17,264 research outputs found
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
Tiling groupoids and Bratteli diagrams
Let T be an aperiodic and repetitive tiling of R^d with finite local
complexity. Let O be its tiling space with canonical transversal X. The tiling
equivalence relation R_X is the set of pairs of tilings in X which are
translates of each others, with a certain (etale) topology. In this paper R_X
is reconstructed as a generalized "tail equivalence" on a Bratteli diagram,
with its standard AF-relation as a subequivalence relation.
Using a generalization of the Anderson-Putnam complex, O is identified with
the inverse limit of a sequence of finite CW-complexes. A Bratteli diagram B is
built from this sequence, and its set of infinite paths dB is homeomorphic to
X. The diagram B is endowed with a horizontal structure: additional edges that
encode the adjacencies of patches in T. This allows to define an etale
equivalence relation R_B on dB which is homeomorphic to R_X, and contains the
AF-relation of "tail equivalence".Comment: 34 pages, 4 figure
Convergence of algorithms for reconstructing convex bodies and directional measures
We investigate algorithms for reconstructing a convex body in from noisy measurements of its support function or its brightness
function in directions . The key idea of these algorithms is
to construct a convex polytope whose support function (or brightness
function) best approximates the given measurements in the directions
(in the least squares sense). The measurement errors are assumed
to be stochastically independent and Gaussian. It is shown that this procedure
is (strongly) consistent, meaning that, almost surely, tends to in
the Hausdorff metric as . Here some mild assumptions on the
sequence of directions are needed. Using results from the theory of
empirical processes, estimates of rates of convergence are derived, which are
first obtained in the metric and then transferred to the Hausdorff
metric. Along the way, a new estimate is obtained for the metric entropy of the
class of origin-symmetric zonoids contained in the unit ball. Similar results
are obtained for the convergence of an algorithm that reconstructs an
approximating measure to the directional measure of a stationary fiber process
from noisy measurements of its rose of intersections in directions
. Here the Dudley and Prohorov metrics are used. The methods are
linked to those employed for the support and brightness function algorithms via
the fact that the rose of intersections is the support function of a projection
body.Comment: Published at http://dx.doi.org/10.1214/009053606000000335 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Sampling of stochastic operators
We develop sampling methodology aimed at determining stochastic operators
that satisfy a support size restriction on the autocorrelation of the operators
stochastic spreading function. The data that we use to reconstruct the operator
(or, in some cases only the autocorrelation of the spreading function) is based
on the response of the unknown operator to a known, deterministic test signal
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