Texture approximation or texture modelling with components represented by the von Mises-Fisher matrix distribution on SO(3) and the Bingham distribution on S-+(4)

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-+(4) subset of R(4) was neither recognized nor appreciated. However, ever, 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 S-4 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.