We develop an approach for estimation with Gaussian Markov processes that imposes a smoothness prior while allowing for discontinuities. In- stead of propagating information laterally between neighboring nodes in a graph, we study the posterior distribution of the hidden nodes as a whole—how it is perturbed by invoking discontinuities, or weakening the edges, in the graph. We show that the resulting computation amounts to feed-forward fan-in operations reminiscent of V1 neurons. Moreover, using suitable matrix preconditioners, the incurred matrix inverse and determinant can be approximated, without iteration, in the same compu- tational style. Simulation results illustrate the merits of this approach.