Ghost Correction

Pole figures do not contain the odd-order harmonic coefficients of an orientation distribution function (ODF). A reconstruction must therefore choose those coefficients without help from the measurements. See Ambiguity of Pole Figure Inversion for the demonstration and Harmonic Representation for the coefficient description.

Setting the unknown odd-order coefficients to zero is a safe mathematical choice, but it has a visible cost. Real texture components become too weak and sit on a uniform background that is too high. Spurious components may also appear beside them. These inversion artefacts are called ghosts; they are not measurement noise.

Matthies' remedy is to estimate the uniform portion of the ODF first. He called this portion the phon. MTEX estimates it from the low-intensity tail of the pole figures, subtracts it, and reconstructs the sharper remainder. Non-negativity then constrains the missing odd-order coefficients more strongly. calcODF applies this correction by default.

Ghost correction adds a physical preference; it does not recover information that diffraction measured. It matters most for weak textures with a substantial uniform portion. Sharp textures have less to gain because non-negativity already constrains them strongly.

This page isolates the effect with a known model that is nine tenths uniform. It assumes the reconstruction workflow from ODF Reconstruction and the ODF normalization from ODF Theory.

Build a deliberately weak ODF

cs = crystalSymmetry('222');
mod1 = orientation.byEuler(0,0,0,cs);
odf = 0.9*uniformODF(cs) + ...
  0.1*unimodalODF(mod1,'halfwidth',10*degree);

Simulate the pole figures

The experiment samples three lattice-plane normals on an antipodal specimen-direction grid with 5 degree spacing. These synthetic pole figures now stand in for measurements, while the true ODF remains known.

% 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);

Each pole figure has one broad maximum on a nearly uniform background. The weak contrast is exactly the situation in which the uniform portion can hide missing odd-order information.

Reconstruct with and without correction

The 'noGhostCorrection' flag disables the default correction.

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

Omitting that flag gives the corrected reconstruction.

recCor = calcODF(pf,'silent');

Compare the fits to the pole figures

calcError returns one RP error per pole figure. First evaluate the reconstruction without correction.

rpUncorrected = calcError(pf,rec,'RP','silent')

rpUncorrected =
    0.0089    0.0088    0.0109

Now evaluate the corrected reconstruction with the same measure.

rpCorrected = calcError(pf,recCor,'RP','silent')

rpCorrected =
    0.0256    0.0246    0.0264

The uncorrected RP values range from 0.0088 to 0.0109, compared with 0.0246 to 0.0264 after correction. This is neither surprising nor a defect. Ghost correction adds an assumption about the uniform portion, and that assumption can only cost fit. Judged on the pole figures alone, the uncorrected reconstruction wins.

Compare the reconstructions with the truth

A measured texture has no known true ODF. This synthetic example does, so calcError can compare the functions directly. First compute the L1 error without correction.

l1Uncorrected = calcError(rec,odf,'L1')

l1Uncorrected =
    0.1255

Then compute the same error with correction.

l1Corrected = calcError(recCor,odf,'L1')

l1Corrected =
    0.0054

Here the uncorrected result loses by a factor of 23.2: 0.1255 against 0.0054. This is the argument for ghost correction in two pairs of numbers. The reconstruction that fits the pole figures roughly two to three times better is more than twenty times farther from the true ODF.

Inspect the ODF sections

Without ghost correction:

plot(rec,'sections',9,'silent','sigma');

The component is broad and weak, and the surrounding density is raised. Now plot the corrected reconstruction on the same type of sections.

plot(recCor,'sections',9,'silent','sigma');

The corrected sections concentrate more density at the component and return the surrounding background towards its true level.

Read a profile through the component

A fibre is the one-dimensional set of orientations that maps a crystal direction onto a specimen direction. The following fibre passes through the model component. Plot the true ODF first.

close all;
f = fibre(Miller(0,1,0,cs),yvector);
plot(odf,f,'linewidth',2,'figSize','small');
hold on;

Add the reconstruction without correction.

plot(rec,f,'linewidth',2);

Finally add the corrected reconstruction as a dashed curve.

plot(recCor,f,'linestyle','--','linewidth',2);
hold off;
legend({'true ODF','without ghost correction','with ghost correction'});

The printed rows follow the legend order. The columns report the maximum and the values at 45, 90, and 270 degrees along the fibre.

profileOri = orientation(f,'points',361);
profileValues = [eval(odf,profileOri),eval(rec,profileOri),...
  eval(recCor,profileOri)].';
profileCheck = [max(profileValues,[],2),profileValues(:,46),...
  profileValues(:,91),profileValues(:,271)];
fprintf(['                    peak   at 45   at 90  at 270\n' ...
  'true ODF          %6.2f  %6.2f  %6.2f  %6.2f\n' ...
  'without correction%6.2f  %6.2f  %6.2f  %6.2f\n' ...
  'with correction   %6.2f  %6.2f  %6.2f  %6.2f\n'],profileCheck.');

peak   at 45   at 90  at 270
true ODF           39.78    0.90    0.90    0.90
without correction 25.23    2.82    3.71    3.71
with correction    37.38    1.06    1.05    1.05

The uncorrected curve has three defects. Its peak reaches 25.23 mrd where the true peak reaches 39.78 mrd. At 45 degrees it is 2.82 mrd instead of 0.90 mrd, so the density missing from the peak has entered the background. Small bumps at 90 and 270 degrees are components conjured out of nothing.

The dashed corrected curve follows the true curve closely. Its peak is 37.38 mrd and its value at 45 degrees is 1.06 mrd. The two extra bumps in the uncorrected curve are the ghosts that give the effect its name.

Inspect the harmonic coefficients

The effect lives in the odd-order coefficients, where it is clearest. First express all three functions as harmonic series through degree 25.

odf = FourierODF(odf,25);
rec = FourierODF(rec,25);
recCor = FourierODF(recCor,25);

The L2 error without ghost correction is:

l2Uncorrected = calcError(rec,odf,'L2')

l2Uncorrected =
    0.3621

With ghost correction it is:

l2Corrected = calcError(recCor,odf,'L2')

l2Corrected =
    0.0312

The values are 0.3621 and 0.0312. Like the L1 comparison, this puts the two reconstructions an order of magnitude apart.

Plot the harmonic spectrum

A harmonic spectrum groups coefficient magnitudes by degree. Plot the true ODF first.

close all;
plotSpektra(odf,'linewidth',2,'figSize','small');
hold on;

Add the uncorrected reconstruction. Its zig-zag is the key feature: odd degrees are pulled towards zero while the even degrees remain close to the truth.

plotSpektra(rec,'linewidth',2);

The corrected reconstruction follows the true spectrum smoothly.

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

A systematic odd-even zig-zag is the diagnostic to look for. It shows that the reconstruction had to guess odd-order information that the pole figures did not measure. The pattern diagnoses this inversion ambiguity; it is not by itself evidence that every feature in a measured ODF is correct after correction.

Further reading

Next

The Santa Fe Example repeats the comparison on a standard model ODF with simulated counting noise. Then continue to The Dubna Example for measured neutron-diffraction pole figures, where the true ODF is no longer available for comparison.