This paper addresses the issue of reconstructing the unknown field of absorption and scattering coefficients from time-resolved measurements of diffused light in a computationally efficient manner. The intended application is optical tomography, which has generated considerable interest in recent times. The inverse problem is posed in the Bayesian framework. The maximum a posteriori (MAP) estimate is used to compute the reconstruction. We use an edge-preserving generalized Gaussian Markov random field to model the unknown image. The diffusion model used for the measurements is solved forward in time using a finite-difference approach known as the alternating-directions implicit method. This method requires the inversion of a tridiagonal matrix at each time step and is therefore of O(N) complexity, where N is the dimensionality of the image. Adjoint differentiation is used to compute the sensitivity of the measurements with respect to the unknown image. The novelty of our method lies in the computation of the sensitivity since we can achieve it in O(N) time as opposed to O(N2) time required by the perturbation approach. We present results using simulated data to show that the proposed method yields superior quality reconstructions with substantial savings in computation.
Keywords: optical tomography, diffusion, Bayesian estimation, MAP estimation, Markov random field, adjoint differentiation, finite-difference technique
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