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42
Reconstruction of coefficients in scalar secondorder elliptic equations from knowledge of their solutions
"... This paper concerns the reconstruction of possibly complexvalued coefficients in a secondorder scalar elliptic equation posed on a bounded domain from knowledge of several solutions of that equation. We show that for a sufficiently large number of solutions and for an open set of corresponding bo ..."
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This paper concerns the reconstruction of possibly complexvalued coefficients in a secondorder scalar elliptic equation posed on a bounded domain from knowledge of several solutions of that equation. We show that for a sufficiently large number of solutions and for an open set of corresponding boundary conditions, all coefficients can be uniquely and stably reconstructed up to a well characterized gauge transformation. We also show that in some specific situations, a minimum number of such available solutions equal to In = 1 2n(n+3) is sufficient to uniquely and globally reconstruct the unknown coefficients. This theory finds applications in several coupledphysics medical imaging modalities including photoacoustic tomography, transient elastography, and magnetic resonance elastography. 1
Recovery of a source term or a speed with one measurement and applications
"... ABSTRACT. We study the problem of recovery the source a.t;x/F.x / in the wave equation in anisotropic medium with a known so that a.0;x / 6D 0 with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the nonlinear ..."
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Cited by 16 (5 self)
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ABSTRACT. We study the problem of recovery the source a.t;x/F.x / in the wave equation in anisotropic medium with a known so that a.0;x / 6D 0 with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the nonlinear problem of recovery the sound speed in the equation uttc2.x/u D 0with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented. 1.
CARLEMAN ESTIMATES FOR GLOBAL UNIQUENESS, STABILITY AND NUMERICAL METHODS FOR COEFFICIENT INVERSE PROBLEMS
, 2012
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Mathematics of Photoacoustic and Thermoacoustic Tomography
, 2009
"... The chapter surveys the mathematical models, problems, and algorithms of the Thermoacoustic (TAT) and Photoacoustic (PAT) Tomography. TAT and PAT represent probably the most developed of the several novel “hybrid ” methods of medical imaging. These new modalities combine different physical types of ..."
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Cited by 11 (3 self)
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The chapter surveys the mathematical models, problems, and algorithms of the Thermoacoustic (TAT) and Photoacoustic (PAT) Tomography. TAT and PAT represent probably the most developed of the several novel “hybrid ” methods of medical imaging. These new modalities combine different physical types of waves (electromagnetic and acoustic in case of TAT and PAT) in such a way that the resolution and contrast of the resulting method are much higher than those achievable using only acoustic or electromagnetic measurements.
THERMOACOUSTIC TOMOGRAPHY ARISING IN BRAIN IMAGING
"... Abstract. We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a conv ..."
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Cited by 10 (7 self)
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Abstract. We study the mathematical model of thermoacoustic and photoacoustic tomography when the sound speed has a jump across a smooth surface. This models the change of the sound speed in the skull when trying to image the human brain. We derive an explicit inversion formula in the form of a convergent Neumann series under the assumptions that all singularities from the support of the source reach the boundary. 1.
Inverse Transport Theory of Photoacoustics
, 908
"... Abstract. We consider the reconstruction of optical parameters in a domain of interest from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency electromagnetic waves into the domain and measures acoustic signals emitted by the resulting thermal expansion. Acoustic signals are ..."
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Abstract. We consider the reconstruction of optical parameters in a domain of interest from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency electromagnetic waves into the domain and measures acoustic signals emitted by the resulting thermal expansion. Acoustic signals are then used to construct the deposited thermal energy map. The latter depends on the constitutive optical parameters in a nontrivial manner. In this paper, we develop and use an inverse transport theory with internal measurements to extract information on the optical coefficients from knowledge of the deposited thermal energy map. We consider the multimeasurement setting in which many electromagnetic radiation patterns are used to probe the domain of interest. By developing an expansion of the measurement operator into singular components, we show that the spatial variations of the intrinsic attenuation and the scattering coefficients may be reconstructed. We also reconstruct coefficients describing anisotropic scattering of photons, such as the anisotropy coefficient g(x) in a HenyeyGreenstein phase function model. Finally, we derive stability estimates for the reconstructions.
Hybrid inverse problems for a system of Maxwell’s equations
 Inverse Problems
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Quantitative photoacoustic tomography with partial data
 Inverse Problem
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IS A CURVED FLIGHT PATH IN SAR BETTER THAN A STRAIGHT ONE?
"... ABSTRACT. In the plane, we study the transformR
f of integrating a unknown function f over circles centered at a given curve. This is a simplified model of SAR, when the radar is not directed but has other applications, like thermoacoustic tomography, for example. We study the problem of recovering ..."
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Cited by 4 (1 self)
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ABSTRACT. In the plane, we study the transformR
f of integrating a unknown function f over circles centered at a given curve. This is a simplified model of SAR, when the radar is not directed but has other applications, like thermoacoustic tomography, for example. We study the problem of recovering the wave front set WF.f /. If the visible singularities of f hit once, we show that the “artifacts ” cannot be resolved. If
is a closed curve, we show that this is still true. On the other hand, if f is known a priori to have singularities in a compact set, then we show that one can recover WF.f /, and moreover, this can be done in a simple explicit way, using backpropagation for the wave equation. 1.