1,375 research outputs found
Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets
We consider algebraic Delone sets in the Euclidean plane and
address the problem of distinguishing convex subsets of by X-rays
in prescribed -directions, i.e., directions parallel to nonzero
interpoint vectors of . Here, an X-ray in direction of a finite
set gives the number of points in the set on each line parallel to . It is
shown that for any algebraic Delone set there are four prescribed
-directions such that any two convex subsets of can be
distinguished by the corresponding X-rays. We further prove the existence of a
natural number such that any two convex subsets of
can be distinguished by their X-rays in any set of
prescribed -directions. In particular, this
extends a well-known result of Gardner and Gritzmann on the corresponding
problem for planar lattices to nonperiodic cases that are relevant in
quasicrystallography.Comment: 21 pages, 1 figur
Regularized Newton Methods for X-ray Phase Contrast and General Imaging Problems
Like many other advanced imaging methods, x-ray phase contrast imaging and
tomography require mathematical inversion of the observed data to obtain
real-space information. While an accurate forward model describing the
generally nonlinear image formation from a given object to the observations is
often available, explicit inversion formulas are typically not known. Moreover,
the measured data might be insufficient for stable image reconstruction, in
which case it has to be complemented by suitable a priori information. In this
work, regularized Newton methods are presented as a general framework for the
solution of such ill-posed nonlinear imaging problems. For a proof of
principle, the approach is applied to x-ray phase contrast imaging in the
near-field propagation regime. Simultaneous recovery of the phase- and
amplitude from a single near-field diffraction pattern without homogeneity
constraints is demonstrated for the first time. The presented methods further
permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval
and tomographic inversion. We demonstrate the potential of this approach by
three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.Comment: (C)2016 Optical Society of America. One print or electronic copy may
be made for personal use only. Systematic reproduction and distribution,
duplication of any material in this paper for a fee or for commercial
purposes, or modifications of the content of this paper are prohibite
50 years of first passage percolation
We celebrate the 50th anniversary of one the most classical models in
probability theory. In this survey, we describe the main results of first
passage percolation, paying special attention to the recent burst of advances
of the past 5 years. The purpose of these notes is twofold. In the first
chapters, we give self-contained proofs of seminal results obtained in the '80s
and '90s on limit shapes and geodesics, while covering the state of the art of
these questions. Second, aside from these classical results, we discuss recent
perspectives and directions including (1) the connection between Busemann
functions and geodesics, (2) the proof of sublinear variance under 2+log
moments of passage times and (3) the role of growth and competition models. We
also provide a collection of (old and new) open questions, hoping to solve them
before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with
expanded and additional material. Small typos corrected throughou
Quantitative photoacoustic imaging in radiative transport regime
The objective of quantitative photoacoustic tomography (QPAT) is to
reconstruct optical and thermodynamic properties of heterogeneous media from
data of absorbed energy distribution inside the media. There have been
extensive theoretical and computational studies on the inverse problem in QPAT,
however, mostly in the diffusive regime. We present in this work some numerical
reconstruction algorithms for multi-source QPAT in the radiative transport
regime with energy data collected at either single or multiple wavelengths. We
show that when the medium to be probed is non-scattering, explicit
reconstruction schemes can be derived to reconstruct the absorption and the
Gruneisen coefficients. When data at multiple wavelengths are utilized, we can
reconstruct simultaneously the absorption, scattering and Gruneisen
coefficients. We show by numerical simulations that the reconstructions are
stable.Comment: 40 pages, 13 figure
Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations
We reconsider the conceptual foundations of the renormalization-group (RG)
formalism, and prove some rigorous theorems on the regularity properties and
possible pathologies of the RG map. Regarding regularity, we show that the RG
map, defined on a suitable space of interactions (= formal Hamiltonians), is
always single-valued and Lipschitz continuous on its domain of definition. This
rules out a recently proposed scenario for the RG description of first-order
phase transitions. On the pathological side, we make rigorous some arguments of
Griffiths, Pearce and Israel, and prove in several cases that the renormalized
measure is not a Gibbs measure for any reasonable interaction. This means that
the RG map is ill-defined, and that the conventional RG description of
first-order phase transitions is not universally valid. For decimation or
Kadanoff transformations applied to the Ising model in dimension ,
these pathologies occur in a full neighborhood of the low-temperature part of the first-order
phase-transition surface. For block-averaging transformations applied to the
Ising model in dimension , the pathologies occur at low temperatures
for arbitrary magnetic-field strength. Pathologies may also occur in the
critical region for Ising models in dimension . We discuss in detail
the distinction between Gibbsian and non-Gibbsian measures, and give a rather
complete catalogue of the known examples. Finally, we discuss the heuristic and
numerical evidence on RG pathologies in the light of our rigorous theorems.Comment: 273 pages including 14 figures, Postscript, See also
ftp.scri.fsu.edu:hep-lat/papers/9210/9210032.ps.
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