ODF Estimation from Pole Figure Data

A pole figure does not measure individual orientations. Each value sums every orientation that places one lattice plane in one specimen direction. Those orientations form a one-dimensional fibre in orientation space. One pole figure therefore cannot be inverted by itself.

Combining several pole figures adds constraints, but it still does not make the inverse problem unique. The practical task is to find an orientation distribution function (ODF) whose recalculated pole figures agree with the measurements, and then to inspect where they do not agree.

This page assumes the pole-figure idea from Pole Figures and corrected measurements from Data Correction. The definition and units of an ODF are introduced in ODF Theory. Here the reconstruction uses calcODF. The Ghost Effect explains what the measurements can never determine.

plottingConvention.default('y↑→x');
mtexdata dubna silent

Inspect the measurements first

The data contain seven neutron-diffraction pole figures from a quartz specimen. One measurement is a superposition of two unresolved reflections. Import explains how the Miller indices and their structure coefficients enter that measurement model.

plot(pf)

The seven panels contain sharp maxima at different specimen directions. The superposed reflection has two Miller indices in its title. Look for missing coverage, isolated points, or a pattern that is inconsistent with the other panels before asking an ODF to explain the data.

Reconstruct an ODF

With no options, calcODF uses MTEX's modified least-squares solver and automatic ghost correction.

tic
odf = calcODF(pf)
defaultTime = toc;

odf = SO3FunRBF (Quartz → y↑→x)
 
  multimodal components
  kernel: de la Vallee Poussin, halfwidth 5°
  center: 19848 orientations, resolution: 5°
  weight: 1

The result is an SO3FunRBF. It represents the ODF as a weighted superposition of unimodal kernel functions on a grid in orientation space. The display reports 19,848 grid centres for this reconstruction. The object can be analysed like any other ODF; continue to ODF Analysis after validating the reconstruction.

Compare recalculated and measured pole figures

The first validation is visual. Recalculate exactly the lattice planes and superpositions that were measured.

plotPDF(odf,pf.allH,'antipodal','silent','superposition',pf.c)

The maxima appear in the same regions and reach comparable intensities. The recalculated pole figures are smoother because the finite-width kernels do not reproduce every fluctuation in the measurement.

calcError makes the same comparison one pole figure at a time. The 'RP' measure divides the absolute difference by the recalculated intensity, but only where that intensity exceeds the threshold supplied after 'RP'. The default threshold is 1.

rpError = calcError(pf,odf,'RP','silent')

rpError =
    0.8258    0.8571    0.3662    0.3629    0.6014    0.6140    0.4803

The seven RP values range from 0.36 to 0.86. Because RP is relative, weak regions can dominate even when their absolute differences are small.

Called without a measure, calcError uses a regularised relative error. It divides by the larger intensity plus half the scale factor, so it does not divide by a value near zero.

regularisedError = calcError(pf,odf,'silent')

regularisedError =
    0.3957    0.3248    0.2381    0.2886    0.2900    0.3619    0.3222

The regularised values range from 0.24 to 0.40. They are not smaller RP values; they answer a different question. Compare reconstructions with the same measure and threshold throughout. The other available measures are 'l1' and 'l2'.

A single error vector hides where the mismatch occurs. plotDiff plots the regularised relative difference for every measured direction.

plotDiff(pf,odf,'silent')

These residuals are not scattered. In every one of the seven panels the mean residual grows from the centre of the disc outward, roughly doubling by the rim.

Scattered residuals would be consistent with measurement noise. A pattern this orderly points to a systematic cause worth checking, and one that grows with specimen tilt is the signature of an omitted defocusing correction. An incorrect background or a pole figure that does not belong with the others would also leave structure. The pattern is a diagnostic clue, not proof of any one cause.

Comparing two ODFs

The calcError overload also compares two ODFs. This is a different calculation from comparing an ODF with pole-figure measurements. To make the distinction visible, build one broad component at the strongest orientation of the reconstruction.

[~,oriPref] = max(odf);
odfModel = unimodalODF(oriPref,'halfwidth',15*degree);

plotPDF(odfModel,pf.allH,'antipodal','superposition',pf.c)

odfDifference = calcError(odfModel,odf)

odfDifference =
    0.5868

The model pole figures retain the main maxima and omit the weaker components. The large ODF difference is therefore expected: one unimodal component cannot represent this multi-component texture.

