Abstract

Image warps -or just warps- capture the geometric deforma- tion existing between two images of a deforming surface. The current approach to enforce a warp’s smoothness is to penalize its second order partial derivatives. Because this favors locally affine warps, this fails to capture the local pro jective component of the image deformation. This may have a negative impact on applications such as image registration and deformable 3D reconstruction. We propose a novel penalty designed to smooth the warp while capturing the deformation’s local pro jective structure. Our penalty is based on equivalents to the Schwarzian deriva- tives, which are pro jective differential invariants exactly preserved by homographies. We propose a methodology to derive a set of Partial Dif- ferential Equations with homographies as solutions. We call this system the Schwarzian equations and we explicitly derive them for 2D functions using differential properties of homographies. We name as Schwarp a warp which is estimated by penalizing the residual of Schwarzian equa- tions. Experimental evaluation shows that Schwarps outperform existing warps in modeling and extrapolation power, and lead to far better re- sults in Shape-from-Template and camera calibration from a deformable surface.