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
a bipolar distribution, if \(k_1 + k_4 > k_2 + k_3\),
a circular distribution, if \(k_1 + k_4 = k_2 + k_3\),
a spherical distribution, if \(k_1 + k_4 < k_2 + k_3\),
a uniform distribution, if \(k_1 = k_2 = k_3 = k_4\),
The general setup of the Bingham distribution in MTEX is done as follows
The bipolar case and unimodal distribution
First, we define some unimodal odf
Next, we simulate individual orientations from this odf, in a scattered axis/angle plot in which the simulated data looks like a sphere
From this simulated EBSD data, we can estimate the parameters of the Bingham distribution,
TODO
where U is the orthogonal matrix of eigenvectors of the orientation tensor and kappa the shape parameters associated with the U.
next, we test the different cases of the distribution on rejection
The spherical test case failed to reject for some level of significance, hence we would dismiss the hypothesis prolate and oblate.
Prolate case and fiber distribution
The prolate case corresponds to a fiber.
As before, we generate some random orientations from a model odf. The shape in an axis/angle scatter plot reminds of a cigar
We estimate the parameters of the Bingham distribution
and test on the three cases
The test clearly rejects the spherical and prolate case, but not the prolate. We construct the Bingham distribution from the parameters, it might show some skewness
Oblate case
The oblate case of the Bingham distribution has no direct counterpart in terms of texture components, thus we can construct it straightforward
The oblate cases in axis/angle space remind on a disk
We estimate the parameters again
and do the tests
the spherical and oblate case are clearly rejected, the prolate case failed to reject for some level of significance