Bingham ODFs

A Bingham orientation distribution function (ODF) is one compact model for a texture with different spreads in different directions. It uses one quadratic form. This form can describe a round or elongated peak, a peak spread along a fibre, or a surface-like distribution in orientation space. Kernel density estimation instead places one kernel at every measured orientation.

This page assumes the normalization introduced in ODF Theory and the model families compared in ODF Modeling. It also uses the unit-quaternion representation of a rotation. Quaternions \(g\) and \(-g\) represent the same rotation. A Bingham density has the same value at both. Its antipodal symmetry is therefore exactly what a density on rotations requires.

MTEX represents the result as an SO3FunBingham. It therefore supports the same evaluation, plotting, and sampling interface as other orientation functions.

plottingConvention.default('y↑→x');

Constructing a General Bingham ODF

A Bingham model needs four mutually orthogonal quaternion axes and four shape parameters. The axes are orthogonal in four-dimensional quaternion space; they are not four physical specimen directions. The example uses the identity quaternion basis as the axes.

cs = crystalSymmetry('1');

kappa = [100 90 80 0];
U = orientation.eye(cs);

odf = BinghamODF(kappa,U)

odf = SO3FunBingham (1 → y↑→x)
 
  kappa: 100 90 80 0
  weight: 1

The printed summary identifies a Bingham component and reports its four shape parameters. BinghamODF normalizes the component, so its mean is 1 multiple of a random distribution (mrd).

Pole figures project the ODF onto specimen directions. The unequal shape parameters make the pole-density features anisotropic rather than round.

h = Miller({0,0,1},{1,0,0},{1,1,1},cs);
plotPDF(odf,h,'antipodal','silent','layout',[1 3]);

The same anisotropy appears in sections through orientation space. The peak changes position between the six panels. Each panel cuts a different part of the three-dimensional ODF.

plot(odf,'sections',6);

Estimating the Parameters

calcBinghamODF estimates a Bingham model from individual orientations. Those orientations may come from EBSD or from a simulation. This compact fit can work with far fewer measurements than a kernel estimate. It also assumes that one Bingham component is adequate.

Start with a fibre ODF about a randomly chosen fibre.

odfTrue = fibreODF(fibre.rand(cs));

plotPDF(odfTrue,h,'antipodal','silent');

Each pole figure contains the ring expected from a fibre texture. The ring position differs between crystal directions, but it belongs to the same orientation-space fibre.

Draw 2000 orientations from that ODF and add their pole points to all three panels with discreteSample.

ori = discreteSample(odfTrue,2000);
plot(ori,'add2all','MarkerEdgeColor','k',...
  'MarkerSize',5,'MarkerFaceColor','none','MarkerEdgeAlpha',0.2);

The black points fluctuate about the pole-density rings. They are a finite random sample, not a second texture component.

Fit a Bingham distribution to the simulated orientations.

odf = calcBinghamODF(ori)

plotPDF(odf,h,'antipodal','silent');

odf = SO3FunBingham (1 → y↑→x)
 
  kappa: 0 2 182 182
  weight: 1

The fitted pole figures recover the rings of the original fibre ODF from the 2000 sampled orientations. Small irregularities in the point overlay are sampling variation; the fitted Bingham model itself is smooth.

Internally, MTEX obtains the four axes from the sample orientation tensor. They are its eigenvectors. MTEX derives odf.kappa from the corresponding eigenvalues with a large-concentration approximation. Thus calcBinghamODF is a moment-based approximate fit. It is not a general maximum-likelihood optimizer. Reach for this model when one compact shape is plausible. Compare the fitted plots with the data before treating its parameters as a complete description.

Three Non-Uniform Cases

The general quaternion Bingham distribution is classified as bipolar, circular, or spherical. For texture work these correspond to a localized orientation peak, a fibre-like peak, and a surface-like distribution. Here "spherical" describes the set of maxima on the quaternion sphere; it does not mean a uniform ODF.

A Bipolar or Unimodal Distribution

A scalar shape parameter and a reference orientation define a unimodal Bingham distribution. BinghamODF expands the scalar 20 to [20 0 0 0]. The reference orientation is the unique maximum, apart from its antipode and symmetry-equivalent descriptions.

cs = crystalSymmetry('321');
oriRef = orientation.byEuler(45*degree,0*degree,0*degree,cs);

odf = BinghamODF(20,oriRef)

plot(odf,'sections',6,'silent','contourf','sigma');

odf = SO3FunBingham (321 → y↑→x)
 
  kappa: 20 0 0 0
  weight: 1

The filled contours form localized patches. Crystal symmetry repeats the same physical peak; those repeated patches are not additional components.

