decision fusion strategies in multisensor environments

1140 IEEE TRANSACTIONS ON SYSTEMS. MAN, AND CYBERNETICS, VOL. 21, NO. 5, SEPTEMBEWOCTOBER 1991

Decision Fusion Strategies in Multisensor Environments

Belur V. Dasarathy, Senior Member, IEEE

Abstract-Efficient decision fusion strategies, for deriving optimal decisions in multisensor target recognitionltracking environments, are postulated and analyzed in this study. This fusion paradigm, which is designed for fusing decisions derived from a suite of parallel sensors, is embedded within a recursive system structure to achieve significant enhancement in the reliability of the fused decisions.

I. INTRODUCTION HE STUDY OF distributed multisensor networks and T the problem of intelligent integration of multisensor data

[24] has gained immense popularity recently with the advent of vigorous sponsorship from DOD of the sensor fusion technology areas. The survey by Luo and Kay [24] that covers more than 200 papers, attempts to draw a distinction between the terms multisensor integration and fusion although in most literature these terms have often been used interchangeably. The distinction however remains murky even within this survey. In general, sensor fusion can be accomplished at different levels: data fusion, feature fusion and decision fusion. A variety of paradigms have been put forth for the study of the sensor fusion at all these levels and were extensively reviewed in a recent survey by Dasarathy [25]. Depending upon the specific sensors used, fusion at the data and feature levels may or may not always be practical. For example, data fusion would require compatible sensors that are appropriately registered to ' permit such data level integration. Further, decision fusion is deemed more robust than fusion at the lower levels, as failure of one of the sensors in the sensor suite does not signify total catastrophic failure of the entire system. These comparative aspects are discussed in more detail in the survey paper [25]. Here, this study addresses the limited problem of fusion at the highest level, namely decision fusion, which minimizes the amount of data to be transmitted between the individual sensors and the central fusion processor by limiting it to just the current decisions derived by the sensors. While there indeed have been a number of studies [1]-[23] in this decision fusion area over the past decade, the approach and the information processing structure developed here are significantly different from the previous studies. The objective of this study is to develop possible strategies for fusion of current decisions derived in parallel from the different sensors in order to ensure that the fused decision derived by the

Manuscript received August 10, 1990; revised March I , 1991. This work was supported in part by U S . Army Strategic Defense Command under Contract DASG-60-89-C- 130.

The author is with Dynetics, Inc., Huntsville, AL 35814-5050. IEEE Log Number 9102024.

process is optimal for the environment and such a reliable fused decision is obtained at the earliest possible time.

Tenney and Sandell [l] were one of the first to study the problem of detection with distributed sensors. The analysis follows classical Bayesian theory, and offers decentralized statistical hypothesis testing at each of the individual sensors in the distributed network to determine the optimal local detection rules. It did not however explicitly tackle the problem of fusion of these decisions in terms of development of optimal data fusion algorithms. Dasarathy [2] studied a limited version of the decision fusion problem in the context of multiple classifiers operating with a single sensor, using a recursive structure that could be viewed as a forerunner to the problem tackled here. Chair and Varshney [ 3 ] attempted the development of an optimal global decision rule through weighting the independent optimal decisions of the individual sensors, by their reliability rates known in terms of false alarm and miss rates, and comparing the weighted sum of these decision probabilities against a likelihood threshold to derive the global decision. The problem considered by them is essentially one of target detection, i.e., a two-class problem. A different type of generalization of the analysis in [l] was presented independently by Sadjadi [4] around the same time as the study by Chair and Varshney [3]. He offered an optimal solution for the general case of m-ary hypotheses testing for 'n sensors but again without explicit optimal fusion. A loss function is defined corresponding to each local decision and a mean loss function averaged over these individual loss functions is minimized resulting in sets of simultaneous inequalities involving the generalized likelihood ratios at the local detectors. The solutions to these inequalities define the optimal decision domains. The study by Reibman and Nolte [5] extended the previous study (31 by simultaneous optimization of the local detectors while deriving the overall optimal fusion design. This involves solution of coupled equations represent- ing the performance of the local detectors and the global fusion processor. In essence, this study combined the ideas of local decision optimization of Tenney and Sandell with the fusion optimization proposed by Chair and Varshney [3].

The same scenario is handled in a more general way by Thomopoulos [6] et al., using the Neyman-Pearson (NP) test both at the individual sensor level as well as at the decision fusion level. The fused decision is shown to have a higher probability of detection than the individual one for cases with three or more sensors under a constant false alarm rate for all the sensors. The method also permits some quality information on each of the decisions to be taken into account.

0018-9472/$01.00 0 1991 IEEE

DASARATHY: DECISION FUSION STRATEGIES 1141

In a second study, Reibman and Nolte [7] looked at the system performance in more detail than before. Thomopoulos et al. [8] revisited the problem of decision fusion in distributed sensor systems considered earlier with Neyman-Pearson test at the fusion center as before but likelihood ratio tests at the individual sensors. Two specific computationally efficient but suboptimal algorithms were proposed and numerically illustrated. Thomopoulos and Okello [9] presented an approach to the problem of detection with consulting sensors and associated communication costs. Here, one of the two available sensors, is viewed as the primary sensor and the other only as a consulting sensor, to be called upon only when deemed cost justifiable in view of the communication costs involved in the consultation process. Various random and non-random request schemes are considered in their analysis along with some numerical results for illustrating the principles. Demirbas [ 101 presented a new maximum a posteriori approach to this problem of recognition in multisensor environments. He showed that in an object detection environment, the maximum a posteriori estimation approach minimizes the mean square error.

Viswanathan and Thomopoulos [ 111 considered the serial problem involving hand-over from one sensor to the other, instead of the parallel decision fusion considered in their earlier studies, once again using the NP test. The j th sensor decision is dependent (and optimal in the NP sense) not only on its own observations but also on the optimal NP decision of the previous ( j - 1)th sensor. It is claimed that the serial scheme has a better performance than the parallel scheme in certain cases of two-sensor suite. However, this is not true if more than two sensors are deployed in the fusion process. Also, such a serial configuration is more susceptible to overall system failures caused by failures in one of the components or the in-between links thereof. From this view point a combination of serial-parallel configuration may be worthy of further exploration. A distributed Bayesian hypothesis testing approach with distributed data fusion was presented by Chair and Varshney [12]. This study, unlike earlier studies wherein the fusion processing was limited to the central location, looks at a system configuration that permits data fusion at each local site. The analysis leads to multiple coupled nonlinear equations and solutions may involve multiple local minima. Identification of global minimum becomes compute intensive and therefore the practicality of the approach in real world applications requires further careful consideration. Thomopou- 10s and Okello [13] dealt with the problem of detection with mismatched sensors. This addresses a very specific type of problem wherein the two sensors each have certain blind spots in their field of view. There are numerous assumptions made in the analysis such as known geometry of the visible and blind spot regions and the like. The study is therefore not of a general nature, although some of the concepts may have applications in other specific contexts as well. This was followed by a study by Barkat and Varshney [14] on the constant false alarm rate (CFAR) problem in the distributed sensor environment. The study examines the scope for adaptive threshold techniques instead of pre-fixed thresholds used in the earlier studies in order to assure a detection with constant false alarm rate. Lee and Chao [15] presented an interesting study on optimum

decision space partitioning in multisensor environments. The study investigates the benefits of transmitting reliability data in addition to the traditional decision only transmission assumed between the local sensors and the central processor in many of the previous studies. This transmission is still limited to just a few additional bits and not complete local likelihood ratio information that represents the other end of the spectrum. This compromise level of information transmission in effect represents a partitioning of the local decision space and entails an optimization of the partitioning problem.

