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In this paper, we propose an attraction-repulsion expectation-maximization (AREM) algorithm for image reconstruction and sensor field estimation. We rely on a new method for density estimation to address the problems of image reconstruction from limited samples and sensor field estimation from randomly scattered sensors. Density estimation methods often suffer from undesirable phenomena such as over-fitting and over-smoothing. Specifically, various density estimation techniques based on a Gaussian mixture model (GMM) tend to cluster the Gaussian functions together, thus resulting in over-fitting. On the other hand, other approaches repel the Gaussian functions and yield over-smooth density estimates. We propose a method that seeks an equilibrium between over-fitting and over-smoothing in density estimation by incorporating attraction and repulsion forces among the Gaussian functions and determining the optimal balance between the competing forces experimentally. We model the attractive and repulsive forces by introducing the Gibbs and inverse Gibbs distributions, respectively. The maximization of the likelihood function augmented by the Gibbs density mixture is solved under the expectation-maximization (EM) method. Computer simulation results are provided to demonstrate the effectiveness of the proposed AREM algorithm in image reconstruction and sensor field estimation.