A Circular or Fibre Distribution

Passing a fibre with one shape parameter expands 20 to [20 20 0 0]. The two equal leading parameters make the density constant along the fibre, while the two zeros control its transverse decay.

f = fibre(Miller(0,0,1,cs),vector3d(1,2,3));
odf = BinghamODF(20,f)

plot(odf,'sections',6,'silent','sigma');

odf = SO3FunBingham (321 → y↑→x)
 
  kappa: 20 20 0 0
  weight: 1

The high-density feature now continues through successive section panels instead of closing around one orientation. That continuity is the visual signature of the fibre.

A Spherical or Surface-Like Distribution

Three equal nonzero parameters give the spherical case. The omitted fourth value is padded with zero, so this call uses [10 10 10 0].

odf = BinghamODF([10 10 10],quaternion.eye,cs)

plot(odf,'sections',6,'silent','sigma');

odf = SO3FunBingham (321 → y↑→x)
 
  kappa: 10 10 10 0
  weight: 1

This plot is not uniform: the density avoids the fourth quaternion axis and spreads its maxima over a two-dimensional set. A uniform distribution requires all four parameters to be equal. Because a common offset is irrelevant, MTEX can represent that case with [0 0 0 0].

Reading the Parameters

Whichever constructor produced the model, odf.A holds its four principal quaternion axes as orientations. odf.kappa holds the shape parameter belonging to each axis. For a fitted model, these axes are the eigenvectors of the orientation tensor and the shape parameters are derived from its eigenvalues.

Reconstruct the general example so the following output has deliberately unequal parameters.

odf = BinghamODF([100 90 80 0],orientation.eye(cs));

odf.A

ans = orientation (321 → y↑→x)
  size: 4 × 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2
     0    0    0
     0  180    0
    90  180  270
   180    0    0

The four displayed orientations are one orthogonal basis of quaternion space. Their row order matches the order of the following parameters.

odf.kappa.'

ans =
   100    90    80     0

The constructor preserves the supplied axis order. A fitted model keeps the eigenvector order returned by the orientation tensor calculation. Do not sort odf.kappa without applying the same permutation to odf.A. The classification below instead labels the parameters in descending order.

The bipolar, circular, and spherical classification applies to the distribution on the quaternion sphere. With nontrivial crystal symmetry, MTEX averages over symmetry-equivalent copies. The resulting ODF may not look like one of the three unsymmetrised cases. The sums \(k_1+k_4\) and \(k_2+k_3\) no longer cleanly separate the visible shapes.

The Maths Behind the Model

The Bingham density on the unit-quaternion sphere is

\[f(\mathbf{g};\mathbf{K},\mathbf{U}) = C(\mathbf{K})^{-1} \exp\!\left\{\mathbf{g}^{\mathrm{T}}\mathbf{U}\mathbf{K} \mathbf{U}^{\mathrm{T}}\mathbf{g}\right\},\qquad \mathbf{g}\in S^3,\]

where \(\mathbf{U}\) is a \(4\times4\) orthogonal matrix. Its columns \(u_{1},\ldots,u_{4}\in S^3\) are the principal quaternion axes. \(\mathbf{K}\) is diagonal, with ordered entries \(k_1\geq k_2\geq k_3\geq k_4\). The normalizing constant is \(C(\mathbf{K})\). It is a confluent hypergeometric function with matrix argument.

Adding the same constant to all four \(k_i\) changes the exponential and its normalizing constant by the same factor. Only three concentration differences are therefore identifiable. The matrix \(\mathbf{U}\) must also be estimated. This is a low-dimensional model, but it does not have only four fitted numbers.

For ordered parameters, the three non-uniform cases satisfy

The distribution is uniform if \(k_1=k_2=k_3=k_4\).

Further Reading

Next

The other model ODFs are Radial ODFs and Fibre ODFs. A nonparametric alternative does not assume one Bingham shape; see Density Estimation. The shared plotting tools are introduced in Plotting an ODF.