Ph.D. thesis

Defended April 27th, 2012.
Title: Taming unstable inverse problems. Mathematical routes toward high-resolution medical imaging modalities. (with distinction) pdf
Adviser: Guillaume Bal.

Research Topics:

Interests in the disciplines of applied functional analysis, differential geometry, partial differential equations, stationary phase and inverse problems.

I. Inverse problems involving the Boltzmann transport equation and application to medical imaging

  • Reconstruction of the scattering coefficient from boundary measurement in the stationary and time-harmonic setting, with applications in Optical Tomography.
  • Numerical work: coding a forward transport solver with image rotation techniques. Application to reconstruction of optical parameters in stationary and time-harmonic settings. Languages used: Matlab and Python.

Related publications

  1. "Inverse Transport with isotropic sources and angularly-averaged measurements" (with Guillaume Bal and Ian Langmore), Inv. Probl. Imaging, Vol.2 (2008), No. 1. pdf
  2. "An accurate solver for forward and inverse transport" (with Guillaume Bal), Journal of Computational Physics, Vol. 229, Issue 13, July, 2010. pdf
  3. "Angular average of time-harmonic transport solutions" (with Guillaume Bal, Alexandre Jollivet and Ian Langmore), Communications in Partial Differential Equations, 1532-4133, Vol. 36, Issue 6, 2011, Pages 1044 - 1070. pdf
  4. "Inverse transport with isotropic time-harmonic sources" (with Guillaume Bal), SIAM J. Math. Anal., Vol. 44, No. 1, pp. 134-161 (2012). pdf

II. Theoretical and numerical aspects of hybrid or coupled-physics medical imaging methods

  • In particular, inverse conductivity problem with internal measurements of power density type, with applications in Ultrasound Modulated- Electrical Impedance Tomography (UMEIT) and Optical Tomography (UMOT). I am concerned with the derivation of reconstruction algorithms and their numerical validation in isotropic and anisotropic settings. Languages used for numerical work: MatLab and FreeFEM++.
  • I also recently started a project involving the reconstruction of the elasticity tensor from internal measurements in linear elasticity, with application in transient elastography.

Related publications

  1. "Inverse diffusion problems with redundant internal information" (with Guillaume Bal), Inv. Probl. Imaging, Vol. 6 (May 2012), No. 2. arXiv
  2. "Inverse anisotropic diffusion from power density measurements in two dimensions" (with Guillaume Bal), Inverse Problems 28 (2012) 084001. arXiv
  3. "Inverse diffusion from knowledge of power densities" (with Guillaume Bal, Eric Bonnetier and Faouzi Triki), Inv. Probl. Imaging, Vol 7 (May 2013), No. 2. arXiv
  4. "Inverse anisotropic conductivity from power densities in dimension $n\ge 3$" (with Guillaume Bal), Comm. PDE, Vol. 38, Issue 7 (2013). arXiv
  5. (submitted) "Linearized internal functionals for anisotropic conductivities" (with Guillaume Bal and Chenxi Guo), submitted to Inv. Probl. and Imaging (2013). arXiv
  6. (submitted) "Inverse anisotropic conductivity from internal current densities" (with Guillaume Bal and Chenxi Guo), submitted to Inverse Problems (2013). arXiv

III. Geodesic X-ray transforms and the tensor tomography problem

  • Numerical implementation of reconstruction algorithms for functions and solenoidal vector fields.
  • In two dimensions, reconstruction of solenoidal tensors of higher order.
  • Study of nonsimple cases.

Related publications

  1. (submitted) "Numerical implementation of geodesic X-ray transforms and their inversion", submitted to SIAM J. on Imaging Sciences. (2013). arXiv

Talks:

  • "Inverse Transport and Tomography", May 2010, Banff:
    Talk title: "An accurate solver for forward and inverse transport", pdf
  • "APAM Research Seminar", March 4, 2011, Columbia University:
    Title: "Inverse Diffusion with power density measurements", pdf
  • "PASI CIPPDE 2012", Jan. 24, 2012, Santiago de Chile:
    Title: "Inverse anisotropic diffusion from power density functionals in two dimensions", pdf
  • Conference on Inverse Problems in honor of Gunther Uhlmann, June 19th, 2012, University of California Irvine:
    Title: "The inverse conductivity problem with power densities in general dimension", pdf
  • University of Helsinki, June 10th, 2013. Title: "Inverse anisotropic conductivity from power densities in two dimensions", pdf
  • AIPC 2013, KAIST (Daejeon, Korea), July 1st, 2013. Title: "Coupled-physics methods for inverse conductivity", pdf
  • Inverse Problems and Geometry, BIRS, Banff, Sept. 2013. Title:"Recent progress on the explicit inversion of geodesic X-ray transforms", pdf

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