Smoothly warp an image. This is probably a diffeomorphism, a very practical one. Can arbitrary diffeomorphisms be approximated by combining simpler objects like the way splines can approximate arbitrary curves and surfaces?
Consider a vector field joining the "before" and "after" location of each pixel. Approximate small patches of the vector field (somehow) by discretely sampling it and interpolating a function over the patch, then glue these patches together in a way that preserves smoothness.