Analysis and Exploitation of Matrix Structure Arising in Linearized Optical Tomographic Imaging

Hyde, Daniel C.; Kilmer, Misha E.; Brooks, Dana H.; Miller, Eric L.

doi:10.1137/060657285

Cited by 2 publications

(1 citation statement)

References 27 publications

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“…In [31], the authors develop analytical formulas for inversion based on Fourier analysis when the sources and detectors are distributed uniformly on the boundary of a regular geometry such as a plane, cylinder, or sphere. In our previous work, we have exploited the structure of the Green's function in regular geometries to decompose the Born operator into a number of sparse easily computed matrices [21]. The approach of compressing the operator H is similar to that derived in [9].…”

Section: Related Workmentioning

confidence: 99%

Fast Algorithms for Hyperspectral Diffuse Optical Tomography

Saibaba¹,

Kilmer²,

Miller³

et al. 2015

SIAM J. Sci. Comput.

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Abstract.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 hyperspectral DOT (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) [1] 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. 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.1. Introduction. Diffuse optical tomography (DOT) is an imaging technique that uses near infrared light to image highly scattering media. A good review has been provided in [3] and an updated version is provided in [2]. The imaging modality has shown great promise as a low-cost alternative or complement to existing medical imaging technology particularly in brain imaging and breast cancer detection. The region of interest is illuminated with near infrared light over a collection of wavelengths and the data are comprised of observations of the resulting scattered diffuse fields at a number of locations surrounding the medium. Given these measurements as well as the partial differential equation governing the interaction of light and tissue (typically, the diffusion equation), we seek to recover space and time-varying maps (i.e. images) of concentrations of physiologically relevant chromophores such as oxygenated and ...

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Section: Related Workmentioning

confidence: 99%

Fast Algorithms for Hyperspectral Diffuse Optical Tomography

Saibaba¹,

Kilmer²,

Miller³

et al. 2015

SIAM J. Sci. Comput.

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A reconstruction algorithm for ultrasound-modulated diffuse optical tomography

Ammari

Bossy

Garnier

et al. 2014

Proc. Amer. Math. Soc.

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The aim of this paper is to develop an efficient reconstruction algorithm for ultrasound-modulated diffuse optical tomography. In diffuse optical imaging, the resolution is in general low. By mechanically perturbing the medium, we show that it is possible to achieve a significant resolution enhancement. When a spherical acoustic wave is propagating inside the medium, the optical parameter of the medium is perturbed. Using cross-correlations of the boundary measurements of the intensity of the light propagating in the perturbed medium and in the unperturbed one, we provide an iterative algorithm for reconstructing the optical absorption coefficient. Using a spherical Radon transform inversion, we first establish an equation that the optical absorption satisfies. This equation together with the diffusion model constitutes a nonlinear system. Then, solving iteratively such a nonlinear coupled system, we obtain the true absorption parameter. We prove the convergence of the algorithm and present numerical results to illustrate its resolution and stability performances.

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Analysis and Exploitation of Matrix Structure Arising in Linearized Optical Tomographic Imaging

Cited by 2 publications

References 27 publications

Fast Algorithms for Hyperspectral Diffuse Optical Tomography

Fast Algorithms for Hyperspectral Diffuse Optical Tomography

A reconstruction algorithm for ultrasound-modulated diffuse optical tomography

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