64,536 research outputs found
Indirect Image Registration with Large Diffeomorphic Deformations
The paper adapts the large deformation diffeomorphic metric mapping framework
for image registration to the indirect setting where a template is registered
against a target that is given through indirect noisy observations. The
registration uses diffeomorphisms that transform the template through a (group)
action. These diffeomorphisms are generated by solving a flow equation that is
defined by a velocity field with certain regularity. The theoretical analysis
includes a proof that indirect image registration has solutions (existence)
that are stable and that converge as the data error tends so zero, so it
becomes a well-defined regularization method. The paper concludes with examples
of indirect image registration in 2D tomography with very sparse and/or highly
noisy data.Comment: 43 pages, 4 figures, 1 table; revise
On random tomography with unobservable projection angles
We formulate and investigate a statistical inverse problem of a random
tomographic nature, where a probability density function on is
to be recovered from observation of finitely many of its two-dimensional
projections in random and unobservable directions. Such a problem is distinct
from the classic problem of tomography where both the projections and the unit
vectors normal to the projection plane are observable. The problem arises in
single particle electron microscopy, a powerful method that biophysicists
employ to learn the structure of biological macromolecules. Strictly speaking,
the problem is unidentifiable and an appropriate reformulation is suggested
hinging on ideas from Kendall's theory of shape. Within this setup, we
demonstrate that a consistent solution to the problem may be derived, without
attempting to estimate the unknown angles, if the density is assumed to admit a
mixture representation.Comment: Published in at http://dx.doi.org/10.1214/08-AOS673 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
A Variational Stereo Method for the Three-Dimensional Reconstruction of Ocean Waves
We develop a novel remote sensing technique for the observation of waves on the ocean surface. Our method infers the 3-D waveform and radiance of oceanic sea states via a variational stereo imagery formulation. In this setting, the shape and radiance of the wave surface are given by minimizers of a composite energy functional that combines a photometric matching term along with regularization terms involving the smoothness of the unknowns. The desired ocean surface shape and radiance are the solution of a system of coupled partial differential equations derived from the optimality conditions of the energy functional. The proposed method is naturally extended to study the spatiotemporal dynamics of ocean waves and applied to three sets of stereo video data. Statistical and spectral analysis are carried out. Our results provide evidence that the observed omnidirectional wavenumber spectrum S(k) decays as k-2.5 is in agreement with Zakharov's theory (1999). Furthermore, the 3-D spectrum of the reconstructed wave surface is exploited to estimate wave dispersion and currents
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
A graph-based mathematical morphology reader
This survey paper aims at providing a "literary" anthology of mathematical
morphology on graphs. It describes in the English language many ideas stemming
from a large number of different papers, hence providing a unified view of an
active and diverse field of research
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