# The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis

@article{Kunze2004TheBD, title={The Bingham Distribution of Quaternions and Its Spherical Radon Transform in Texture Analysis}, author={K. Kunze and H. Schaeben}, journal={Mathematical Geology}, year={2004}, volume={36}, pages={917-943} }

AbstractSpherical geometry of quaternions is employed to characterize the Bingham distribution on the 3-dimensional sphere
$${\mathbb{S}}^3 \subset {\mathbb{R}}^4 $$
as being uniquely composed of a bipolar, a circular and a spherical component. A new parametrization of its dispersion parameters provides a classification of patterns of crystallographic preferred orientations (CPO, or textures). It is shown that the Bingham distribution can represent most types of ideal CPO patterns; in… Expand

#### 51 Citations

Symmetrized Bingham Distribution for Representing Texture: Parameter estimation with respect to crystal and sample symmetries

- Materials Science, Physics
- 2016

The quaternion Bingham distribution has been used to model preferred crystallographic orientation, or crystallographic texture, in polycrystalline materials in the materials science and geological… Expand

Representation of Crystallographic Texture using the Symmetric Bingham Distribution

- Mathematics
- 2016

Crystalline materials often exhibit a strong anisotropy, or directionally dependent response, with respect to their properties. Most structural materials are polycrystalline, composed of many… Expand

Unsupervised Learning for Efficient Texture Estimation From Limited Discrete Orientation Data

- Materials Science
- Metallurgical and Materials Transactions A
- 2013

The estimation of orientation distribution functions (ODFs) from discrete orientation data, as produced by electron backscatter diffraction or crystal plasticity micromechanical simulations, is… Expand

Hyperspherical harmonics for the representation of crystallographic texture

- Materials Science
- 2008

The feasibility of representing crystallographic textures as quaternion distributions by a series expansion method is demonstrated using hyperspherical harmonics. This approach is refined by… Expand

Monte Carlo Pose Estimation with Quaternion Kernels and the Bingham Distribution

- Computer Science
- Robotics: Science and Systems
- 2011

A new method to model the spatial distribution of oriented local features on an object is presented, which is used to infer object pose given small sets of observed local features. Expand

Inferential statistics of electron backscatter diffraction data from within individual crystalline grains

- Physics
- 2010

Highly concentrated distributed crystallographic orientation measurements within individual crystalline grains are analysed by means of ordinary statistics neglecting their spatial reference. Since… Expand

A concise quaternion geometry of rotations

- Mathematics
- 2005

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two-way correspondence… Expand

Tracking 3-D Rotations with the Quaternion Bingham Filter

- Mathematics
- 2013

A deterministic method for sequential estimation of 3-D rotations is presented. The Bingham distribution is used to represent uncertainty directly on the unit quaternion hypersphere. Quaternions… Expand

Recursive Bingham filter for directional estimation involving 180 degree symmetry

- Computer Science
- 2014

This work presents a recursive filter based on the Bingham distribution, which is defined on the considered domains and can be applied to circular filtering problems with 180 degree symmetry and to estimation of orientations in three dimensional space. Expand

Material Behavior: Texture and Anisotropy

- 2010

This contribution is an attempt to present a self-contained and comprehensive survey of the mathematics and physics of material behavior of rocks in terms of texture and anisotropy. Being generally… Expand

#### References

SHOWING 1-10 OF 25 REFERENCES

Kriging of Regionalized Directions, Axes, and Orientations I. Directions and Axes

- Mathematics
- 2002

The problem to predict a direction, axis, or orientation (rotation) from corresponding geocoded data is discussed and a general solution by virtue of embedding a sphere/hemisphere in a real vector… Expand

A concise quaternion geometry of rotations

- Mathematics
- 2005

This communication compiles propositions concerning the spherical geometry of rotations when represented by unit quaternions. The propositions are thought to establish a two-way correspondence… Expand

Texture Approximation or Texture Modelling with Components Represented by the von Mises–Fisher Matrix Distribution on SO(3) and the Bingham Distribution on S4++

- Mathematics
- 1996

Several model orientation density functions and corresponding pole density functions have been intuitively introduced into quantitative texture analysis, particularly with respect to `component fit'… Expand

Orientation Statistics Without Parametric Assumptions

- Mathematics
- 1986

SUMMARY Maximum likelihood estimation using the matrix von Mises-Fisher distribution in orientation statistics leads to unacceptably complicated likelihood equations, essentially because of the… Expand

ESTIMATION OF THE CONCENTRATION PARAMETERS OF THE FISHER MATRIX DISTRIBUTION ON 50(3) AND THE BINGHAM DISTRIBUTION ON Sq, q≥ 2

- Mathematics
- 1993

Summary
Two procedures are considered for estimating the concentration parameters of the Fisher matrix distribution for rotations or orientations in three dimensions. The first is maximum… Expand

Application of the Bingham distribution function in paleomagnetic studies

- Geology
- 1980

Fisherian statistical parameters are frequently published for paleomagnetic data that form elongate directional distributions, despite the fact that they are strictly applicable to circularly… Expand

Rotations, Quaternions, and Double Groups

- Mathematics
- 1986

Quaternions are a simple and powerful tool for handling rotations and double groups. This book gives a complete treatment of finite point groups as subgroups of the full rotation group and emphasizes… Expand

The Radon Transform

- Mathematics
- 1980

The first edition of this book has been out of print for some time and I have decided to follow the publisher's kind suggestion to prepare a new edition. Many examples with explicit inversion… Expand

Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace and Virtual Reality

- Engineering
- 2002

Ever since the Irish mathematician William Rowan Hamilton introduced quaternions in the 19th century - a feat he celebrated by carving the founding equations into a stone bridge - mathematicians and… Expand

Some notes on the fisher–bingham family on the sphere

- Mathematics
- 1988

The fisher–Bingham family is potentially useful class of spherical distributions, but its practical application has been hindered by various problems, including; the identification of the form of the… Expand