Local Tomography

Faridani, Adel; Ritman, Erik L.; Smith, Kennan T.

doi:10.1137/0152026

Cited by 190 publications

(170 citation statements)

References 9 publications

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“…As we do not know an inversion formula for F we do not try to reconstruct n directly from its integrals g = F n over ellipses. Instead, we employ ideas from Lambda tomography [9] and define the reconstruction operator Λ = ∆F * ΦF where Φ = Φ(s, t) is a smooth compactly supported cutoff function, ∆ is the Laplacian, and F * is the formal (weighted) L 2 -adjoint of F satisfying…”

Section: 3mentioning

confidence: 99%

“…Let F denote this elliptic Radon transform. Then, our imaging operator is Λ = ∆F * ΦF where ∆ is the Laplacian, F * is the formal adjoint in an L 2 -space with a smooth weight, see (9) below, and Φ is a smooth cutoff function such that F * ΦF is well defined. We argue that Λ is a pseudodifferential operator of order 1 and hence emphasizes some singularities, see [16], that is, Λf images some of the discontinuities of f .…”

Section: Introductionmentioning

confidence: 99%

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Approximate inverse for the common offset acquisition geometry in 2D seismic imaging

Grathwohl¹,

Kunstmann²,

Quinto³

et al. 2017

Inverse Problems

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Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Approximate inverse for the common offset acquisition geometry in 2D seismic imaging

Grathwohl¹,

Kunstmann²,

Quinto³

et al. 2017

Inverse Problems

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“…The derivation of Lamda CT is explained in references [1], [2], [3], and [4]. Since the primary difference between Global and Local CT concerns the design of the convolution filter, two versions of a Local CT filter are discussed here.…”

Section: Introductionmentioning

confidence: 99%

Development and Application of Local 3-D CT Reconstruction Software for Imaging Critical Regions in Large Ceramic Turbine Rotors

Sivers

Holloway

Ellingson

1993

Review of Progress in Quantitative Nondestructive Evaluation

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“…This is followed by a brief outline of the FBP and WBP methods. We then move on to our algorithm, electron lambdatomography (ELT), which is based on two-dimensional lambda tomography (12)(13)(14). However, ELT is also valid for a broad range of three-dimensional data acquisition geometries.…”

mentioning

confidence: 99%

Electron lambda-tomography

Quinto

Skoglund²,

Öktem³

2009

Proc. Natl. Acad. Sci. U.S.A.

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Filtered back-projection and weighted back-projection have long been the methods of choice within the electron microscopy community for reconstructing the structure of macromolecular assemblies from electron tomography data. Here, we describe electron lambda-tomography, a reconstruction method that enjoys the benefits of the above mentioned methods, namely speed and ease of implementation, but also addresses some of their shortcomings. In particular, compared to these standard methods, electron lambdatomography is less sensitive to artifacts that come from structures outside the region that is being reconstructed, and it can sharpen boundaries.local tomography | limited angle tomography | microlocal analysis T his article describes a reconstruction method applicable to electron tomography (ET). The rigorous mathematical description of the method and its application to ET is given in ref.1. Here we concentrate on the functionality of the method in an experimental setting with tests on real ET data. Furthermore, we derive a heuristic explanation for its advantages and guidelines for its usage. Finally, we compare it with the most widely used methods in the field, namely filtered back-projection (FBP) and weighted back-projection (WBP). In this context, it should be mentioned that other reconstruction methods have also been developed and applied to ET. Iterative methods, such as algebraic reconstruction technique (ART) and simultaneous iterative reconstruction technique (SIRT) (2, 3), became practically applicable to ET only after regularization through early stopping. A clever discretization, based on Kaiser-Bessel window functions (blobs), was combined with strongly over-relaxed ART and then applied to ET data in refs. 4-6. Another approach is based on variational regularization where in refs. 7 and 8 relative entropy regularization is applied to ET. For more on these other approaches and their merits, we refer to refs. 9, (section 10.2), 10, and 11.We begin with a very brief introduction to ET, including a discussion of the various data collection geometries and a mathematical formulation of the structure determination problem in ET. This is followed by a brief outline of the FBP and WBP methods. We then move on to our algorithm, electron lambdatomography (ELT), which is based on two-dimensional lambda tomography (12)(13)(14). However, ELT is also valid for a broad range of three-dimensional data acquisition geometries. It is a method that maintains the main benefits of the FBP and WBP methods, namely speed and ease of implementation, while addressing some of the shortcomings. In particular, ELT is generally less sensitive to artifacts that come from structures outside the region of interest (ROI) than these other methods. We conclude by providing examples of reconstructions obtained by ELT from real and simulated ET data.Basic Notation. We now introduce notation used throughout the paper. We let R denote the set of real numbers and R + the set of positive real numbers. The three-dimensional space is denoted by ...

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Local Tomography

Cited by 190 publications

References 9 publications

Approximate inverse for the common offset acquisition geometry in 2D seismic imaging

Approximate inverse for the common offset acquisition geometry in 2D seismic imaging

Development and Application of Local 3-D CT Reconstruction Software for Imaging Critical Regions in Large Ceramic Turbine Rotors

Electron lambda-tomography

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