  In an influential paper, Tadmor, Nezzar and Vese (Multiscale Model. Simul.
(2004)) introduced a hierarchical decomposition of an image as a sum of
constituents of different scales. Here we construct analogous hierarchical
expansions for diffeomorphisms, in the context of image registration, with the
sum replaced by composition of maps. We treat this as a special case of a
general framework for multiscale decompositions, applicable to a wide range of
imaging and nonlinear inverse problems. As a paradigmatic example of the
latter, we consider the Calder\'on inverse conductivity problem. We prove that
we can simultaneously perform a numerical reconstruction and a multiscale
decomposition of the unknown conductivity, driven by the inverse problem
itself. We provide novel convergence proofs which work in the general abstract
settings, yet are sharp enough to settle an open problem on the hierarchical
decompostion of Tadmor, Nezzar and Vese for arbitrary functions in $L^2$. We
also give counterexamples that show the optimality of our general results.
