Manipulating Focal Sets on the Unit Simplex: Application to Plastic Sorting

Abstract

Belief functions are quite generic models when it comes to represent uncertain data, as it extends a wide range of uncertainty models (possiblity and probability distributions, among others). Usually, belief functions are defined over finite spaces, however many real word problems require to deal with beliefs over a continuous space while maintaining computational efficiency. This paper discusses the case of focal sets on the unit simplex, and proposes efficient inference tools to manipulate them. Such sets can be used to represent unknown proportions that one may face in various fields like soil contamination managing, plastic sorting or image reconstruction. In this paper, we illustrate their use on an industrial problem of plastic sorting, where the proportion of material impurities must not go over a limit while minimizing the rejection of sorted materials, whose nature is uncertain.

Handling Mixture Optimisation Problem Using Cautious Predictions and Belief Functions

“…The information resulting from adding sf 2 with sf 1 is a mass function denoted m 1⊕2 and defined as follows for I ⊂ U [7]:…”

“…Note that in case where imprecise information are convex sets, e.g. intervals, only the lower and the upper bounds of the intervals are involved in the determination of J K [7].…”

“…Thus we need to make inferences over such event. Given focal elements I = I 1 × ... × I n , in case where I i = [ i , u i ] are intervals it was shown [7] that…”

“…For computational reasons, we stopped the sorting procedure at the 25th fragment for each simulation. Note that the complexity of the sorting procedure is exponential, i.e., O((2 |S| ) nb of pieces ) [7]. Figure 2, 3 and 4 show respectively the evolution of the weight of materials in the two first containers, the belief that the constraints are respected and the real proportions of impurities.…”

“…As we add pieces to a given container, we will also have to compute the global uncertainty of a container by adding mass functions of different weights. To do so, we will adapt the technique proposed in [7] for general intervals to the case of discrete proportions.…”