Homometric model sets and window covariograms

Abstract

Two Delone sets are called homometric when they share the same autocorrelation or Patterson measure. A model set LAMBDA within a given cut and project scheme is a Delone set that is defined through a window W in internal space. The autocorrelation measure of LAMBDA is a pure point measure whose coefficients can be calculated via the so-called covariogram of W. Two windows with the same covariogram thus result in homometric model sets. On the other hand, the inverse problem of determining LAMBDA from its diffraction image ultimately amounts to reconstructing W from its covariogram. This is also known as Matheron's covariogram problem. It is well studied in convex geometry, where certain uniqueness results have been obtained in recent years. However, for non-convex windows, uniqueness fails in a relevant way, so that interesting applications to the homometry problem emerge. We discuss this in a simple setting and show a planar example of distinct homometric model sets.

10 pages, 6 figures

References in corpus (4)

• The covariogram determines three-dimensional convex polytopes
• Discrete Tomography of Planar Model Sets
• A Note on Shelling
• Combinatorial problems of (quasi-)crystallography

Cited by in corpus (17)

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• Kinematic Diffraction from a Mathematical Viewpoint
• The covariogram determines three-dimensional convex polytopes
• Aperiodic crystals and beyond
• Homometric Point Sets and Inverse Problems
• Stationary processes with pure point diffraction
• Extinctions and Correlations for Uniformly Discrete Point Processes with Pure Point Dynamical Spectra
• Surprises in aperiodic diffraction
• Random point sets and their diffraction
• The covariogram and Fourier-Laplace transform in ${\mathbb{C}}^n$
• The cross covariogram of a pair of polygons determines both polygons, with a few exceptions
• Covariograms generated by valuations
• How model sets can be determined by their two-point and three-point correlations
• On the reconstruction of planar lattice-convex sets from the covariogram
• Homometry and direct-sum decompositions of lattice-convex sets
• Confirmation of Matheron's conjecture on the covariogram of a planar convex body
• Some comments on the inverse problem of pure point diffraction