John C. Schotland

Optical tomography: forward and inverse problems

This is a review of recent mathematical and computational advances in optical tomography. We discuss the physical foundations of forward models for light propagation on microscopic, mesoscopic and macroscopic scales. We also consider direct and numerical approaches to the inverse problems that arise at each of these scales. Finally, we outline future directions and open problems in the field.

Quantitative optical tomography of sub-surface heterogeneities using spatially modulated structured light

We present a wide-field method for obtaining three-dimensional images of turbid media. By projecting patterns of light of varying spatial frequencies on a sample, we reconstruct quantitative, depth resolved images of absorption contrast. Images are reconstructed using a fast analytic inversion formula and a novel correction to the diffusion approximation for increased accuracy near boundaries. The method provides more accurate quantification of optical absorption and higher resolution than standard diffuse reflectance measurements.

Photon hitting density

Optical and near-IR spectroscopy and imaging of highly scattering tissues require information about the distribution of photon-migration paths. We introduce the concept of the photon hitting density, which describes the expected local time spent by photons traveling between a source and a detector. For systems in which photon transport is diffusive we show that the hitting density can be calculated in terms of diffusion Greens functions. We report calculations of the hitting density in model systems.

Inverse problem in optical diffusion tomography. I. Fourier-Laplace inversion formulas.

We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. Inversion formulas based on the Fourier-Laplace transform are used to establish the existence and uniqueness of solutions to this problem in planar, cylindrical, and spherical geometries.

Imaging complex structures with diffuse light.

We use diffuse optical tomography to quantitatively reconstruct images of complex phantoms with millimeter sized features located centimeters deep within a highly-scattering medium. A non-contact instrument was employed to collect large data sets consisting of greater than 10(7) source-detector pairs. Images were reconstructed using a fast image reconstruction algorithm based on an analytic solution to the inverse scattering problem for diffuse light.

The Bad Truth about Laplace's Transform

Inverting the Laplace transform is a paradigm for exponentially ill-posed problems. For a class of operators, including the Laplace transform, we give forward and inverse formulae that have fast implementations using the fast Fourier transform. These formulae lead easily to regularized inverses whose effects on noise and filtered data can be precisely described. Our results give cogent reasons for the general sense of dread most mathematicians feel about inverting the Laplace transform.

Inverse problem in optical diffusion tomography. II. Role of boundary conditions

We consider the inverse problem of reconstructing the absorption and diffusion coefficients of an inhomogeneous highly scattering medium probed by diffuse light. The role of boundary conditions in the derivation of Fourier-Laplace inversion formulas is considered. Boundary conditions of a general mixed type are discussed, with purely absorbing and purely reflecting boundaries obtained as limiting cases. Four different geometries are considered with boundary conditions imposed on a single plane and on two parallel planes and on a cylindrical and on a spherical surface.

Inverse problem in optical diffusion tomography. III. Inversion formulas and singular-value decomposition

We continue our study of the inverse scattering problem for diffuse light. In particular, we derive inversion formulas for this problem that are based on the functional singular-value decomposition of the linearized forward-scattering operator in the slab, cylindrical, and spherical geometries. Computer simulations are used to illustrate our results in model systems.

Inverse scattering for near-field microscopy

We derive the analytic singular value decomposition of the linearized scattering operator for scalar waves. This representation leads to a robust inversion formula for the inverse scattering problem in the near zone. Applications to near-field optics are described.

Experimental demonstration of an analytic method for image reconstruction in optical diffusion tomography with large data sets

We report the first experimental test of an analytic image reconstruction algorithm for optical tomography with large data sets. Using a continuous-wave optical tomography system with 10(8) source-detector pairs, we demonstrate the reconstruction of an absorption image of a phantom consisting of a highly scattering medium containing absorbing inhomogeneities.