Motion estimation and vector splines

Many formulations of visual reconstruction problems (e.g. optic flow, shape from shading, biomedical motion estimation from CT data) involve the recovery of a vector field. Often the solution is characterized via a generalized spline or regularization formulation using a smoothness constraint. This paper introduces a decomposition of the smoothness constraint into two parts: one related to the divergence of the vector field and one related to the curl or vorticity. This allows one to “tune” the smoothness to the properties of the data. One can, for example, use a high weighting on the smoothness imposed upon the curl in order to preserve the divergent parts of the field. For a particular spline within the family introduced by this decomposition process, we derive an exact solution and demonstrate the approach on examples