A novel method for the estimation of an orientation density function (odf) from diffraction pole figures is presented which is especially well suited for sharp textures and high spatial resolution pole figures measured with respect to arbitrarily scattered specimen directions, e.g. with an area detector. The method may be seen as a compromise of the two different approaches to approximate an odf and its pole figures suggested by the Darboux differential equation governing pole figures. Correspondingly, an odf is approximated with kernels which are well localized in spatial and frequency domain, more specifically with functions which are radially symmetric in spatial domain and with Fourier coefficients which vanish smoothly and sufficiently fast. This approach allows for multi-scale representation of the orientation and the pole density functions, respectively.

The estimated odf is computed as the solution of a minimization problem which is based on a model of the diffraction counts as a Poisson process. The algorithm applies dicretisation with radially symmetric basis functions approximated by finite harmonic series expansions and fast Fourier techniques to guarantee smooth approximation and high performance. An implementation of the algorithm is available as open source Matlab toolbox MTEX. MTEX provides all the properties of the estimated odf as C-coefficients, volume portions, texture index, entropy, etc., which are of interest.

The kernel approach is equally well appropriate to estimate an odf and its characteristic properties from individual orientation measurements by non-parametric density estimation. Choosing the Dirichlet kernel for this estimation, unbiased estimates of the C-coefficients up to any reasonably given finite order may be computed.

MTEX is not only a versatile toolbox for texture analysis and modeling but also provides a unique way to analyse texture based on integral or individual orientation measurements.

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