Boundary Approach to Characterize the Inner and Outer Approximation of the Image of a Disk

Calculating directly the inner and outer approximation of the image of a set by a function can be challenging. Then, it is sometimes preferred to compute the image of the boundary of the set instead. However, boundary-based methods are subject to the apparition of fake boundaries in the image set. These fake boundaries add pessimism when characterizing the inner approximation of the image set. This paper then introduces the notion of Box Chains to simplify the detection and the suppression of the fake boundaries. The characterization of the inner and outer approximation of the image set in the case of a function from the unit disk D to R2 will be considered, with two examples.