1,375 research outputs found

    Solution of a uniqueness problem in the discrete tomography of algebraic Delone sets

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    We consider algebraic Delone sets Λ\varLambda in the Euclidean plane and address the problem of distinguishing convex subsets of Λ\varLambda by X-rays in prescribed Λ\varLambda-directions, i.e., directions parallel to nonzero interpoint vectors of Λ\varLambda. Here, an X-ray in direction uu of a finite set gives the number of points in the set on each line parallel to uu. It is shown that for any algebraic Delone set Λ\varLambda there are four prescribed Λ\varLambda-directions such that any two convex subsets of Λ\varLambda can be distinguished by the corresponding X-rays. We further prove the existence of a natural number cΛc_{\varLambda} such that any two convex subsets of Λ\varLambda can be distinguished by their X-rays in any set of cΛc_{\varLambda} prescribed Λ\varLambda-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

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    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

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    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

    Ghosts in Discrete Tomography

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    Quantitative photoacoustic imaging in radiative transport regime

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    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

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    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 d≥3d \ge 3, these pathologies occur in a full neighborhood {β>β0, ∣h∣<ϵ(β)}\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \} of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension d≥2d \ge 2, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension d≥4d \ge 4. 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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