Tomography Through the Transport Equation

Аниконов, Д. С.; Kovtanyuk, Andrey E.; Прохоров, И. В.

doi:10.1007/978-1-4612-1550-9_3

Cited by 16 publications

(23 citation statements)

References 1 publication

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“…For the first time, similar ideas were used by the author in [3,4]. Then this method was developed in the study of problems of X-ray tomography and integral geometry [5][6][7][8][9][10]. The statement of the problems with unknown boundaries is not rare these days.…”

Section: Remark 32mentioning

confidence: 97%

A Method for Studying Singular Integral Equations

Аниконов

2010

Sib Math J

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We examine a singular integral equation of the first kind on a bounded open set of an n-dimensional space. Open subsets with a common (contact) (n − 1)-dimensional piecewise smooth part of boundaries are selected. The underdetermined case is treated in which the unknown part of the integrand depends on 2n independent variables whereas a given integral depends only on n variables. In this situation we pose the problem of finding the contact part of the boundaries and prove unique solvability of the problem. We use the following notation: ρ(A, B) is the distance between sets A and B in R n ; ∂A is the boundary ofConsider a function g(x, y, ω) measurable on T ×T ×Ω and bounded almost everywhere in the following sense: There exist T and Ω ( T ∈ T and Ω ∈ Ω) coinciding in measure with T and Ω and such that |g(x, y, ω)| ≤ const for (x, y, ω) ∈ T × T × Ω. Here and in what follows, the notation const is used to denote a positive constant. In order to reduce the number of denotations, the bounded functions are often written below as O(1). We put T i = T i ∩ T and T 0 = T 1 ∪ T 2 , i = 1, 2. A point of the unit sphere Ω is denoted by ω if it designates an independent variable. If this point is expressed through other variables then we employ the letter s; for example, s = (y − x)/|y − x|. Consider the integral(1.1)The functions V i (x) for x ∈ T i , i = 1, 2, are understood to be singular integrals. We assume that T i , i = 1, 2, and g(x, y, ω) satisfy the following condition: for every u ∈ T i ,Of course, this restriction means that the integrand is a summable function. For convenience, g(x, y, ω) is assumed to be extended by zero in y beyond T . Note that if we use the spherical coordinates y − u = tν, ν ∈ Ω, dy = t n−1 dtdν then (1.2) is a consequence of the condition Ω dν ∞ 0 |g(x, u + tν, ω) − g(x, u, ω)| t dt ≤ const, ω ∈ Ω, x ∈ T .

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Section: Remark 32mentioning

confidence: 97%

A Method for Studying Singular Integral Equations

Аниконов

2010

Sib Math J

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“…They have a variety of applications in optical tomography, theory of nuclear reactors, etc. (see, e.g., the book of Anikonov, Kovtanyuk and Prokhorov [1] and references therein). This paper addresses the question of the global uniqueness for a CIP for the non-stationary single-speed transport equation with the extra lateral boundary data.…”

Section: Introductionmentioning

confidence: 99%

“…Stability, uniqueness and existence results and references to such results for CIPs for the stationary transport equation can be found, e. e.g., in [1] and in the book of Romanov [25]. Uniqueness and existence results for CIPs for the non-stationary transport equation were obtained in the works of Prilepko and Ivankov [22], [23] and [24].…”

Section: Introductionmentioning

confidence: 99%

Global Uniqueness for a Coefficient Inverse Problem for the Non-Stationary Transport Equation via Carleman Estimate

Klibanov¹,

Pamyatnykh²

2006

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“…This work is dedicated to applicability in positron emission tomography of the inhomogeneity indicator proposed in [15][16][17][18][19] to determine the boundaries of some unknown radiation sources. The inhomogeneity indicator was introduced in [15] for determining the boundaries of inclusions that are contained inside an inhomogeneous medium, within the framework of transmission tomography.…”

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confidence: 99%

Numerical experiments with the inhomogeneity indicator in positron emission tomography

Yarovenko

2012

J. Appl. Ind. Math.

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By means of numerical methods, the question is studied of applicability of the inhomogeneity indicator in positron emission tomography. The signal registered by the tomograph is described in terms of an imitation model using the Monte Carlo method. The possibility is demonstrated of the effective use of the inhomogeneity indicator for solving the problem under consideration. Some numerical results are presented in graphical form for reconstructing the boundaries of unknown activity sources.The foundation of positron emission tomography (PET) are based on the idea of registration of two oppositely directed gamma-quanta appearing as a result of annihilation of the positrons emitted by a special agent introduced into the patient's body. Under this approach, the functioning of the tomograph's detectors function, depending on one another. If two detectors simultaneously register a signal (i.e., the so-called coincidence happens) then it is assumed that the annihilation point is situated on the line joining the detectors. This line is called the line of response. The registration of a pair of photons allows us to proceed without the collimators in determining the direction of the gamma-quanta movement and is a specific filter which helps distinguishing exactly those trajectories of photons that carry the necessary information on the distribution of the activity source in the matter [1].The majority of scanners can function both in the slice-by-slice mode (when the axial collimation is created with the help of the special tungsten plates called the septa), and in the three-dimensional mode (when the septa are pulled in and the coincidences are registered between all possible pairs of detectors). The usage of collimation leads to diminishing the random components in the detected signal. At the same time, a three-dimensional PET system, allowing registering the coincidences between any rings of detectors, leads to an eight-fold increase of the number of registered coincidences in comparison with the slice-by-slice regime [2], which amplifies the sensitivity of the tomograph and allows faster accumulating the necessary volume of information. In connection with this, the considered research area is most promising [1]. However, increase of the total number of registered events leads inevitably to increase of the quantity of random events whose effect on the reconstruction quality is negative. For example, registration of the scattered photons results in distinguishing some false response lines. This circumstance, in turn, leads to the necessity of development of new information processing methods [1].The main body of research in this area is directed to the construction of various empirical formulas and methods to allow estimating the contribution of the scattered radiation into the signal registered by the detectors and distinguishing the nonscattered (ballistic) portion of radiation [3][4][5][6][7][8][9][10][11]. After that the tomography problem is reduced to a simple inversion of the Radon transform, and the theory in th...

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Tomography Through the Transport Equation

Cited by 16 publications

References 1 publication

A Method for Studying Singular Integral Equations

A Method for Studying Singular Integral Equations

Global Uniqueness for a Coefficient Inverse Problem for the Non-Stationary Transport Equation via Carleman Estimate

Numerical experiments with the inhomogeneity indicator in positron emission tomography

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