Three-Dimensional Image Reconstruction from Compton Camera Data

Kuchment, Peter; Terzioglu, Fatma

doi:10.1137/16m107476x

Cited by 34 publications

(41 citation statements)

References 40 publications

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“…• Practical soundness of the derived inversion techniques is shown by their numerical implementation in the most interesting dimensions two and three. One should also notice, that the new algorithm of the 3D cone transform inversion works much faster than some of the ones developed in [22]. The reason is that a much coarser mesh (1.8K nodes) on the sphere suffices, rather than 30K used in [22].…”

Section: Conclusion and Remarksmentioning

confidence: 99%

Inversion of weighted divergent beam and cone transforms

Kuchment¹,

Terzioglu²

2017

Inverse Problems &Amp; Imaging

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In this paper, we investigate the relations between the Radon and weighted divergent beam and cone transforms. Novel inversion formulas are derived for the latter two. The weighted cone transform arises, for instance, in image reconstruction from the data obtained by Compton cameras, which have promising applications in various fields, including biomedical and homeland security imaging and gamma ray astronomy. The inversion formulas are applicable for a wide variety of detector geometries in any dimension. The results of numerical implementation of some of the formulas in dimensions two and three are also provided.1991 Mathematics Subject Classification. 44A12, 53C65, 92C55.

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Section: Conclusion and Remarksmentioning

confidence: 99%

Inversion of weighted divergent beam and cone transforms

Kuchment¹,

Terzioglu²

2017

Inverse Problems &Amp; Imaging

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“…Lately, impressive progress has been achieved on cone Radon transforms on cylinder or spherical surfaces by M Haltmeier and co-workers [18]. Several inverse formula for three dimensional Compton cameras reconstruction have been established by S Moon [19,20], V Maxim et al [21] and P Kuchment and F Terzioglu [22,23]. They surely will induce further results both useful for integral geometry as well as for next generation imaging processes.…”

Section: Three-dimensional Compton Scattered Radiation Imaging and Vamentioning

confidence: 99%

The Development of Radon Transforms Associated to Compton Scatter Imaging Concepts

Nguyen¹,

Truong²

2018

EJMCA

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For more than three decades, the exploitation of Compton scattering has led to significant progress in imaging the inner parts of an optically opaque object. In this imaging process, the Compton scattering angle plays an essential role as it labels the recorded data, suppressing the motion of the scanning camera, as well as generates new geometric manifolds on which radiation measurements are made and thereby introducing new relevant Radon transforms. The present paper is aimed to synthetize the conceptual development of Compton scattering imaging in order to gain a constructive overview and possibly induce a further innovative growth to it. We believe this may lay foundation to new mathematical tools as well as lead to a great diversity of applications.

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“…Ever since the Compton camera, also known as an electronically collimated -camera, was first introduced for use in Single Photon Emission Computed Tomography (SPECT), a Radon-type transform that assigns a function to its surface integral over various sets of cones has attracted substantial attention [2,3,5,8,9,12,13,16,17,20,23,27,[29][30][31]. For the generalized Radon transform including toric Radon transform, see [22].…”

Section: Introductionmentioning

confidence: 99%

“…Terzioglu derived several inversion formulas and also obtained the relation between the regular Radon transform and the conical Radon transform with the vertices on the whole space in [28]. In a sequential work [13], the authors developed new analytical reconstruction techniques that are valid for an arbitrary geometry of detectors satisfying an admissibility condition.…”

Section: Introductionmentioning

confidence: 99%

An inversion of the conical Radon transform arising in the Compton camera with helical movement

Kwon

2019

Biomed. Eng. Lett.

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Since the Compton camera was first introduced, various types of conical Radon transforms have been examined. Here, we derive the inversion formula for the conical Radon transform, where the cone of integration moves along a curve in threedimensional space such as a helix. Along this three-dimensional curve, a detailed inversion formula for helical movement will be treated for Compton imaging in this paper. The inversion formula includes Hilbert transform and Radon transform. For the inversion of Compton imaging with helical movement, it is necessary to invert Hilbert transform with respect to the inner product between the vertex and the central axis of the cone of the Compton camera. However, the inner product function is not monotone. Thus, we should replace the Hilbert transform by the Riemann-Stieltjes integral over a certain monotone function related with the inner product function. We represent the Riemann-Stieltjes integral as a conventional Riemann integral over a countable union of disjoint intervals, whose end points can be computed using the Newton method. For the inversion of Radon transform, three dimensional filtered backprojection is used. For the numerical implementation, we analytically compute the Hilbert transform and Radon transform of the characteristic function of finite balls. Numerical test is given, when the density function is given by a characteristic function of a ball or three overlapping balls.

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Three-Dimensional Image Reconstruction from Compton Camera Data

Cited by 34 publications

References 40 publications

Inversion of weighted divergent beam and cone transforms

Inversion of weighted divergent beam and cone transforms

The Development of Radon Transforms Associated to Compton Scatter Imaging Concepts

An inversion of the conical Radon transform arising in the Compton camera with helical movement

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