Thermoacoustic tomography for an integro-differential wave equation modeling attenuation

Acosta, Sebastián; Palacios, Benjamin

doi:10.1016/j.jde.2017.10.012

Cited by 24 publications

(24 citation statements)

References 34 publications

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“…A different class of algorithms extends the time reversal technique to the attenuated case (see [1,3,10,31,35,40,59,72]). Note that the time reversal of the attenuated wave equation yields a noise amplifying equation.…”

Section: Comparison To Previous and Related Workmentioning

confidence: 99%

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Iterative methods for photoacoustic tomography in attenuating acoustic media

2017

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The development of efficient and accurate reconstruction methods is an important aspect of tomographic imaging. In this article, we address this issue for photoacoustic tomography. To this aim, we use models for acoustic wave propagation accounting for frequency dependent attenuation according to a wide class of attenuation laws that may include memory. We formulate the inverse problem of photoacoustic tomography in attenuating medium as an illposed operator equation in a Hilbert space framework that is tackled by iterative regularization methods. Our approach comes with a clear convergence analysis. For that purpose we derive explicit expressions for the adjoint problem that can efficiently be implemented. In contrast to time reversal, the employed adjoint wave equation is again damping and, thus has a stable solution. This stability property can be clearly seen in our numerical results. Moreover, the presented numerical results clearly demonstrate the efficiency and accuracy of the derived iterative reconstruction algorithms in various situations including the limited view case.

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Section: Comparison To Previous and Related Workmentioning

confidence: 99%

“…This equation has been considered, for example, in [68,67,76]. For tissue, the exponent γ in (2.9) is in the range (1,2]. Note that for positive γ that is not an integer the power law model is weakly causal.…”

Section: Example 25 (Power Law)mentioning

confidence: 99%

Iterative methods for photoacoustic tomography in attenuating acoustic media

2017

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“…In order to change the situation, we propose a new mathematical concept-generalized positive definite kernel, and present its basic properties (in section 3). Then, with the help of GPDK and its properties derived here, we prove a criterion of the polynomial stability and the integrability of the energy for (1) with such a general but more complicated and useful memory. It is, as far as we know, the first and efficient criterion of the polynomial stability for these evolution equations.…”

mentioning

confidence: 99%

“…The global solutions to the equation of thermoelasticity with fading memory are presented in [29]) when the kernels are strongly positive definite. The existence and stability of solutions to the inverse problem of thermoacoustic tomography is obtained in [1] when the kernels are strongly positive definite. For more recent works involving the strongly positive definite kernels or completely positive definite kernels, we refer the reader to, e.g., [19,20].…”

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confidence: 99%

Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

Jin¹,

Liang²,

Xiao³

2021

DCDS-S

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<p style='text-indent:20px;'>We are concerned with the polynomial stability and the integrability of the energy for second order integro-differential equations in Hilbert spaces with positive definite kernels, where the memory can be oscillating or sign-varying or not locally absolutely continuous (without any control conditions on the derivative of the kernel). For this stability problem, tools from the theory of existing positive definite kernels can not be applied. In order to solve the problem, we introduce and study a new mathematical concept – <i>generalized positive definite kernel</i> (GPDK). With the help of GPDK and its properties, we obtain an efficient criterion of the polynomial stability for evolution equations with such a general but more complicated and useful memory. Moreover, in contrast to existing positive definite kernels, GPDK allows us to directly express the decay rate of the related kernel.</p>

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“…For qualitative photoacoustic tomography, the goal is to form an image of the initial state of the pressure field using boundary measurements of the transient pressure waves. Most of the reconstruction methods assume that the actual pressure field (Dirichlet data) can be measured at the boundary [2,3,4,5,11,22,32,35,36,37,38,42,44,51,52,55,59,60,61,63]. In reality, ultrasound sensors are not able to measure the pressure field directly.…”

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confidence: 99%

Well-Posedness for Photoacoustic Tomography with Fabry--Perot Sensors

Acosta

2019

SIAM J. Imaging Sci.

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In the mathematical analysis of photoacoustic imaging, it is usually assumed that the acoustic pressure (Dirichlet data) is measured on a detection surface. However, actual ultrasound detectors gather data of a different type. In this paper, we propose a more realistic mathematical model of ultrasound measurements acquired by the Fabry-Perot sensor. This modeling incorporates directional response of such sensors. We study the solvability of the resulting photoacoustic tomography problem, concluding that the problem is well-posed under certain assumptions. Numerical reconstructions are implemented using the Landweber iterations, after discretization of the governing equations using the finite element method.

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Thermoacoustic tomography for an integro-differential wave equation modeling attenuation

Cited by 24 publications

References 34 publications

Iterative methods for photoacoustic tomography in attenuating acoustic media

Iterative methods for photoacoustic tomography in attenuating acoustic media

Uniform polynomial stability of second order integro-differential equations in Hilbert spaces with positive definite kernels

Well-Posedness for Photoacoustic Tomography with Fabry--Perot Sensors

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