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A Uniqueness Theorem for Thermoacoustic Tomography in the Case of Limited Boundary Data
Dustin Steinhauer
TL;DR
Thermoacoustic tomography seeks to recover energy absorption from boundary pressure measurements, but the paper focuses on variable wave speed and partial-boundary data. It combines domain-of-dependence arguments with Tataru's unique continuation to prove uniqueness under stated geometric and time conditions.
Problem
The paper asks whether energy absorption can be uniquely recovered for variable acoustic speed from measurements on only part of the boundary.
Source: §1 · p. 1Method
The proof establishes vanishing in exterior domains of dependence and then applies Tataru's unique continuation theorem.
Source: §2 · pp. 4–5, §3 · p. 6Results
Under the theorem's conditions, vanishing near the boundary propagates to f ≡ 0, yielding uniqueness.
Source: §2 · p. 2, §3 · pp. 6–7Takeaways & Limitations
For sufficiently extensive partial-boundary measurements, uniqueness can be concluded even though exact reconstruction may be unlikely for general measurement surfaces.
Source: §1 · p. 1The theorem assumes smooth initial data supported inside Ω and an observation time T greater than d(x,Γ) for every x ∈ Ω.
Source: §2 · p. 2
Abstract
from arXiv · showhide
We prove a uniqueness theorem for compactly supported initial data for the variable speed wave equation arising in models of thermoacoustic tomography, given measurements on a part of the boundary. The proof is based on domain of dependence arguments and D. Tataru's unique continuation theorem.
1. Introduction
TAT seeks to recover energy absorption from pressure measurements, combining electromagnetic contrast with ultrasound resolution. This paper addresses uniqueness for variable acoustic speed when measurements cover only part of the boundary.
1. Introduction
- TAT aims to recover the energy absorption f(x) from pressure variations measured at the object's surface.A brief microwave pulse generates the pressure variation u(x,t).Source: §1 · p. 1
- Mathematically, the task is to recover f(x) from boundary values of u(x,t), possibly on only part of the boundary.Source: §1 · p. 1
- For variable acoustic speed c(x), less is known than for constant speed, where reconstruction methods exist for complete boundary data.A reconstruction formula for variable speed was obtained using Laplacian eigenfunctions.Source: §1 · p. 1
- With sufficiently extensive partial-boundary measurements, the paper seeks uniqueness for variable c, while noting exact reconstruction may be unlikely for general measurement surfaces.Source: §1 · p. 1
2. Domains of Dependence
The section proves an exterior domain-of-dependence result and uses it as the first step toward uniqueness under a geometric condition on the observed boundary and a sufficiently large observation time.
2. Domains of Dependence
- The wave-speed function c induces the Riemannian metric gij(x) = c^-2(x)δij, whose curve lengths define geodesic distance.The distance is the infimum of lengths over curves between the points.Source: §2 · p. 2
- Theorem 2.1 assumes the geometric condition (P) and an observation time T exceeding d(x,Γ) for every x ∈ Ω.Source: §2 · p. 2
- Green's theorem and Tataru's unique continuation yield f ≡ 0 when the measured-boundary condition and observation-time requirement hold.The proof first establishes vanishing in an exterior domain of dependence, then continues the vanishing into Ω.Source: §2 · p. 2
- The proof approximates a Lipschitz domain of dependence by piecewise-smooth regions whose boundary meets ∂Ω only within Γ.Green's theorem then shows v vanishes on the approximating regions and, by continuity, on the domain of dependence.Source: §2 · pp. 4–5
3. Tataru’s Theorem and Unique Continuation
The section applies Tataru’s unique-continuation result to expand vanishing from a metric-space cylinder into a distance-time cone; repeated propagation then yields f≡0.
3. Tataru’s Theorem and Unique Continuation
- Tataru’s uniqueness theorem states that vanishing on one side of a noncharacteristic hypersurface forces vanishing in a neighborhood of that point.The theorem is stated for a differential operator with smooth coefficients.Source: §3 · p. 6
- If u vanishes on a metric ball cylinder and D is below the metric’s injectivity radius at z, then u vanishes where d(x,z)+|t|<D.The result uses balls defined by the metric g induced by c(x).Source: §3 · p. 6
- The proof builds smoothed regions whose boundaries are noncharacteristic, then uses unique continuation to extend the region where u vanishes.The boundaries are smooth except at (z,±D), and the regions exhaust X.Source: §3 · pp. 6–7
- Bounded curvature of the induced metric provides a uniform radius for geodesic normal coordinates used in the continuation argument.The stated conditions are that c is bounded away from zero and its first and second derivatives are bounded.Source: §3 · p. 7
- Vanishing on B(p,r)×[−H,H] extends to points satisfying d(x,z)<r+min(H−|t|,δ).The proof applies the local propagation lemma near the time-cylinder ends and in its interior.Source: §3 · p. 7
- Repeated propagation shows u vanishes wherever d(x,z)+|t|≤H, and therefore the initial data satisfy f≡0.The argument applies the cylinder-propagation corollary repeatedly.Source: §3 · p. 7