High Resolution Texture Analysis with Spherical Wavelets

What is it about?

The representations of both orientation and pole density functions in terms of spherical wavelets are exposed and their properties and relationship are discussed. The main idea of wavelet analysis is to obtain a multiscale representation of the functions or the data which allows localization in space and frequency. This mainstream idea of modern data analysis is adapted to texture analysis. Initially the pole figures are scanned on a coarse grid and a first rough approximation of the orientation density function by low-order wavelets is calculated. Next, areas of specific interest in orientation space, e.g. large values, large gradients, etc. of the first approximation, are interactively defined or automatically detected, and the assumption is imposed that the first approximation is sufficiently good everywhere else. Next, the pole figures patches corresponding to the distinguished area of orientation space are determined employing the geometry of the orientation-to-pole projection. Then the initial coarse grid is refined and additional intensities are sampled for the refined grid locations within the distinguished surface patches. Outside the surface patches, intensities associated to the grid locations are numerically determind form the first approximation of the orientation density function. Based on the additional experimental intensity data and the calculated intensities a new wavelet approximation of larger order is calculated. It improves the approximation of the orientation density function in the areas of specific interest and leaves the first rough approximation almost unchanged elsewhere. Thus, it approximates the experimental data according to both the coarse and the refined grid. This procedure may be iterated until the improvement can be neglected. The method seems appropriate to evaluate synchrotron radiation data and to resolve the “orientation distribution within a single crystal”. Also, the approach by spherical wavelets provides a means to control the sampling process of pole density functions to gradually adapt automatically to a local refinement of the spatial resolution.

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