Demirbas [ 161 presented a nonparametric binary decision trees approach to the fusion problem, which could be viewed more as a feature fusion tool. In that sense, this is some what outside the main stream of the studies discussed here, but does contain some interesting elements of possible extrapolatory value. Thomopoulos et al. [17] provided some additional results in terms of proofs of the combined NP test (at the fusion center) and likelihood ratio criterion (at the individual sensors) based solutions offered for the distributed decision problem considered earlier in [6], [8]. Recently, Hoballah and Varshney [ 181 investigated this problem from an information theoretic point of view. They employ an entropy-based cost function for the optimization of the overall decision system, which is intended to maximize the amount of information transfer from the input side to the output side. In another paper [19] in the same issue, they dealt with the problem of deriving a global decision by combining local binary decisions through a Bayesian formulation. A special case of identical local detection sensors with independent observations is also considered. Some specific fusion scenarios were explored recently by Polychronopoulos and Tsitsiklis [20]. A finite valued function of the local observations is transmitted to the fusion processor wherein the M-ary decision is to be made. The asymptotic scenario of very large number of sensors is analyzed to conclude that large number of sensors transmitting coarse (few bits of) information yield better result than fewer number of sensors offering more detailed (large number of bits) data. This conclusion does not take into consideration the cost of such large network of sensors.

Very recently, Baraghimian and Klinger [21] proposed a preferential voting scheme for the fusion process. This comes closest in terms of the concepts being presented here in this study, in that the fusion process is based on a voting scheme. However, it does not include the novel concept of recursive structure (within which the voting is embedded) that is being offered here. A more recent example of these decision fusion studies is that of Krzysztofowicz and Long, once again on a Bayesian detection model [22]. Three specific schemes for fusing information are analyzed. They are: fusion of observations (data fusion), fusion of local decisions (decision fusion), and lastly fusion of probabilities (which can be looked upon as partially decision fusion, with some data from the local sensors also being taken into account in the fusion process). This study, unlike most of the earlier ones, attempts to relate the analysis to non-DOD applications although in reality the extent of applicability of this study to other applications is no more or no less than most of the prior studies in the area. Unfortunately, the study does not clearly bring out its

1142 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 21, NO. 5, SEPTEMBERIOCTOBER 1991

TRffiER APPROPRIATE

y ~ s ACTION C0NSlSTEN-r +

DECISION ACROSS THE SUITE?

Fig. 1. A recursive processing system structure for optimal decisions with a parallel sensor suite.

contribution relative to that made by the other studies cited here. The study by Tang et al. [23] represents the most recent of these efforts. They deal with the problem first enunciated by Tenney and Sandell [ l ] and relate their efforts to those of Chair and Varshney [3] as well as Reibman and Nolte [4]. However, none of the other efforts cited here [5]-[22] are included in their discussions. The analysis being presented here is most akin to studies on voting schemes [21] but goes further by offering a recursive structure for enhancing the results of such voting. This is a significant departure from the other studies, none of which looked at the scope for improving performance with a recursive structure triggered on the basis of results of decision fusion.

In the study presented here, the problem is viewed as one of parallel fusion of the individual independent decisions with a recursive structure that permits enhancement of the reliability of the fused decision. Instead of a voting based fusion approach, this recursive structure can be combined with any one of the other optimal fusion techniques [1]-[23] as well. While these and other alternatives are still under investigation, an analysis based on a simple voting or consistency measure dependent fusion scheme is being presented here to mainly illustrate the recursive model and its benefits. In the problem environment considered here, it is assumed that if a reliable decision cannot be made at any given stage, then we seek a new set of decisions from the sensors and completely disregard the earlier decisions that are deemed suspect due to the lack of required degree of consistency. Accordingly, the fusion strategies developed here are based only on current decisions from the sensor suite rather than on all the cumulative decisions. However, a discussion of the cumulative strategy is included in the concluding section to illustrate the scope for widening the application potential of the concepts proposed here in this study.

11. A PARALLEL SENSORS SUITE ANALYSIS

Fig. 1 portrays a general system structure of information fusion processing for optimal decision making in multi-sensor environments. The sensor suite consists of t sensors each of which generates a data set Zj processed by its own customized set of algorithms either locally in a distributed sense or in the centralized parallel processor (depending on the application environment and its constraints). The output Oj of each sensor-processor pair is a decision Dj with

an integer value in the range (1, m) corresponding to a choice among m possible decisions. In the domain of target recognition/multi-target tracking problems, this output can be either a target identification or target data association label. The function of the fusion processor is to determine the consistency of the decisions across the sensor suite at a given time. While it is theoretically possible to visualize an alternative strategy of determining the consistency across the temporal scale, i.e., over a sequence of decisions by each of the sensors, the motivation here is mainly to determine the instantaneous consistency across the sensor suite. Accordingly, the strategy proposed here is to accept the resulting decision and end the recursions if such consistency is reached, or else discard the local decisions as suspect and continue the recursions till an acceptable level of consistent decision is attained. Various measures of consistency can be visualized ranging from consensus (i.e., all decisions are identical) to at least some s of the t sensors are in agreement. The model presented here permits recursive operations directed towards enhancement of the reliability of the derived decision. This is dictated by the decision fusion processor that evaluates the consistency of the decisions across the different sensors and sends a signal to continue the recursive process (of acquiring a new set of signals and reprocessing them) whenever a lack of consistency is detected by the processor. There is no cumulative record of these decisions to be maintained as the consistency assessment is only across the current set of local decisions. While the implications of the alternative strategy of a cumulative consistency check are briefly considered in the closing section, a detailed analysis of such alternatives is decidedly outside the scope of this study.

For simplicity of presentation of the concepts and associated strategies, we shall consider the case wherein the output of each sensor-processor pair is a simple decision among a choice of m decisions, i.e., the output is an integer value in the range (1, m). Let (ej : j = 1, . . . , t ) represent the efficiency or probability of correctness of the decision Dj reached by sensor-processor pair j . Implicit in this statement is an assumption that the probability of correctness of the decision Dj is the same for all possible m values of Dj. This assumption is being made mainly to keep the analysis simple without in anyway sacrificing its applicability. Even if this assumption were to be not strictly valid, one could approximate it by using the weighted average of the probabilities over all

DASARATHY: DECISION FUSION STRATEGIES

decision choices corresponding to each sensor as a measure of e,j. This weighting would be as per the a priori probabilities of the decision choices available in the environment. The value (1 - e J ) can be used to represent the corresponding misrecognition or error rate for the sensor-processor pair assuming that it always provides a decision (right or wrong) i.e., no possibility of a neutral or unknown decision by an individual sensor is allowed under this scenario. Also, this error rate is the weighted average of both types of wrong decisions in the context of a target detection problem, namely false alarms and misses. The more general scenario, where the sensor-processor pair can have a "no decision" option, can also be analyzed in a fairly similar fashion. This is considered beyond the scope of this presentation in view of the publication space limitations and results of on going studies of this nature will be published in due course separately.

