1,489 research outputs found
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
The Orlicz version of the Minkowski problem on for
An Orlicz version of the -Minkowski problem on is discussed
corresponding to the case
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
The covariogram and Fourier-Laplace transform in
The covariogram of a convex body in is the
function which associates to each the volume of the
intersection of with . Determining from the knowledge of is
known as the Covariogram Problem. It is equivalent to determining the
characteristic function of from the modulus of its Fourier transform
in , a particular instance of the Phase Retrieval
Problem.
We connect the Covariogram Problem to two aspects of the Fourier transform
seen as a function in . The first connection is with
the problem of determining from the knowledge of the zero set of
in . To attack this problem T. Kobayashi studied the
asymptotic behavior at infinity of this zero set. We obtain this asymptotic
behavior assuming less regularity on and we use this result as an essential
ingredient for proving that when is sufficiently smooth and in any
dimension , is determined by in the class of sufficiently smooth
bodies.
The second connection is with the irreducibility of the entire function
. This connection also shows a link between the Covariogram Problem
and the Pompeiu Problem in integral geometry.Comment: Version accepted on Proc. London Math. Soc. With respect to version 1
some parts of the proof of the asymptotic behavior have been clarified and
new details have been adde
Convergence in shape of Steiner symmetrizations
There are sequences of directions such that, given any compact set K in R^n,
the sequence of iterated Steiner symmetrals of K in these directions converges
to a ball. However examples show that Steiner symmetrization along a sequence
of directions whose differences are square summable does not generally
converge. (Note that this may happen even with sequences of directions which
are dense in S^{n-1}.) Here we show that such sequences converge in shape. The
limit need not be an ellipsoid or even a convex set.
We also deal with uniformly distributed sequences of directions, and with a
recent result of Klain on Steiner symmetrization along sequences chosen from a
finite set of directions.Comment: 11 page
Covariogram of non-convex sets
The covariogram of a compact set A contained in R^n is the function that to
each x in R^n associates the volume of A intersected with (A+x). Recently it
has been proved that the covariogram determines any planar convex body, in the
class of all convex bodies. We extend the class of sets in which a planar
convex body is determined by its covariogram. Moreover, we prove that there is
no pair of non-congruent planar polyominoes consisting of less than 9 points
that have equal discrete covariogram.Comment: 15 pages, 7 figures, accepted for publication on Mathematik
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