Conditional cyclic Markov random fields

Grenander et al proposed a conditional cyclic Gaussian Markov random field model for the edges of a closed outline in the plane. In this paper the model is recast as an improper cyclic Gaussian Markov random field for the vertices. The limiting behaviour of this model when the vertices become closely spaced is also described and in particular its relationship with the theory of ‘snakes’ (Kass et al) is established. Applications are given in Grenander et al, Mardia et al and Kent et al.