Wavelet Representation of Diffraction Pole Figures

ABSTACT Experimental diiraction pole gure data are thought of as being discretely sampled from pole density functions P(h; r), i.e. even probability density functions deened 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 speciic 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 brieey reviewing the basics of wavelets and the speciics of spherical wavelets, we introduce another representation of pole gures in terms of spherical wavelets. We will show that spherical wavelets are well suited to render pole gures, 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 gure 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 reenement of the spatial resolution. A diiraction pole gure is mathematically represented as the projection of an orientation density function f : SO(3) 7 ! IR 1 + basically provided by the integral operator (P h f)(r) = 1 2 Z fg2SO(3) jh=grg f(g)dv(g) = P(h; r) (1) where the function P(h; r) : S 2 S 2 7 ! IR 1 + for a given crystallographic direction h 2 S 2 IR 3 may be referred to as hyperspherical X{ray transform of f with respect to h. The path of integration fg 2 SO(3)jh = grg in (1) is a great{circle of the three-dimensional sphere S 3 IR 4 parametrized by h and r, cf. 8]. A crystallographic pole density function is the superposition of X-ray transforms with respect There are several representations of X-ray transforms and hence pole density functions, including the representation by