Recovering the Initial Data of the Wave Equation from Neumann Traces

Dreier, Florian; Haltmeier, Markus

doi:10.1137/20m1362322

Cited by 2 publications

(7 citation statements)

References 38 publications

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“…Since we have proven in theorem 1 that K Ω = 0 is equivalent to Γ = ∂Ω being an ellipsoid, we conclude that the Neumann data version of theorem 1 is also true, i.e. inversion formula [3,4] is exact for ellipsoids only.…”

Section: Discussionmentioning

confidence: 67%

“…• In [3,4], inversion formulas for ellipsoids were obtained for the problem of reconstructing f (x) from the Neumann data ∂ ∂ν u| Γ×[0,∞) . In particular, it has been shown that, as in the case of Dirichlet data, if these formulas are applied to an arbitrary convex domain Ω with a smooth boundary, then an error terms K Ω appears, still given by equations ( 4), ( 8), (9).…”

Section: Discussionmentioning

confidence: 99%

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On the exactness of the universal backprojection formula for the spherical means Radon transform

Agranovsky

Kunyansky

2023

Inverse Problems

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The spherical means Radon transform f(x, r) is deﬁned by the integral of a function f in ℝn over the sphere S(x, r) of radius r centered at a x, normalized by the area of the sphere. The problem of reconstructing f from the data f(x, r) where x belongs to a hypersurface Γ⊆ℝn and r ∈ (0, ∞) has important applications in modern imaging modalities, such as photo- and thermoacoustic tomography. When Γ coincides with the boundary ∂Ω of a bounded (convex) domain Ω⊆ℝn, a function supported within Ω can be uniquely recovered from its spherical means known on Γ. We are interested in explicit inversion formulas for such a reconstruction. If Γ=∂Ω, such formulas are only known for the case when Γ is an ellipsoid (or one of its partial cases). This gives rise to the natural question: can explicit inversion formulas be found for other closed hypersurfaces Γ? In this article we prove, for the so-called "universal backprojection inversion formulas", that their extension to non-ellipsoidal domains Ω is impossible, and therefore ellipsoids constitute the largest class of closed convex hypersurfaces for which such formulas hold.

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Section: Discussionmentioning

confidence: 67%

Section: Discussionmentioning

confidence: 99%

On the exactness of the universal backprojection formula for the spherical means Radon transform

Agranovsky

Kunyansky

2023

Inverse Problems

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“…Specifically, we assume a finite end time T ∈ (0, ∞) for measuring the acoustic signals and consider mixed data on the time interval (0, T) as an output signal of the transducers. As opposed to the previous articles [38,39], where we assumed that Neumann traces in arbitrary dimensions as well as mixed data in two dimensions are available on the whole time interval (0, ∞), this justifies truncating data at finite values of T. The problem of recovering the initial data ( f , 0) from Dirichlet traces on a finite time interval has already been discussed in [12], where an inversion formula in dimension two has been developed. More precisely, they used Abel transform inversion for recovering the spherical means with centers lying on a circle in R 2 from its Dirichlet traces.…”

Section: Inversion From Finite Time Intervalsmentioning

confidence: 99%

“…The subsequent calculations are based on our previous work in [38,39], where we derived the reconstruction formulas…”

Section: Previous Workmentioning

confidence: 99%

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Photoacoustic inversion formulas using mixed data on finite time intervals*

Dreier

Haltmeier

2022

Inverse Problems

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We study the inverse source problem in photoacoustic tomography (PAT) for mixed data, which denote a weighted linear combination of the acoustic pressure and its normal derivative on an observation surface. We consider in particular the case where the data are only available on finite time intervals, which accounts for real-world usage of PAT where data are only feasible within a certain time interval. Extending our previous work, we derive explicit formulas up to a smoothing integral on convex domains with a smooth boundary, yielding exact reconstruction for circular or elliptical domains. We also present numerical reconstructions of our new exact inversion formulas on finite time intervals and compare them with the reconstructions of our previous formulas for unlimited time wave measurements.

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Recovering the Initial Data of the Wave Equation from Neumann Traces

Cited by 2 publications

References 38 publications

On the exactness of the universal backprojection formula for the spherical means Radon transform

On the exactness of the universal backprojection formula for the spherical means Radon transform

Photoacoustic inversion formulas using mixed data on finite time intervals*

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