Let pk, yk. and rk represent the incremental probabilities of the fused correct, incorrect and nondecisions at the kth stage of recursive process. By definition

p l + 41 + r1 = 1. (1)

As only the undecided cases are reprocessed:

P2 + Y2 + 7-2 = r1. (2)

Generalizing (2) we get

pk + q k + rk = T k - 1 . (3 )

Using (3) recursively in (1) gives

k k

CPz +CY* + r k = 1 (4) z=1 z = 1

i.e.,

with

Qrc = i=l

Assuming e j 's are constant in the range of measurements made for a specific target, the instantaneous correct and incorrect fused decision rates, which are functions of e3 's, can be assumed to be also constant and hence the incremental rates can be written as

This assumes that local decisions from the same sensor are conditionally independent in time. Under such an assumption one could admittedly visualize a sequential test using multiple

1143

this analysis is to explore the benefits of using multiple nonidentical sensors that may derive comparable decisions that are not necessarily based on the same type of information content in the environment. Accordingly, we concentrate on a recursive strategy involving multiple sensors, rather than a sequential strategy involving a single sensor. While a fusion based on all the local decisions accumulated thus far can be visualized under this multisensor scenario, the objective here is limited to fusion of current decisions at each stage, the implication being that whenever a reliable fused decision is not derived at any stage, the whole set of local decisions at that stage are considered suspect and disregarded. The current analysis is accordingly limited in principle to a noncumulative fusion strategy, with only limited discussions of the cumulative fusion strategy being included in the concluding section. Substitution of (9) and (10) in (8) (with IC incremented by 1) and use of (1) leads to

r k + l = r k r l (11)

rk = r: (12)

i.e.,

k

= p1 CrP-') (13)

Q~ = q1 Err-'). (14)

a = 1 k

a = 1

Now, k Er!'-') = (1 - r f ) / ( 1 - r l ) ; 0 5 r1 < 1 (15)

z = 1

i.e.,

p k = pl (1 - .:)/(I - TI)

Q k = ~ i ( 1 - r:)/(1 - T I )

rk = 1 - (4 + Q k ) r1 = 1 - ( p 1 + 41):

(16)

(17) (18) (19) 0 5 r1 < 1.

It is obvious from (16) and (17) that P k and Q k monotonically increase with k and Tk decreases monotonically with k unless r1 is zero at the first stage itself. For the case of m = 2, i.e., a decision with only two choices (for example, a target detection problem), r1 can be zero if t is odd and a majority decision is used as the measure of consistency. This makes the recursive process a single shot affair. Examining (16) and (17) further, it can be observed that the asymptotic limit values (as k tends to infinity) of P k and Q k are given by

P k l m a x = Pl/(Pl + 91)

Q k i m , , = Y I / ~ + YI)

(20)

(21)

i.e.,

Use of (5 ) in (22) shows that the nondecision rate 7-k tends observations of a single sensor. However, the objective of to zero asymptotically, i.e., eventually if we keep processing,

1144 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 21, NO. 5 , SEPTEMBERIOCTOBER 1991

we always will end up with a decision (be it correct or not). Comparing (20) and (21), we can see that the asymptotic fused correct decision rate will exceed the incorrect decision rate if and only if the initial correct decision rate pl is greater than initial incorrect decision rate 41. On the other hand, for the initial nonrecognition rate to be not less than the initial misrecognition rate, we can show that

41 I (1 - Pl)/2. (23)

We now examine the relationship of e3’s to pl and 41 by initially considering certain specific values of t , the number of sensors in the environment. While one could conceivably look at the general case directly, the specific cases are indeed of particular interest as in the real world one may have only very few sensors and the application engineer would be benefited from a thorough and detailed analysis of these special cases at a level that is not practical in the general case. Also, the start from the special cases makes the analysis easier to follow especially for those interested more in the results of specific scenarios than in the analytical elegance of generalities.

A. Two Sensors Case (t = 2)

Here, the consensus is given by an agreement between the decisions generated by the two sensor-processor pairs. As- suming independence of the probabilities of correct decisions arising under the two sensors (although this may not be strictly true, it permits estimation of bounds on the values of p l and 41 1

P I = e1e2 I 1 41 = (1 - e l ) ( l - e2)/(m - 1) 5 1 T I = [ (m - 2) + (el + e2) - mele2]/(m - 1) < 1. (26)

Here, if one were to permit nondecision rates or partial ignorance about sensor performance and employ an evidential reasoning approach, p l would be expressed in terms of cJ (correct recognition rates) sand 41 would be expressed in terms of wj (misrecognition rates) resulting in a similar but far more complex expression for T I and the subsequent analysis would also get correspondingly changed. These are deemed beyond the scope of this study and as such are not presented here. Reviewing (24) and (25), it is obvious that both the resultant initial correct and incorrect decision rates are bounded on the upper side by the lesser of the corresponding pairs of the individual sensor correct and incorrect decision rates, i.e.,

(24) (25)

Pl I min(e11e2) (27) 41 I min((1 - e l ) , (1 - e2)). (28)

We shall now consider special cases that deal with interesting scenarios before returning to this general case of two sensors. These cases provide some useful insights into the decision fusion and associated recursive strategy.

Subcase AI:-e, = 1 (1 5 i 5 2 ) : At this juncture, it would be interesting to consider the scenario wherein one of the two sensors is always perfect. Of course, if we were to know which sensor is the perfect one, then we could disregard the other sensor and accept the decision of the perfect one without sensor fusion even being required. However, if we

were ignorant as to which of the two is the perfect sensor at any given time but could expect that at least one of the two would be almost perfect most of the time (which is not an unreasonable assumption if redundancy approaches are employed in sensor system design), then expressions (24) through (26) would lead us to

Pl = e 41 = 0 r 1 = 1 - e

where e is the efficiency of the imperfect sensor. These are independent of the value of m. Use of these values of p l and q1 in (20) and (21) leads to

Expressions (32) and (33) denote that the fused decision asymptotically approaches the correct decision irrespective of number decision choices even though one of the sensors may remain imperfect throughout and further, the system (user) does not know as to which of these two sensors is the imperfect one. We next consider the opposite case of one sensor being always wrong. While this is definitely not a desirable state to be in, it is interesting from the view point of learning its consequences to the fusion process and hence the look at this special but unacceptable case.

Subcase A2:-ei = 0 (1 5 i 5 2): Here, expressions (24) and (26) reduce to

Pl = 0 (34) 41 = (1 - .)/(m - 1) (35) TI = (m - 2 + e ) / ( m - 1) (36)

where e is the efficiency of the other sensor. Under these initial conditions, the recursive fusion process leads to the asymptotic state

P/c 1 max = 0

Q k 1 max = 1

(37)

(38)

i.e., the fusion process leads to a total failure of the system under this scenario. It is therefore important to ensure that no totally faulty sensor is allowed to operate indefinitely in a two sensor fusion system of this type. Under this scenario, if the other sensor is perfect, i.e., e = I, (34)-(36) further reduce to

Pl = 41 = 0 r1 = 1.

