Closed-form connectivity-preserving solutions for motion compensation using 2-D meshes

Motion compensation using two-dimensional (2-D) mesh models requires computation of the parameters of a spatial transformation for each mesh element (patch). It is well known that the parameters of an affine (bilinear or perspective) mapping can be uniquely estimated from three (four) point correspondences (at the vertices of a triangular or quadrilateral mesh element). On the other hand, overdetermined solutions using more than the required minimum number of point correspondences provide increased robustness against correspondence-estimation errors; however, this necessitates special consideration to preserve mesh-connectivity. This paper presents closed-form, overdetermined solutions for least squares estimation of affine motion parameters for a triangular mesh, which preserve mesh-connectivity using patch-based or node-based connectivity constraints. In particular, four new algorithms are presented: patch-constrained methods using point correspondences or spatio-temporal intensity gradients, and node-constrained methods using point correspondences or spatio-temporal intensity gradients. The methods using point correspondences can be viewed as postprocessing of a dense motion field for best representation in terms of a set of irregularly spaced samples. The methods that are based on spatio-temporal intensity gradients offer closed-form solutions for direct estimation of the best node-point motion vectors (equivalently the best transformation parameters). We show that the performance of the proposed closed-form solutions are comparable to those of the alternative search-based solutions at a fraction of the computational cost.