Control the discretization

The solver places unimodal components on a grid in orientation space. The 'resolution' option sets the grid spacing, while each default de la Vallee Poussin kernel has the same halfwidth as that spacing.

The calcODF reference describes its default grid as 1.5 times the pole- figure resolution. The current MLSSolver constructor instead uses the stored resolution of the first pole figure. Pass 'resolution' explicitly when that distinction matters to reproducibility.

A 15 degree grid demonstrates the speed-resolution trade-off.

tic
odfCoarse = calcODF(pf,'resolution',15*degree,'silent');
coarseTime = toc;

plotPDF(odfCoarse,pf.allH,'antipodal','silent','superposition',pf.c)

defaultPeak = max(odf);
coarsePeak = max(odfCoarse);
meanDefaultRP = mean(rpError);
meanCoarseRP = mean(calcError(pf,odfCoarse,'RP','silent'));
fprintf(['Default: %.2f s, peak %.1f mrd, mean RP %.2f; ',...
  '15 degree: %.2f s, peak %.1f mrd, mean RP %.2f\n'],...
  defaultTime,defaultPeak,meanDefaultRP,...
  coarseTime,coarsePeak,meanCoarseRP);

Default: 2.41 s, peak 94.0 mrd, mean RP 0.59; 15 degree: 0.16 s, peak 27.1 mrd, mean RP 0.75

The 15 degree result keeps the main maxima but broadens them. The timing printed above shows the speed-up. Its peak is 27.1 mrd instead of 94.0 mrd, and the mean RP error rises from 0.59 to 0.75. A coarse grid cannot represent a sharp texture. A finer grid costs more time and, once it exceeds the information in the measurements, can give a false impression of resolved detail.

Two options control the kernel directly. The 'kernel' option accepts an SO3Kernel object. The 'halfwidth' option keeps the default de la Vallee Poussin kernel and changes its width.

The zero range method

If a measured pole figure is genuinely zero in a region, every orientation contributing there must also have zero density. The zero range method removes those orientation-grid nodes before solving. It can make a sharp reconstruction with large empty regions both faster and finer.

tic
odfZero = calcODF(pf,'zeroRange','silent');
zeroTime = toc;

plotPDF(odfZero,pf.allH,'antipodal','silent','superposition',pf.c)

zeroPeak = max(odfZero);
meanZeroRP = mean(calcError(pf,odfZero,'RP','silent'));
fprintf(['Zero range: %.2f s, peak %.1f mrd, mean RP %.2f; ',...
  'default: %.2f s, peak %.1f mrd, mean RP %.2f\n'],...
  zeroTime,zeroPeak,meanZeroRP,...
  defaultTime,defaultPeak,meanDefaultRP);

Zero range: 1.12 s, peak 110.4 mrd, mean RP 0.51; default: 2.41 s, peak 94.0 mrd, mean RP 0.59

On this data set the method does change the reconstruction. The peak rises from 94.0 to 110.4 mrd and the mean RP error falls from 0.59 to 0.51. The timings printed above show that it also runs faster here, because the solver keeps far fewer grid nodes. The sharper recalculated maxima are the visible difference to look for. Try the method rather than assuming it will help. The zeroRangeMethod reference lists the threshold and smoothing options that decide what counts as zero.

Ghost correction

The odd-order harmonic coefficients of an ODF do not appear in its pole figures, so the measurements do not determine them. Setting them to zero produces the ghost effect: a raised uniform background, weakened real components, and sometimes spurious components.

Ghost Correction explains Matthies' remedy, which MTEX applies by default and which matters most for weak textures. The Santa Fe Example measures the benefit on a model ODF for which the true answer is known. A good fit to the pole figures alone does not establish that a reconstructed ODF is unique or true.

The maths behind the estimator

In simplified notation, calcODF minimises a modified least-squares functional over non-negative combinations of the kernel components:

\[f_{\mathrm{est}} = \mathrm{argmin}_{f} \sum_{i=1}^{N} \sum_{j=1}^{N_i} \frac{|\alpha_i Rf(h_i,r_{ij}) - I_{ij}|^2}{I_{ij}}.\]

Dividing by \(I_{ij}\) makes the functional modified least squares. It weights each measurement by its own intensity, so a bright point cannot dominate a dark one. The factor \(\alpha_i\) absorbs the unknown scale of each pole figure, which allows unnormalised intensity data to be used.

Further reading