(39) (40)

Although the perfect sensor makes the misrecognition rate go to zero, the recognition rate remains to be zero because of the other faulty sensor, with the result no agreement is ever possible between the two sensors making nondecision as the only possible result with a probability 1.

Subcase A3:-ei = e (V i : i = 1,2): For this case, (24) through (26) reduce to

pl = e2 5 1 (41)

DASARATHY: DECISION FUSION STRATEGIES

0.6 L 7

U m U - Q- 0 4

0.2

0.0 0.2 0.4 e 0.6 0 8 1 .o 0.0

Fig. 2. Fused correct, incorrect and nondecision rates versus individual sensor efficiency.

q1 = (1 - e)'/(m - 1) I 1 (42) (43) 7-1 = (1 - e ) [m( l + e ) - 2]/(m - 1) < 1.

Differentiating T I with respect to e, we get

dr l /de = 2(1 - me) / (m - 1). (44)

Setting the derivative to zero, the nondecision rate attains an extremum value at

e = I/m. (45)

Differentiating (44), the second derivative of r1 is a negative value independent of e and thus the extremum reached is a maximum. Substituting (45) in (43) the maximum value of T I

is

T l l m a x = (m - l ) /m (46)

attained at e = l / m . As m, the number of decision choices, increases the maximum value of T I , i.e., the maximum initial nondecision rate, increases and asymptotically approaches 1. This is because the error rate decreases with increase in the number of decision choices and with the correct decision rate remaining constant, the nondecision rate increases with m as the sum of these rates is always unity by definition.

Further, for m = 2 under the subcase A3, (41) through (43) reduce to

PI = p l = e' (47) Q I = 41 = (1 - e)' (48) r1 = 2e( l - e ) . (49)

Fig. 2 shows the plot of p l ,q1, and rlversus e. As e increases, p1 increases, 41 decreases, and 7-1 first increases and then decreases. As expected from the previous analysis, r1 peaks at e = l /m = 0.5 with a value of (m - l ) /m = 0.5 for the case m = 2 . Fig. 3 shows the plot of the asymptotic values Pk 1 max and Qk 1 max (obtained by substituting (47) and (48) in (20) and (21)) versus e for this same case (the sum of the two is always unity as T k reaches zero asymptotically).

The crossover point of the recognition and misrecognition rates is at e = 0.5 for m = 2 . The initial and the final (asymptotic) fused correct decision rates are shown superimposed

1 I45

1 0

0.8 -

- 0.6 - E

a 0.4

m - 1 -

0.2 -

- 0.8

- 0 6

E" - -0.4 Cr

- 0 2

0.0 0.2 0.4 e 0.6 0.8 1 .o

Fig. 3. Asymptotic fused decision rates versus individual sensor efficiency.

- U

Q 7

- X

E - n y

1 .o

0.8

0.6

0.4

0.2

n n _ _ 0.0 0 2 0 4 0 6 0 8 1 0

e

Fig. 4. Initial and final fused decision rates versus individual sensor efficiency levels.

in Fig. 4 to bring out the effective benefits of the recursive process, Fig. 5(a) shows the progressive improvement in performance of with increasing values of k . It can be seen from this illustration that as early as the end of three recursions ( k = 4) the performance reaches close to asymptotic levels. Given the fact that the summed correct decision rate increases as the recursive process continues, it would be interesting to determine if this summed rate ever exceeds the initial individual sensor correct decision rate e. This corresponds to a 45 degree line through the origin in Figs. 3 and 5(a), i.e., this event can occur only for e > 0.5 ( l /m) . (This is based on the assumption of zero nondecision rate for the individual sensor. Otherwise, the results are correspondingly different). Fig. 5(b) shows an alternative portrayal of the dependence of the fused decision rate Pk on k at different values of e. These would be of value in determining an optimal number of recursions for the process. Using (16) and (41) the condition when PI, exceeds e (although the general case itself can be analyzed, the special case of equally efficient sensors is considered here for simplicity of expressions derived by the analysis) can be written as

i.e.,

1146 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 21, NO. 5, SEPTEMBER‘OCTOBER 1991

1 0

- k = l - k=2 k=3

0 0 U 0 0 2 0 4 0 6 0 8 1 0

e

1 .o - e=0.2 - e=0.3 - e = 0.4 - e = 0 . 5

I e = 0 . 6 - e = 0 . 7

P e = 0 . 8

0.8

0.6

nr

0 4

0 2

0.0

k

Fig. 5. (a) Fused correct decision rate versus sensor efficiency at different stages of the recursion process. (b) Fused correct decision rate versus recursion number at different sensor efficiency rates.

Equation (51) can be rewritten using (43) (since TI < 1) as

IC 2 ln[(l - e)(me - l)/e(m - I)] / ln[(l - e)(m + me - 2)/(m - l)] (52)

Thus, (52) (with e > l /m) gives the minimum number of recursions needed for the fused decision to be better than the decision of the individual sensor. Deriving the limiting values for Pk and Qk through substitution of (41) and (42) in (20) and (21)

Pk 1 max = (m - I)e2/(me2 - 2e + I) (53)

(54) Q~ I max = (1 - e)’/(me2 - 2e + 1)

Fig. 6 shows a plot of this asymptotic correct fused decision rate as a function of the individual sensor efficiency e for different values of the number of decision choices m (target classes). As m increases, the system performance improves because the likelihood of fused incorrect decision rate drops off. For (53) to reach a value greater than e, it can be shown that

(1 - e)(me - 1) > 0 (55)

i.e.,

l / m < e < 1. (56)

This is also the range in which (52) leads to a positive value for k . These lower and upper bounds are consistent with the physical facts of the multisensor environment under consideration. The individual sensor correct decision rate has to be better than the success rate ( l /m) obtainable under a purely random decision process with m decision choices. On the other hand, perfect sensors (e = 1) would obviate the need for the recursive process. The min value of e at which the correct fused decision rate Pk would exceed the incorrect fused decision rate Qk can be found (by comparing (16) and (17)) to be

Substituting (41) and (42) in (57) and solving for e in terms of m gives elmin as

Fig. 7 shows a plot of this minimum value of e (for which the correct fused decision rate would equal or exceed the incorrect fused decision rate) versus m. This once again is valid under the assumption of zero nondecision rate for the individual sensor. It also provides for comparison purposes the trend of e = l /m, the random-choice correct decision rate, at which the nondecision rate attains its maximum.

At m = 2, the two values of e are equal. However, as m increases, the minimum value of e (for equal correct and incorrect fused decision rates) decreases but at a much slower pace than the correct decision rate obtained under random assignment.

Substitution of this minimum value of e in (41)-(43) gives

p1 = 41 = l /[m + 2(m - 1p2] (59) TI = [(m - 2) + 2(m - l)”’]l/[m + 2(m - 1)’/2]. (60)

For this value of e, as m increases both pl and 41 decrease and hence TI increases since from (1) their sum is always unity. Substitution of this minimum value of e (for which pk is equal to Qk for all values of k ) from (58) in (53) and (54) leads to the limiting values of both pk and &k as 1/2 which agrees perfectly with results derivable directly by setting (20) equal to (21) and using (22).

