Harmonic Representation of Rotational Functions

A rotational function can be stored as a finite list of coefficients instead of as values on an orientation grid. This is the harmonic representation used by SO3FunHarmonic. It is useful for analysing smoothness, filtering by angular scale, and turning many operations on functions into algebra on their coefficients.

This page assumes the definition and normalization of an orientation distribution function (ODF) and the common interface introduced in Orientation Dependent Functions. An ODF is one important rotational function, but the harmonic representation applies to any scalar function on the rotation group.

plottingConvention.default('y↑→x');

The Series and Its Bandwidth

Fourier series replace a complicated function by a weighted sum of known basis functions. On a circle those basis functions are sines and cosines. On the rotation group they are Wigner-D functions. Thus a rotational function \(f\colon \mathrm{SO}(3) \to \mathbb C\) can be written as

\[ f(\mathbf{R}) = \sum_{n=0}^N \sum_{k,l=-n}^n \hat f_n^{k,l}\,\mathrm{D}_n^{k,l}(\mathbf{R}). \]

Here \(\hat f_n^{k,l}\) are the Fourier coefficients and \(\mathrm{D}_n^{k,l}\) are the Wigner-D functions. The largest retained degree \(N\) is the bandwidth. Increasing the bandwidth permits narrower detail, but it does not recover information already absent from the source data. A sharp cutoff removes fine scales and therefore smooths the function, although it can also cause ringing near sharp features.

Before symmetry is taken into account, degree \(n\) contributes \((2n+1)^2\) coefficients. A series through bandwidth \(N\) therefore has

\[ \sum_{n=0}^N(2n+1)^2 = \frac{(N+1)(2N+1)(2N+3)}{3} \]

coefficients. This cubic growth is why bandwidth affects storage and computation much more strongly than its name may suggest.

Several normalizations of Wigner-D functions occur in the literature. MTEX uses an orthonormal basis with respect to the normalized measure on \(\mathrm{SO}(3)\), so

\[ \|\mathrm{D}_n^{k,l}\|_2 = 1. \]

Crystal and specimen symmetry restrict which coefficient combinations are admissible. MTEX applies those restrictions through the symmetries attached to the function. The basis convention and its relation to other conventions are detailed under Wigner-D Functions.

Converting an ODF

Start with an ODF reconstructed from the Dubna pole figure data. The zero-range flag belongs to that reconstruction; the page on the Dubna example explains why it is used.

mtexdata dubna silent
odf = calcODF(pf,'resolution',5*degree,'zeroRange','silent');

Converting any SO3Fun to SO3FunHarmonic computes a finite harmonic approximation. Here the requested bandwidth is 32. The conversion keeps the crystal and specimen symmetries of the ODF.

f = SO3FunHarmonic(odf,'bandwidth',32);

Coefficient Storage

The complex Fourier coefficients are stored in f.fhat as one vector. The degree-zero block is f.fhat(1). The nine entries f.fhat(2:10) form the degree-one block, and the next 25 entries form the degree-two block. Within a block, rows correspond to \(k=-n,\ldots,n\) and columns to \(l=-n,\ldots,n\). MATLAB stores the columns one after another. The degree-two coefficient matrix is therefore

reshape(f.fhat(11:35),5,5)

ans =
  Columns 1 through 4
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0990 - 0.2350i   0.9290 - 0.4718i   1.1372 + 0.0000i   0.9290 + 0.4718i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i   0.0000 + 0.0000i
  Column 5
   0.0000 + 0.0000i
   0.0000 + 0.0000i
   0.0990 + 0.2350i
   0.0000 + 0.0000i
   0.0000 + 0.0000i

Up to its bandwidth, this vector completely characterizes the harmonic function. The compact representation also makes many calculations algebraic. For example, orientation averages of symmetric second-rank single-crystal properties such as thermal expansion or conductivity need only low harmonic degrees. Fourth-rank elastic properties additionally involve degree four. The effective property still depends on the single-crystal tensor; the ODF coefficients describe the texture part of the average.

