Inner and outer approximation of the image of a set by a nonlinear function

This paper proposes an original method to compute an inner and an outer approximation of the image $f\left(X\right)$ of a subset $X$ of $R^n$ (e.g., $X$ is the unit ball) by a smooth nonlinear function $f:R^n\to R^n$. A boundary approach is proposed. More precisely, the boundary $\partial f\left(X\right)$ will be covered by parallelepipeds in order to get an accurate enclosure of $f\left(X\right)$.