Algorithm of finding a body projection within an absorbing and scattering medium

Abstract

The problem of an arbitrary medium probing by use of radiation flux on the bases of the monoenergetic transport equation is considered. The medium is assumed to comprise some bodies, which radiation characteristics differ from ones of the medium. The outgoing radiation density flux is assumed to be measured at a plane nonintersecting the bodies whereas the boundaries of the body projections (shadows) on the plane are to be found. The case when the direct imaging of the objects is embarrassing owing to a strong scattering and absorption within the medium. The current work is being carried out under a special project so the limitations imposed have a specific character. More definite, these limitations are adopted to the problem of the world ocean near-bottom zone probing.

Problem of two-beam tomography

“…Let G and G 1 be convex bounded domains in ‫ޒ‬ 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.…”

“…In the problem under study, we use a tested indicator of inhomogeneity (see [1]). Denote by ∇ P f(x, ω P ) and ∇ Q f(y, ω Q ) the two dimensional gradients with respect to r of the traces of f(r, ω) at r = x, x ∈ P 1 , ω = ω P and at r = y, y ∈ Q 1 , ω = ω Q .…”

“…Under certain constraints, which are assumed to hold in this paper, the behavior of indicators (3) was studied in [1]. Specifically, it was proved that Ind(x) ∞ and Ind(y) ∞ if and only if x ∈ D P , x ∂D P , y ∈ D Q , y ∂D Q , respectively.…”

“…3. The values of (a) f(x, ω P ) and (b) Ind(x) found at nodes of a uniform square 101 × 101 grid covering the square P1 for the plane P at α = 0.9. The test conditions (i.e., the absorption and scattering of the medium) are such that direct visu alization (see (a)) does not produce an image of the desired object, while the reconstruction with the help of the indicator (see (b)) yields a clear boundary of the shadow of G 1 .…”