# Set inversion

In mathematics, set inversion is the problem of characterizing the preimage X of a set Y by a function f, i.e., X = f−1(Y) = {xRn | f(x) ∈ Y}.

In most applications, f is a function from Rn to Rp and the set Y is a box of Rp (i.e. a Cartesian product of p intervals of R).

When f is nonlinear the set inversion problem can be solved [1] using interval analysis combined with a branch-and-bound algorithm. [2]

The main idea consists in building a paving of Rp made with non-overlapping boxes. For each box [x], we perform the following tests:

1. if f([x]) ⊂ Y we conclude that [x] ⊂ X;
2. if f([x]) ∩ Y = ∅ we conclude that [x] ∩ X = ∅;
3. Otherwise, the box [x] the box is bisected except if its width is smaller than a given precision.

To check the two first tests, we need an interval extension (or an inclusion function) [f] for f. Classified boxes are stored into subpavings, i.e., union of non overlapping boxes. The algorithm can be made more efficient by replacing the inclusion tests by contractors.

## Example

The set X = f−1([4,9]) where f(x1, x2) = x12 + x22 is represented on the figure. For instance, since [-2,1]2+[4,5]2=[0,4]+[16,25]=[16,29] does not intersect the interval [4,9], we conclude that the box [-2,1]×[4,5] is outside X. Since [-1,1]2+[2,√5]2=[0,1]+[4,5]=[4,6] is inside [4,9], we conclude that the whole box [-1,1]×[2,√5] is inside X.

A ring defined as a set inversion problem

## Application

Set inversion is mainly used for path planning, for nonlinear parameter set estimation [3] [4] , for localization [5] [6] or for the characterization of stability domains of linear dynamical systems. [7] .

## References

1. ^ Jaulin, L.; Walter, E. (1993). "Set inversion via interval analysis for nonlinear bounded-error estimation". Automatica 29 (4).
2. ^ Jaulin, L.; Kieffer, M.; Didrit, O.; Walter, E. (2001). Applied Interval Analysis. Berlin: Springer. ISBN 1-85233-219-0.
3. ^ Jaulin, L.; Godet, J.L; Walter, E.; Elliasmine, A.; Leduff, Y. (1997). "Light scattering data analysis via set inversion". Journal of Physics A: Mathematical and General 30.
4. ^ Braems, I.; Berthier, F.; Jaulin, L.; Kieffer, M.; Walter, E. (2001). "Guaranteed Estimation of Electrochemical Parameters by Set Inversion Using Interval Analysis". Journal of Electroanalytical Chemistry 495 (1).
5. ^ Colle, E.; Galerne, S. (2013). "Mobile robot localization by multiangulation using set inversion". Robotics and Autonomous Systems 66 (1).
6. ^ Drevelle, V.; Bonnifait, Ph. (2011). "A set-membership approach for high integrity height-aided satellite positioning". GPS Solutions 15 (4).
7. ^ Walter, E.; Jaulin, L. (1994). "Guaranteed characterization of stability domains via set inversion". IEEE Trans. on Autom. Control 39 (4).