WAVELET REPRESENTATION OF DIFFRACTION POLE FIGURES

摘要

Experimental diffraction pole figure data are thought of as being discretely sampled from pole density functions P(h, r), i.e. even probability density functions defined on the cross-product S~2 × S~2 of two unit spheres. Several useful representations of pole density functions exist which are usually related to specific purposes: (ⅰ) series expansion into spherical harmonics, (ⅱ) series expansion into (unimodal) radial basis functions, (ⅲ) series expansion into piecewise constant functions. Their critical parameter may to some extent be adjusted to the total number and/or the spatial arrangement of the intensity measurements. However, they are in no way involved in the sampling process itself. After briefly reviewing the basics of wavelets and the specifics of spherical wavelets, we introduce another representation of pole figures in terms of spherical wavelets. We will show that spherical wavelets are well suited to render pole figures, and to resolve the inverse problem. Moreover, we will demonstrate that wavelets are well apt to allow for locally varying spatial resolution, thus providing a digital device to zoom into pole figure areas of special interest. Considering a measuring time of roughly 1 hour for 1000 intensity values, such a device seems to be required to increase the spatial resolution by a factor of 1000 or greater locally. Eventually, we shall present a promising prospect that wavelets provide the means to control the texture goniometer and the sampling process to gradually adapt automatically to a local refinement of the spatial resolution.