Optical tomography: inversion based on adjoint differentiation

Abstract

Because of its unique characteristics for imaging tissue, there is growing medical interest in the diagnostic procedure of optical tomography. In this procedure, one attempts to reconstruct the properties of a tissue sample from measurements of how infrared light pulses are propagated through the sample. The reconstructions show the spatial distribution in the sample of the diffusion coefficient, and sometimes the absorption coefficient. Because infrared photons scatter nearly isotropically, their propagation in tissue is fairly well modeled in terms of the time-dependent diffusion equation. The goal of the reconstruction algorithm is to invert measurements of the diffusion process. This phenomenon often can not be described analytically, but only by numerically solving the forward simulation process, e.g., by using a finite-difference technique.

A standard approach to inversion (or reconstruction) is to use a gradient-based optimization procedure to find the parameters that best match the data, as summarized by a scalar function, e.g., chi-squared, a term familiar to physicists. However, this approach becomes intractable when the data can only be numerically calculated, as in the present case, because the gradient of the objective function would typically require numerous forward calculations. However, this problem can be solved with an efficient calculation of the gradient provided by the Adjoint Differentiation In Code Technique (ADICT). ADICT involves applying the chain rule for differentiation to the computational simulation code, in effect, "differentiating" the forward code. The calculation of the gradient proceeds in the reverse (adjoint) direction. The advantage of ADICT is that the required gradient with respect to all the parameters on which the physical process depends can be calculated in a time comparable to that required for the (forward) calculation. In the present infrared examples, the gradients are calculated with respect to around 10000 parameters.

I will present several examples of 2D and 3D optical tomography reconstructions. Other potential application areas for this methodology include modeling of the ocean, atmosphere, fluid flow, shock-wave phenomena, and complex imaging situations.

Keywords: optical tomography, diffusion of infrared light, adjoint differentiation, inversion of numerical simulations

References

• "Model-based image reconstruction from time-resolved diffusion data," S. S. Saquib, et al., Proc. SPIE 3034, pp. 369-380 (1997)
• "Inversion based on computational simulations" K. M. Hanson, et al., in Maximum Entropy and Bayesian Methods, pp. 121-135 (Kluwer Academic, Dordrecht, 1998)
• "Tomographic imaging of biological tissue by time-resolved, model-based, iterative image reconstruction," A. H. Hielscher et al., in Proc. Adv. Opt. Imag. and Phot. Migr (OSA, 1998)
• "Gradient-based iterative image reconstruction scheme for time-resolved optical tomography," A.H. Hielscher et al., IEEE Trans. Med. Imag. 18, pp. 1-10, 1999
• "Two- and three-dimensional optical tomography of finger joints for diagnostics of rheumatoid arthritis," A. D. Klose et al., Proc. SPIE 3566 (1998)

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