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On singularities and instability of reconstruction in thermoacoustic tomography
Linh V. Nguyen
TL;DR
TAT seeks to recover the initial pressure of a wave from measurements on an observation surface, and reconstruction stability depends on whether relevant geometric rays meet that surface. The paper proves that when they do not, reconstruction is not Hölder stable, covering incomplete observations and trapping.
Problem
TAT reconstructs the initial pressure from measured wave pressure, and the stability of that recovery depends on visibility of the relevant geometric rays.
Source: §1 · p. 1Method
Results
When a nonzero direction has rays from every point in U avoiding V, reconstruction from g is not Hölder stable.
Source: §3 · p. 3Takeaways & Limitations
Incomplete observation surfaces and trapped rays can both violate visibility and yield the same instability.
Source: §1 · p. 2The theorem assumes the measurement operator T is injective and considers recovery on a bounded region of interest Ω.
Source: §3 · p. 3
Abstract
from arXiv · showhide
We consider the problem of thermoacoustic tomography (TAT), in which one needs to reconstruct the initial value of a solution of the wave equation from its value on an observation surface. We show that if some geometric rays for the equation do not intersect the observation surface, the reconstruction in TAT is not Hölder stable.
1 Introduction
TAT reconstructs the initial pressure from measurements on an observation surface, with the pressure linked to absorbed electromagnetic energy. This paper shows that, for variable sound speed, failure of visibility makes reconstruction not Hölder stable.
1 Introduction
- TAT recovers initial pressure f from measured pressure g, using a wave-equation model with initial pressure f and zero initial velocity.The model is commonly accepted; the initial pressure is roughly proportional to absorbed electromagnetic energy.Source: §1 · p. 1
- For constant sound speed, visibility requires every straight line through each point in the region of interest to intersect the observation surface; a closed surrounding surface satisfies this condition.Under this condition, reconstruction is only mildly unstable, similarly to inversion of the standard Radon transform [14].Source: §1 · pp. 1–2
- The paper focuses on reconstruction stability in a bounded region and presents the instability result after recalling wavefront-set propagation.Stability in this setting had been partially addressed in earlier work [10] [14] [17] [21].Source: §1 · pp. 1–2
- If variable-speed geometric rays fail to intersect the observation surface, reconstruction is not Hölder stable; incomplete data and trapping produce the same instability.The visibility condition can fail when the surface does not completely surround the region or when rays are trapped.Source: §1 · p. 2
2 Singularities and Wavefront set
The section connects image singularities with wavefront-set propagation: singularities travel along geometric rays, and their absence from a region implies smoothness of the pressure there. This supplies the propagation framework for the paper’s stability analysis.
2 Singularities and Wavefront set
- The wavefront set records where a distribution is not microlocally smooth, including boundary points and normal directions for a smooth characteristic-function boundary.Such singularities correspond to sharp details such as object boundaries, density jumps, and tissue interfaces.Source: §2 · p. 2
- Each wavefront-set element of f propagates in two opposite directions along bicharacteristics, whose spatial projections form geometric rays.The bicharacteristics are well defined for positive smooth sound speed bounded above and below.Source: §2 · pp. 2–3
- If the rays associated with every singularity of f avoid an open spatial region V, then the pressure is smooth on V.The corollary follows from the result that when the forward and backward bicharacteristics avoid an open space-time set, p is smooth there.Source: §2 · p. 3
3 Instability of reconstruction in thermoacoustic tomography
Theorem 3.1 shows that reconstruction is not Hölder stable when a fixed nonzero direction generates rays from every point in U that miss V. The proof constructs data with wavefront directions avoiding V, establishes boundedness of the measurement operator on that subspace, and uses singular-value estimates to rule out a Hölder estimate.
3 Instability of reconstruction in thermoacoustic tomography
- When rays in a fixed nonzero direction from every point of U miss V, reconstruction of f from g is not Hölder stable.The theorem rules out any estimate with positive Hölder exponent under the stated condition.Source: §3 · p. 3
3.1 Proof of Lemma 3.2
- The proof uses an infinite-dimensional closed subspace X whose functions have wavefront directions with rays that avoid V, yielding a bounded measurement operator on X.The ray condition implies smoothness of the measured wave on V, and the lemma establishes boundedness in every H^s for s ≥ 0.Source: §3 · p. 3, §3.1 · p. 4
- Energy conservation, bounds on c(x), and restriction to the observation set establish boundedness of the measurement operator on X in Sobolev spaces.The proof uses the conserved energy of an auxiliary wave and the bounded restriction operator to the measurement surface.Source: §3.1 · p. 4
3.2 Proof of Theorem 3.1
- Theorem 3.1 is proved by assuming a Hölder estimate and deriving singular-value bounds incompatible with the embedding estimates as j tends to infinity.The argument restricts to a subspace X_s1 and combines upper and lower bounds for singular values of natural Sobolev embeddings.Source: §3.2 · pp. 5–6
Appendix
A bounded linear operator from X to Y whose range lies in a continuously embedded Banach space Z is bounded as an operator from X to Z.
Appendix
- If Z is continuously embedded in Y and a bounded linear operator T:X→Y has T(X)⊂Z, then T:X→Z is bounded.The proof applies the closed graph theorem, using continuity of the embedding and uniqueness of limits in Y.Source: Appendix · p. 6