Support

Support from NSF grant DMS-1712790 (formerly 1634544, formerly 1514820) is gratefully acknowledged.

Research Topics

My broad research interests are in Inverse problems and their applications to imaging sciences (e.g., medical and geophysical imaging). The analysis of an inverse problem divides into the following aspects: injectivity and stability assessments (i.e. is the unknown uniquely characterized by the data ? if yes, in what function spaces is the inverse continuous and what is its modulus of continuity ?); derivation of inversion formulas and their numerical implementation; strategies for dealing with and reducing noise in corrupted data. The main fields involved in my work are Analysis of PDE's (which is also the arXiv tag most often used for inverse problems papers, as "inverse problems" is not itself an arXiv classification), differential geometry, applied functional analysis and numerical methods.

I. Inverse problems in Boltzmann transport and applications to medical imaging

  • Reconstruction of the scattering coefficient from angularly averaged boundary measurements in the stationary and time-harmonic setting, with applications in Optical Tomography.
  • Numerical work (in Matlab and Python): coding a forward transport solver with image rotation techniques and controlled transverse diffusion to tackle the so-called "ray effect". Application to reconstruction of optical parameters in the stationary setting.
  • Reconstruction of both attenuation and isotropic scattering from knowledge of ballistic and single scattering measurements, with applications to SPECT.
Related publications: [J1], [J2], [J3], [J4], [J16]

II. Theoretical and numerical aspects of coupled-physics (a.k.a. hybrid) medical imaging methods

  • In particular, inverse conductivity problem with internal measurements of power density and current density type, with applications to some coupled-physics medical imaging modalities (UMEIT, UMOT, ImpACT, CDII). 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++.
  • Reconstruction of isotropic and anisotropic elasticity tensors from internal displacement fields, with application in transient elastography.
Related publications: [J5], [J6], [J7], [J8], [J9], [J10], [J11], [J12], [J17], [J26]

III. Integral geometry, geodesic X-ray transforms, tensor tomography

  • Numerical implementation of reconstruction algorithms for functions and solenoidal vector fields from knowledge of their ray transform. Theoretical and numerical generalization to other types of integrands (symmetric differentials and their transverse derivative).
  • In two dimensions, reconstruction of solenoidal tensors of higher order.
  • Study of non-simple metrics (e.g. metrics with conjugate points) and the detrimental effects of caustics on the qualitative properties of the X-ray transform. Study of cases with trapping and nontrivial topology.
  • Inversion in the attenuated case.
Related publications: [J13], [J14], [J15], [J18], [J19], [J20], [J21], [J22], [J23], [J24], [J25]

Talks

(Hover on the title for some of the later abstracts)