  The image reconstruction of chromophore concentrations using Diffuse Optical
Tomography (DOT) data can be described mathematically as an ill-posed inverse
problem. Recent work has shown that the use of hyperspectral DOT data, as
opposed to data sets comprising of a single or, at most, a dozen wavelengths,
has the potential for improving the quality of the reconstructions. The use of
hyperspectral diffuse optical data in the formulation and solution of the
inverse problem poses a significant computational burden. The forward operator
is, in actuality, nonlinear. However, under certain assumptions, a linear
approximation, called the Born approximation, provides a suitable surrogate for
the forward operator, and we assume this to be true in the present work.
Computation of the Born matrix requires the solution of thousands of large
scale discrete PDEs and the reconstruction problem, requires matrix-vector
products with the (dense) Born matrix. In this paper, we address both of these
difficulties, thus making the Born approach a computational viable approach for
hyDOT reconstruction. In this paper, we assume that the images we wish to
reconstruct are anomalies of unknown shape and constant value, described using
a parametric level set approach, (PaLS) on a constant background. Specifically,
to address the issue of the PDE solves, we develop a novel recycling-based
Krylov subspace approach that leverages certain system similarities across
wavelengths. To address expense of using the Born operator in the inversion, we
present a fast algorithm for compressing the Born operator that locally
compresses across wavelengths for a given source-detector set and then
recursively combines the low-rank factors to provide a global low-rank
approximation. This low-rank approximation can be used implicitly to speed up
the recovery of the shape parameters and the chromophore concentrations.
