Discrete conformal mappings of planar triangle meshes, also known as the As-Similar-As-Possible (ASAP) mapping, involve the minimization of a quadratic energy function, thus are very easy to generate and are popular in image warping scenarios. We generalize this classical mapping to the case of quad meshes, taking into account the mapping of the interior of the quad, and analyze in detail the most common case - the unit grid mesh. We show that the generalization, when combined with barycentric coordinate mappings between the source and target polygons, spawns an entire family of new mappings governed by quadratic energy functions, which allow to control quite precisely various effects of the mapping. This approach is quite general and applies also to arbitrary planar polygon meshes. As an application of generalized ASAP mappings of the unit grid mesh, we demonstrate how they can be used to warp digital photographs to achieve a variety of effects. One such effect is modifying the perspective of the camera that took a given photograph (without moving the camera). A related, but more challenging, effect is re-photography - warping a contemporary photograph in order to reproduce the camera view present in a vintage photograph of the same scene - taken many years before with a different camera from a different viewpoint. We apply the generalized ASAP mapping to these images, discretized to a unit grid. Using a quad mesh (as opposed to a triangle mesh) permits biasing towards affine maps of the unit squares. This allows the introduction of an As-Affine-As-Possible (AAAP) mapping for a good approximation of the homographies present in these warps, achieving quite accurate results. We demonstrate the advantages of the AAAP mapping on a variety of synthetic and real-world examples.
All Science Journal Classification (ASJC) codes
- Computer Graphics and Computer-Aided Design