# Homometric Point Sets and Inverse Problems - Mathematics > Metric Geometry

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Abstract: The inverse problem of diffraction theory in essence amounts to thereconstruction of the atomic positions of a solid from its diffraction image.From a mathematical perspective, this is a notoriously difficult problem,even in the idealised situation of perfect diffraction from an infinitestructure.Here, the problem is analysed via the autocorrelation measure of theunderlying point set, where two point sets are called homometric when theyshare the same autocorrelation. For the class of mathematical quasicrystalswithin a given cut and project scheme, the homometry problem becomes equivalentto Matheron-s covariogram problem, in the sense of determining the window fromits covariogram. Although certain uniqueness results are known for convexwindows, interesting examples of distinct homometric model sets already emergein the plane.The uncertainty level increases in the presence of diffuse scattering.Already in one dimension, a mixed spectrum can be compatible with structures ofdifferent entropy. We expand on this example by constructing a family of mixedsystems with fixed diffraction image but varying entropy. We also outline howthis generalises to higher dimension.

Author: ** Uwe Grimm Milton Keynes, Michael Baake Bielefeld**

Source: https://arxiv.org/