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Remarks on propagation of discontinuities in stationary radiative transfer

Daisuke Kawagoe

TL;DR

This work analyzes discontinuities in the stationary radiative transfer equation with incoming boundary data, examining how discontinuities in the solution arise from boundary data or from discontinuities in the coefficients. The model uses $\xi \cdot \nabla_x I + (\mu_t) I = \mu_s \int p I' d\sigma'$, with $\mu_t = \mu_a + \mu_s$, and studies propagation along characteristics under generalized convexity. Under the generalized convexity condition, boundary-induced discontinuities propagate along forward characteristics and the inverse X-ray transform recovers $\mu_t$ from boundary measurements; without convexity, coefficient-induced discontinuities can appear in 2D but do not necessarily affect the transform under certain geometric restrictions, and 3D can be reduced to 2D. A numerical experiment illustrates reconstruction of $\mu_t$ from boundary data, validating the theoretical predictions and highlighting practical implications for optical tomography.

Abstract

We consider the stationary transport equation with the incoming boundary condition. We are interested in discontinuities of the solution. Under the generalized convexity condition, it is known that it has only boundary-induced discontinuities, which are discontinuities arising from discontinuous boundary data, they propagate along positive characteristic lines, and we can reconstruct the attenuation coefficient from boundary measurements by the inverse X-ray transform. In this article, we observe that coefficient-induced discontinuities, discontinuities of the solution arising from discontinuous coefficients, would also appear without the generalized convexity condition. If the set of discontinuous points of the coefficients contains at most finite number of flat parts, coefficient-induced discontinuities do not affect the inverse X-ray transform. We also remark that, under the generalized convexity condition, a three dimensional inverse problem can be reduced to the two dimensional one. A numerical experiment is exhibited.

Remarks on propagation of discontinuities in stationary radiative transfer

TL;DR

This work analyzes discontinuities in the stationary radiative transfer equation with incoming boundary data, examining how discontinuities in the solution arise from boundary data or from discontinuities in the coefficients. The model uses , with , and studies propagation along characteristics under generalized convexity. Under the generalized convexity condition, boundary-induced discontinuities propagate along forward characteristics and the inverse X-ray transform recovers from boundary measurements; without convexity, coefficient-induced discontinuities can appear in 2D but do not necessarily affect the transform under certain geometric restrictions, and 3D can be reduced to 2D. A numerical experiment illustrates reconstruction of from boundary data, validating the theoretical predictions and highlighting practical implications for optical tomography.

Abstract

We consider the stationary transport equation with the incoming boundary condition. We are interested in discontinuities of the solution. Under the generalized convexity condition, it is known that it has only boundary-induced discontinuities, which are discontinuities arising from discontinuous boundary data, they propagate along positive characteristic lines, and we can reconstruct the attenuation coefficient from boundary measurements by the inverse X-ray transform. In this article, we observe that coefficient-induced discontinuities, discontinuities of the solution arising from discontinuous coefficients, would also appear without the generalized convexity condition. If the set of discontinuous points of the coefficients contains at most finite number of flat parts, coefficient-induced discontinuities do not affect the inverse X-ray transform. We also remark that, under the generalized convexity condition, a three dimensional inverse problem can be reduced to the two dimensional one. A numerical experiment is exhibited.
Paper Structure (4 sections, 6 theorems, 51 equations, 4 figures)

This paper contains 4 sections, 6 theorems, 51 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Propagation of boundary-induced discontinuities
  3. Coefficient-induced discontinuities in 2D
  4. Remarks on discontinuities in 3D

Key Result

Proposition 2.1

Suppose that a boundary data $I_0$ is bounded and that it satisfies at least one of the following two conditions. Then, there exists a unique solution $I$ to the boundary value problem eq:STE-eq:BC, and we have

Figures (4)

  • Figure 2.1: An example of the generalized convexity condition
  • Figure 4.1: The section of the example at $x_3 = 0.50$
  • Figure 4.2: The reconstructed image of $\mu_\text{t}$
  • Figure 4.3: The section of the reconstructed image

Theorems & Definitions (9)

  • Proposition 2.1: CK
  • Remark 1
  • Remark 2
  • Proposition 2.2: CK
  • Proposition 2.3: CK
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Theorem 3.3