Given a diffusion $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, if one applies the transform
$f(x) = \int_a^x \frac{1}{\sigma(u)}du$, one can use Ito's lemma to show that
$df(X_t) = \left( \frac{\mu(X_t)}{\sigma(X_t)} - \frac{\sigma'(X_t)}{2} \right)dt + dW_t$.
Is there a way to explicitly invert this transform in the general case, assuming $\sigma$ is sufficiently nice?