5,620 research outputs found

    An integral geometry lemma and its applications: the nonlocality of the Pavlov equation and a tomographic problem with opaque parabolic objects

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    As in the case of soliton PDEs in 2+1 dimensions, the evolutionary form of integrable dispersionless multidimensional PDEs is non-local, and the proper choice of integration constants should be the one dictated by the associated Inverse Scattering Transform (IST). Using the recently made rigorous IST for vector fields associated with the so-called Pavlov equation vxt+vyy+vxvxy−vyvxx=0v_{xt}+v_{yy}+v_xv_{xy}-v_yv_{xx}=0, we have recently esatablished that, in the nonlocal part of its evolutionary form vt=vxvy−∂x−1 ∂y [vy+vx2]v_{t}= v_{x}v_{y}-\partial^{-1}_{x}\,\partial_{y}\,[v_{y}+v^2_{x}], the formal integral ∂x−1\partial^{-1}_{x} corresponding to the solutions of the Cauchy problem constructed by such an IST is the asymmetric integral −∫x∞dx′-\int_x^{\infty}dx'. In this paper we show that this results could be guessed in a simple way using a, to the best of our knowledge, novel integral geometry lemma. Such a lemma establishes that it is possible to express the integral of a fairly general and smooth function f(X,Y)f(X,Y) over a parabola of the (X,Y)(X,Y) plane in terms of the integrals of f(X,Y)f(X,Y) over all straight lines non intersecting the parabola. A similar result, in which the parabola is replaced by the circle, is already known in the literature and finds applications in tomography. Indeed, in a two-dimensional linear tomographic problem with a convex opaque obstacle, only the integrals along the straight lines non-intersecting the obstacle are known, and in the class of potentials f(X,Y)f(X,Y) with polynomial decay we do not have unique solvability of the inverse problem anymore. Therefore, for the problem with an obstacle, it is natural not to try to reconstruct the complete potential, but only some integral characteristics like the integral over the boundary of the obstacle. Due to the above two lemmas, this can be done, at the moment, for opaque bodies having as boundary a parabola and a circle (an ellipse).Comment: LaTeX, 13 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1507.0820

    Histogram Tomography

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    In many tomographic imaging problems the data consist of integrals along lines or curves. Increasingly we encounter "rich tomography" problems where the quantity imaged is higher dimensional than a scalar per voxel, including vectors tensors and functions. The data can also be higher dimensional and in many cases consists of a one or two dimensional spectrum for each ray. In many such cases the data contain not just integrals along rays but the distribution of values along the ray. If this is discretized into bins we can think of this as a histogram. In this paper we introduce the concept of "histogram tomography". For scalar problems with histogram data this holds the possibility of reconstruction with fewer rays. In vector and tensor problems it holds the promise of reconstruction of images that are in the null space of related integral transforms. For scalar histogram tomography problems we show how bins in the histogram correspond to reconstructing level sets of function, while moments of the distribution are the x-ray transform of powers of the unknown function. In the vector case we give a reconstruction procedure for potential components of the field. We demonstrate how the histogram longitudinal ray transform data can be extracted from Bragg edge neutron spectral data and hence, using moments, a non-linear system of partial differential equations derived for the strain tensor. In x-ray diffraction tomography of strain the transverse ray transform can be deduced from the diffraction pattern the full histogram transverse ray transform cannot. We give an explicit example of distributions of strain along a line that produce the same diffraction pattern, and characterize the null space of the relevant transform.Comment: Small corrections from last versio

    3D Coronal Density Reconstruction and Retrieving the Magnetic Field Structure during Solar Minimum

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    Measurement of the coronal magnetic field is a crucial ingredient in understanding the nature of solar coronal phenomena at all scales. We employed STEREO/COR1 data obtained during a deep minimum of solar activity in February 2008 (Carrington rotation CR 2066) to retrieve and analyze the three-dimensional (3D) coronal electron density in the range of heights from 1.5 to 4 Rsun using a tomography method. With this, we qualitatively deduced structures of the coronal magnetic field. The 3D electron density analysis is complemented by the 3D STEREO/EUVI emissivity in the 195 A band obtained by tomography for the same CR. A global 3D MHD model of the solar corona was used to relate the reconstructed 3D density and emissivity to open/closed magnetic field structures. We show that the density maximum locations can serve as an indicator of current sheet position, while the locations of the density gradient maximum can be a reliable indicator of coronal hole boundaries. We find that the magnetic field configuration during CR 2066 has a tendency to become radially open at heliocentric distances greater than 2.5 Rsun. We also find that the potential field model with a fixed source surface (PFSS) is inconsistent with the boundaries between the regions with open and closed magnetic field structures. This indicates that the assumption of the potential nature of the coronal global magnetic field is not satisfied even during the deep solar minimum. Results of our 3D density reconstruction will help to constrain solar coronal field models and test the accuracy of the magnetic field approximations for coronal modeling.Comment: Published in "Solar Physics

