21,315 research outputs found
On the reconstruction of convex sets from random normal measurements
We study the problem of reconstructing a convex body using only a finite
number of measurements of outer normal vectors. More precisely, we suppose that
the normal vectors are measured at independent random locations uniformly
distributed along the boundary of our convex set. Given a desired Hausdorff
error eta, we provide an upper bounds on the number of probes that one has to
perform in order to obtain an eta-approximation of this convex set with high
probability. Our result rely on the stability theory related to Minkowski's
theorem
Fitting Tractable Convex Sets to Support Function Evaluations
The geometric problem of estimating an unknown compact convex set from
evaluations of its support function arises in a range of scientific and
engineering applications. Traditional approaches typically rely on estimators
that minimize the error over all possible compact convex sets; in particular,
these methods do not allow for the incorporation of prior structural
information about the underlying set and the resulting estimates become
increasingly more complicated to describe as the number of measurements
available grows. We address both of these shortcomings by describing a
framework for estimating tractably specified convex sets from support function
evaluations. Building on the literature in convex optimization, our approach is
based on estimators that minimize the error over structured families of convex
sets that are specified as linear images of concisely described sets -- such as
the simplex or the spectraplex -- in a higher-dimensional space that is not
much larger than the ambient space. Convex sets parametrized in this manner are
significant from a computational perspective as one can optimize linear
functionals over such sets efficiently; they serve a different purpose in the
inferential context of the present paper, namely, that of incorporating
regularization in the reconstruction while still offering considerable
expressive power. We provide a geometric characterization of the asymptotic
behavior of our estimators, and our analysis relies on the property that
certain sets which admit semialgebraic descriptions are Vapnik-Chervonenkis
(VC) classes. Our numerical experiments highlight the utility of our framework
over previous approaches in settings in which the measurements available are
noisy or small in number as well as those in which the underlying set to be
reconstructed is non-polyhedral.Comment: 35 pages, 80 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
Phase Retrieval via Matrix Completion
This paper develops a novel framework for phase retrieval, a problem which
arises in X-ray crystallography, diffraction imaging, astronomical imaging and
many other applications. Our approach combines multiple structured
illuminations together with ideas from convex programming to recover the phase
from intensity measurements, typically from the modulus of the diffracted wave.
We demonstrate empirically that any complex-valued object can be recovered from
the knowledge of the magnitude of just a few diffracted patterns by solving a
simple convex optimization problem inspired by the recent literature on matrix
completion. More importantly, we also demonstrate that our noise-aware
algorithms are stable in the sense that the reconstruction degrades gracefully
as the signal-to-noise ratio decreases. Finally, we introduce some theory
showing that one can design very simple structured illumination patterns such
that three diffracted figures uniquely determine the phase of the object we
wish to recover
Near Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
Suppose we are given a vector in . How many linear measurements do
we need to make about to be able to recover to within precision
in the Euclidean () metric? Or more exactly, suppose we are
interested in a class of such objects--discrete digital signals,
images, etc; how many linear measurements do we need to recover objects from
this class to within accuracy ? This paper shows that if the objects
of interest are sparse or compressible in the sense that the reordered entries
of a signal decay like a power-law (or if the coefficient
sequence of in a fixed basis decays like a power-law), then it is possible
to reconstruct to within very high accuracy from a small number of random
measurements.Comment: 39 pages; no figures; to appear. Bernoulli ensemble proof has been
corrected; other expository and bibliographical changes made, incorporating
referee's suggestion
- …