Generalized Thin-Plate Spline Warps

The Thin-Plate Spline warp has been shown to be a very effective parameterized model of the optic flow field between images of various types of deformable surfaces, such as a paper sheet being bent. Recent work has also used such warps for images of a smooth and rigid surface. Standard Thin-Plate Spline warps are however not rigid, in the sense that they do not comply with the epipolar geometry. They are also intrinsically affine, in the sense of the affine camera model, since they are not able to simply model the effect of perspective projection. We propose three types of warps based on the Thin-Plate Spline. The first one is a rigid flexible warp. It describes the optic flow field induced by a smooth and rigid surface, and satisfies the affine epipolar geometry constraint. The second and third proposed warps extend the standard Thin-Plate Spline warp and the proposed rigid flexible warp to the perspective camera model. The properties of these warps are studied in details and a hierarchy is defined. Experimental results on simulated and real data are reported.