A boundary approach for set inversion

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Abstract

In the paper, we present a new interval-based set inversion algorithm which takes into account the continuity of the problem. In the case where the set Y to be inverted has some volume, we show that inverting the boundary Y of Y is sufficient to reconstruct the preimage X=f1(Y). The inversion of Y separates the domain of f into two regions : one inside X and one outside. To detect which part is inside or outside, we show that we can retro-propagate the information coming from Y at negligible cost. The efficiency of the approach is illustrated on a localization problem.

Introduction

NotationIn this paper, a vector x of Rn and a vector-valued function f will be written in bold font. An interval [x] or an axis aligned box [x] will be written between brackets. The image by f of a box [x] will be written as f([x])=y|x[x],y=f(x).

Given a function f:RnRm and a set YRn. Set inversion is the problem of characterizing the set X of all x inside a prior domain X(0)Rn such that f(x)Y. We have X=X(0)f1(Y)=xX(0)f(x)Y.Note that the function f is not necessarily invertible. For instance if f(x)=x2, Y=[4,16] and X(0)=[5,20] then X=[5,20]f1([4,16])=[3,20]([4,2][2,4])=[3,2][2,4]Moreover, the solution set maybe empty. For instance, if X(0)=[1,1] instead of [5,20] then X=[1,1]f1([4,16])=.The set membership formalism is thus well suited to ill-posed problems involving uncertainties.

It is used in many engineering domains such as localization (Colle and Galerne, 2013, Drevelle and Bonnifait, 2013), parameter estimation (Jaulin et al., 1997, Kreinovich et al., 1997), control (Vinas et al., 2006), calibration (Daney et al., 2006), robotics (Rohou et al., 2019), etc. Most guaranteed algorithms for set inversion are based on interval constraint propagation methods (van Emden, 1999, van Hentenryck et al., 1997). They use a forward–backward contractor (Araya et al., 2008, Chabert and Jaulin, 2009) such as the HC4-revise algorithm (Benhamou et al., 1999a). Now, contractors have mainly be developed for solving equations (Cébério and Granvilliers, 2001, Neumaier, 2004) or global optimization (Hansen, 1992) where the solution set is reduced to some isolate points or surfaces and has no interior.

For set inversion, the solution set has an interior and exterior. Existing contractors are not able to retro-propagate properly the information coming from Y. This leads to a poor outer approximation and unnecessary computations. It is sometimes possible to use the complementary set Y¯ to find an inner approximation (Benhamou et al., 1999c), but, a part of the work done for contracting with respect to Y has to be repeated for Y¯, which should be avoided.

In this paper, we propose to inverse the boundary Y of Y instead of inverting Y. Due to the fact that YY, the corresponding contractor will more efficient. This inversion will provide an outer approximation for the boundary X of the solution set X. For the parts of the search space which are not in the boundary approximation, we need to show either they are inside or outside X. We show that this information can be obtained with almost no extra computation cost by retro-propagating some binary information from Y. To our knowledge, this is new in the community of constraint propagation.

The paper is organized as follows. Section 2 recalls the principle of set inversion algorithms and motivates the need of a more powerful propagation and the use of the boundaries. Section 3 presents in a new way, inspired from control theory, to present the forward–backward propagation. Section 4 gives the boundary approach for set inversion which corresponds to the main contribution of this paper. Section 5 proposes an application to robot localization based on a Time Difference Of Arrival (TDOA). Section 6 concludes the paper.

Section snippets

Motivation

Fig. 1 illustrates the behavior of a typical contractor-based algorithm (Jaulin and Desrochers, 2014) to characterize a set X (magenta). The algorithm uses two types of contractors: a contractor CX for X and a contractor CX¯ for its complementary X¯. To characterize the part of X which is inside a box [x0] the principle of the algorithm is the following

  • 1.

    We have a set of boxes (green in the picture) which have to be studied. These green boxes are qualified as the new boxes. They have been

Principle

We recall here a result given in Jaulin et al. (2001) which can be see as an abstraction of the algorithm presented in Montanari and Rossi (1991). Consider the constraint f(x)Y,xX(0)where f is a composition of functions: f=fnf2f1. The forward–backward sequence given by Algorithm 1 computes exactly the set X=X(0)f1(Y).

For each k, we have X(k)=f1:k(X(0))X(k)=f1:k(X(0))fk+1:n1(Y)X=X(0)where fk:=ffk. The proof of this result is a consequence of the fact that the chain structure of

Principle

The boundary X of a subset X of Rn is the set of points which can be approached both from the inside of X and from the outside of X. It is also the set of points in the closure of X not belonging to its interior. We have the following proposition.

Proposition 4

Consider a continuous function f:RnRp defined everywhere. If X=f1(Y), we have Xf1(Y).

Proof

We will show that if xX then y=f(x)Y. Denote by Bη(x) the open ball with radius η and center x. We have xXη>0,a,bBη(x)|aX,bXη>0,a,bBη(x)|f(a)Y,f

Test-cases

In this Section, we give two examples. The first example illustrates an academic set inversion problem involving a nonlinear function. The Python code is also given. The second example is a localization problem using a TDOA technique (Time Difference Of Arrival). The problem is known to be ill-conditioned and highly nonlinear (Lee, 1975).

Conclusion

In this paper, we have proposed a new forward–backward contraction procedure for set inversion. The principle is to inverse the boundary of the set Y to be inverted and to retro-propagate a color on the interval bounds in order to decide if the win boxes are inside or outside the solution set.

This approach has several interesting features:

  • 1

    Contrarily to existing contractor-based set inversion algorithm, we do not have to retro-propagate twice (once for inside Y, once for outside), but only once

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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