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Diffuse optical tomography (DOT) is a sensitive and relatively low cost imaging modality. However, the inverse problem of reconstructing optical parameters from scattered light measurements is highly nonlinear due to the nonlinear coupling between the optical coefficients and the photon flux in the diffusion equation. Even though nonlinear iterative methods have been commonly used, such iterative processes are computationally expensive especially for the three dimensional imaging scenario with massive number of detector elements. The main contribution of this paper is a novel non-iterative and exact inversion algorithm when the optical inhomogeneities are sparsely distributed. We show that the problem can be converted into simultaneous sparse representation problem with multiple measurement vectors from compressed sensing framework. The exact reconstruction formula is obtained using simultaneous orthogonal matching pursuit (S-OMP) and a simple two step approach without ever calculating the diffusion equation. Simulation results also confirm our theory.