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Ge Wang - One of the best experts on this subject based on the ideXlab platform.
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Gel'fand-Graev's Reconstruction Formula in the 3D real space.
Medical Physics, 2011Co-Authors: Ge WangAbstract:Purpose: Gel’fand and Graev performed classical work on the inversion of integral transforms in different spaces [Gel’fand and Graev, Funct. Anal. Appl. 25(1) 1–5 (1991)]. This paper discusses their key results for further research and development.Methods: The Gel’fand–Graev inversion Formula reveals a fundamental relationship between projection data and the Hilbert transform of an image to be reconstructed. This differential backprojection (DBP)/backprojection filtration (BPF) approach was rediscovered in the CT field, and applied in important applications such as Reconstruction from truncated projections, interior tomography, and limited-angle tomography. Here the authors present the Gel’fand–Graev inversion Formula in a 3D setting assuming the 1D x-ray transform.Results: The pseudodifferential operator is a powerful theoretical tool. There is a fundamental mathematical link between the Gel’fand–Graev Formula and the DBP (or BPF) approach in the case of the 1D x-ray transform in a 3D real space.Conclusions: This paper shows the power of mathematics for tomographic imaging and the value of a pure theoretical finding, which may appear quite irrelevant to daily healthcare at the first glance.
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Gel'fand-Graev's Reconstruction Formula in the 3D real space.
Medical physics, 2011Co-Authors: Ge WangAbstract:Gel'fand and Graev performed classical work on the inversion of integral transforms in different spaces [Gel'fand and Graev, Funct. Anal. Appl. 25(1) 1-5 (1991)]. This paper discusses their key results for further research and development. The Gel'fand-Graev inversion Formula reveals a fundamental relationship between projection data and the Hilbert transform of an image to be reconstructed. This differential backprojection (DBP)∕backprojection filtration (BPF) approach was rediscovered in the CT field, and applied in important applications such as Reconstruction from truncated projections, interior tomography, and limited-angle tomography. Here the authors present the Gel'fand-Graev inversion Formula in a 3D setting assuming the 1D x-ray transform. The pseudodifferential operator is a powerful theoretical tool. There is a fundamental mathematical link between the Gel'fand-Graev Formula and the DBP (or BPF) approach in the case of the 1D x-ray transform in a 3D real space. This paper shows the power of mathematics for tomographic imaging and the value of a pure theoretical finding, which may appear quite irrelevant to daily healthcare at the first glance.
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On a Derivative-Free Fan-Beam Reconstruction Formula
IEEE Transactions on Image Processing, 2010Co-Authors: Ge WangAbstract:We clarify that the derivative-free fan-beam Reconstruction Formula [IEEE Trans. Image Process. 2, 543-547, 1993] only allows exact Reconstruction of an object for a circular trajectory or at the origin of the coordinate system for a radially symmetric noncircular trajectory.
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Half-scan cone-beam X-ray microtomography Formula
Scanning, 2008Co-Authors: Ge Wang, T. H. Lin, Y. Liu, Ping Chin ChengAbstract:An x-ray shadow projection microtomographic system using a scannable point source is under development at AMIL-ARTS, SUNY at Buffalo. To overcome the limitations of the commonly used Feldkamp's cone-beam Reconstruction Formula, we have developed a generalized Feldkamp-type cone-beam Reconstruction Formula. In the generalized Feldkamp-type cone-beam Reconstruction, a scanning locus can be either planar or helix-like, and a transaxial slice is reconstructed using projection data collected from a 360 degrees angular range (full scan). In this paper, the full-scan cone-beam Formula is modified to require only projection data of approximate 180 degrees plus two fan-angles (half scan). First, a half-scan derivative-free noncircular fan-beam Reconstruction Formula is Formulated. Then, a half-scan cone-beam Reconstruction Formula is derived as an extension of the half-scan fan-beam Reconstruction Formula using Feldkamp's procedure. Typical numerical simulation results are given for both half-scan Formulae. Compared with the full-scan cone-beam Formula, the half-scan cone-beam Formula reduces the involved angular range of projection data and allows better longitudinal/temporal resolution.
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Exact Reconstruction for cone-beam scanning along nonstandard spirals and other curves
Developments in X-Ray Tomography IV, 2004Co-Authors: Shiying Zhao, Ge WangAbstract:In this article we consider cone-beam CT projections along a nonstandard 3-D spiral with variable radius and variable pitch. Specifically, we generalize an exact image Reconstruction Formula by Zou and Pan (2004a) and (2004b) to the case of nonstandard spirals, by giving a new, analytic proof of the Reconstruction Formula. Our proof is independent of the shape of the spiral, as long as the object is contained in a region inside the spiral, where there is a PI line passing through any interior point. Our generalized Reconstruction Formula can also be applied to much more general situations, including cone-beam scanning along standard (Pack, et al. 2004) and nonstandard saddle curves, and any smooth curve from one endpoint of a line segment to the other endpoint, for image Reconstruction of that line segment. In other words, our results can be regarded as a generalization of Orlov’s classical papers (1975) to cone-beam scanning.
