We present \emph{telescoping} recursive representations for both continuousand discrete indexed noncausal Gauss-Markov random fields. Our recursions startat the boundary (a hypersurface in $\R^d$, $d \ge 1$) 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. Underappropriate conditions, the recursions for the random field are linearstochastic differential/difference equations driven by white noise, for whichwe derive recursive estimation algorithms, that extend standard algorithms,like the Kalman-Bucy filter and the Rauch-Tung-Striebel smoother, to noncausalMarkov random fields.
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机译：我们为连续和离散索引的非因果高斯-马尔可夫随机场提供\ emph {telescoping}递归表示。我们的递归从边界（$ \ R ^ d $，$ d \ ge 1 $的超曲面）开始，然后向内望远镜，例如对于图像，伸缩表示将递归从$ d = 2 $减少到$ d = 1 $ ，即单维度递归。在适当的条件下，随机场的递归是由白噪声驱动的线性随机微分/差分方程，为此，我们推导了递归估计算法，将标准算法（例如Kalman-Bucy滤波器和Rauch-Tung-Striebel平滑器）扩展到非因果马尔可夫随机领域。
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