The elliptic systems method (ESM), previously developed by the second and third authors, is extended to the reconstruction of the diffusion coefficient of an inverse problem for the parabolic equation in the n-dimensional case (n = 2,3). This inverse problem has applications to optical imaging of small abnormalities hidden in a random media, such as biological tissues, foggy atmospheres, murky water, etc. Results of numerical experiments are presented in the two-dimensional case, for realistic ranges of the parameters.