General Case A4: Relaxing the assumption of equally efficient sensors, let

where a( 5 1) is the ratio of the efficiency of the less perfect sensor to that of the better one. The case a = 1 represents the equally efficient sensors scenario discussed earlier (Subcase A3). Expressions (24) through (26) can be rewritten using


- m = 3 - m = 4 - m = 5 - m = 6

I m = 7

m = E

0 0 0 2 0 4 0 6 0 8 1 0

Fig. 6. Asymptotic fused correct decision rate versus sensor efficiency for different number of decision choices.

2 4 6 E 1 0 m

Fig. 7. Minimum required sensor efficiency versus number of decision choices.

(61) and (62) as

rate, we can write using (63) and (64) in (20) and (21), as

amin 2 (1 - e ) / e [ l + ( m - 2)eI. (67)

Since a 5 1 by definition, we can derive from (67) the condition

Equations (67) and (68) together represent the conditions for asymptotic fused recognition rate to be greater than the corresponding misrecognition rate. It is to be noted here that (68) is the same as the one obtained earlier for the special case of equally efficient sensors. This effectively supports the intuitive reasoning that the system performs optimally when both the sensors are equally efficient, i.e., a = 1. For the special case of m = 2 , i.e., for example, the detection problem, this condition reduces to

(69) amin 2 (1 - e)/.; with emin 2 0.5.

p l = ae2 5 1

TI = [ (m - 2 ) + (1 + a)e - mae2] / (m - 1) < 1

(63) Differentiation of (65) with respect to e gives

dr l /de = ((1 + a ) - 2mae]/(m - I). (70) q1 = (1 - e ) ( l - ae)/(m - 1) 5 1 (64) (65)

Setting (70) equal to zero gives the value of e, at which r1

It can be shown by substitution of (63) and (64) in (23) that for m 2 3 the nondecision rate T I will be higher than the incorrect decision rate 41 irrespective of the value of e. and a.The asymptotic performance of this sensor suite (using (63) and (64) in (20)) is

Pk1 max = (m - l)ae2/[1 - (1 + a)e + mae2] (66)

Fig. 8(a) shows a plot of Pk 1 max versus a for different values of e for the case m = 2 . Higher the value of e faster is the improvement in the performance with increase in a. Fig. 8(b) shows the complementary plot of Pk I max versus e for different

attains its maximum, as

e = (1 + a) /2ma; l/(2m - 1) 5 a 5 1 (71) = 1; 0 5 a 5 1/(2m - 1). (72)

Equation (71) reduces to (45) for a = 1. Substitution of (71) in (65) gives

r1 I max = [4am2 - 8am + (1 + a)’]/4a(m - 1)m. (73)

As is to be expected (73) reduces to (46) for a = 1, the maximum admissible value of a. At the transition value of a, (a = 1/(2m - l ) ) , (73) reduces to

T I 1 max = 2(m - 1)/(2m - 1). (74) values of a. From this it can be observed that low values of a has a significant impact on the system performance even for high values of e. Thus it is necessary that the minimum performance level among the sensors be as high as feasible for the overall system performance to be acceptably efficient. Once again, for the asymptotic correct decision rate to be better than (or at least equal to) the corresponding incorrect decision

Equation (71) is plotted in Fig. 9 to show the effect of the sensor efficiency ratio on the value of e at which the nondecision rate peaks out for different number of decision choices. Higher the ratio a, lower is the level of e at which T I

peak is reached. As the number of decision choices increases,

1148 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 21, NO. 5, SEPTEMBERIOaOBER 1991

a! 0.5

0.3

x

2 I

ax

1 0

0 8

0 6

0 4

0 2

0 0 U 0 0 2 0 4 0 6 0 8 1 0

1 0

0 8

0 6

2 nr 0 4

0 2

0 0 0 2 0 4 0 6 0 8 1 0 0 0

Z e = 0.1 - e = 0 . 2 - e = 0 . 3 - e = 0 . 4 - e = 0 . 5 - e = 0 . 6 - e = 0.7

I e = 0 . 8 - e = 0 . 9

I a =0.1

-t a = 0 . 2 - a = 0 . 3 - a = 0 . 4 - a = 0 . 5 - a = 0 . 6 - a = 0 7 - a=0.8 - a=0.9

-t. a = 1 . 0

e

(b)

Fig. 8. (a) Asymptotic fused correct decision rate versus sensor performance ratio at different sensor efficiencies. (b) Asymptotic fused correct decisionrate versus sensor efficiency at different sensor performance ratios.

0.11 " " " " . ' 0.0 0.2 0 4 0.6 0.8 1 0

a

Fig. 9. Sensor performance versus efficiency ratio for different number of decision choices.

the level of e (at which r1 peaks) reduces further and further. The transition value of a (i.e., at e = 1) in this context decreases inversely as m increases.

B. Three Sensors Case ( t = 3)

Extending the analysis to the case of three sensors, we can conceive of two alternative scenarios of agreement or consistency of decisions across the sensor suite. The first, as before, is the consensus strategy where the decisions reached

by all the three sensor-processor pairs are identical to one another. A second scenario is one where there is at least a majority agreement. In this case of three sensors, this works out to at least two of the three sensor-processor pairs having to agree on their decisions.

Subcase B l ) Consensus strategy of fusion: Under the consensus scenario, the initial joint correct, incorrect and nondecision rates are similar to the two sensor case and are given by

3

P I = n e j I 1

q1 = J-J (1 - ej)/(m - 1)' I 1

(75)

(76)

j=1 3

j=1

As before, these fused correct and incorrect decision rates have upper bounds that are the corresponding minimums of the individual correct and incorrect decision rates. Once again, we shall look at some special cases portraying scenarios of possible interest.


L Y

U m W

r

a-

Fig. 10.

0.0 0.2 0 4 0.6 0 8 1 0 e

e

Fused decision rates versus sensor efficiency for three sensors (consensus strategy) case.

Subcase Bla-ei = 1 ( 1 5 i 5 3 ) ) As before, consider the scenario with at least one (almost) perfect sensor in the sensor suite. Then, (79-07) reduce to

where 1 5 ( j , k ) 5 3; j = k . Once again, use of these values of pl and 41 in (20) and (21) brings home the fact that the multisensor system asymptotically achieves the correct decision even if only one sensor remains perfect during the recursive process.

Subcase Blb-ei = 0 (1 5 i 5 3 ) ) Once again, considering the opposite case of one always totally faulty sensor, (75)-(77) reduce to

Pl = 0 (81) (82) q1 = (1 - e ; ) ( l - e k ) / ( m - 112

TI = [m(m - 2 ) + (e; + ek) - e jek] / (m - l)', (83)

15 ( j , k ) 5 3 ; j = k. As before this implies that asymptotically the fused mis-

recognition rate reaches unity leading to failure of the system. Even if one (or both) of the other two sensors is (are) perfect and the misrecognition rate given by (82) goes to zero, the recognition rate remains zero but the nondecision rate goes to 1 . Thus, under the consensus strategy, the three-sensor suite also needs to have all of its sensors periodically tested against catastrophic failures. We shall now examine the equally imperfect sensors scenario.

