On the inversion of the geodesic Radon transform on the hyperbolic plane

Lissianoi, Sergei; Пономарев, И. Н.

doi:10.1088/0266-5611/13/4/010

Cited by 18 publications

(13 citation statements)

References 13 publications

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“…For example, it was shown that the Radon transform on co-planar circles whose centers are restricted to a bounded domain is invertible [1]. Radon transforms on other smooth curves have also been considered [2][3][4]. Recently, a series of papers have explored a circular-arc transform which arises when the signal is generated by first-order Compton scattering of X-rays [5,6].…”

Section: Introductionmentioning

confidence: 99%

Inversion formulas for the broken-ray Radon transform

2011

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We consider the inverse problem of the broken ray transform (sometimes also referred to as the V-line transform). Explicit image reconstruction formulas are derived and tested numerically. The obtained formulas are generalizations of the filtered backprojection formula of the conventional Radon transform. The advantages of the broken ray transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium simultaneously and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is typically discarded.

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Section: Introductionmentioning

confidence: 99%

Inversion formulas for the broken-ray Radon transform

2011

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“…Here the geodesic Radon transform integrates along geodesics in H 2 with respect to the measure induced by the Riemannian metric on H 2 . Methods of harmonic analysis (Fourier and Radon transforms and their inversions) on the hyperbolic plane are well developed (e.g., [16,66,67,87,89,92]). One hopes to use them to invert the geodesic Radon transform, to de-convolve, and as the result recover β.…”

Section: Reconstruction Algorithms and The Hyperbolic Integral Geometrymentioning

confidence: 99%

“…In particular, the papers mentioned above contain explicit inversion formulas for the hyperbolic geodesic Radon transform. The formula obtained in [92] was numerically implemented in [48] and works as nicely and stably as the standard inversions of the regular Radon transform 8 .…”

Section: Reconstruction Algorithms and The Hyperbolic Integral Geometrymentioning

confidence: 99%

Generalized transforms of Radon type and their applications

Kuchment¹

2006

The Radon Transform, Inverse Problems, and Tomography

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These notes represent an extended version of the contents of the third lecture delivered at the AMS Short Course "Radon Transform and Applications to Inverse Problems" in Atlanta in January 2005. They contain a brief description of properties of some generalized Radon transforms arising in inverse problems. Here by generalized Radon transforms we mean transforms that involve integrations over curved surfaces and/or weighted integrations. Such transformations arise in many areas, e.g. in Single Photon Emission Tomography (SPECT), Electrical Impedance Tomography (EIT) thermoacoustic Tomography (TAT), and other areas. 1 Numerous other reasons to study this transform are known, e.g. Radar and Sonar imaging, approximation theory, PDEs, potential theory, complex analysis, etc. [2, 93]. Although in dimensions higher than two one should probably use the word "spherical" rather than "circular," we will use for simplicity the latter. This should not create any confusion.

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“…The difficulty of numerically implementing (4.11) lies in the fact that it is complicated to numerically implement a twodimensional non-Euclidean convolution on the hyperbolic space. In [26], Lissianoi and Ponomarev focus on the problem of numerically inverting the geodesic Radon transform by developing an algorithm, and the problem regarding the deconvolution is also considered there. For this purpose, they consider the inversion formula (4.6) and use it to derive an inversion formula for the geodesic Radon transform that it is more suitable for computations.…”

Section: ∂(∂U)mentioning

confidence: 99%

Continuous and discrete inverse conductivity problems

Baras¹,

Berenstein²,

Gavilanez³

2004

Partial Differential Equations and Inverse Problems

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ISR develops, applies and teaches advanced methodologies of design and analysis toAbstract Tomography using CT scans and MRI scans is now well-known as a medical diagnostic tool which allows for detection of tumors and other abnormalities in a noninvasive way, providing very detailed images of the inside of the body using low dosage X-rays and magnetic fields. They have both also been used for determination of material defects in moderate size objects. In medical and other applications they complement conventional tomography. There are many situations where one wants to monitor the electrical conductivity of different portions of an object, for instance, to find out whether a metal object, possibly large, has invisible cracks. This kind of tomography, usually called Electrical Impedance Tomography or EIT, has also medical applications like monitoring of blood flow. While CT and MRI are related to Euclidean geometry, EIT is closely related to hyperbolic geometry. A question that has arisen in the recent past is whether there is similar "tomographic" method to monitor the "health" of networks. Our objective is to explain how EIT ideas can in fact effectively be used in this context.

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On the inversion of the geodesic Radon transform on the hyperbolic plane

Cited by 18 publications

References 13 publications

Inversion formulas for the broken-ray Radon transform

Inversion formulas for the broken-ray Radon transform

Generalized transforms of Radon type and their applications

Continuous and discrete inverse conductivity problems

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