Multiwave imaging in an enclosure with variable wave speed

Abstract

In this paper we consider the mathematical model of thermo-and photoacoustic tomography for the recovery of the initial condition of a wave field from knowledge of its boundary values. Unlike the free-space setting, we consider the wave problem in a region enclosed by a surface where an impedance boundary condition is imposed. This condition models the presence of physical boundaries such as interfaces or acoustic mirrors which reflect some of the wave energy back into the enclosed domain. By recognizing that the inverse problem is equivalent to a statement of boundary observability, we use control operators to prove the unique and stable recovery of the initial wave profile from knowledge of boundary measurements. Since our proof is constructive, we explicitly derive a solvable equation for the unknown initial condition. This equation can be solved numerically using the conjugate gradient method. We also propose an alternative approach based on the stabilization of waves. This leads to an exponentially and uniformly convergent Neumann series reconstruction when the impedance coefficient is not identically zero. In both cases, if well-known geometrical conditions are satisfied, our approaches are naturally suited for variable wave speed and for measurements on a subset of the boundary.

A Dissipative Time Reversal Technique for Photoacoustic Tomography in a Cavity

“…We refer the reader to [BLR92] for a detailed discussion of GCC. This condition was shown in [SY15,AM15] to be sufficient for the stability of the inversion of Λ. Our results are also based on the assumption that GCC is satisfied.…”

“…Contraction properties of the time reversal operator. Similarly to [SY15,AM15], our analysis of the problem relies on the known results on propagation of singularities of the wave equation (e.g., [AM77,MS78]) and on the known solution of the wave stabilization problem [BLR92].…”

“…While we only considered the simplest wave equation in the Euclidean spaces, our analysis can be extended to problems formulated on Riemannian manifolds (as in [SY15]) and/or to problems with a potential (as in [AM15]). …”

“…The first author is grateful to C. Bardos, G. Nakamura, and P. Stefanov for helpful discussions and comments. He also thanks S. Acosta for the slides of his talk on the paper [AM15]. The second author thanks L. Friedlander for a helpful discussion.…”

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

“…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.…”