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Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography
Mark Agranovsky, Peter Kuchment
TL;DR
TAT uses RF absorption to generate pressure waves that are measured for reconstruction, while PAT initiates the signal with a laser pulse. The paper develops an inversion formula for arbitrary observation-surface geometry and variable sound speed under a non-trapping condition, and establishes uniqueness even when the source extends beyond the observation surface.
Problem
Explicit inversion formulas were not known for non-spherical surfaces or variable sound speed, and earlier formulas could reconstruct incorrectly when the source extended outside the observation surface.
Source: §1 · p. 4Method
The paper develops a general operator-theory inversion formula for arbitrary observation-surface geometry and variable non-trapping sound speed.
Source: §1 · p. 4Results
Theorem 7 states that under the non-trapping conditions, the data uniquely determine any compactly supported f(x), without requiring its support to lie inside S.
Source: §3.3 · p. 14Takeaways & Limitations
The formulas apply with variable non-trapping sound speed where the method of fails because the whole-space Green's function cannot be written explicitly.
Source: §4 · p. 15Although recovery is unique when support extends beyond S, reconstruction can be unstable for some parts of f outside the observation surface.
Source: §4 · p. 17
Abstract
from arXiv · showhide
The paper contains a simple approach to reconstruction in Thermoacoustic and Photoacoustic Tomography. The technique works for any geometry of point detectors placement and for variable sound speed satisfying a non-trapping condition. A uniqueness of reconstruction result is also obtained.
1 Introduction
TAT combines electromagnetic absorption contrast with ultrasound pressure-wave detection to recover the absorption distribution. The paper develops a general inversion approach for arbitrary observation-surface geometry and variable sound speed under a non-trapping condition, and establishes uniqueness of reconstruction.
1 Introduction
- TAT uses RF absorption to generate thermoelastic pressure waves, which transducers measure to recover the absorption distribution.Absorption supplies contrast, while ultrasound detection supplies high resolution.Source: §1 · pp. 1–2
- Earlier explicit inversion formulas were limited to spherical measurement surfaces, and some gave incorrect interior reconstructions when the source extended outside S [10, 11, 24, 42].The introduction also notes that explicit formulas were not known for non-spherical surfaces or variable sound speed.Source: §1 · pp. 3–4
- The proposed operator-theory inversion applies to arbitrary observation-surface geometry, variable non-trapping sound speed, and sources not confined inside the observation surface.The formula involves functions of the Dirichlet Laplacian inside S, and the paper shows how it works in special cases.Source: §1 · p. 4
- The paper establishes uniqueness under non-trapping conditions, even when a compactly supported source extends beyond the observation surface, though the inversion recovers only its interior part.The paper develops the general formula through a constant-speed example, local energy decay estimates, and an abstract hyperbolic-equation result.Source: §1 · p. 4
2 Mathematical model of TAT
The mathematical model treats the unknown absorption distribution as the initial pressure and uses pressure measurements on a detector surface as data. The paper addresses inversion and uniqueness for this recovery problem.
2 Mathematical model of TAT
- The reconstruction target is a compactly supported function f(x), and point detectors on a closed surface S measure pressure p(y,t).The model assumes the sound speed c(x) is known.Source: §2 · p. 5
- The data g(y,t) are the lateral pressure measurements used to recover the initial value f(x) at t = 0.The model states p(x,0) = f(x), p_t(x,0) = 0, and p(y,t) = g(y,t) on S × R+.Source: §2 · p. 5
- Although the data are recorded on S × R+, the whole-space wave-equation constraint can make them sufficient for recovery of f.The paper addresses uniqueness of reconstruction and inversion, not the full list of tomographic questions such as stability or incomplete data.Source: §2 · p. 6
3 Discussion of the problem and the main results
The paper develops reconstruction from detector data under compact-support and non-trapping assumptions, including for variable sound speed and without requiring the object to lie inside the observation surface. It gives an interior reconstruction procedure and proves uniqueness for the whole compactly supported function.
