Integral Geometry in Hyperbolic Spaces and Electrical Impedance Tomography

Berenstein, Carlos A.; Tarabusi, Enrico Casadio

doi:10.1137/s0036139994277348

Cited by 39 publications

(30 citation statements)

References 10 publications

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“…In this way, we can see the analogy with the well-known inverse conductivity problem and we could try to profit from the large body of mathematical research in this area. The analogy with this particular inverse problem indicates that if one were to pursue this "abstract" approach the "correct" geometry is closer to be hyperbolic than to be Euclidean [7]. On the other hand, as of this moment, we have found that those tomographic analogies are more useful for providing directions of research and methods to consider these problems than providing an exact correspondence between the two phenomena.…”

Section: Introductionmentioning

confidence: 86%

“…In [6] and [7] it is shown that the hyperbolic Radon transform is involved in the problem of reconstructing the conductivity distribution on a plate by using electrical impedance tomography EIT.…”

Section: The Hyperbolic Radon Transform and Eitmentioning

confidence: 99%

“…In [7] is shown that the linearized problem can in fact be described explicitly in the context of hyperbolic geometry using R H and a radial convolution operator with kernel k. Let k be given by…”

Section: The Approximate Solution To the Eit Problemmentioning

confidence: 99%

See 2 more Smart Citations

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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Section: Introductionmentioning

confidence: 86%

Section: The Hyperbolic Radon Transform and Eitmentioning

confidence: 99%

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Continuous and discrete inverse conductivity problems

Baras¹,

Berenstein²,

Gavilanez³

2004

Partial Differential Equations and Inverse Problems

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show abstract

“…It implies that the natural domain might be a hyperbolic space of dimension higher than 2. One can see that Munzner's suggestion leads to a question closely resembling EIT, and it is natural to consider it a problem in hyperbolic tomography [7], [8]. On the other hand, we have just obtained a significant result on the inversion of the Neumann-Dirichlet problem by studying it directly on "weighted" graphs [11].…”

Section: The Weighted Graph Modelmentioning

confidence: 99%

“…On the other hand, we have just obtained a significant result on the inversion of the Neumann-Dirichlet problem by studying it directly on "weighted" graphs [11]. Similarly, the Radon transform in the hyperbolic plane has been studied in [7], [8], and [27].…”

Section: The Weighted Graph Modelmentioning

confidence: 99%

Network Tomography

Gavilanez¹,

Berenstein²,

Baras³

2005

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ISR develops, applies and teaches advanced methodologies of design and analysis to solve complex, hierarchical, heterogeneous and dynamic problems of engineering technology and systems for industry and government. ISR is a permanent institute of the University of Maryland AbstractWhile conventional tomography is associated to the Radon transform in Euclidean spaces, electrical impedance tomography or EIT is associated to the Radon transform in the hyperbolic plane. We discuss some recent work on network tomography that can be associated to a problem similar to EIT on graphs and indicate how in some sense it may be also associated to the Radon transform on trees.

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Sets that maximize probability and a related variational problem

Salamanca

2020

Can J Statistics

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Let  be a random variable of a Riemannian manifold. We assume that the C 2 -probability density function of  exists. This research addresses two variational questions. The first concerns sets that maximize their probability among those that have a fixed volume. We prove that such a set must have a probability density function that is constant along its boundary; equivalently, such a set must be a density level set. We also obtain the equations related to the maximization property (the stability of the solutions). The other variational problem is the inverse of the first question, namely which sets minimize their volume among those sets which have a predetermined probability? The solution of this problem will define a notion of a quantile set. We show that the solutions of both variational problems coincide (the critical point equation and the stability condition). As theoretical applications, we consider a decision-making problem and fuzzy sets. As practical applications, we first explore how to locate a powerplant, and subsequently develop a model for a distribution of cheetahs.

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Integral Geometry in Hyperbolic Spaces and Electrical Impedance Tomography

Cited by 39 publications

References 10 publications

Continuous and discrete inverse conductivity problems

Continuous and discrete inverse conductivity problems

Network Tomography

Sets that maximize probability and a related variational problem

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