Mark A. Anastasio - One of the best experts on this subject based on the ideXlab platform.
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A simple Fourier transform-based Reconstruction Formula for photoacoustic computed tomography with a circular or spherical measurement geometry
Photons Plus Ultrasound: Imaging and Sensing 2013, 2013Co-Authors: Kun Wang, Mark A. AnastasioAbstract:Photoacoustic computed tomography (PACT), also known as optoacoustic tomography or thermoacoustic tomography, is an emerging biomedical imaging technique that combines optical absorption contrast with ultrasound detection principles. Recently, a novel analytic image Reconstruction Formula has been proposed that operates on a data function expressed in the temporal frequency and spatial domains. The validity the Formula has been demonstrated for a two-dimensional (2D) circular measurement geometry. In this study, computer simulation studies are conducted to validate the Reconstruction Formula for a three-dimensional (3D) spherical measurement geometry. This Formula provides new insights into how the spatial frequency components of the sought-after object function can be explicitly determined by the temporal frequency components of the data function measured with a 2D circular or 3D spherical measurement geometry in PACT. Comparing with existing Fourier transform-based Reconstruction Formulas, the Reconstruction Formula possesses a simple structure that requires no computation of series expansions or multi-dimensional interpolation in Fourier space.
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A simple Fourier transform-based Reconstruction Formula for photoacoustic computed tomography with a circular or spherical measurement geometry
Physics in medicine and biology, 2012Co-Authors: Kun Wang, Mark A. AnastasioAbstract:Photoacoustic computed tomography (PACT), also known as optoacoustic tomography, is an emerging imaging modality that has great potential for a wide range of biomedical imaging applications. In this note, we derive a hybrid Reconstruction Formula that is mathematically exact and operates on a data function that is expressed in the temporal frequency and spatial domains. This Formula explicitly reveals new insights into how the spatial frequency components of the sought-after object function are determined by the temporal frequency components of the data function measured with a circular or spherical measurement geometry in two- and three-dimensional implementations of PACT, respectively. The structure of the Reconstruction Formula is surprisingly simple compared with existing Fourier-domain Reconstruction Formulae. It also yields a straightforward numerical implementation that is robust and two orders of magnitude more computationally efficient than filtered backprojection algorithms.
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Compensation of shear waves in photoacoustic tomography with layered acoustic media
Photons Plus Ultrasound: Imaging and Sensing 2012, 2012Co-Authors: Robert W. Schoonover, Mark A. AnastasioAbstract:An image Reconstruction Formula is presented for photoacoustic computed tomography (PCT) that is valid for a layered medium in which some of the layers may be solids and detection is performed on a planar measurement surface. It is assumed that the optical absorber is embedded in a single fluid layer and any elastic solid layers present are separated by one or more fluid layers. Computer-simulation studies are used to validate the proposed Reconstruction Formula.
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Compensation of shear waves in photoacoustic tomography with layered acoustic media
Journal of the Optical Society of America. A Optics image science and vision, 2011Co-Authors: Robert W. Schoonover, Mark A. AnastasioAbstract:An image Reconstruction Formula is presented for photoacoustic computed tomography that accounts for conversion between longitudinal and shear waves in a planar-layered acoustic medium. We assume the optical absorber that produces the photoacoustic wave field is embedded in a single fluid layer and any elastic solid layers present are separated by one or more fluid layers. The measurement aperture is assumed to be planar. Computer simulation studies are conducted to demonstrate and investigate the proposed Reconstruction Formula.
Kun Wang - One of the best experts on this subject based on the ideXlab platform.
