a visual search model: the waiting time distribution of the number of fixations until detection

A Visual Search Model: The Waiting Time Distribution of the Number of Fixations until Detection

Tony Lin Industrial and Manufacturing Systems Engineering, GMI Engineering and

Management Institute, Flint, Michigan 48504

Visual search is an important aspect of human tasks in industrial and military applications. Physically, a visual search process consists of a sequence of eye fixations. It has been hypothesized, based on search patterns found in early studies, that it is possible for fixations to follow both random and systematic patterns. Some early research works have been done in visual search. Analysis of human visual search involves examining experimental data and fitting the search time distribution. Some investigations have found that search times are described well by exponential distribution. This article discusses a visual search performance model based on derived search time distributions. The investigation of such a model is helpful in computing the probability of detecting a target, given a specified duration of search.

1. INTRODUCTION

Let us consider the following situation. An observer is to find k critical targets among M targets which are randomly placed in the visual search field. Assume that the observer will complete the search in M fixations, with each fixation covering a single target and requiring a fixed amount of time. In the process of searching, the observer will fixate one target after another without returning to a previously fixated target, a process reasonably described in terms of sampling without replacement. Of interest is the waiting time, in terms of either the number of fixations or the actual search time, required to locate the first critical target.

2. MODEL FOR THE CASE OF TWO CRITICAL TARGETS

Let p,( i = I , 2) be the probability that the observer locates the ith critical target, given that the target is fixated. All M targets are in specified locations. Hence, in a sequence of M fixations, only two among M fixations will cover the first and second critical targets. We may think of the two critical targets as being randomly assigned to the M fixations. Therefore, it is appropriate to consider a set Z = {(x, y): x = 1, . . . , M, y = 1, . . . , M ; x # y} of M(M - 1) equally likely ordered pairs, with (x , y) indicating that critical targets 1 and 2 are fixated on the xth and the yth fixations, respectively. For example, if M = 10, then (5 ,

Naval Research Logistics, Vol. 38, pp. 157-170 (1991) Copyright 0 1991 by John Wiley & Sons, Inc. CCC 0028-1441/91/020157-14$04.00

158 Naval Research Logistics, Vol. 38 (1991)

3) indicates that critical target 1 is fixated on the fifth fixation and the critical target 2 is fixated on the third fixation in a complete scan of ten fixations. Consider a random vector ( X , Y ) such that the density function of ( X , Y ) is given by

(1) 1

f(X,Y) = M ( M - 1)’ v (X,Y) E z,

where

X = number of fixations needed to fixate critical target 1

and

Y = number of fixations needed to fixate critical target 2;

then Min( X , Y ) = the number of fixations needed to fixate a critical target for the first time, so that the event {Min(X, Y ) = X = i} is the event that the first critical target fixated is critical target 1, and is in fact fixated on the ith fixation. Also, the event {Min(X, Y ) = Y = i} is the event that the first critical target fixated is critical target 2, and is in fact fixated on the ith fixation. Therefore,

P(critica1 target 1 is fixated first, and then on the ith fixation)

= P(Min(X, Y ) = X = i)

= P({(i , i + m) E Z: m = 1, 2, . . . , M - i})

( M - i) M ( M - 1)

- -

and

P(critica1 target 2 is fixated first, and then on the ith fixation)

= P(Min(X, Y ) = Y = i)

= P({(i + m, i) E Z : m = 1, 2, . . . , M - i})

- ( M - i) - M ( M - 1)’ (3)

Similarly, Max( X , Y) = the number of fixations needed to fixate both critical target 1 and critical target 2, so that the event {Max(X, Y ) = X = i} is the event that the second critical target fixated is critical target 1, and is in fact fixated on the ith fixation. Also, the event {Max(X, Y ) = Y = i} is the event

Lin: A Visual Search Model 159

that the second critical target fixated is critical target 2, and is in fact fixated on the ith fixation. Therefore,

P(Max(X, Y ) = X = i)

= P({(i, m) E Z : m = 1, 2 , . . . , i - l})

(i - 1) - - M ( M - 1)

and

P(Max(X, Y ) = Y = i)

= P({(m, i) E Z: m = 1, 2, . . . , i - l})

(i - 1) - - M ( M - 1 ) .

(4)

Now it is possible to compute the following

P(critica1 target 1 is fixated first, and detected, and in fact on the ith fixation)

= p1 . P(Min(X, Y ) = X = i)

(M - i) - - M ( M - l ) P 1

and

P(critica1 target 2 is fixated first, and detected, and in fact on the ith fixation)

( M - i) - - M(M - 1 ) ” (7)

Similarly, it is possible to compute

P(detect critical target 1 on the ith fixation, having mixed critical target 2)

= p 1 * (1 - p2) . P(Max(X, Y ) = X = i)

(i - 1) . PI * (1 - P2) - -

M ( M - 1)


and

P(detect critical target 2 on the ith fixation, having mixed critical target 1)

p2(l - pz) . P(Max(X, Y ) = Y = i)

(i - 1) * (1 - PI) *P2. M(M - 1) (9)

be the number of fixations needed to locate a critical target for the first -

time. Then from (6)-(9),

1 M(M - 1) P(W = i) = . ((M - i)pl + (M - i ) ~ ?

+ (i - llPl(1 - Pz) + (i - 1)(1 - PJPZ)

1 1 2(i - 1) p1p2.

