Reconstructions from Doppler Radon transforms

Stråhlén, Kent

doi:10.1109/icip.1996.561009

Cited by 8 publications

(11 citation statements)

References 6 publications

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“…This theorem can be found in [8] and in [6]. Once the divergence is known the potential part of the vector field is obtained from Theorem 4.…”

Section: Theorem 6 Iff E S(mentioning

confidence: 98%

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Exponential vector field tomography

Stråhlén¹

1997

Image Analysis and Processing

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In the exponential Radon transform in 1~2, the integrals of a scalar function f over lines, with exponential weight functions, are determined. In this paper we demonstrate how to define two different kinds of exponential Radon transforms for vector fields in R 2 in a natural way. It is shown that having data from these transforms it is possible to reconstruct the vector field uniquely. The motivation to study this problem is ultrasound measurements of flows, from which velocity spectra along lines can be determined. The first moment of these can be interpreted by means of one of the exponential vectorial Radon transforms.

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“…This theorem can be found in [8] and in [6]. Once the divergence is known the potential part of the vector field is obtained from Theorem 4.…”

Section: Theorem 6 Iff E S(mentioning

confidence: 98%

“…The theory of how to define X-ray and Radon transforms for vector fields and reconstruction from these transforms has been addressed in among others [3], [1], [7], [5], [6], [8], [9] and [10]. Given one X-ray like transform the solenoid (rotational) part of the flow can be found.…”

Section: Introductionmentioning

confidence: 99%

Exponential vector field tomography

Stråhlén¹

1997

Image Analysis and Processing

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“…A variety of numerical solvers for (2.2) have been developed in recent years using a connection between D and the Radon transform like Stråhlén [28], Andersson [2], [22]. Others rely on least squares methods (Derevtsov and Kashina [5]), the singular value decomposition (Kazantsev and Bukhgeim [12]) or iterative schemes (Wernsdörfer [30]).…”

Section: An Inversion Scheme For Dmentioning

confidence: 99%

“…by Stråhlén [28], Andersson [2], Derevtsov and Kashina [5], Kazantsev and Bukgheim [12] and Wernsdörfer [30] among others. by Stråhlén [28], Andersson [2], Derevtsov and Kashina [5], Kazantsev and Bukgheim [12] and Wernsdörfer [30] among others.…”

Section: Introductionmentioning

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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“…In fact, there exists a relationship between the two transforms (see section 2). Hence a number of algorithms for the two-dimensional case are based on this relation, leading to a method of filtered backprojection type for reconstructing the curl of f (see Stråhlén [24], Sparr et al [22], Winters and Rouseff [28]). Desbat [4] treats the two-dimensional vector field problem with direct algebraic methods and gives some efficient parallel sampling schemes.…”

Section: Introductionmentioning

confidence: 99%

An efficient mollifier method for three-dimensional vector tomography: convergence analysis and implementation

Schuster

2001

Inverse Problems

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We consider the problem of three-dimensional vector tomography, that means the reconstruction of vector fields and their curl from line integrals over certain components of the field. It is well known that only the solenoidal part of the field can be recovered from these data. In this paper the method of approximate inverse is modified for vector fields and applied to this problem, leading to an efficient solver of filtered backprojection type. We prove convergence of the reconstructed solution, if the number of data tends to infinity, which means the method is exact. Finally, numerical results are presented for a straight flow through a cylinder.

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Reconstructions from Doppler Radon transforms

Cited by 8 publications

References 6 publications

Exponential vector field tomography

Exponential vector field tomography

Error estimates for defect correction methods in Doppler tomography

An efficient mollifier method for three-dimensional vector tomography: convergence analysis and implementation

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