Invertibility and stability for a generic class of radon transforms with application to dynamic operators

RabieniaHaratbar, Siamak

doi:10.1515/jiip-2018-0014

Cited by 5 publications

(3 citation statements)

References 27 publications

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“…This has been shown in the literature, for instance in [39] for generic integration curves, in [18] for dynamic CT or in [5] for dynamic PAT. The proof can be extended by analogy to our case and thus is omitted here.…”

Section: The Corresponding Canonical Relation Is Given Bymentioning

confidence: 53%

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An efficient reconstruction approach for a class of dynamic imaging operators

Hahn

Garrido

2019

Inverse Problems

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This article deals with the problem of recovering an image from tomographic data corrupted by the motion of an object. Using generalized Radon transforms as modeling for the dynamic imaging operators, we present a novel approach for the reconstruction process. The proposed method, motivated by results on microlocal analysis, approximates the solution via a filtered backprojection-type algorithm which has the advantage of being quick to implement. The numerical results applied on dynamic photoacoustic tomography (PAT) data corroborate the efficiency and relevance of the proposed method.

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Section: The Corresponding Canonical Relation Is Given Bymentioning

confidence: 53%

“…In order to verify that L Γ,P is well-defined, one typically assumes a periodic structure on Γ, a and Ψ regarding ϕ (see, for example, [5,12,18]). A setting of generic curves without periodicity assumptions has been studied in [39].…”

Section: Analysis Of the Reconstruction Formulamentioning

confidence: 99%

An efficient reconstruction approach for a class of dynamic imaging operators

Hahn

Garrido

2019

Inverse Problems

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“…For a theoretical analysis, the motion model Φ is typically assumed to satisfy the following additional conditions, cf. [8,20,21,38]:…”

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confidence: 99%

Using the Navier-Cauchy equation for motion estimation in dynamic imaging

Hahn¹,

Kienle-Garrido²,

Klingenberg³

et al. 2020

Preprint

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Tomographic image reconstruction is well understood if the specimen being studied is stationary during data acquisition. However, if this specimen changes during the measuring process, standard reconstruction techniques can lead to severe motion artefacts in the computed images. Solving a dynamic reconstruction problem therefore requires to model and incorporate suitable information on the dynamics in the reconstruction step to compensate for the motion.Many dynamic processes can be described by partial differential equations which thus could serve as additional information for the purpose of motion compensation. In this article, we consider the Navier-Cauchy equation which characterizes small elastic deformations and serves, for instance, as a model for respiratory motion. Our goal is to provide a proof-of-concept that by incorporating the deformation fields provided by this PDE, one can reduce the respective motion artefacts in the reconstructed image. To this end, we solve the Navier-Cauchy equation prior to the image reconstruction step using suitable initial and boundary data. Then, the thus computed deformation fields are incorporated into an analytic dynamic reconstruction method to compute an image of the unknown interior structure. The feasibility is illustrated with numerical examples from computerized tomography.

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