This paper investigates forward and inverse problems in fluorescence
optical tomography, with the aim to devise stable methods for the
tomographic image reconstruction.
We analyze solvability of a standard nonlinear forward model and two
approximations by reduced models, which provide certain advantages
for a theoretical as well as numerical treatment of the inverse
problem. Important properties of the forward operators, that map the
unknown fluorophore concentration on virtual measurements, are
derived; in particular, the ill-posedness of the reconstruction
problem is proved, and uniqueness issues are discussed.
For the stable solution of the inverse problem, we consider
Tikhonov-type regularization methods, and we prove that the forward
operators have all the properties, that allow to apply standard
regularization theory. We also investigate the applicability of
nonlinear regularization methods, i.e., TV-regularization and a
method of levelset-type, which are better suited for the
reconstruction of localized or piecewise constant solutions.
The theoretical results are supported by numerical tests, which demonstrate the viability of the reduced models for the treatment of the inverse problem, and the advantages of nonlinear regularization methods for reconstructing localized fluorophore distributions.
Mathematics Subject Classification: 35R30, 47J06, 49N45, 65M32, 92C55.