6 research outputs found
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
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
Optimal rates of convergence for convex set estimation from support functions
We present a minimax optimal solution to the problem of estimating a compact,
convex set from finitely many noisy measurements of its support function. The
solution is based on appropriate regularizations of the least squares
estimator. Both fixed and random designs are considered.Comment: Published in at http://dx.doi.org/10.1214/11-AOS959 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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 allow for limited 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 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
On the estimation of a convex set with corners
In robotic vision using laser-radar measurements, noisy data on convex sets with corners are derived in terms of the set's support function. The corners represent abutting edges of manufactured items, and convey important information about the items' shape. However, simple methods for set estimation, for example based on fitting random polygons or smooth sets, either add additional corners as an artifact of the algorithm, or approximate corners by smooth curves. It might be argued, however, that corners have special significance in the interpretation of a set, and should not be introduced as an artifact of the estimation procedure. In this paper we suggest a corner-diagnostic approach, in the form of a three-step algorithm which (a) identifies the number and positions of corners, (b) fits smooth curves between corners, and (c) splices together the smooth curves and the corners, to produce an over-all estimate of the convex set. The corner-finding step is parametric in character, and although it is based on detecting change points in high-order derivatives of the support function, it produces root-n consistent estimators of the locations of corners. On the other hand, the smooth-curve fitting step is entirely nonparametric. The splicing step marries these two disparate approaches into a single, practical methodP. Hall and B.A. Turlac