8.1   Introduction

The concept of distributed detection fusion^† or decision fusion has evidenced wide interest over the past 25 years, beginning with applications ranging from multimode signal detection and distributed multisensor decision theory to multiple change detection algorithms for surveillance imaging systems to subordinate team decision managers or decision makers (DMs) within a command and control formulation. Recent applications have involved team-decision concepts for wireless networks and particle swarm theory for biometric security systems.

Distributed detection fusion problems are often represented as optimization problems involving a composite collection of local DMs (involving sensors, algorithms, control processes, and persons) feeding a centralized (global) DM. As we embrace the ever-growing challenge of network-centric systems, the need and the opportunity now exist to communicate between any number of local DMs, providing distributed interaction between local DMs within a region or between global managers. With this emphasis on network-centric development, demand for optimal decision techniques would need to grow as well.

This chapter provides the basic concepts of distributed detection fusion, beginning with the concepts of classical detection theory portrayed by Van Trees.¹ We will discuss the most basic architecture—a parallel fusion architecture—and in doing so, lay out the means to develop and design system decision fusion rules. Finally, we will provide a discussion on fusion rules and the assumptions placed on local DMs.

The structure of this chapter is as follows: Section 8.2 provides the fundamental concepts of detection fusion theory, beginning with a single DM. We build on this approach to extend to multiple DMs operating within a parallel fusion system in Section 8.3, based on the concepts of team-decision theory. Section 8.4 provides an analysis of some of the common fusion rules, how they are constructed, and their performance.

8.2   Single Node Detection Fundamentals

In developing an optimal detection approach for distributed DMs, we consider the binary hypothesis testing problem for N local DMs. Consider each DM having to select between two hypotheses, H₁ and H₀, with known prior probabilities for each hypothesis. These hypotheses represent observations, indicating acceptance (H₁) or rejection (H₀) of the existence of, say, a target versus noise, correct versus false target classification, or detecting relevant anomaly detections over that of random activity.

Consider Figure 8.1. When sensors monitor their environment, either they observe an entity of interest, Z₁, and the local DM declares H₁ (e.g., detect the target), or they detect noise (no target), with the local DM declaring H₀ with some statistical basis. It is assumed that these choices represent the total observation space as the union of the observations, Z = Z₁ ∪ Z₀, with no overlap between the observation subspaces, that is, Z₁ ∩ Z₀ = ϕ.

To develop the concepts of minimum risk associated with making a decision on a target, consider an individual DM generating a decision, u_i, representing one of the two hypotheses based on its observation, y_i:

H₀:y_i = n_i (noise only)

H₁:y_i = r + n_i (target plus noise)

The noise and signals observed by each DM are considered to be statistically independent from any other DM. Decisions characterized by each DM, u_i, are dependent on both the observation and a threshold. We can denote these decisions as follows:

This detection performance is characterized by the local DM based on a probability conditioned on choosing the true hypothesis, P(u_i|H_j). Bayes formulation attributes the cost to each possible choice with decision rules designed to minimize the cost of making a decision, that is, minimizing the expected risk, R.

For a single DM, we can express this risk as follows:

where C_ij is the cost of declaring hypothesis i, given that hypothesis j is true and P_j is the prior probability representing the choice that hypothesis j is true based on the observation y. The conditional probability represents the probability that integrating over a region, Z_i, will generate a correct response for a target.

Expanding over the summation gives

which reduces to

by taking advantage of the fact that Z represents the total observation space, with unity probability. Minimizing risk is equivalent to minimizing the integrand, that is, assigning y to H₁ if the integrand is positive or to H₀ for a negative integrand. Alternatively, we can write the integrand in the form of a likelihood ratio as follows:

where the left-hand side indicates the likelihood ratio and the right-hand side the detection threshold.

We can rewrite the risk in terms of the following well-known probabilities:

• Probability of a false alarm, P_F = ∫_Z1 p(y|H₀)dy

• Probability of a detection, P_D = ∫_Z1 p(y|H₁)dy

• Probability of miss detection, PM=∫Z0p(y|H1)dy=1−PD

Also note that the prior probability of H₁ can be written as P₁ = 1 – P₀.

The risk becomes

which can be reduced to

where we define the costs to be C_F = P₀(C₁₀ – C₀₀), C_D = (1 – P₀)(C₀₁ – C₁₁), and C = P₀C₀₀ + (1 – P₀)C₀₁, assuming the cost of an incorrect decision to be higher than the cost of a correct decision, that is, C₁₀ > C₀₀ and C₀₁ > C₁₁.

