FAST ALGORITHMS FOR HYPERSPECTRAL DIFFUSE OPTICAL TOMOGRAPHY

Arvind K. Saibaba

doi:10.1137/140990073

2015-01-01

DOI：10.1137/140990073

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摘要

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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 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 hyperspectral DOT 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 [A. Aghasi, M. Kilmer, and E. L. Miller, SIAM J. Imaging Sci., 4 (2011), pp. 618-650] 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 the 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. We provide a detailed analysis of the accuracy and computational costs of the resulting algorithms and demonstrate the validity of our approach by detailed numerical experiments on a realistic geometry for breast imaging.

Keyword:

diffuse optical tomography

inverse problems

recycling Krylov subspaces

parametric level set

recursive SVD