  This paper is concerned with the ubiquitous inverse problem of recovering an
unknown function u from finitely many measurements possibly affected by noise.
In recent years, inversion methods based on linear approximation spaces were
introduced in [MPPY15, BCDDPW17] with certified recovery bounds. It is however
known that linear spaces become ineffective for approximating simple and
relevant families of functions, such as piecewise smooth functions that
typically occur in hyperbolic PDEs (shocks) or images (edges). For such
families, nonlinear spaces [Devore98] are known to significantly improve the
approximation performance. The first contribution of this paper is to provide
with certified recovery bounds for inversion procedures based on nonlinear
approximation spaces. The second contribution is the application of this
framework to the recovery of general bidimensional shapes from cell-average
data. We also discuss how the application of our results to n-term
approximation relates to classical results in compressed sensing.
