Problem of two-beam tomography

Abstract

3 with smooth C 2 boundaries such that ⊂ G and G 2 = G\ ; Ω = {ω : ω ∈ ‫ޒ‬ 3 |ω| = 1}; L(r, ω) = {r + tω, t ∈ ‫ޒ‬ 1 } for (r, ω) ∈ ‫ޒ‬ 3 × Ω; d(r, ω) be the distance from a point r (r ∈ G) to the boundary of G in the direction ω; ∂T be the boundary of an arbitrary set T, T ⊂ ‫ޒ‬ 3 ;and C(T, ω) be a cylinder in ‫ޒ‬ 3 with the base T and the axis ω, i.e., the union of points lying on all straight lines L(τ, ω), τ ∈ T. A mathematical model of a probing signal is the following integrodifferential radiation transfer equation for points (r, ω) ∈ G × Ω:(1)Recall that f(r, ω) is the particle flux density at the point r in the direction ω, μ(r) is the extinction coef ficient, k(r, ωω') is the scattering phase function, and J(r, ω) is the density of internal sources. Assume that the functions μ(r), k(r, ωω'), and J(r, ω) have continuous and bounded first partial derivatives with respect to all their variables for r ∈ G i , i = 1, 2, ω ∈ Ω, and ω' ∈ Ω. Note that jump discontinuities in r are admitted at points r ∈ ∂G 1 . As a result, G 1 can be interpreted as an inhomogeneity in G enclosed by the boundary G 2 .A conventional boundary condition for Eq. (1) is specified as (2) where h is the incident flux radiation density. Under fairly general assumptions, given μ, k, J, and h, the problem of determining f(r, ω) from Eq. (1) and condition (2) has a unique solution [3][4][5].The general aim of our study is to derive information on G 1 from the values of f(r, ω) for r belonging to two plane cross sections of G 2 and for directions ω orthogonal to these cross sections. For a more rigorous) G Ω, × ∈ = Abstract-An idea was developed suggested in a number of studies dealing with the search for inho mogeneous inclusions inside an unknown medium given the radiation measured in a plane outside the desired body. Specifically, the medium was proposed to be probed in two directions (at two angles) in contrast to previous works, where a single direction was used. Accordingly, the probing results became more informative: the determination of the object's shadow on the measurement area (antenna) was supplemented with the possibility of localizing the desired body in space. A tomographic location algorithm was proposed that can underlie a new orientation method in arbitrary absorbing and scat tering media. As before, the case was considered where direct visualization (photograph) fails to pro duce a distinguishable structure of the medium. The problem was solved by analyzing signals passing through the medium. A number of numerical experiments were performed by applying computer sim ulation. The numerical results were illustrated by plots and tomograms.