Subcase Blc-ei = e (V i : i = 1,3) ) For this case, (75) through (77) reduce to

pl = e3 (84) q1 = ( I - e13/(m - 112 (85) T I = (1 - e ) [ m ( m - 2 ) ( 1 + e + e 2 ) + 3 e ] / ( m - 1)2. (86)

Using (23), for m > 2 , r1 2 41. Differentiation of (86) with respect to e gives

(87) drl/de = 3[m(2 - m)e2 - 2e + l] /(m - 1)2.

Fig. 11. Asymptotic fused decision rates versus sensor efficiency for three sensor (consensus) case.

Setting (87) to zero and solving the quadratic in e in terms of m, we get the two values of e as either l / m or 1 / (2 - m). The second value is not feasible as m 2 2 and hence the nondecision rate again attains the maximum at e = l / m the same as was obtained for the case of two sensors. Substituting e = l / m in (86),

r1 I max = (m2 - 1)/m2 1 [(m + l)/m][(m - I)/"]. (88)

Comparison of (88) and (46) shows that the maximum nondecision rate increased with an increase of the number of sensors, which is to be expected as there is more scope for disagreement with more number of sensors. For m = 2 under the subcase B1 c, the expressions (84) through (86) reduce to

p l = e3 (89) ql = (1 - el3 (90) r1 = 3e(l - e ) . (91)

Fig. 10 shows the plot of pl, q1 and r1 versus e . Once again, as e increases, pl increases, q1 decreases and TI first increases and then decreases. As is to be expected from the above analysis r1 peaks out at e = l / m = 0.5 with a peak value of 0.75 for the case m = 2 . Comparing Fig. 10 with Fig. 2, we see that the initial fused decision rates (both correct and incorrect) are lower under the three sensors case than under the two sensors case because of the higher nonrecognition rate in the former case.

However, the asymptotic performance, as shown in Fig. 11, is better under the three sensors case for e > 0.5 (for m = 2 ) as can be observed by comparing this with Fig. 3. This is because there is more of the nonrecognition cases from which a correct decision can be retrieved through the recursive process under the three sensors case. Expression for Pk, the summed correct decision rate after k recursions, can again be derived and evaluated for exceeding the individual rate e as

= e 3 ( 1 - r f ) / ( ~ - T I ) 2 e

r t 5 (e' - 1 + r l ) / e 2 .

(92)

I.e.,

(93)

1150 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. 21, NO. 5, SEPTEMBEWOCTOBER 1991

0.0 0 2 0 4 0 6 0 8 1 .o e

Fig. 12. Fused decision rate versus sensor efficiency at different stages of the recursive process for three sensors (consensus strategy) case.

As before noting that T I < 1, an expression for IC can be derived as

IC 2 ln{[m(m - 2)e2 + 2e - 11[1- e ] / ( m - I ) ~ ~ ~ } / ~ D [ T ~ I .

(94)

Once again, we can determine at what value of e would the fused correct decision rate p k equal or better the fused incorrect decision rate Q k . Substitution of (84) and (85) in (57) gives

e lmin > - 1/[1+ (m - 1)2/3]. (95)

Comparing (95) with (58) we can see that for corresponding values of m, the minimum required e values are going to be lower for the three sensor case relative to the two sensor case. This is a measure of the benefit of having the extra sensor in the decision process.

Plugging in this minimum value of e in (84) and (85), we get

(96) 213 3

p1 = 41 = 1/[1+ ( m - 1) ] . As in the two sensors case, p1 and q1 decrease with increase in m, but at a faster rate than before.

Fig. 12 shows the progress in the fused correct decision rate with increase in e at different stages of the recursion process. Comparison of Fig. 12 with Fig. 5(a) shows that while the fused correct decision rate starts off lower under the three sensors case as compared to two sensors case, it reaches higher values eventually (even if with perhaps more recursions) for the same level of individual sensor efficiency.

Subcase B2) Majority decision strategy of fusion: Instead of consensus, majority decision may be used as the standard of consistency across the sensor suite. In this case, the analysis detailed above has to be modified as follows.

3 2 3

j = k + l k=l j=1

3 2

e3ek + 2 fi e3] / ( m - (99) j=k+l k=l 3=1

3 3 2

r l = ( m - 2 ) ( m - 3 ) + 2 C e J - ( m + 1 ) c e , e k [ ,=1 j = k + l k = l

+ 2m fi eJl / ( m - 112. ( 100) ,=1

Comparing the correct fused decision rates under the consensus and majority decision strategies (expressions (75) and the expanded form of (97)) we can see that the correct decision rate under the majority strategy is always greater than under the consensus, because of the fact that lack of agreement and hence nondecision rate is likely to be greater under the consensus strategy. Comparing (76) and the expanded version of (98) we can see that, this is true for the misrecognition rate also but to a lesser extent. The extent, by which the majority decision yields a higher misrecognition rate over the consensus decision, depends on the number of decision choices m. We now consider some special cases of interest.

Subcase B2a-e, = ek = 1 (1 5 ( j , I C ) 5 3) ) Unlike under the consensus strategy, here we consider the scenario wherein at least two of the three sensors are viewed as (nearly) perfect. The previous scenario of only one perfect sensor does not necessarily lead to a zero misrecognition rate under the majority scenario as two imperfect sensors can when in agreement, even if i t be wrongly so, can overrule the perfect sensor decision and lead to a wrong fused decision. Under the scenario of two perfect sensors, (97) reduces to

p1 = 1 . (101)

This implies that with two perfect parallel sensors and a majority decision, the fused decision is correct at the very first stage and no recursion is required irrespective of the efficiency of the third sensor.

Subcase B2b-e, = ek = 0 (1 5 ( j , k ) 5 3) ) Considering the case of two of the three sensors operating in essentially failure mode (i.e., offering wrong decisions), (97)-(99) leads to

Pl = 0 (102) 41 = [ (3m - 5 ) + 2 ( 2 - m ) e ] / ( m - 1)’ (103) T I = (m - 2 ) [ ( m - 3) + 2 e ] / ( m - 1)’. (104)

This effectively means that once again the decision fusion process ends in failure. If the remaining sensor is perfect, i.e., e = 1 in (102) and (103), then interestingly, because of the complexities of the majority decision strategy (such as an agreement between the decisions of the faulty sensors overruling the decision of the perfect sensor), the expression for 41, (98), does not reduce to zero. Instead, (102) and (103) reduce to

41 = l / (m - 1) T1 = (m - 2)/(m - 1).

(105) ( 106)

However, this does not prevent the fusion process from asymptotic total failure as p l is still zero.

DASARAWY: DECISION FUSION STRATEGIES

Subcase B2c-m = 2 ) For this case, (97) through (99) reduce to

3 2 3

j=k+l k=l j=l

r1 = 0. (109)

Here, we can see that r1 becomes zero for m = 2 , irrespective of the values of e j . This is becauGe at least two sensors will always be in agreement when there are only two choices facing each of the three sensors and under majority rule this is enough to avoid the possibility of a nondecision. Thus the asymptotic behavior expressed by (20) and (21) is attained explicitly at the first instance ( k = 1) itself as the possibility of recursions does not even arise under these circumstances.

