Reconstructions in impedance and optical tomography with singular interfaces

  title={Reconstructions in impedance and optical tomography with singular interfaces},
  author={Guillaume Bal},
  journal={Inverse Problems},
  • G. Bal
  • Published 1 February 2005
  • Mathematics
  • Inverse Problems
Singular layers modelled by a tangential diffusion process supported on an embedded closed surface (of co-dimension 1) have found applications in tomography problems. In optical tomography they may model the propagation of photons in thin clear layers, which are known to hamper the use of classical diffusion approximations. In impedance tomography they may be used to model thin regions of very high conductivity profile. In this paper we show that such surfaces can be reconstructed from boundary… 
We consider the reconstruction of singular surfaces from the over-determined boundary conditions of an elliptic problem. The problem arises in optical and impedance tomography, where void-like
Fréchet derivative with respect to the shape of a strongly convex nonscattering region in optical tomography
The aim of optical tomography is to reconstruct the optical properties inside a physical body, e.g. a neonatal head, by illuminating it with near-infrared light and measuring the outward flux of
Factorization method and irregular inclusions in electrical impedance tomography
In electrical impedance tomography, one tries to recover the conductivity inside a physical body from boundary measurements of current and voltage. In many practically important situations, the
Numerical implementation of the factorization method within the complete electrode model of electrical impedance tomography
In electrical impedance tomography, one tries to recover the spatial conductivity distribution inside a body from boundary measurements of current and voltage. In many practically important
Application of a weaker formulation of the factorization method to the characterization of absorbing inclusions in optical tomography
In the framework of diffuse tomography, i.e. optical tomography and electrical impedance tomography, the factorization method of Andreas Kirsch provides a tool for locating inhomogeneities inside an
Detecting Interfaces in a Parabolic-Elliptic Problem from Surface Measurements
This work deduces an equivalent variational formulation for the parabolic-elliptic problem and gives a new proof of the unique solvability based on Lions’s projection lemma and develops an adaptation of the factorization method to this time-dependent problem.
Locating Transparent Regions in Optical Absorption and Scattering Tomography
This work shows, both theoretically and numerically, that under suitable conditions the factorizatio...
Reconstruction of the photon diffusion coefficient in optical tomography
In this paper, we consider the recovery of photon diffusion coefficient in the case of piecewise constant coefficients in optical tomography from boundary measurements of light propagation within a
Probing for electrical inclusions with complex spherical waves
Let a physical body Ω in ℝ2 or ℝ3 be given. Assume that the electric conductivity distribution inside Ω consists of conductive inclusions in a known smooth background. Further, assume that a subset Γ


Generalized diffusion model in optical tomography with clear layers.
  • G. Bal, Kui Ren
  • Physics, Mathematics
    Journal of the Optical Society of America. A, Optics, image science, and vision
  • 2003
A generalized diffusion equation that models the propagation of photons in highly scattering domains with thin nonscattering clear layers and is shown numerically to be very accurate in two- and three-dimensional idealized cases is introduced.
Explicit Characterization of Inclusions in Electrical Impedance Tomography
It is shown that this procedure is conceptually similar to a recent method proposed by Kirsch in inverse scattering theory and holds true if and only if the dipole singularity lies inside the inhomogeneity.
Characterizing inclusions in optical tomography
In optical tomography, one tries to determine the spatial absorption and scattering distributions inside a body by using measured pairs of inward and outward fluxes of near-infrared light on the
Transport Through Diffusive and Nondiffusive Regions, Embedded Objects, and Clear Layers
  • G. Bal
  • Mathematics
    SIAM J. Appl. Math.
  • 2002
This paper deals with the diffusion approximation of transport equations in diffusive domains with nondiffusive inclusions, such as embedded objects and clear layers, where classical diffusion is not
Optical tomography in medical imaging
We present a review of methods for the forward and inverse problems in optical tomography. We limit ourselves to the highly scattering case found in applications in medical imaging, and to the
An investigation of light transport through scattering bodies with non-scattering regions.
A novel approach to calculating the light transport was developed, using diffusion theory to analyze the scattering regions combined with a radiosity approach to analyzed the propagation through the clear region, which found that the presence of a clear layer had a significant effect upon the light distribution.
The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions.
An improved implementation using a finite element method (FEM) that is direct to introduce extra equations into the standard diffusion FEM to represent nondiffusive light propagation across a nonscattering region.
Boundary conditions for light propagation in diffusive media with nonscattering regions
The diffusion approximation proves to be valid for light propagation in highly scattering media, but it breaks down in the presence of nonscattering regions, so results from an integral method based on the extinction theorem boundary condition are contrasted with both Monte Carlo and finite-element-method simulations.
Reconstruction of the potential from partial Cauchy data for the Schroedinger equation
In this paper we prove in dimension d ≥ 3 that the knowledge of the partial Cauchy data for the Schrodinger equation on any open subset Γ of the boundary determines uniquely the potential q provided
Nonuniqueness in diffusion-based optical tomography.
The main result applies to steady-state (dc) diffusion-based optical tomography, wherein it is demonstrated that simultaneous unique recovery of diffusion and absorption coefficients cannot be achieved.