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A simple Fourier transform-based Reconstruction Formula for photoacoustic computed tomography with a circular or spherical measurement geometry
Photons Plus Ultrasound: Imaging and Sensing 2013, 2013Co-Authors: Kun Wang, Mark A. AnastasioAbstract:Photoacoustic computed tomography (PACT), also known as optoacoustic tomography or thermoacoustic tomography, is an emerging biomedical imaging technique that combines optical absorption contrast with ultrasound detection principles. Recently, a novel analytic image Reconstruction Formula has been proposed that operates on a data function expressed in the temporal frequency and spatial domains. The validity the Formula has been demonstrated for a two-dimensional (2D) circular measurement geometry. In this study, computer simulation studies are conducted to validate the Reconstruction Formula for a three-dimensional (3D) spherical measurement geometry. This Formula provides new insights into how the spatial frequency components of the sought-after object function can be explicitly determined by the temporal frequency components of the data function measured with a 2D circular or 3D spherical measurement geometry in PACT. Comparing with existing Fourier transform-based Reconstruction Formulas, the Reconstruction Formula possesses a simple structure that requires no computation of series expansions or multi-dimensional interpolation in Fourier space.
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A simple Fourier transform-based Reconstruction Formula for photoacoustic computed tomography with a circular or spherical measurement geometry
Physics in medicine and biology, 2012Co-Authors: Kun Wang, Mark A. AnastasioAbstract:Photoacoustic computed tomography (PACT), also known as optoacoustic tomography, is an emerging imaging modality that has great potential for a wide range of biomedical imaging applications. In this note, we derive a hybrid Reconstruction Formula that is mathematically exact and operates on a data function that is expressed in the temporal frequency and spatial domains. This Formula explicitly reveals new insights into how the spatial frequency components of the sought-after object function are determined by the temporal frequency components of the data function measured with a circular or spherical measurement geometry in two- and three-dimensional implementations of PACT, respectively. The structure of the Reconstruction Formula is surprisingly simple compared with existing Fourier-domain Reconstruction Formulae. It also yields a straightforward numerical implementation that is robust and two orders of magnitude more computationally efficient than filtered backprojection algorithms.
Eugene B Postnikov - One of the best experts on this subject based on the ideXlab platform.
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On alternative wavelet Reconstruction Formula: a case study of approximate wavelets
Royal Society open science, 2014Co-Authors: Elena A Lebedeva, Eugene B PostnikovAbstract:The application of the continuous wavelet transform to study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies due to the admissibility condition. We propose an alternative Reconstruction Formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard Morlet wavelet, which is an important example of such analyzing functions, is discussed.
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on alternative wavelet Reconstruction Formula a case study of approximate wavelets
Royal Society Open Science, 2014Co-Authors: Elena A Lebedeva, Eugene B PostnikovAbstract:The application of the continuous wavelet transform to the study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies owing to the admissibility condition. We propose an alternative Reconstruction Formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard reduced Morlet wavelet, which is an important example of such analysing functions, is discussed.
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wavelet Reconstruction Formula that does not require the admissibility condition
2014Co-Authors: Elena A Lebedeva, Eugene B PostnikovAbstract:The application of the continuous wavelet transform to study of a wide class of physical processes with oscillatory dynamics is restricted by large central frequencies due to the admissibility condition. We propose an alternative Reconstruction Formula for the continuous wavelet transform, which is applicable even if the admissibility condition is violated. The case of the transform with the standard Morlet wavelet, which is an important example of such analyzing functions, is discussed.
Jun Xian - One of the best experts on this subject based on the ideXlab platform.
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Reconstruction from convolution random sampling in local shift invariant spaces
Inverse Problems, 2019Co-Authors: Jinming Wen, Jun XianAbstract:In this paper, we consider the problem of reconstructing functions in local multiply generated shift invariant spaces from convolution random samples. The sampling set is randomly chosen with one kind of probability distribution over a bounded cube and the sampled values are the convolution of the original function on sampling set. We obtain an explicit Reconstruction Formula. This Reconstruction Formula succeeds with overwhelming probability when the sampling size is sufficiently large.
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Sampling and Reconstruction in time-warped spaces and their applications
Applied Mathematics and Computation, 2004Co-Authors: Jun Xian, Wei LinAbstract:In this paper, we discuss the Reconstruction and sampling in time-warped shift-invariant spaces. The Reconstruction Formula is obtained in time-warped weighted shift-invariant spaces. As an example or application, we apply it to spline subspaces and some band-limited spaces. In spline subspaces, we show some special features and give a Reconstruction Formula in time-warped spline subspaces through another method. For some band-limited spaces and its time-warped version, we give numerical examples of Benedetto and Heller theorem.
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Reconstruction in time-warped weighted shift-invariant spaces with application to spline subspaces
International Journal of Mathematics and Mathematical Sciences, 2003Co-Authors: Jun XianAbstract:We discuss the reproducing kernel structure in shift-invariant spaces and the weighted shift-invariant spaces, and obtain the Reconstruction Formula in time-warped weighted shift-invariant spaces, then apply them to a spline subspace. In the spline subspace, we give a Reconstruction Formula in a time-warped spline subspace.