2,048 research outputs found
Recovering convex boundaries from blurred and noisy observations
We consider the problem of estimating convex boundaries from blurred and
noisy observations. In our model, the convolution of an intensity function
is observed with additive Gaussian white noise. The function is assumed to
have convex support whose boundary is to be recovered. Rather than directly
estimating the intensity function, we develop a procedure which is based on
estimating the support function of the set . This approach is closely
related to the method of geometric hyperplane probing, a well-known technique
in computer vision applications. We establish bounds that reveal how the
estimation accuracy depends on the ill-posedness of the convolution operator
and the behavior of the intensity function near the boundary.Comment: Published at http://dx.doi.org/10.1214/009053606000000326 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Reconstruction of a piecewise constant conductivity on a polygonal partition via shape optimization in EIT
In this paper, we develop a shape optimization-based algorithm for the
electrical impedance tomography (EIT) problem of determining a piecewise
constant conductivity on a polygonal partition from boundary measurements. The
key tool is to use a distributed shape derivative of a suitable cost functional
with respect to movements of the partition. Numerical simulations showing the
robustness and accuracy of the method are presented for simulated test cases in
two dimensions
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Automatic, computer aided geometric design of free-knot, regression splines
A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality
On the shape-from-moments problem and recovering edges from noisy Radon data
We consider the problem of reconstructing a planar convex set from noisy observations of its moments. An estimation method based on pointwise recovering of the support function of the set is developed. We study intrinsic accuracy limitations in the shape-from-moments estimation problem by establishing a lower bound on the rate of convergence of the mean squared error. It is shown that the proposed estimator is near-optimal in the sense of the order. An application to tomographic reconstruction is discussed, and it is indicated how the proposed estimation method can be used for recovering edges from noisy Radon data
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
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