Constructing a Function from Coefficients

A harmonic function can also be constructed directly from its coefficient vector. In the following example \(\hat f_0^{0,0}=0.5\) and the degree-one block is

\[ \hat f_1 = \left(\begin{array}{rrr} 1 & 4 & 7 \\ 2 & 5 & 8 \\ 3 & 6 & 9 \end{array}\right). \]

f2 = SO3FunHarmonic([0.5,1:9]');
f2.isReal

ans =
  logical
   0

The printed false value is deliberate. Real entries in f.fhat do not by themselves define a real-valued rotational function; paired coefficients must satisfy the conjugate symmetry imposed by the Wigner-D basis. This distinction matters before plotting, because a plot of a complex function can show only one real-valued part.

Identity matrices satisfy the reality relation and give a small example through degree two. The vector order is degree first, then \(l\), then \(k\).

cs = crystalSymmetry('1');
fhat = [1;reshape(eye(3),[],1);reshape(eye(5),[],1)];
fExample = SO3FunHarmonic(fhat,cs)

minimumValue = min(fExample)

fExample = SO3FunHarmonic (1 → y↑→x)
  bandwidth: 2
  weight: 1
 
minimumValue =
   -1.2645

The summary shows that MTEX inferred bandwidth 2 from the vector length. The negative minimum shows why a coefficient vector does not automatically define an ODF. An ODF must be real, nonnegative, and normalized to mean one. Truncating the series of a sharp positive ODF can likewise create small negative undershoots, just as a truncated ordinary Fourier series can ring near a sharp edge.

plot(fExample,'sections',6,'silent','sigma');
mtexColorbar('title','function value');

The sigma sections show the full rotational function. The colorbar spans negative and positive values, which is the visible consequence of choosing coefficients without enforcing ODF positivity.

plotPDF(fExample,[Miller(1,0,0,cs),Miller(1,1,0,cs)],'antipodal');

These pole figures are projections of the same coefficient series. A plausible-looking projection is not evidence that the function itself is a valid ODF; the reality, positivity, and normalization checks still apply in orientation space.

Reading a Spectrum

plotSpektra groups the coefficients by degree. At each degree it plots the square root of their summed squared magnitudes, not the individual coefficients.

close all;
plotSpektra(f,'figSize','small');

Read the curve from left to right as the amount of progressively finer angular structure. Fast decay means that a lower bandwidth may preserve the function well. A slow or irregular tail means that high-degree content remains. In a pole-figure inversion, that tail can also indicate the ghost effect; see Ghost Correction.

Harmonic Approximation of a General SO3Fun

Nothing in the harmonic representation requires the function to be an ODF. As an example, take the misorientation angle to a fixed orientation. It is nonnegative, but it is not normalized to mean one and is therefore not an ODF.

cs = crystalSymmetry('432');
oriRef = orientation.byEuler(30*degree,50*degree,10*degree,cs);
fAngle = SO3FunHandle(@(ori) angle(ori,oriRef)./degree,cs);

quadrature evaluates the function on a quadrature grid and computes its Fourier coefficients. Request the same bandwidth 32 used above.

fAngleHarm = SO3FunHarmonic.quadrature(fAngle,'bandwidth',32);

The maximum pointwise error below measures this random sample only. It is not a guaranteed global error bound.

ori = orientation.rand(1000,cs);

maxApproximationError = max(abs(fAngle.eval(ori) - fAngleHarm.eval(ori)))

maxApproximationError =
    1.0487

The nonzero error is a truncation effect. The angle function has a cusp at oriRef and further cusps where two symmetry-equivalent distance branches tie. It is continuous but not differentiable there, so its coefficients decay slowly and the finite cutoff is felt away from the cusps as well.

close all;
plotSpektra(fAngleHarm,'figSize','small');

Unlike the reconstructed ODF spectrum, this spectrum retains appreciable power near the cutoff. That slow decay is the spectral signature of the nonsmooth angle function. In general, smoother functions have faster coefficient decay and need less bandwidth for the same accuracy.

plotSection(fAngleHarm,'sigma');
mtexColorbar('title','misorientation angle in degree');

The section plot is smooth over most of orientation space but changes direction sharply along the symmetry cut loci. Those ridges are why a finite smooth series cannot reproduce the angle function exactly.

References

Next

The basis behind this representation is developed in Wigner-D Functions. The calculations enabled by the common function interface are collected in Operations on Rotational Functions. For the inverse problem that produced the measured ODF above, continue with Reconstructing an ODF. Approximation from scattered values, including regularization and overfitting, is covered in Harmonic Approximation from Discrete Data. The next page in this chapter, Importing an ODF, explains how MTEX reconstructs function objects from common file representations.