8.3   Parallel Fusion Network

Now consider a number of DMs within the parallel decision fusion network shown in Figure 8.1. Extending the Bayesian minimum risk concepts to a set of distributed detectors, each DM provides a localized decision and transmits results to a centralized fusion system, where they are combined into a composite result. The fusion system combines the local results by an appropriate fusion rule, and provides a final decision, u₀, representing the viability that the global hypothesis represents the true hypothesis. Fused optimization of the results depends on an evaluation of the transmitted decisions from each of the N DMs and developing the appropriate fusion rule to define how individual local decisions are combined.

Varshney^{2, 3, 4 and 5} provides an excellent derivation. We follow a similar approach. We define the overall fused probabilities as follows:

• Fused probability of a false alarm, P_F = Σ_uP(u₀ = 1|u)P(u|H₀)

• Fused probability of a detection, P_D = Σ_uP(u₀ = 1|u)P(u|H₁)

where ∑_u is the summation over the N DMs. Substituting these system probabilities into Equation 8.4 gives the overall risk as follows:

Tsitsiklis^{6, 7 and 8} points out that decentralized detection problems of this type fall within the class of team-decision problems. Once a decision rule is optimized for each local DM, the ensemble set of decisions transmitted to the fusion center represents a realization from all DMs. That is, each DM transmits u_i = γ(y_i) contributing to a global ensemble set u₀ = γ₀(u₁, u₂, u₃, …, u_N) to be evaluated at the fusion center. For conditionally independent conditions, observations of one DM are unaffected by the choice of the other DMs. Tsitsiklis shows that once the individual strategies are optimized, the overall system fusion can be represented by a person-by-person optimization approach.

Person-by-person optimization can be performed in two steps:

1. Likelihood rules are generated for each local DM, holding all other DMs fixed, including the fusion decision.

2. Fusion rule is obtained in a similar way, assuming all local DMs are fixed.

8.4   Fusion rules

The combined team-decision solution, involves solving the N local DM thresholds in Equation 8.8 simultaneously with the 2^N fusion likelihoods in Equation 8.9, coupling the local and fused solutions. Thus, to achieve system optimization, the decision fusion process optimizes the thresholds for each local DM by holding the other DMs fixed, and repeating the process until all local DM thresholds are determined. The resulting thresholds are coupled to the global likelihood and solved simultaneously. Their simultaneous solution provides the basis for determining the set of fusion rules, a total set of 2^2N rules. Selection of the appropriate fusion rule is determined by an evaluation of all possible decisions.

Table 8.1 shows the explosive growth required to develop a complete set of fusion rules—for instance, four DMs give rise to more than 65,500 rules. However, an examination of the rule sets shows that a great many of the rules can be eliminated through the principle of monotonicity,^{2,17, 18, 19 and 20} while even further reduction can be made with some knowledge of the DMs.

The assumption of monotonicity implies that for every DM, P_Di > P_Fi, and as such not all fusion rules satisfy threshold optimality. For example, consider a condition where K sensors select the primary hypothesis, H₁. Under a subsequent test, if M > K sensors select hypothesis H₁, where the set of M sensors includes the set of K sensors, then it is expected that the global decision provides a higher likelihood for the M sensors than for the K sensors. Choosing the fusion rule with the largest likelihood value ensures an optimal result. Alternatively, as Table 8.1 shows, any fusion rule that contradicts this process can be removed, significantly saving on the number of necessary rules to evaluate.

Figure 8.2 shows a simple but common example of a complete set of fusion rules generated for two DMs. We can gain considerable insight by examining these rules further. For example, fusion rule d1 rejects H₁ regardless of the decisions of the two detectors (d1 = 0 for all combinations of u₁ and u₂), whereas fusion rule d16 accepts H₁ under any condition (d16 = 1 for all combinations of u₁ and u₂). Rules d9 through d16 declare H₁ when both local detectors reject H₁. Also rules d3, d5, d7, d9, d11, d13, and d15 all reject H₁ even though both detectors declare H₁ (i.e., u₁ = 1, u₂ = 1). Thus monotonicity maintains that we keep six rules: γ(u) = {d1, d2, d4, d6, d8, d16}.

Figure 8.3 shows the final fusion rule selection using two local detectors. Note that the fusion rules can be reduced even further, since d1 and d16 just bound the set of rules and are not practical. We are left with four rules: the well-known AND and OR rules, as well as rules for relying on one or the other single DM.

Much of the literature has emphasized this particular set, preferring to study the impact of AND and OR rules for their particular development. However, the case for three detectors shows a great deal more flexibility in the rule set.

