
% Fourier coefficients without ghost correction oscillates much more than
% the Fourier coefficients with ghost correction

%%
% true ODF
close all;
plotSpektra(odf,'linewidth',2)

%%
% keep plotting windows and add next plots
hold all

%%
% Without ghost correction:
plotSpektra(rec,'linewidth',2)

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