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.

Letg be a compact Riemmanian manifold with non-empty boundary. Consider the second order hyperbolic initial-boundary value problemis a first-order perturbation of the Laplace operator − g on g . Here b and q are a covector field and a potential, respectively, in . We prove Hölder type stability estimates near generic simple Riemannian metrics for the inverse problem of recovering simultaneously g, b, and q from the hyperbolic Dirichletto-Neumann (DN) map associated, f → u − i b g u × 0 T modulo a class of transformations that fixed the hyperbolic DN map.

In this paper, we develop a general approach to prove stability for the non linear second step of hybrid inverse problems. We work with general functionals of the form σ|∇u| p , 0 < p ≤ 1, where u is the solution of the elliptic partial differential equation ∇ · σ∇u = 0 on a bounded domain Ω with boundary conditions u| ∂Ω = f . We prove stability of the linearization and Hölder conditional stability for the non-linear problem of recovering σ from the internal measurement.

We describe a technique to generate imagery with improved sharpness for individuals having refractive vision problems. Our method can reduce their dependence on corrective eyewear. It also benefits individuals with normal vision by improving visual acuity at a distance and of small details. Our approach does not require custom hardware. Instead, the calculated images can be shown on a standard computer display, on printed paper, or superimposed on a physical scene using a projector. Our technique uses a constrained total-variation method to produce a deconvolution result which upon observation appears sharp at the edges. We introduce a novel relative total variation term that enables controlling ringing reduction, contrast gain and sharpness. The end result is the ability to generate sharper appearing images, even for individuals with refractive vision problems including myopia, hyperopia, presbyopia, and astigmatism. Our approach has been validated in simulation, in camera-screen experiments, and in a study with human observers.

In this paper we consider a mathematical model for photoacoustic imaging which takes into account attenuation due to thermodynamic dissipation. The propagation of acoustic (compressional) waves is governed by a scalar wave equation coupled to the heat equation for the excess temperature. We seek to recover the initial acoustic profile from knowledge of acoustic measurements at the boundary.We recognize that this inverse problem is a special case of boundary observability for a thermoelastic system. This leads to the use of control/observability tools to prove the unique and stable recovery of the initial acoustic profile in the weak thermoelastic coupling regime. This approach is constructive, yielding a solvable equation for the unknown acoustic profile. Moreover, the solution to this reconstruction equation can be approximated numerically using the conjugate gradient method. If certain geometrical conditions for the wave speed are satisfied, this approach is well-suited for variable media and for measurements on a subset of the boundary. We also present a numerical implementation of the proposed reconstruction algorithm.