Homometric point sets and inverse problems

Grimm U, Baake M (2008)
Zeitschrift für Kristallographie 223(11-12): 777-781.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Grimm, Uwe; Baake, MichaelUniBi
Abstract / Bemerkung
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.
Stichworte
Quasicrystal; Inverse problem; Model set; Homometry; Entropy; Diffraction
Erscheinungsjahr
2008
Zeitschriftentitel
Zeitschrift für Kristallographie
Band
223
Ausgabe
11-12
Seite(n)
777-781
ISSN
0044-2968
Page URI
https://pub.uni-bielefeld.de/record/1634235

Zitieren

Grimm U, Baake M. Homometric point sets and inverse problems. Zeitschrift für Kristallographie. 2008;223(11-12):777-781.
Grimm, U., & Baake, M. (2008). Homometric point sets and inverse problems. Zeitschrift für Kristallographie, 223(11-12), 777-781. https://doi.org/10.1524/zkri.2008.1043
Grimm, U., and Baake, M. (2008). Homometric point sets and inverse problems. Zeitschrift für Kristallographie 223, 777-781.
Grimm, U., & Baake, M., 2008. Homometric point sets and inverse problems. Zeitschrift für Kristallographie, 223(11-12), p 777-781.
U. Grimm and M. Baake, “Homometric point sets and inverse problems”, Zeitschrift für Kristallographie, vol. 223, 2008, pp. 777-781.
Grimm, U., Baake, M.: Homometric point sets and inverse problems. Zeitschrift für Kristallographie. 223, 777-781 (2008).
Grimm, Uwe, and Baake, Michael. “Homometric point sets and inverse problems”. Zeitschrift für Kristallographie 223.11-12 (2008): 777-781.

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