[Paper Review] Stability estimates for the Radon transform with restricted data and applications

Pedro Caro, David Dos Santos Ferreira+1 more|arXiv (Cornell University)|Nov 8, 2012

Numerical methods in inverse problems5 references4 citations

TL;DR

This paper establishes stability estimates for the Radon transform with restricted data, proving that the $L^p$ norm of a function can be controlled by its Radon transform over limited angular and spatial domains under support conditions. The key contribution is a quantitative stability result that enables improved stability estimates for inverse boundary value problems with partial data, particularly in Calderón's inverse conductivity problem and related Schrödinger inverse problems.

ABSTRACT

In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the $L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obtain stability estimates for an inverse boundary value problem with partial data.

Motivation & Objective

• To derive stability estimates for the Radon transform when data is restricted to limited angular and spatial domains, under support conditions on the function.
• To apply these estimates to inverse boundary value problems with partial data, particularly in the context of Calderón's inverse conductivity problem.
• To improve the stability of reconstructions in inverse problems where measurements are only available on subsets of the boundary.
• To establish quantitative bounds linking the difference in Dirichlet-to-Neumann maps to the difference in underlying potentials or conductivities.
• To extend results on partial data inverse problems to higher dimensions using microlocal and integral geometry techniques.

Proposed method

• Derive a stability estimate for the local Radon transform using a two-plane transform representation and weighted $L^2$ estimates.
• Use a partition of unity and local coordinates to reduce global estimates to local ones on the sphere $\mathbf{S}^2$.
• Apply Hölder's inequality and change of variables to control the $L^1$ norm of the Radon transform over restricted planes.
• Relate the Radon transform to the two-plane transform via averaging over balls in the plane, enabling control of the transform in terms of $L^2$ norms.
• Combine the local stability estimate with a global theorem (Theorem 2.5) to obtain a global stability estimate for the Radon transform with restricted data.
• Use the logarithmic dependence of the stability constant on the difference of Dirichlet-to-Neumann maps to derive the final quantitative estimate.

Experimental results

Research questions

• RQ1 Can stability estimates be established for the Radon transform when data is restricted to a limited set of planes, under support conditions?
• RQ2 How does the restricted data affect the stability of the inverse problem for the Schrödinger equation or conductivity equation?
• RQ3 What is the quantitative dependence of the reconstruction error on the difference of Dirichlet-to-Neumann maps in partial data settings?
• RQ4 Can the stability estimate be extended to higher dimensions using geometric and microlocal analysis?
• RQ5 What is the role of the convex hull and supporting planes in controlling the stability of the Radon transform with restricted data?

Key findings

• A stability estimate is established for the Radon transform with restricted data: $\int_{-\alpha/2}^{\alpha/2}\int_{\mathbf{S}^2}|\mathcal{R}q(s,\omega)|\,d\sigma(\omega)\,ds \leq C\left(\tau^{-1/4}\|q\|^{1/2}_{L^\infty} + e^{c\tau}\|\Lambda_{q_1}-\Lambda_{q_2}\|^{1/4}\right)^{1/3}$, where $\mathcal{R}$ denotes the Radon transform.
• The estimate is derived via a two-plane transform representation and local coordinate analysis, with the use of averaging over balls in the plane to relate the Radon transform to the two-plane transform.
• The stability constant depends logarithmically on the difference of the Dirichlet-to-Neumann maps, with $\tau = \frac{1}{8c}|\log\|\Lambda_{q_1}-\Lambda_{q_2}\||$.
• The result implies that the $L^p$ norm of the potential $q$ is controlled by the difference of the Dirichlet-to-Neumann maps, under partial data conditions.
• The method applies to the inverse problem of determining $q$ from partial boundary measurements, yielding a logarithmic stability estimate in the $L^\infty$ norm.
• The stability result extends to the inverse conductivity problem via the reduction to the Schrödinger equation, providing a quantitative bound for partial data recovery in dimensions $n \geq 3$.