Taming unstable inverse problems: Mathematical routes toward highresolution medical imaging modalities

Title:

Taming unstable inverse problems: Mathematical routes toward highresolution medical imaging modalities

Author(s):

Monard, Francois

Date:

2012

Type:

Dissertations

Department:

Applied Physics and Applied Mathematics

Permanent URL:

http://hdl.handle.net/10022/AC:P:13162

Notes:

Ph.D., Columbia University.

Abstract:

This thesis explores two mathematical routes that make the transition from some severely illposed parameter reconstruction problems to betterposed versions of them. The general introduction starts by defining what we mean by an inverse problem and its theoretical analysis. We then provide motivations that come from the field of medical imaging. The first part consists in the analysis of an inverse problem involving the Boltzmann transport equation, with applications in Optical Tomography. There we investigate the reconstruction of the spatiallydependent part of the scattering kernel, from knowledge of angularly averaged outgoing traces of transport solutions and isotropic boundary sources. We study this problem in the stationary regime first, then in the timeharmonic regime. In particular we show, using techniques from functional analysis and stationary phase, that this inverse problem is severely illposed in the former setting, whereas it is mildly illposed in the latter. In this case, we deduce that making the measurements depend on modulation frequency allows to improve the stability of reconstructions. In the second part, we investigate the inverse problem of reconstructing a tensorvalued conductivity (or diffusion) coefficient in a secondorder elliptic partial differential equation, from knowledge of internal measurements of power density type. This problem finds applications in the medical imaging modalities of Electrical Impedance Tomography and Optical Tomography, and the fact that one considers power densities is justified in practice by assuming a coupling of this physical model with ultrasound waves, a coupling assumption that is characteristic of socalled hybrid medical imaging methods. Starting from the famous Calderon's problem (i.e. the same parameter reconstruction problem from knowledge of boundary fluxes of solutions), and recalling its lack of injectivity and severe instability, we show how going from DirichlettoNeumann data to considering the power density operator leads to reconstruction of the full conductivity tensor via explicit inversion formulas. Moreover, such reconstruction algorithms only require the loss of either zero or one derivative from the power density functionals to the unknown, depending on what part of the tensor one wants to reconstruct. The inversion formulas are worked out with the help of linear algebra and differential geometry, in particular calculus with the Euclidean connection. The practical payoff of such theoretical improvements in injectivity and stability is twofold: (i) the lack of injectivity of CalderÃ³n's problem, no longer existing when using power density measurements, implies that future medical imaging modalities such as hybrid methods may make anisotropic properties of human tissues more accessible; (ii) the improvements in stability for both problems in transport and conductivity may yield practical improvements in the resolution of images of the reconstructed coefficients.

Subject(s):

Mathematics
Medical imaging and radiology
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