Binary fuzzy measures and Choquet integration for multi-source fusion

Countless challenges in engineering require the intelligent combining (aka fusion) of data or information from multiple sources. The Choquet integral (ChI), a parametric aggregation function, is a well-known tool for multi-source fusion, where source refers to sensors, humans and/or algorithms. In particular, a selling point of the ChI is its ability to model and subsequently exploit rich interactions between inputs. For a task with N inputs, the ChI has 2^N interaction variables. Therefore, the ChI becomes intractable quickly in terms of storage and its data-driven learning. Herein, we study the properties of an efficient to store, compute, and ultimately optimize version of the ChI based on a binary fuzzy measure (BFM). The BFM is further motivated by empirical observations in the areas of multi-sensor fusion and hyperspectral image processing. Herein, we provide a deeper understanding of the inner workings, capabilities and underlying philosophy of a BFM ChI (BChI). We also prove that two fuzzy integrals, the ChI and the Sugeno integral, are equivalent for a BFM. Furthermore, only a small subset of BFM variables need be stored, which reduces the BChI to a relatively simple look up operation.

I. Introduction

Engineering applications like Big Data, remote sensing, unmanned vehicles, robotics, signal and image processing, and machine learning, to name a few, all have something in common. They all typically require the intelligent combining (aka fusion) of multiple sources. Here, the term source is used to refer to the “generator” of data or information, e.g., sensors, humans and/or algorithms. In general, the idea of fusion is to obtain a “better” result than if we only used the individual inputs. However, better is not a well defined concept. In some applications, better might mean taking a set of inputs and reducing them into a single result that can be more efficiently or effectively used for visualization. Better could also refer to obtaining more desirable properties such as higher information content or lower conflict. In areas like machine learning and pattern recognition, theories that power the majority of engineering applications mentioned above, better often refers to some desirable property like more robust and generalizable solutions (e.g., classifiers). Regardless of the task at hand or the particular application, fusion is a core tool at the heart of numerous modern scientific thrusts.