[Paper Review] Stability estimates in stationary inverse transport

Guillaume Bal, Alexandre Jollivet|ArXiv.org|Apr 8, 2008

Numerical methods in inverse problems7 references4 citations

TL;DR

This paper establishes an $L^1$-stability estimate for the reconstruction of spatially dependent scattering and absorption coefficients in stationary linear transport equations from the albedo operator, using a geometric transformation that maps boundary data to planes orthogonal to velocity vectors. The key contribution is a rigorous stability result in dimensions $n \geq 3$, extending prior work limited to spatially independent scattering coefficients.

ABSTRACT

We study the stability of the reconstruction of the scattering and absorption coefficients in a stationary linear transport equation from knowledge of the full albedo operator in dimension $n\geq3$. The albedo operator is defined as the mapping from the incoming boundary conditions to the outgoing transport solution at the boundary of a compact and convex domain. The uniqueness of the reconstruction was proved in [M. Choulli-P. Stefanov, 1996 and 1999] and partial stability estimates were obtained in [J.-N. Wang, 1999] for spatially independent scattering coefficients. We generalize these results and prove an $L^1$-stability estimate for spatially dependent scattering coefficients.

Motivation & Objective

• To prove $L^1$-stability estimates for the reconstruction of spatially dependent scattering and absorption coefficients in stationary linear transport equations.
• To extend prior results that only established partial stability for spatially independent scattering coefficients.
• To develop a geometric framework where the albedo operator maps incoming conditions on a plane to outgoing measurements on another plane, simplifying analysis.
• To establish well-posedness and boundedness of the transport operator in $L^1$ spaces under subcriticality conditions.
• To provide a foundation for stable inverse scattering and absorption coefficient reconstruction from boundary measurements in realistic imaging geometries.

Proposed method

• Transform the domain using a radial extension to map the original boundary $\Gamma_\pm$ to new planes $F_\pm$ orthogonal to velocity vectors, simplifying the analysis of single-scattering contributions.
• Define the albedo operator $\mathcal{A}$ as a mapping from $L^1(F_-)$ to $L^1(F_+)$, representing the forward transport solution from incoming to outgoing data.
• Decompose the albedo operator into singular components using a functional analytic approach, isolating single-scattering contributions via test functions with shrinking support.
• Use a perturbation argument based on the operator $\mathbf{T} = \mathbf{T}_1 + A_2$, where $\mathbf{T}_1$ is the free transport operator and $A_2$ represents scattering, to prove bounded invertibility.
• Apply the Fredholm alternative and Neumann series to show that $\mathbf{T}^{-1}$ exists and is bounded in $L^1(O, |v|\,dxdv)$, ensuring well-posedness of the transport equation.
• Establish stability by bounding the difference in coefficients via the operator norm of $\mathcal{A}$ in $\mathcal{L}(L^1(F_-), L^1(F_+))$.

Experimental results

Research questions

• RQ1 Can $L^1$-stability estimates be established for the reconstruction of spatially dependent scattering and absorption coefficients in stationary linear transport?
• RQ2 How does the geometry of the measurement setup affect the stability of inverse transport reconstructions?
• RQ3 What is the role of single-scattering contributions in the stability analysis, and how can they be isolated using test functions with shrinking support?
• RQ4 Under what conditions is the transport operator $\mathbf{T}$ invertible in $L^1$-based function spaces?
• RQ5 Can the albedo operator be analyzed in a transformed geometry where incoming and outgoing data are defined on planes orthogonal to velocity vectors?

Key findings

• The paper establishes an $L^1$-stability estimate for the reconstruction of both scattering and absorption coefficients when they are spatially dependent, generalizing prior results limited to spatially independent scattering.
• The albedo operator $\mathcal{A}$ is shown to be a bounded operator from $L^1(F_-)$ to $L^1(F_+)$ under appropriate subcriticality assumptions on the coefficients.
• The solution to the transport equation is uniquely solvable in $L^1(O, |v|\,dxdv)$, and the solution operator is bounded with norm controlled by $\|\sigma\|_{\infty}$ and $\|\sigma_p\|_{\infty}$.
• The inverse of the transport operator $\mathbf{T}$ is bounded in $L^1(O, |v|\,dxdv)$, which ensures stability of the forward problem and enables the derivation of stability estimates.
• The stability estimate is derived via a Neumann series expansion of $\mathbf{T}^{-1} = \mathbf{T}_1^{-1}(I + A_2\mathbf{T}_1^{-1})^{-1}$, with the operator norm of $A_2\mathbf{T}_1^{-1}$ bounded by $1 - e^{-2R\mathrm{v}_0^{-1}\|\sigma_p\|_{\infty}} < 1$, ensuring convergence.
• The final stability bound is expressed as $\|f\|_{L^1(O,|v|dxdv)} \leq C_0 \|f_-–\|_{F_-}$, where $C_0$ depends on $R$, $\mathrm{v}_0^{-1}$, $\|\sigma\|_{\infty}$, and the norm of the inverse of the perturbation operator.