    Gaussian process tomography for soft x-ray spectroscopy at WEST without equilibrium information

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    International audienceGaussian process tomography (GPT) is a recently developed tomography method based on the Bayesian probability theory [J. Svensson, JET Internal Report EFDA-JET-PR(11)24, 2011 and Li et al., Rev. Sci. Instrum. 84, 083506 (2013)]. By modeling the soft X-ray (SXR) emissivity field in a poloidal cross section as a Gaussian process, the Bayesian SXR tomography can be carried out in a robust and extremely fast way. Owing to the short execution time of the algorithm, GPT is an important candidate for providing real-time reconstructions with a view to impurity transport and fast magnetohydrodynamic control. In addition, the Bayesian formalism allows quantifying uncertainty on the inferred parameters. In this paper, the GPT technique is validated using a synthetic data set expected from the WEST tokamak, and the results are shown of its application to the reconstruction of SXR emissivity profiles measured on Tore Supra. The method is compared with the standard algorithm based on minimization of the Fisher information

    Tomographic reconstruction of quantum states in N spatial dimensions

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    Most quantum tomographic methods can only be used for one-dimensional problems. We show how to infer the quantum state of a non-relativistic N-dimensional harmonic oscillator system by simple inverse Radon transforms. The procedure is equally applicable to finding the joint quantum state of several distinguishable particles in different harmonic oscillator potentials. A requirement of the procedure is that the angular frequencies of the N harmonic potentials are incommensurable. We discuss what kind of information can be found if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite time measurements. As a further example of quantum state reconstruction in N dimensions we consider the two related cases of an N-dimensional free particle with periodic boundary conditions and a particle in an N-dimensional box, where we find a similar condition of incommensurability and finite recurrence time for the one-dimensional system.Comment: 8 pages, 1 figur

    Direct and Inverse Computational Methods for Electromagnetic Scattering in Biological Diagnostics

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    Scattering theory has had a major roll in twentieth century mathematical physics. Mathematical modeling and algorithms of direct,- and inverse electromagnetic scattering formulation due to biological tissues are investigated. The algorithms are used for a model based illustration technique within the microwave range. A number of methods is given to solve the inverse electromagnetic scattering problem in which the nonlinear and ill-posed nature of the problem are acknowledged.Comment: 61 pages, 5 figure

    Visual Quality Enhancement in Optoacoustic Tomography using Active Contour Segmentation Priors

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    Segmentation of biomedical images is essential for studying and characterizing anatomical structures, detection and evaluation of pathological tissues. Segmentation has been further shown to enhance the reconstruction performance in many tomographic imaging modalities by accounting for heterogeneities of the excitation field and tissue properties in the imaged region. This is particularly relevant in optoacoustic tomography, where discontinuities in the optical and acoustic tissue properties, if not properly accounted for, may result in deterioration of the imaging performance. Efficient segmentation of optoacoustic images is often hampered by the relatively low intrinsic contrast of large anatomical structures, which is further impaired by the limited angular coverage of some commonly employed tomographic imaging configurations. Herein, we analyze the performance of active contour models for boundary segmentation in cross-sectional optoacoustic tomography. The segmented mask is employed to construct a two compartment model for the acoustic and optical parameters of the imaged tissues, which is subsequently used to improve accuracy of the image reconstruction routines. The performance of the suggested segmentation and modeling approach are showcased in tissue-mimicking phantoms and small animal imaging experiments.Comment: Accepted for publication in IEEE Transactions on Medical Imagin

    Singular value decomposition for the 2D fan-beam Radon transform of tensor fields

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    In this article we study the fan-beam Radon transform Dm{\cal D}_m of symmetrical solenoidal 2D tensor fields of arbitrary rank mm in a unit disc D\mathbb D as the operator, acting from the object space L2(D;Sm){\mathbf L}_{2}(\mathbb D; {\bf S}_m) to the data space L2([0,2π)×[0,2π)).L_2([0,2\pi)\times[0,2\pi)). The orthogonal polynomial basis sn,k(±m){\bf s}^{(\pm m)}_{n,k} of solenoidal tensor fields on the disc D\mathbb D was built with the help of Zernike polynomials and then a singular value decomposition (SVD) for the operator Dm{\cal D}_m was obtained. The inversion formula for the fan-beam tensor transform Dm{\cal D}_m follows from this decomposition. Thus obtained inversion formula can be used as a tomographic filter for splitting a known tensor field into potential and solenoidal parts. Numerical results are presented.Comment: LaTeX, 37 pages with 5 figure
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