Defect correction in vector field tomography: detecting the potential part of a field using BEM and implementation of the method

Schuster, Thomas

doi:10.1088/0266-5611/21/1/006

Cited by 13 publications

(8 citation statements)

References 14 publications

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“…Again we see that the biggest part of the error occurs at the boundary, a phenomenon which was observed in other measure geometries, too, see e.g. [Sch05]. The reasons for this are not entirely clarified.…”

Section: Numerical Experimentsmentioning

confidence: 67%

An exact inversion formula for cone beam vector tomography

Katsevich

Schuster

2013

Inverse Problems

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In this article we present an improved exact inversion formula for the 3D cone beam transform of vector fields. It is well known that only the solenoidal part of a vector field can be determined by the longitudinal ray transform of a vector field in cone beam geometry. The exact inversion formula, as it was developed in A. Katsevich and T. Schuster, An exact inversion formula for cone beam vector tomography, Inverse Problems 29 (2013), consists of two parts. the first part is of filtered backprojection type, whereas the second part is a costly 4D integration and very inefficient. In this article we tackle this second term and achieve an improvement which is easily to implement and saves one order of integration. The theory says that the first part contains all information about the curl of the field, whereas the second part presumably has information about the boundary values. This suggestion is supported by the fact that the second part vanishes if the exact field is divergence free and tangential at the boundary. A number of numerical tests, that are also subject of this article, confirm the theoretical results and the exactness of the formula.1991 Mathematics Subject Classification. Primary 44A12, 65R10, 92C55.

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Section: Numerical Experimentsmentioning

confidence: 67%

An exact inversion formula for cone beam vector tomography

Katsevich

Schuster

2013

Inverse Problems

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“…The inhomogeneous case, where µ is non-constant, is studied more in detail in [5], where an injectivity theorem is proved for divergence free fields. Recent advances in the reconstruction of solenoidal fields include [15,18,19] (cf [24,25,6,13,10]).…”

Section: An Overview Of Known Resultsmentioning

confidence: 99%

The Doppler moment transform in Doppler tomography

Андерссон¹

2005

Inverse Problems

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The problem of reconstructing two-dimensional vector fields from tomographic measurements based on Doppler techniques is addressed. Measurements are made along lines, from which distributions of the field component parallel to the line are obtained. A new transform, the Doppler moment transform, which extracts local information from such data, is introduced. In contrast to most previous approaches in vector tomography that are only able to treat solenoidal fields, the Doppler moment transform provides information also of the irrotational part of the field. Specifically, it gives rise to partial differential equations in the field components. Properties of the new transform and the corresponding equations are discussed, shedding light on questions of uniqueness. It is also shown that the differential equations can be combined to form algebraic equations in the field components. Finally, a reconstruction scheme is constructed and some numerical experiments are provided.

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“…We have compared two approaches for defect correction in Doppler tomography which have been outlined in [6] and [23]. Both methods compute potentials p ( ) , = 1, 2, by solving appropriate Dirichlet and Neumann problems, respectively, and subtract ∇p ( ) from the approximate reconstruction f app .…”

Section: Discussionmentioning

confidence: 99%

Error estimates for defect correction methods in Doppler tomography

Schuster

2006

Journal of Inverse and Ill-posed Problems

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The aim of Doppler tomography is the reconstruction of velocity fields from ultrasonic measurements. The linearized model of this inverse problem is the Doppler transform which integrates projections of the fields along the lines the ultrasonic beams are sent off. Unfortunately the Doppler transform has a non-trivial null space. As a consequence only the solenoidal part of the field can be recovered from the data. This is the reason to develop and investigate methods of defect corrections, which are to improve the reconstruction accuracy. In [J. Inv. Ill-Posed Prob., 12(6), 597-626, 2004] and [Inv. Prob., 21,75-91, 2005] two approaches for the defect correction are considered which rely on the Helmholtz decomposition of vector fields. They consist of the solution of corresponding boundary value problems. This article is concerned with a comparison of these two approaches. We state error estimates based on orthogonal decompositions of vector fields, which indicate the improvement in accuracy we may hope for.

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Defect correction in vector field tomography: detecting the potential part of a field using BEM and implementation of the method

Cited by 13 publications

References 14 publications

An exact inversion formula for cone beam vector tomography

An exact inversion formula for cone beam vector tomography

The Doppler moment transform in Doppler tomography

Error estimates for defect correction methods in Doppler tomography

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