The presentation will be divided in two parts: (1) We will briefly introduce the linear inverse problem setting in which we seek to approximate an unknown function from measurements by an element of a linear space. However, 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). We will then analyze which conditions can give us certified recovery bounds for inversion procedures based on nonlinear approximation spaces. (2) We will apply this framework to the recovery of general bidimensional shapes from cell-average data (high-resolution images from coarser cell averages). Then we will show how to build fast higher order methods to reconstruct interfaces as well as two strategies to deal with non-smooth interfaces presenting corners.