Motion estimation and vector splines

Many formulations of visual reconstruction problems (e.g., optic pow, shape from shading, biomedical motion estimation from CT da ta ) 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 t o “tune” the smoothness to the properties of the da ta . One can, f o r 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 b y this decomposition process, we derive an exact solution and demonstrate the approach on, examples.