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Many important problems in statistical signal processing can be formulated as function estimation from randomly scattered sensors in a multidimensional space, e.g., image reconstruction from photon-limited images and field estimation from scattered sensors. We present a novel approach to the study of signal reconstruction from random samples in a multidimensional space. In particular, we study a classical iterative reconstruction method and demonstrate that it forms a sequence of unbiased estimates for band-limited signals, which converge to the true function in the mean-square sense. We subsequently rely on the iterative estimation method for multidimensional image reconstruction and field estimation from sensors scattered according to a multidimensional Poisson and uniform distribution. Computer simulation experiments are used to demonstrate the efficiency of the iterative estimation method in image reconstruction and field estimation from randomly scattered sensors.