- _ M(M - 1) - MP1 + &P2 -

Similar arguments also give the evident relations

P(detect critical target 1 on the ith fixation)

1 M(M - 1)

1 M(M - 1)

1

- ((M - i>Pl + (i - 1)Pdl - PZ) + (i - llPlP2) - -

(M - 1)Pl - -

- _ - MP1,

and

P(detect critical target 2 on the ith fixation)

Returning to (lo), we have as well the relation

P(detect a critical target on or before the jth fixation)

i = 2 P(W = i)

i = 1

PIPZ, j = 1, 2, . . . , M (13) i(i - 1)


and (11) implies that

P(detect critical target 1 on or before the jth fixation)

- _ - M p I , j = 1 , 2 , . . . , M ;

also, (12) implies that

P(detect critical target 2 on or before the jth fixation)

Moreover, we have

P(comp1ete detection of both critical targets at precisely the ith fixation)

and

P(detecting both critical target 1 and target 2 on or before the jth fixation)

3. GENERALIZATION TO SEVERAL CRITICAL TARGETS

If there are k critical targets on the search field, let p,(i = 1, . . . , k ) be defined analogously to the definition of p, in the previous section. Now consider the following events:

(i)

(ii)

(iii) E',2. .k = n E.j, k 5 j ,

and

Ej = critical target i is fixated during the first j fixations; alternatively,

F{ = critical target i detected in the course of the first j fixations, critical target i is among the j targets fixated first,

k

{ = I


Targets are, of course, in given locations, and are fixated in random sequence; however, we may equivalently and advantageously think in terms of a given sequence cl) of location fixations, with targets randomly assigned to locations. Under this latter model, the event E: is the event that target i is assigned, under the random assignment of targets to locations, to a location among the j locations fixated first in the given sequence of locations GI). Consequently, the probability P(E:) of E: is the hypergeometric probability H ( M , j ; 1, 1) that a single ball randomly designated in a population of j green balls and M - j red balls, is in fact green; hence

i p ( E j ) = - M

Analogously, the probability P(E{,z, . k ) of , k is the hypergeometric probability H ( M , j ; k , k ) that k balls randomly designated in a population of j green balls and M - j red balls, are in fact all green; hence,

Furthermore, the probabilities of the events F a r e related to those of the events E through the multipliers p i , i.e.,

and

P(F: ,Z ..... k ) = (PIP2 ' . ' Pk) ' P ( E ( . 2 ..... k ) , (21)

where 0 < p i 5 1 (i= 1, . . . , k) .

then If W is the waiting time analogous to that defined in the previous section,

so that Boole's formula yields


Therefore,

Formula (23) pertains to the waiting time until first alert, Le., to the time required to first detet a critical target. For k = 2, formula (23) reduces to formula ( 1 3 ) of the previous section. According to Morawski et al. (see [9]), their Eq. (8) is only an approximation to (23).

We complete this section by extending (23) to cover a random number of critical targets. Suppose that it is known that a random number i , 1 5 f 5 R, of critical targets of type i might be present in the search field. Define

j['l = j ( j - 1 ) . . . . . ( j - r + I),

where I , j are integers with 0 5 r 5 j , and j [ O l = 1 . Further, let

and for j = M , we have

Now let

with

= ( 1 - Pi)'.

4;' = P ( i = r ) ,


It follows from (23) that

P(a critical target of type i is detected by the end of the jth fixation)

r

R

r = l

where p i is the probability of detection, given that a critical target of type i is fixated; hence,

B, (M) = B, = P(detect a critical target of any type by the end of a complete scan)

+ q , R [RIA - ( ; ) p ; + ($, + . . * + (-1) R + I p , R ] ,

According to Morawski et al. (see [9]), their Eq. (9) is only an approximation to (26).


It is now assumed that at the end of one complete scan, the observer will recommence scanning so that

P(detect at least one critical target of type i by the end of the jth fixation of the (n + 1)st scan)

= 1 - (1 - Bi)"1 - Bi(j))

For more than one type of critical target present, let 0 be the set of all types of critical targets. Then,

P(detect at least one critical target of type i by the end of the jth fixation of the ( n + 1)st scan)

= 1 - n (1 - B,)"(1 - Bi(j ) ) IEO

3.1. Examples

Let us consider the following examples.

