The Mathematics of Computerized Tomography

Natterer, Frank

doi:10.1118/1.1429631

Cited by 743 publications

(397 citation statements)

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“…Step 3 (Interpolation): This step is necessary in order to make use of a fast Fourier transform (FFT) algorithm; otherwise the Fourier algorithm cannot compete in computational efficiency with other reconstruction algorithms. The FFT cannot be used on the polar grid G p,q [3]. Thus we must interpolate to a suitable Cartesian coordinate grid.…”

Section: )mentioning

confidence: 99%

“…At this point we might ask if it is possible to recover a function from its Radon transform using the backprojection operator. The next theorem, which can be found in [2,3], suggests that any attempt to reconstruct f using this strategy will yield poor results.…”

Section: Filtered Backprojectionmentioning

confidence: 99%

“…Step 3 (Discrete Backprojection): We now require a quadrature rule on S n−1 , based on the choice of θ 1 , θ 2 , ..., θ j , with positive weights α p j . Furthermore we assume that this quadrature rule is exact in H ′ 2m , the even spherical harmonics of degree 2m, for some m (see [3]). Hence…”

Section: Filtered Backprojection Algorithmmentioning

confidence: 99%

“…These integrals are collectively called the Radon transform. Radon's pioneering work was later generalized and extended to higher dimensions and remains an active area of research at the confluence of mathematical tomography, integral geometry, and inverse problems; see [2,3,4]. His work later led to some extraordinary technologies including CT scans.…”

Section: Introductionmentioning

confidence: 99%

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Recovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform

Becnel

Riser-Espinoza

2019

AJPAS

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The Radon transform maps a function on n-dimensional Euclidean space onto its integral over a hyperplane. The fields of modern computerized tomography and medical imaging are fundamentally based on the Radon transform and the computer implementation of the inversion, or reconstruction, techniques of the Radon transform. In this work we use the Radon transform with a Gaussian measure to recover random variables from their conditional expectations. We derive reconstruction algorithms for random variables of unbounded support from samples of conditional expectations and discuss the error inherent in each algorithm.

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Section: )mentioning

confidence: 99%

Section: Filtered Backprojectionmentioning

confidence: 99%

Section: Filtered Backprojection Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform

Becnel

Riser-Espinoza

2019

AJPAS

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“…The Radon transform is of substantial practical importance as it describes for example the map of a cross‐section through a patient's body onto the detector space in computer or emission tomography. For detailed information on computerized tomography see for example Natterer (1986). In a statistical framework, emission tomography was studied by Vardi et al (1985), Johnstone and Silverman (1990) and Goldenshluger and Spokoiny (2006), among others.…”

Section: Inverse Regression Modelsmentioning

confidence: 99%

Testing for Lack of Fit in Inverse Regression—with Applications to Biophotonic Imaging

Bissantz

Claeskens

Holzmann

et al. 2008

Journal of the Royal Statistical Society Series B: Statistical Methodology

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We propose two test statistics for use in inverse regression problems "Y"="K""&thgr;"+"ϵ", where "K" is a given linear operator which cannot be continuously inverted. Thus, only noisy, indirect observations "Y" for the function "&thgr;" are available. Both test statistics have a counterpart in classical hypothesis testing, where they are called the order selection test and the data-driven Neyman smooth test. We also introduce two model selection criteria which extend the classical Akaike information criterion and Bayes information criterion to inverse regression problems. In a simulation study we show that the inverse order selection and Neyman smooth tests outperform their direct counterparts in many cases. The theory is motivated by data arising in confocal fluorescence microscopy. Here, images are observed with blurring, modelled as convolution, and stochastic error at subsequent times. The aim is then to reduce the signal-to-noise ratio by averaging over the distinct images. In this context it is relevant to decide whether the images are still equal, or have changed by outside influences such as moving of the object table. Copyright (c) 2009 Royal Statistical Society.

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Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors

et al. 2015

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We present a novel ionospheric tomography reconstruction method. The method is based on Bayesian inference with the use of Gaussian Markov random field priors. We construct the priors as a system of stochastic partial differential equations. Numerical approximations of these equations can be represented with linear systems with sparse matrices, therefore providing computational efficiency. The method enables an interpretable scheme to build the prior distribution based on physical and empirical information on the structure of the ionosphere. We show through synthetic test cases in a two‐dimensional setup of latitude‐altitude slices how this method can be applied to satellite‐based ionospheric tomography and how information about the structure of the ionosphere can be implemented in the prior. The technique is capable of being easily extended to multifrequency tomographic analysis or used for the inclusion of other data sets of ionospheric electron density, such as ground‐based observations by radars or ionosondes.

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The Mathematics of Computerized Tomography

Cited by 743 publications

References 0 publications

Recovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform

Recovering a Random Variable from Conditional Expectations Using Reconstruction Algorithms for the Gauss Radon Transform

Testing for Lack of Fit in Inverse Regression—with Applications to Biophotonic Imaging

Ionospheric tomography in Bayesian framework with Gaussian Markov random field priors

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