  We investigate the problem of estimating a smooth invertible transformation f
when observing independent samples X_1, ..., X_n ~ P \circ f, where P is a
known measure. We focus on the two dimensional case where P and f are defined
on R^2. We present a flexible class of smooth invertible transformations in two
dimensions with variational equations for optimizing over the classes, then
study the problem of estimating the transformation f by penalized maximum
likelihood estimation. We apply our methodology to the case when P \circ f has
a density with respect to Lebesgue measure on R^2 and demonstrate improvements
over kernel density estimation on three examples.
