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Attempts to recover images from objects which diffuse radiation pose an especially challenging problem in terms of defining a suitable reconstruction algorithm and with regard to identifying an appropriate computing environment for efficient processing. In this paper we consider both issues and, in particular, describe results of an algebraic technique for imaging the interior of objects which diffuse penetrating radiation using a new multicomputer environment. Two important issues which arise when considering the numerical solution of ultra large problems are the numerical precision achieved and the overall computing efficiency. Our interest in this problem concerns the possibility of obtaining 3-D optical images of tissue which could identify the availability of oxygen by evaluating oxygen- dependent changes in the near infrared spectrum of hemoglobin. These studies were motivated by recent reports from our group and others, which showed promising results for imaging in dense scattering media given only diffusely scattered signals. In our model we assume the use of an NIR laser to provide the input radiation and suitable detectors to measure both transmission and backscatter. In our present work we assume a simple Markov process model for the way in which the energy travels in the medium, but it should be noted that the reconstruction technique we propose can use any model, including nonlinear as well as linear effects, and higher order processes. Current simulations are in 2-D but the methods are easily extended to 3-D. The algorithms we propose are more closely related to algebraic reconstruction algorithms such as ART, SIRT, and SART than to algorithms based on the Born and Rytov approximations such as used for tomographic imaging with diffracting sources. Our algorithms are a significant departure from those based on these standard algebraic methods. We assume only a probabilistic knowledge of the path of the radiation, and minimal knowledge of the absorption profile of the medium. In more traditional algebraic methods, a matrix, w, is assumed, where wij represents the fractional area of the jth image cell intercepted by the ith ray. The equation which is solved is w]*f] equals p], where f] represents the absorption of each of the cells and p] the detector readings. It is assumed that [w] is known. Typically the dimension of p] is M and the dimension of f] is N, where M < N in most cases of practical interest. Standard methods are available for the solution of such equations such as least squares, linear programming, or the Kaczmarz method. We propose a different model of the physical problem. We assume that the radiation entering the medium travels through the medium according to some well defined probabilistic model which can be simulated using relaxation techniques. An example of a relaxation technique is the solution of Laplace's equation using the standard five point grid template. That computational model is based on a simple discrete approximation to the partial differential equation.
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