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We consider Suter's (see Proc. CVPR94, Seattle, p.939-948, 1994) DIV-CURL optical flow methods, wherein the problem of computing a velocity field from an image sequence is regularized using smoothness conditions based on the divergence and curl of the field. In particular, we develop stochastic formulations of DIV-CURL splines using the linear smoothing theory of Adams, Willsky, and Levy. Our models are shown to be well posed and thus can be used in both simulating and estimating velocity fields having known stochastic properties. As a special case, our stochastic model reduces to that developed by Rougee, Levy, and Willsky (1984) for the classical Horn and Schunck's (1981) optical flow.