The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces

Haltmeier, Markus; Pereverzyev, Sergiy

doi:10.1016/j.jmaa.2015.04.018

Cited by 24 publications

(29 citation statements)

References 49 publications

(103 reference statements)

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“…Elliptical cylinders and paraboloids are natural candidates for carrying out these generalizations. Such investigations are the subject of our work in [16].…”

Section: Discussionmentioning

confidence: 98%

Recovering a Function from Circular Means or Wave Data on the Boundary of Parabolic Domains

Haltmeier

Pereverzyev

2015

SIAM J. Imaging Sci.

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Determining a function from its circular means with restricted centers is crucial for many modern medical imaging and remote sensing applications. Examples include photo-and thermoacoustic tomography, ultrasound imaging, and synthetic aperture radar. In this paper we derive an explicit inversion formula for the case where the centers of the circles of integration are restricted to the boundary of parabolic domains. A similar result is obtained for recovering the initial data of the wave equation from data on the boundary of a parabolic domain. These results show that the universal back-projection formulas, previously known to be exact for half-spaces, discs, and ellipsoids, are also exact for parabolic domains. We further present numerical simulations supporting our theoretical results. There, we also include two previously proposed strategies (dynamic aperture length correction and weight factor correction) accounting for the truncation of the parabola in practical applications.

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“…Elliptical cylinders and paraboloids are natural candidates for carrying out these generalizations. Such investigations are the subject of our work in [16].…”

Section: Discussionmentioning

confidence: 98%

Recovering a Function from Circular Means or Wave Data on the Boundary of Parabolic Domains

Haltmeier

Pereverzyev

2015

SIAM J. Imaging Sci.

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“…For each fixed ε, wave η ε (τ − ω · x) is in the form (12) and satisfies (13). Therefore, due to Proposition 1, density ϕ ω,ε (t,ŷ) vanishes on W ∩ (R × S 2 ), where W is defined by (16), with T (ω) = (−1, 1). As in the 2D case, in the limit ε → 0, plane waves η ε (τ − ω · x) converge to δ (τ − ω · x) .…”

Section: Explicit Expression For the Densitymentioning

confidence: 89%

“…Given a Lipschitz domain Ω − and plane wave µ ω (t, x) satisfying (12), (13), there exists a unique tempered distribution ξ ω (t, ·) supported on T (ω) with values in H −1/2 (Γ), such that µ ω (t, x) is represented in T (ω) × Ω − by the single layer potential in the form (14). Moreover, density ξ ω vanishes on W ∩ (R×Γ), where W is defined by (16).…”

Section: Representing Plane Waves By Single Layer Potentialsmentioning

confidence: 99%

Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data

Kunyansky

2018

Inverse Problems

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We investigate the inverse source problem for the wave equation, arising in photo-and thermoacoustic tomography. There exist quite a few theoretically exact inversion formulas explicitly expressing solution of this problem in terms of the measured data, under the assumption of constant and known speed of sound. However, almost all of these formulas require data to be measured either on an unbounded surface, or on a closed surface completely surrounding the object. This is too restrictive for practical applications. The alternative approach we present, under certain restriction on geometry, yields theoretically exact reconstruction of the standard Radon projections of the source from the data measured on a finite open surface. In addition, this technique reduces the time interval where the data should be known. In general, our method requires a pre-computation of densities of certain single-layer potentials. However, in the case of a truncated circular or spherical acquisition surface, these densities are easily obtained analytically, which leads to fully explicit asymptotically fast algorithms. We test these algorithms in a series of numerical simulations.

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“…In [20], it was shown, that the same result also holds for parabolic domains Ω with d = 2. The formula G d in arbitrary spatial dimension d ≥ 2 on certain quadric hypersurfaces, including the parabolic ones, has been analyzed in [21].…”

Section: Explicit Inversion Formulamentioning

confidence: 99%

“…Comparing the error estimates (13), (14) for the learned approximations with n ≥ 1 and the error estimates (20), (21) for the approximations using zero extension of the limited view wave data, one sees that these error estimates differ regarding the following factors:…”

Section: Approximate Reconstructions and Their Error Analysismentioning

confidence: 99%

Operator Learning Approach for the Limited View Problem in Photoacoustic Tomography

Dreier

Pereverzyev

Haltmeier

2018

Computational Methods in Applied Mathematics

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In photoacoustic tomography, one is interested to recover the initial pressure distribution inside a tissue from the corresponding measurements of the induced acoustic wave on the boundary of a region enclosing the tissue. In the limited view problem, the wave boundary measurements are given on the part of the boundary, whereas in the full view problem, the measurements are known on the whole boundary. For the full view problem, there exist various fast and robust reconstruction methods. These methods give severe reconstruction artifacts when they are applied directly to the limited view data. One approach for reducing such artefacts is trying to extend the limited view data to the whole region boundary, and then use existing reconstruction methods for the full view data. In this paper, we propose an operator learning approach for constructing an operator that gives an approximate extension of the limited view data. We consider the behavior of a reconstruction formula on the extended limited view data that is given by our proposed approach. Approximation errors of our approach are analyzed. We also present numerical results with the proposed extension approach supporting our theoretical analysis.

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The universal back-projection formula for spherical means and the wave equation on certain quadric hypersurfaces

Cited by 24 publications

References 49 publications

Recovering a Function from Circular Means or Wave Data on the Boundary of Parabolic Domains

Recovering a Function from Circular Means or Wave Data on the Boundary of Parabolic Domains

Theoretically exact photoacoustic reconstruction from spatially and temporally reduced data

Operator Learning Approach for the Limited View Problem in Photoacoustic Tomography

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