3 Discussion of the problem and the main results
- The observation surface may consist of finitely many transversally intersecting smooth pieces, and f need not be supported inside it.The paper also assumes smooth sound speed and initial function for simplicity, noting that this regularity restriction can be weakened.Source: §3 · pp. 6–7
- The setup assumes compact support of f, positive sound speed equal to 1 outside a bounded region, and non-trapping rays.The paper notes that sufficiently fast decay can replace compact support; without sufficiently fast decay, uniqueness fails.Source: §3 · p. 7
- The inverse problem is treated as observability: although measurements are on S, the wave solution holds in the whole space.Compact support and non-trapping yield local energy decay inside B, with exponential decay in odd dimensions and power decay in even dimensions.Source: §3 · pp. 7–8
3.1 A test case: constant sound speed in an odd dimension
- In the odd-dimensional constant-speed test case, the data yield an L2(B)-convergent reconstruction of f inside B.The derivation uses Huygens’ principle and assumes a smooth, compactly supported initial function.Source: §3.1 · pp. 8, 10
3.2 An abstract theorem
- An abstract result extends the representation and coefficient-recovery procedure to a Hilbert space with a positive self-adjoint operator having discrete spectrum.Its stated conditions include decay of u and its derivative and integrability of α(t).Source: §3.2 · pp. 11–12
3.3 An application to TAT in inhomogeneous media
- Local-energy decay and Sobolev trace estimates ensure that the inhomogeneous-medium problem satisfies the abstract theorem’s conditions.The paper applies these estimates to u(t) and E(gtt).Source: §3.3 · p. 12
- For inhomogeneous sound speed, f is reconstructed inside B from g by an L2(B)-convergent series.The theorem applies under the sound-speed conditions imposed at the start of Section 3.Source: §3.3 · pp. 12–13
- Under the stated non-trapping conditions, the data uniquely determine any compactly supported f, without requiring its support to lie inside S.The uniqueness result covers the whole function, not just its restriction to the interior B.Source: §3.3 · p. 14
4 Further discussion and remarks
The discussion explains how the inversion handles sources outside the observation surface, relates the formulas to earlier approaches, and notes implementation choices and qualifications. It also identifies a possible link between decay of solutions and range descriptions of the data.
4 Further discussion and remarks
- In the one-dimensional constant-speed example, substituting the boundary data into formula (14) recovers the standard Fourier coefficients of f.The example uses B = [0, 1] and the eigenfunction system ψk.Source: §4 · pp. 14–15
- The paper's formulas recover those of after adapting the data, but do not require the whole-space Green's function that needs.They also apply to variable non-trapping sound speed, where the passage says the method of fails.Source: §4 · p. 15
- The inversion does not require f to be supported inside S, and its use of all times avoids the finite-time information loss associated with earlier backprojection formulas.Those earlier formulas use data only up to the diameter of B; for support extending outside S, the passage says reconstruction is then not unique.Source: §4 · p. 15
- The initial velocity can also be reconstructed using formulas (5) and (16), although the paper assumes it is zero throughout.Those formulas were derived without the zero-initial-velocity assumption.Source: §4 · p. 15
- The paper's accepted nonhomogeneous-medium models neglect density variations, but its Hilbert-space formalism can include them through a more general operator A.The passage says this requires little change to the formulas and methods.Source: §4 · p. 16
- The integral formulas use functions of the Dirichlet Laplacian in L2(B), which the paper implements with eigenfunction expansions; a spherical-surface backprojection conversion remains only a possibility.The authors say the route from these formulas to analytic backprojection is unclear.Source: §4 · pp. 16–17
- Theorem 7 gives unique recovery for compactly supported f even when its support is not confined to B, although some exterior parts may be unstable.The passage relates this instability to visible or audible and non-visible or non-audible zones of the wave front set.Source: §4 · p. 17
- The authors trace their approach to special features of the boundary data and connect solution decay to range descriptions for constant speed, with possible analogs for variable speed.The constant-speed link is stated as something that can be shown but is not demonstrated in the paper.Source: §4 · pp. 17–18