A Reconstruction Approach for Imaging in  3D Cone Beam Vector Field Tomography

Schuster, Thomas; Theis, David; Louis, Alfred K.

doi:10.1155/2008/174283

Cited by 19 publications

(25 citation statements)

References 28 publications

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“…Hence, the highest possible convergence order is achieved when 1 = (1 − ν)/ν ⇔ ν = 1/2. But for ν = 1/2, we immediately get (36) from (24). .…”

Section: Lemma 6 From (34) We Deduce Thatmentioning

confidence: 92%

“…These reconstruction kernels are solutions of a dual equation that is associated with a socalled mollifier that implies the regularization property. The method is well established in Hilbert space settings and was successfully applied to different problems such as computerized tomography, Louis, Schuster [12], inverse scattering, Abdullah, Louis [1], Doppler tomography [19,20,24], x-ray diffractometry [23] and even image processing, Louis [11]. A concise monograph on that method is Schuster [22].…”

Section: Solving Linear Operator Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse

Schuster¹,

Schöpfer²

2010

Inverse Problems

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The method of approximate inverse is a mollification method for stably solving inverse problems. In its original form it has been developed to solve operator equations in L 2-spaces and general Hilbert spaces. We show that the method of approximate inverse can be extended to solve linear, ill-posed problems in Banach spaces. This paper is restricted to function spaces. The method itself consists of evaluations of dual pairings of the given data with reconstruction kernels that are associated with mollifiers and the dual of the operator. We first define what we mean by a mollifier in general Banach spaces and then investigate two settings more exactly: the case of L p-spaces and the case of the Banach space of continuous functions on a compact set. For both settings we present the criteria turning the method of approximate inverse into a regularization method and prove convergence with rates. As an application we refer to x-ray diffractometry which is a technique of non-destructive testing that is concerned with computing the stress tensor of a specimen. Since one knows that the stress tensor is smooth, x-ray diffractometry can appropriately be modelled by a Banach space setting using continuous functions.

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“…Hence, the highest possible convergence order is achieved when 1 = (1 − ν)/ν ⇔ ν = 1/2. But for ν = 1/2, we immediately get (36) from (24). .…”

Section: Lemma 6 From (34) We Deduce Thatmentioning

confidence: 92%

Section: Solving Linear Operator Equationsmentioning

confidence: 99%

Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse

Schuster¹,

Schöpfer²

2010

Inverse Problems

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“…Let γ ϕ,s : [τ ϕ,s , 0] → Ω ⊂ R 2 be the coordinate representation of a geodesic on (M, g). The family (γ ϕ,s ) s∈[−r,r] can be interpreted as isocurves of a function (14) in the sense that the geodesics are implicitly defined by (15) φ…”

Section: Motivation and Mathematical Setup We Consider A Fluid Contamentioning

confidence: 99%

“…Schuster [12,13,14] computed reconstruction kernels for the 3D Doppler transform, leading to a highly efficient inversion method of filtered backprojection type. This method has since been extended to cone beam geometry; see [15]. A filtered backprojection algorithm for 3D vector tomography can also be found in Lade, Paganin, and Morgan [10].…”

Section: Introductionmentioning

confidence: 99%

Tomographic Reconstruction of the Curl and Divergence of 2D Vector Fields Taking Refractions into Account

Pfitzenreiter¹,

Schuster²

2011

SIAM J. Imaging Sci.

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This article is concerned with the problem of reconstructing the curl and divergence of two-dimensional (2D) vector fields from ultrasound time-of-flight measurements taking the refraction of the emitted ultrasound rays into account. Usually, in 2D vector field tomography the refraction of the ultrasound signal is neglected; i.e., one assumes that the signal propagates along straight lines. We address refractive effects, assuming that the ultrasound signal propagates along the geodesics of a Riemannian metric that is associated with the refractive index which is due to Fermat's principle. The investigated vector field, however, is still defined on a domain in R 2 that is equipped with the Euclidean metric. We show a connection between the ray transforms and the Radon transform along geodesic curves which generalizes a well-known result from the Euclidean case. Relying on this relation, we develop an asymptotic reconstruction formula for computing the curl and divergence of the vector field using Fourier integral operators. This article concludes with detailed numerical tests proving on the one hand a good performance of our method and showing on the other hand an improvement compared with computations that assume a propagation along straight lines.

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“…Vector field tomography is more complicated than scalar field tomography whose purpose is to recover a scalar‐valued function, and it has been achieved by researchers, eg, see Finch and Tuy. () Some works in vector field tomography have been conducted in both 2D and 3D scenarios, (see previous studies()). In Kazantsev and Schuster, the 3D asymptotic inversion formula for a vector field f was presented, whose main purpose was to recover its solenoidal part by using the data

\int_{S^{2}} \int_{0}^{\infty} \underline{θ} \cdot boldf false(\underline{a} + t \underline{θ} false) d t δ^{'^{'}} false(\underline{θ} \cdot \underline{η} false) d \underline{η},

where a is an emanated point on a curve surrounding the object f satisfying the Tuy's condition of order 3 and θ is the direction of the integrating line.…”

Section: Introductionmentioning

confidence: 99%

Vector field reconstruction via quaternionic setting

Wongsason

2017

Math Methods in App Sciences

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We propose the reconstruction of the solenoidal part of a vector field supported in the unit ball in 3 dimensions by using cone beam data from a curve surrounding it, and this curve satisfies the Tuy's condition of order 3. We use the quaternionic inversion formula to decompose the solenoidal part of a vector field into 2 parts. To recover the first one, which is the main part of the solenoidal component, another definition of a cone beam transform containing both Doppler and transverse data will be introduced. The second part will be reconstructed by using information from the first part as in Katsevich and Schuster's work with less data. KEYWORDScone beam data, Doppler-type transform, quaternions, Radon transform, transverse-type transform, vector field tomography 684

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A Reconstruction Approach for Imaging in 3D Cone Beam Vector Field Tomography

Cited by 19 publications

References 28 publications

Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse

Solving linear operator equations in Banach spaces non-iteratively by the method of approximate inverse

Tomographic Reconstruction of the Curl and Divergence of 2D Vector Fields Taking Refractions into Account

Vector field reconstruction via quaternionic setting

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