*Medical Imaging: Image Processing*, K.M. Hanson, ed.,
*Proc. SPIE ***3034**, pp. 369-380, 1997

## Model-based image reconstruction from time-resolved diffusion data

Suhail S. Saquib, Kenneth M. Hanson, and Gregory S. Cunningham

*Los Alamos National Laboratory*
### Abstract

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(N^{2}) 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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