Multi-Dimensional Image Reconstruction and Field Estimation from Randomly Scattered Sensors

Many important problems in statistical signal processing can be formulated as function estimation from randomly scattered sensors in a multi-dimensional space, e.g. image reconstruction from photon-limited images. We model the problem of image reconstruction and field estimation from scattered sensors as multi-dimensional function reconstruction from random samples. We present a novel approach to the study of signal reconstruction from random samples in a multi-dimensional space. In particular, we show that sampling using constant-mean point processes yields an unbiased estimate. Moreover, 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. Finally, we rely on the iterative estimation method for multi-dimensional image reconstruction and field estimation from sensors scattered according a multi-dimensional Poisson and uniform distribution. Computer simulation experiments demonstrate the efficiency of the iterative estimation method in image reconstruction and field estimation from randomly scattered sensors.