Bingham Distribution

plottingConvention.default('y↑→x');

Theory

The Bingham distribution has the density function

\[ f(g;K,U) = _1\!F_1 \left(\frac{1}{2},2,K \right)^{-1} \exp \left\{ g^T UKU g \right\},\qquad g\in S^3, \]

where \(U\) is an \(4 \times 4\) orthogonal matrix with unit quaternions \(u_{1,..,4}\in S^3\) in the columns and \(K\) is a \(4 \times 4\) diagonal matrix with the entries \(k_1,..,k_4\) describing the shape of the distribution. \(_1F_1(\cdot,\cdot,\cdot)\) is the hypergeometric function with matrix argument normalizing the density.

The shape parameters \(k_1 \ge k_2 \ge k_3 \ge k_4\) give

The general setup of the Bingham distribution in MTEX is done as follows

cs = crystalSymmetry('1');

kappa = [100 90 80 0];   % shape parameters
U     = orientation.eye(cs);  % principle axes as orthogonal orientations

odf = BinghamODF(kappa,U)

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

Lets visualize the ODF as pole figures

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

and in Euler angle space

plot(odf,'sections',6)

Estimating the parameters of a Bingham distribution

The importance of the Bingham distribution is that it is a quite low dimensional model for an orientation distribution function that still is flexible enough to represent different kinds of textures like fibers and unimodal distributions. Furthermore, we may estimate Bingham distribution from a set of individual orientations, coming e.g. from an EBSD measurement or a plasticity simulation. In contrast to kernel density estimation estimating the parameters of the Bingham distributions requires much less data. Lets demonstrate the process of fitting a Bingham ODF to experimental data. To this end we start with a randomly aligned fibre ODF

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

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

Next we use this fibre ODF to simulate only 2000 random orientations using the command discreteSample

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

To those simulated orientation data we can now fit a Bingham distribution using the command calcBinghamODF

odf = calcBinghamODF(ori)

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

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

We observe an almost perfect fit between the original fibre ODF and the Bingham distribution estimated from only 2000 randomly drawn orientations. Internally the estimate is read off the orientation tensor of the data - odf.A is its matrix of eigenvectors and odf.kappa holds the shape parameters associated with them.

Specific Bingham distributions

In the following we present the three corner cases of the Bingham distribution: the unimodal distribution, the fibre distribution, and the spherical distribution.

The unimodal case A unimodal Bingham distribution with reference orientation oriRef and kappa=40 is constructed by

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

% the corresponding Bingham ODF
odf = BinghamODF(20,oriRef)

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

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

The fibre case For a fibre symmetric Bingham distribution we simply specify the fibre and the first kappa parameter. The first two kappa parameters are always equal while the third and fourth are zero.

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 spherical case The spherical case is characterized by the fact that we have 3 equal non zero kappa coefficients.

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

Reading the estimated parameters

Whichever way the parameters were obtained, odf.A holds the four principal axes as orientations and odf.kappa the shape parameters belonging to them - A is the matrix of eigenvectors of the orientation tensor of the data and kappa is derived from its eigenvalues.

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

odf.A

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

odf.kappa.'

ans =
   100    90    80     0

The classification into the bipolar, circular and spherical case listed at the top of this page is stated for the parameters in this order. Note that it applies to the distribution on the quaternion sphere - once a nontrivial crystal symmetry is imposed, the estimated parameters describe the symmetrised distribution and the three cases are no longer cleanly separated by the sums \(k_1+k_4\) and \(k_2+k_3\).