Subcase B2d-e; = e (V i : i = 1,3)) For this case, (97) through (99) reduce to

41 = [(3m - 5 ) + 6(2 - m)e + 3(m - 3)e2 + 2 e 3 ] / ( m - 1 ) 2

(111)

r1 = (m-2)[(m-3)+6e-3(m+l)e2+2me3]/(m- 1)'.

(112)

Differentiation of (111) with respect to e gives

drl /de = 6 ( m - 2)[me2 - (m + 1)" + l ] / ( m - 1 ) 2 .

(113)

Setting (112) to zero to find the extremum point for r1 gives two values for e as 1 and l / m . Substituting these values in the second derivative shows that e = 1 represents the minimum value and e = l /m leads to the maximum value, which agrees well with physical considerations. The maximum and minimum values are therefore:

The summed correct decision rate Pk can be derived and evaluated for the threshold at which it exceeds the individual rate e as

Pk = e2(3 - 2 e ) ( 1 - r;) /( l- T I ) > e (116)

from which an expression for IC can be derived in terms of e and m similar to (94). For m = 2 under the subcase B2d, the expressions (109) through (111) reduce to

p l = e2(3 - 2e) (117) 41 = I - 3 2 + 2e3 (118)

T1 = 0. (119)

Cl- U 5 Q-

0.6 -

0.4 -

0.0 0.2 0.4 0.6 0.8 e

1151

I

0

Fig. 13. Fused decision rates versus sensor efficiency for three sensors (majority strategy) case.

These can be derived as a special case of subcase B2c also with all the sensors being equally imperfect. Fig. 13 shows the plot of p l and 41 versus e . Comparison of Figs. 10 and 13 shows that as e increases (from zero), the fused correct decision rate p l increases (from zero) more rapidly under the majority strategy than under the consensus strategy with the latter catching up to its value (of 1) only at the very end (e = 1). On the other hand, as e increases (from zero), the fused misrecognition rate 41 decreases (from 1) more slowly under the majority strategy than under the consensus strategy catching down to its value (of zero) once again at the very end (e = 1). These differences represent the extent of nondecision rate r1 under the consensus strategy that increases from zero in the beginning, peaks at e = 0.5, and reduces back to zero at the end. The value of under the majority strategy remains zero for all values of e when m = 2. Although these plots are valid only for m = 2 , the trends are similar for other values of m also.

C. General Case - I Sensors

Generalizing the analysis to t sensors, the expressions for the consensus strategy can be written as

t

t

q1 = n (1 - e j > / ( m - ~ ) ( ~ - l ) 5 1 (121) j=1

and TI is given by (19). As before, we consider some special subcases here also.

Subcase C l -e, = 1 (1 5 i 5 t ) ) The scenario of at least one perfect sensor under consensus leads to

which, when substituted into (20), leads to the conclusion that asymptotically the multisensor system performance reaches perfection as long as one sensor-processor pair in the system keeps deriving the correct decision (this is limited to the consensus strategy only). This demonstrates the benefits of the multisensor fusion process.


1 .o

0.8

I 0.6 X

E v

0.4

0.2

0.0

- e=02 - e=03

-----b e=04 - e=o5

e=06

e=07

e=08

1 2 5 6 3 t

Fig. 14. Asymptotic fused decision efficiency versus number of sensor at different sensors efficiencies.

Subcase C2-e, = 0 (1 5 i 5 t ) ) The scenario of one totally faulty (wrong decision always!) sensor leads once again to the fusion process failure under the consensus strategy irrespective of the other sensors. If one of the other remaining sensors is perfect, this makes a consensus impossible and hence the nondecision rate reaches unity while both recognition and misrecognition rate become zero. This is strictly a result of the consensus strategy and hence is one of its main detractions requiring constant monitoring of sensors. A majority (or s out of t sensors) strategy would be beneficial under these circumstances.

Subcase C3-ei = e (V i : i = 1 , . . . , t ) ) For this case of equally imperfect sensors, the expressions (1 1 l) , (1 12), and (19) reduce to

p l = e t (123) q1 = (1 - e ) ' / ( m - I)(~- ') (124) r1 = 1 - et - ( 1 - e)'/(m - l)(t-l). (125)

Differentiation of 7-1 in (124) with respect to e gives

drl/de = -te(t-l) + t(1 - e ) ( t - l ) / ( m - l)(t-'). (126)

Setting (125) to zero gives the same value for extremum point as before: e = l /m. Substitution of this value of e in (124), after some algebraic simplification, gives

It is easy to check that for t = 2 and t = 3 , (126) reduces to (46) and (88) respectively thus confirming the analysis. As m, the number of decision choices, increases, the maximum value of the nonrecognition rate given by (126) approaches unity asymptotically but this maximum value occurs at lower and lower values of e approaching a value zero asymptotically. The summed correct decision rate can be derived as before and compared to the individual rate e as

~k = et( l - r f ) / ( 1 - T I ) 2 e. (128)

As earlier, noting that r1 < 1, the threshold for k at which the summed rate exceeds the individual rate, can be expressed as

2 In{(l- e)[ l - [(I - e ) / ( m - ~ ) e ] ( ~ - ' ) ] ) / l n [ r l ] . (129)

As is to be expected, expression (128) reduces to (52) and (94) for t = 2 and t = 3 respectively. Again, the min value of e at which the correct fused decision rate exceeds the incorrect fused decision rate is determined (substituting (122) and (123) in (57)) as

(130)

As can easily be verified, (129) reduces to (58) and (95) for t = 2 and t = 3 respectively. As t increases, the minimum value of e tends closer and closer to l /m, the random-choice based correct decision rate. The gap between the two curves shown in Fig. 7 for t = 2 becomes smaller with increasing t values. This means that as more sensors are added to the decision process, less accurate they need to be for the correct fused decision rate to be better than incorrect fused decision rate. Substitution of this minimum value of e in (122) and (123) leads to

As before increasing values of m decrease both p l and 41. Also increasing values of t decrease both p l and 41 at the minimum value of e. Instead of determining the minimum value of e at which the correct fused decision rate exceeds the incorrect fused decision rate, one could determine the minimum number of sensors required for this scenario to be true. This can be accomplished by again substituting (122) and (123) in (57) and solving for t (instead of e) as

tlmin 2 ln[m - 1]/1n[(m - l )e / ( l - e ) ] ;

m > 2 ; O < e < l . (132)

Thus, for a given value of m, higher the sensor effectiveness e lesser is the number of sensors needed for the fused correct decision rate to better the incorrect decision rate. This is consistent with intuitive reasoning. The variations in asymptotic fused correct decision rate as t increases is shown in Fig. 14 for different values of e. This clearly illustrates the benefit of having multiple sensors whenever e > 0.5. As can be seen therein, higher the value of e lower is the number of sensors required to achieve performance close to perfect decision rates.