Figure 8.4 shows the set of fusion rules for three detectors after we apply monotonicity, reducing the number of rules from 256 to 20. Two important additional rules become evident: “Majority Logic” and “Sensor Dominance.” For the former rule, the global decision rule requires at least two or more DMs to agree on H₁ before declaring H₁ at the fusion center. For Sensor Dominance, we assume that a single detector, say u₂, operates with sufficient quality to be accepted by the fusion system whenever it declares H₁; however, when u₂ rejects the hypothesis, the other two DMs must both have positive decision before making a global declaration of H₁. This condition could occur, for example, when a high quality imaging radar detects a target by itself, while two poor resolution sensors (detectors u₁ and u₃) must have consensus for the fusion system to declare H₁.

As such, we can use this approach to eliminate some of these rules on the basis of the quality of performance of the local DMs and their sensors. For the example just discussed, we would not necessarily keep rule d34 (u₁ dominates) or rule d88 (u₃ dominates), since their performance quality is poorer than u₂. To assess the rule set further, we would need to examine the impact of each individual detector performance on the environment and set the system fusion rules accordingly.

Functionally, each rule applies a different set of constraints on the basis of individual detection performance to provide the fusion system with a complete set of representative rules. Selecting the rule with the highest likelihood ensures improved target detection while reducing false alarms. If we have knowledge of the individual sensors, we can select those fusion rules that best fit their individual characteristics. Figure 8.5 shows a representative set of fusion performance curves that compare performance from three detectors against that of an individual detector. For convenience, we have assumed that all three detectors perform equally well. We can use the graph to examine the effects of the fusion rules for both fused system probability of detection (P_D) as well as system probability of false alarm (P_F). That is, for probability values ≥0.5 along the x-axis, we can attribute these to individual detector probability of detection (P_Di), and those probability values ≤0.5 can represent individual detector probability of false alarm (P_Fi). The same can be considered for the fusion system performance along the y-axis. For instance, consider the two vertical lines at P_Di = 0.88 and P_Fi = 0.10. The P_D line intersects the AND rule giving a fused value of 0.6815; intersection with the OR rule occurs at P_Di = 1 – (1 – 0.88)³ for a system value of 0.9983. Similarly for P_Fi, the AND rule gives system P_F = 0.001 and the OR rule gives P_F = 0.2710. From this, it appears that the OR rule can significantly outperform all other rules for P_D, but at the cost of higher false alarms, whereas the AND fusion rule appears to be an excellent means of reducing P_F, but provides poor overall system performance.

Now let us consider each of the four fusion rules shown in Figure 8.5 in more detail:

1. The AND rule relaxes constraints on the individual detectors, allowing each detector to pass a large number of both detections and false alarms, relying on the fusion process to improve performance by eliminating most false reports. In this way, relaxed individual performance conditions will ensure improved target reportability from each detector, even though a larger set of false reports may also be developed.

2. The OR rule provides the reversed effect, relying on individual detectors to operate at higher detection thresholds to significantly eliminate false alarms for the fused system. This rule assumes the detectors are quite adept at distinguishing target reports from false alarms. If the detectors produce too many false reports, this rule can be detrimental. As such, a higher set of thresholds are applied to each detection source.

3. The Majority Logic rule falls somewhere between AND and OR performance curves, exhibiting some of the best characteristics from both strategies. It is based on the assumption that not as many false alarms would be generated by the individual detectors as in the AND case, whereas the potential to distinguish target reports from false alarms is not as stringent as in the OR case.

4. The Sensor Dominance rule presents an interesting case, relaxing constraints only for the single, dominant detector, expecting a lower number of false alarms for that detector, but relying on the fused element to reduce false reports from the other two poorer performing detectors. Depending on the degree of improvement of the dominant detector over the other two, system performance can improve, but false alarm performance is determined by the performance of the individual dominant detector. This rule relies on the assumption that the dominant detector is of very high quality.

Recently, particle swarm optimization theory has been suggested as an alternative to person-by-person optimization^21,22 when we consider the explosive growth of rules. Here, multiple classifiers are fused at the decision level, and the particle swarm optimization techniques determine the optimal local decision thresholds for each classifier and the fusion rule, replacing the person-by-person strategy discussed in Section 8.3. Other recent trends have focused on the application of distributed fusion to wireless sensor networks.²³

8.5   Summary

We have presented an introduction into the details involving the development of distributed detection fusion, beginning with the fundamentals of classical Bayesian likelihoods and extending these concepts to a parallel distributed fusion detection system, building on the concepts of system fusion rules to combine the local detectors. Each fusion rule applies a coupled decision statistic that adjusts detection performance thresholds for each DM on the basis of the combined performance of the other local detectors. We have also discussed how global fusion rules perform in terms of individual (local) detection performance, pointing out how detections and false alarms are treated on the basis of different fusion rules.

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