13 research outputs found
The polar of convex lattice sets
Let be a convex lattice set in containing the origin as the interior of its convex hull. In this paper, the definition of the polar of a convex lattice set is given both in and . Some properties and inequalities about the convex lattice sets and their polar are established
Homometric Point Sets and Inverse Problems
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.Comment: 8 page
Some open problems regarding the determination of a set from its covariogram
We present and discuss some open problems related to the determination of a set K from its covariogram, i.e. the function which provides the volumes of the intersections of K with all its possible translates
On the reconstruction of planar lattice-convex sets from the covariogram
A finite subset of is said to be lattice-convex if is
the intersection of with a convex set. The covariogram of
is the function associating to each u \in
\integer^d the cardinality of . Daurat, G\'erard, and Nivat and
independently Gardner, Gronchi, and Zong raised the problem on the
reconstruction of lattice-convex sets from . We provide a partial
positive answer to this problem by showing that for and under mild extra
assumptions, determines up to translations and reflections. As a
complement to the theorem on reconstruction we also extend the known
counterexamples (i.e., planar lattice-convex sets which are not
reconstructible, up to translations and reflections) to an infinite family of
counterexamples.Comment: accepted in Discrete and Computational Geometr
The cross covariogram of a pair of polygons determines both polygons, with a few exceptions
The cross covariogram g_{K,L} of two convex sets K and L in R^n is the
function which associates to each x in R^n the volume of the intersection of K
and L+x.
Very recently Averkov and Bianchi [AB] have confirmed Matheron's conjecture
on the covariogram problem, that asserts that any planar convex body K is
determined by the knowledge of g_{K,K}.
The problem of determining the sets from their covariogram is relevant in
probability, in statistical shape recognition and in the determination of the
atomic structure of a quasicrystal from X-ray diffraction images.
We prove that when K and L are convex polygons (and also when K and L are
planar convex cones) g_{K,L} determines both K and L, up to a described family
of exceptions. These results imply that, when K and L are in these classes, the
information provided by the cross covariogram is so rich as to determine not
only one unknown body, as required by Matheron's conjecture, but two bodies,
with a few classified exceptions.
These results are also used by Bianchi [Bia] to prove that any convex
polytope P in R^3 is determined by g_{P,P}.Comment: 26 pages, 9 figure
Phase retrieval for characteristic functions of convex bodies and reconstruction from covariograms
We propose strongly consistent algorithms for reconstructing the
characteristic function 1_K of an unknown convex body K in R^n from possibly
noisy measurements of the modulus of its Fourier transform \hat{1_K}. This
represents a complete theoretical solution to the Phase Retrieval Problem for
characteristic functions of convex bodies. The approach is via the closely
related problem of reconstructing K from noisy measurements of its covariogram,
the function giving the volume of the intersection of K with its translates. In
the many known situations in which the covariogram determines a convex body, up
to reflection in the origin and when the position of the body is fixed, our
algorithms use O(k^n) noisy covariogram measurements to construct a convex
polytope P_k that approximates K or its reflection -K in the origin. (By recent
uniqueness results, this applies to all planar convex bodies, all
three-dimensional convex polytopes, and all symmetric and most (in the sense of
Baire category) arbitrary convex bodies in all dimensions.) Two methods are
provided, and both are shown to be strongly consistent, in the sense that,
almost surely, the minimum of the Hausdorff distance between P_k and K or -K
tends to zero as k tends to infinity.Comment: Version accepted on the Journal of the American Mathematical Society.
With respect to version 1 the noise model has been greatly extended and an
appendix has been added, with a discussion of rates of convergence and
implementation issues. 56 pages, 4 figure