1 0 information or could be adaptive, based on a tracking of performance in the operational phase. An additional possibility, which was in fact suggested by one of the reviewers also, is that of making the decision at the kth stage, based not only on votes derived at the kth stage but also on all of the prior votes of these sensors. While this cumulative strategy is a valid alternative worthy of independent exploration, it changes to

0.4 some extent, the initial assumption made in this study, namely, that if a fused decision of desired consistency (consensus or

0.2 - t = 3 majority) was not possible, then this set of decisions are to be deemed suspect and rejected from further consideration. While detailed analysis of these alternatives is deemed outside the scope of this current study, we can make some general

0 8

0.6

1.0 0.0 0.0 0.2 0 . 4 0.6 0.8

e observations regarding these cumulative strategies. Under such

Fig. 15. Asymptotic fused decision efficiency versus sensor efficiency with different number of sensors.

Asymptotic values of the fused decision rates can be for- mally expressed for this case of t equally efficient sensors by substituting (122) and (123) in (20) and (21) as

Pk I max = e f ( e , m ) / s ( e , m )

Q k I max (1 - e I t / d e , m) (134)

g(e, m) = [e f ( e . m) + (1 - elt] f ( e , m ) = [(m - ~ ) e ] ( ~ - l )

(133)

(135) (136)

The scenario wherein at least s of the t sensors agree (s 2 t /2 ) can also be worked out in the same way the majority decision scenario was developed for the three sensors case. However, the resulting expressions tend to be rather complex and does not necessarily lead to any better understanding of the concepts and hence are omitted here in the interests of brevity of presentation.

In closing, the benefits of this multisensor based recursive fusion process can be best summarized by looking at the classical distributed detection problem (i.e., the case m = 2 ) and plotting (as shown in Fig. 15) the asymptotic performance of the fusion processor given by (132) as a function of the individual sensor detect rate e with different number of sensors ( t ) in the sensor suite. As the number of sensors increases, the fused performance improves and reaches close to perfect performance at lesser and lesser levels of e as long as this basic individual sensor efficiency e is greater than half. The potential for application of these concepts and strategies to more complex scenarios as well as other alternative avenues of analysis are now discussed in the next section.

a cumulative decision strategy, a consensus approach to fusion becomes impractical, as even a single instance of dissent at any recursion will rule out the possibility of ever reaching a fused consensus decision. On the other hand, in general, the majority decision approach under cumulative strategy will always tend to be more conservative than a corresponding decision under the noncumulative strategy followed here. Let ni be the number of votes received in the ith recursion by any specific decision say to class j , and N be the sum of all these n, votes received over R recursions. It is easy to verify that

then N = En, 5 Rt/2. R

if n, 5 t /2 V a = 1,. . .. R, a=1

That is, any time a majority is achievable in a cumulative sense, a majority status will have to be achieved at least in one of the recursions, while the reverse is not necessarily true. In some simple cases, for example, with say two classes and two sensors, the two strategies become essentially equivalent, as majority in a cumulative sense is achieved at the same time as a majority in a noncumulative sense is achieved. However, if instead of consensus or majority based fusion, one visualizes a maximum (even if less than majority) vote based fusion rule, the cumulative strategy may open up a whole range of potential decision fusion opportunities, not available under a noncumulative strategy. This becomes particularly obvious whenever the number of classes m, is larger than the number of sensors, t. These scenarios would require more detailed analysis, which is well beyond the scope of the current effort for reasons stated at the outset. Also, in this study, the costs of continuation (i.e., costs of postponing the fused decision) were not taken into account. Additional studies, wherein such costs are taken into consideration, are on the anvil.

More complex multisensor scenarios, such as serial and combined serial/parallel sensor suites in lieu of the parallel sensor suite studied here, are also under investigation in

111. OTHER ONGOING EFFORTS AND AVENUES OF DEVELOPMENT

The recursive model and the associated strategies presented here are being applied to a wider class of problems than the ones analyzed here. The scenario of each sensor having an option of nondecision based on a reliability test is an example of this, but has not been presented here due to space considerations. Also, in place of the simple count voting scheme, weighted voting schemes can be applied within the same context. Such weighting can be on the basis of a priori

the context of specific DOD applications. A decision fusion simulator (261 has been developed for the analysis of these complex scenarios under a variety of conditions. Instead of combining the decisions at each stage on a collective voting basis, any one of the optimal (Bayesian, Neyman-Pearson and the like) fusion approaches discussed in the earlier works [1]-[24] may be substituted within the same recursive structure developed here. More or less equivalent analyses with


similar performance benefits result from such substitutions. These details are however well beyond the scope of this paper whose main objective has been to present the central concept of a recursive structure and its benefits. Extensions to cases wherein the individual outputs are not necessarily integer valued decisions but continuous valued scalars or vectors (which facilitate data and feature fusion) have also been visualized. Also, instead of using classical probability theory, the recently popular evidential reasoning approach can be employed to develop a more realistic analysis of this recursive fusion strategy that would permit us to express our uncertainties about the sensor-processor environments. Irrespective of the potential for such applications to specific problem environments, this study provides an insight into some interesting aspects of sensor fusion, such as the advantages of having at least one perfect (or more realistically speaking, nearly perfect) sensor and at the other end of the spectrum, the catastrophic risk (under the consensus decision strategy) of even a single faulty sensor within the system. While these additional alternative avenues continue to be investigated, the present analysis is reported here as a forerunner to bring out the innovation and usefulness of the recursive model and the associated fusion strategies.

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pp. 536-544, July 1989.

Dr. Belur V. Dasarathy received the doctorate degree in engineering at the Indian Institute of Science, Bangalore, India, where he later served as a founding member of the faculty of the School of Automation.

He is a Senior Principal Engineer, on staff to the Executive Vice-president at Dynetics, Inc., Huntsville, AL, where he is currently engaged in research and development in the areas of sensor fusion, pattern recognition, image analysis, machine intelligence and related topics in the design and

development of automated optimal decision systems as applied to a broad spectrum of multispectral and multisensor environments arising in various strategic and tactical DOD applications. His two decades of experience includes NASA, and civilian applications of these technologies as well. Previously, as senior technical manager at Intergraph Corporation, Huntsville, he was responsible for the design and development of their first commercial symbolicharacter recognition (SCR) and image processing (IDEALS) systems as well as a target recognition system (TARECS) for MICOM. His other associations include Southern Methodist University, Dallas, TX, Computer Sciences Corporation, Huntsville, AL, where he worked in two stints on remote sensing applications.

Dr. Dasarathy has written more than 100 technical papers in his varied fields of interest over the years, which have also included nonlinear, adaptive, optimal and learning systems analysis. He is the author of the recently published IEEE Computer Press tutorial entitled, “Nearest Neighbor (NN) Norms: NN pattern classification techniques.” He has been a reviewer for several international journals including several IEEE publications in these areas. He is the editor of the IEEE Huntsville Section newsletter, LIVEWIRE. Earlier, he was the chairman of the Huntsville IEEE Computer Fair and the Rocket City seminars. He is listed in Who’s Who in Computer Graphics (Marquis, 1984), Personalities of the South (American Biographical Institute, 1986), Who’s Who in Technology Today, (Dick Publishing, 5th Edition), and Who’s Who in the South and Southwest, (22nd Edition, 1991).

decision fusion strategies in multisensor environments

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