EXAMPLE 1 (Morawski et al. [9]): An inspector faces only a single type of critical target on a search field of 50 targets, with the probability of detection, given a fixation, being 0.8. Each fixation takes 300 ms. Suppose that the fault can occur up to three-times on the field, with probabilities of occurrence ql l =

0.7, qI2 = 0.2, q13 = 0.1. Hence

B, = (0.7)(0.8) + (0.2)(0.8 + 0.8 - 0.64) + (0.1)

x (3 x 0.8 - 3 x 0.8' + 0.83) = 0.8512,

B,(j) = (0.7)(0.016j) + (0.2)(0.032/) - O.O002612j(j - 1))

+ (0.1)(0.048j - 3 x O.O002612j(j - 1)

+ 0.000004354(j)(j - l)( j - 2))


= (0.7 x 0.016; + 0.2 x 0.032; + 0.2 x 0.0002612; -t 0.1 X 0.048;

+ 0.1 x 3 x 0.0002612; + 0.1 x 2 x 0.000004354)j

- (0.2 x 0.0002612 + 0.3 x 0.0002612 + 0.3

x 0.000004354)j2 + 0.0000004354j3

= 0.0225314; - 0.0001319j7 + 0.0000004354j3.

Table 1. time of Example 1.

Cumulative distributions of search

t (seconds) P!,.,* 1.50 3.00 4.50 6.00 7.50 9.00

10.50 12.00 13.50 15.00 16.50 18.00 19.50 21.00 22.50 24.00 25.50 27.00 28.50 30.00 31.50 33.00 34.50 36.00 37.50 39.00 40.50 42.00 43.50 45.00 46.50 48.00 49.50 51.00 52.50 54.00 55.50 57.00 58.50 60.00

0.10941 0.21255 0.30974 0.40132 0.48762 0.56895 0.64564 0.71803 0.78643 0.85118 0.86748 0.88283 0.89729 0.91092 0.92376 0.93586 0.94727 0.95804 0.96822 0.97785 0.98028 0.98256 0.98472 0.98674 0.98866 0.99046 0.99215 0.99376 0.99527 0.99670 0.99707 0.9974 1 0.99773 0.99803 0.99831 0.99858 0.99883 0.99907 0.99930 0.9995 1

* n = [t/15]; j = [t/0.3] - [t/15] x 50, where [ ] is the greatest integer function.


.05--

Formula (28) make it possible to compute the cumulative distribution functions

pfl,, = 1 - (1 - BJ"(1 - B,( j ) )

= 1 - (0.1488)"(1 - B,(j)). (31)

Data from the cumulative search time distribution, Pn,, is shown in Table 1 and plotted in Figure 1.

EXAMPLE 2: If there is only a single type of critical target on a field of 10 targets and the critical target occurs up to three times with probabilities of occurrence being the same as in Example 1; then

B, = 0.8512

and

B,(j) = (0.7)(0.08j) + (0.2)(0.16j - 0.00711j(j - 1))

+ (0.1)(0.24j - 3(0.00711)j(j - 1)

+ 0.000711j(j - l ) ( j - 2))

= 0.11569721' - 0.0037683/' + 0.0000711~. (32)

Search Time (seconds)

Figure 1. Cumulative distribution of search time for Example 1.

D


Table 2. Cumulative distributions of search time of Example 2.

t (seconds) P, ,* 0.30 0.60 0.90 1.20 1.50 1.80 2.10 2.40 2.70 3.00 3.30 3.60 3.90 4.20 4.50 4.80 5.10 5.40 5.70 6.00 6.30 6.60 6.90 7.20 7.50 7.80 8.10 8.40 8.70 9.00 9.30 9.60 9.90

10.20 10.50 10.80 11.10 11.40 11.70 12.00

0.11200 0.21689 0.31510 0.40705 0.49317 0.57388 0.64962 0.7208 1 0.78787 0.85124 0.86787 0.88347 0.89809 0.91177 0.92458 0.93659 0.94786 0.95846 0.96844 0.97786 0.98034 0.98266 0.98484 0.98687 0.98878 0.99057 0.99224 0.99382 0.99530 0.99671 0.99707 0.99742 0.99774 0.99805 0.99833 0.99860 0.99885 0.99908 0.99930 0.99951

* n = [t/3]; j = [t/0.3] - [t/3] x 10, where [ ] is the greatest integer function.

Data from the cumulative search time distribution, Pni is shown in Table 2 and plotted in Figure 2.

4. CONCLUSIONS

In this article we developed a visual search performance model based on fitting the search time distribution. It permits a realistic representation of typical visual


.05--

C 0

0 a, a,

.- ..-

..- n L 0

’:: .85

Search Time (seconds)

Figure 2. Cumulative distribution of search time for Example 2.

search tasks. Our approach establishes the mathematical basis of a general method of quantifying visual search.

ACKNOWLEDGMENT

The author would like to thank the referee and the associate editor for a number of helpful comments which have significantly improved the quality of the article.

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[9] Morawski, T., Drury, C.G., and Karwan, M.H., “Predicting Search Performance for Multiple Targets,” Human Factors, 22, 707-718 (1980).

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a visual search model: the waiting time distribution of the number of fixations until detection

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