What nice basis sets are there for the space of invertible transformations in $\mathbb R^2$?
Say I have a manifold $\mathcal M$ (in fact it is either the Euclidean plane $\mathbb R^2$ or the sphere $S^2$), and two different coordinate charts $\varphi:U_\varphi \rightarrow\mathbb R^2$ and $\psi:U_\psi\rightarrow\mathbb R^2 $ where $U_\varphi,U_\psi\subset \mathcal M$ (moreover let these charts each map to some rectangle $\mathbb I\subset \mathbb R^2$ in coordinate space). Furthermore, let the image of these two charts overlap in some region $U_\varphi \cap U_\psi$ (which happens to be a strict subset of $U_\varphi$ and of $U_\psi$). The transformation $\varphi \circ \psi^{-1}$ thus is an invertible function which maps $\mathbb R^2 \rightarrow\mathbb R^2$ (albeit only defined for some bounded domain and range).
I'd like a basis series, i.e. let $\varphi \circ \psi^{-1} = \lim_{n\rightarrow\infty}\sum^n_i w_i\, f_i$ where $\left\{ f_i:\mathbb R^2 \rightarrow \mathbb R^2 \right\}$, such that the transformation map can be approximated with a (small) finite number of terms. Perhaps something resembling a cosine expansion or spherical harmonics (enabling separation of fast-varying components). Ideally, I'd prefer the basis expansion to have some kind of symmetry with respect to its inverse, such as for any given $n$, that $\sum^n_i w_i\, f_i = \left( \sum^n_i w'_i\, f_i \right)^{-1}$.
An application for this is that the manifold is a (potential) panoramic image scene, each photo is a chart, and by automatic feature matching (such as the SIFT computer vision algorithm) for each overlapping pair of photos I have a large number of points $x$ for which I already know both $\varphi(x)$ and $\psi(x)$. I want to interpolate and extrapolate these datapoints (e.g. analytically approximate $\varphi \circ \psi^{-1}$ and then smoothly extend it) in order to align and stitch the photos together. The simplest ways to do this would be to start by choosing one master image (e.g. only consider $\varphi \circ \psi^{-1}$ not $\psi \circ \varphi^{-1}$) or incorporate special knowledge about the camera lens function $\varphi_{(\Theta,\Phi)}$. However, I want to avoid brittle assumptions about the cameras, and I also want to avoid treating any one photo as more privileged than the others (so that I can instead try to average out their peculiar view geometries).