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

Several model orientation density functions and corresponding pole density functions have been intuitively introduced into quantitative texture analysis, particularly with respect to `component fit' methods. Their relation to the von Mises–Fisher matrix distribution on SO(3) or equivalently to the Bingham distribution of axes on S<sup>4</sup><sub>+</sub> ⊂ was neither recognized nor appreciated. However, most of the model functions suggested and applied in component-fit methods actually reduce to special cases of the von Mises–Fisher matrix distribution, particularly to the unimodal and the circular (`fibre') case. Thus it provides a general mathematical model orientation density function without, however, any theoretical justification in terms of texture formation. Its one-one correspondence with the Bingham distribution on S⁴ is discussed in terms of implications for future applications in texture analysis. Also, a critical appraisal of component-fit methods is given including a generalization towards a full inversion method.