Telescoping Recursive Representations and Estimation of Gauss-Markov Random Fields

We present telescoping recursive representations for both continuous and discrete indexed noncausal Gauss-Markov random fields. Our recursions start at the boundary (a hypersurface in R-d, d >= 1) and telescope inwards. For example, for images, the telescoping representation reduce recursions from d = 2 to d = 1, i.e., to recursions on a single dimension. Under appropriate conditions, the recursions for the random field are linear stochastic differential/difference equations driven by white noise, for which we derive recursive estimation algorithms, that extend standard algorithms, like the Kalman-Bucy filter and the Rauch-Tung-Striebel smoother, to noncausal Markov random fields.