The Open UniversitySkip to content

Homometric point sets and inverse problems

Grimm, Uwe and Baake, Michael (2008). Homometric point sets and inverse problems. Zeitschrift für Kristallographie, 223(11-12) pp. 777–781.

Full text available as:
PDF (Accepted Manuscript) - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (129kB)
DOI (Digital Object Identifier) Link:
Google Scholar: Look up in Google Scholar


The inverse problem of diffraction theory in essence amounts to the reconstruction 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 infinite structure.

Here, the problem is analysed via the autocorrelation measure of the underlying point set, where two point sets are called homometric when they share the same autocorrelation. For the class of mathematical quasicrystals within a given cut and project scheme, the homometry problem becomes equivalent to Matheron´s covariogram problem, in the sense of determining the window from its covariogram. Although certain uniqueness results are known for convex windows, interesting examples of distinct homometric model sets already emerge in the plane.

The uncertainty level increases in the presence of diffuse scattering. Already in one dimension, a mixed spectrum can be compatible with structures of different entropy. We expand on this example by constructing a family of mixed systems with fixed diffraction image but varying entropy. We also outline how this generalises to higher dimension.

Item Type: Journal Item
Copyright Holders: 2008 Oldenbourg Wissenschaftsverlag, Mu¨nchen
ISSN: 0044-2968
Project Funding Details:
Funded Project NameProject IDFunding Body
Combinatorics of Sequences and Tilings and its ApplicationsEP/D058465/1EPSRC (Engineering and Physical Sciences Research Council)
Academic Unit/School: Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
Faculty of Science, Technology, Engineering and Mathematics (STEM)
Item ID: 14992
Depositing User: Uwe Grimm
Date Deposited: 23 Feb 2009 14:08
Last Modified: 08 Dec 2018 03:51
Share this page:


Altmetrics from Altmetric

Citations from Dimensions

Download history for this item

These details should be considered as only a guide to the number of downloads performed manually. Algorithmic methods have been applied in an attempt to remove automated downloads from the displayed statistics but no guarantee can be made as to the accuracy of the figures.

Actions (login may be required)

Policies | Disclaimer

© The Open University   contact the OU