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Thermal image target acquisition probabilities in the presence of vibrations
Infrared Phys. Technol. Vol. 36, No. 3. pp. 691-702. 1995 Copyright ,~ 1995 Elsevier Science Ltd 1350-4495(94)00108-1 Printed in Great Britain. All rights reserved 1350-4495,'95 $9.50 + 0.00
Pergamon
THERMAL PROBABILITIES
IMAGE IN THE
TARGET ACQUISITION PRESENCE OF VIBRATIONSt
O. HADAR, S. R. ROTMAN and N. S. KOPEIKA Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel
(Received 2 August 1994) Abstract--The effects of low frequency mechanical vibrations on thermal imaging target acquisition are considered. A model is described that takes as input mechanical vibration data such as amplitude and frequency as well as the physical characteristics of the target. The output is the probability of detection as a function of time. Analysis indicates that even low amplitude vibration greatly affects the predicted target detection times and ranges and, hence, the utility of IR systems in realistic scenarios.
I.
INTRODUCTION
There are numerous factors which influence an observer's ability to acquire a target. One of the common problems in real time imaging systems is image motion and vibration. Image motion as a result of vibrations is often the limiting factor in image resolution for moving systems, such in reconnaissance, robotics, computer vision, etc. The reduction of image resolution due to image motion influences the ability of an observer to detect the target. This paper presents a modification to the standard military model for IR target acquisition of the Center for Night Vision and Electro-Optics (CNVEO)J ~ The modification is for dynamic imaging systems where image motion or vibrations are involved in the acquisition process. Motion may derive from vehicular motors, airplanes turbines, or even the refrigeration system cooling the IR receiver. The development is based on a new method for modeling the motion degradation as an optical transfer function (OTF) appropriate to any kind of image motionJ 2"3~This paper deals with low frequency vibrations where the OTF is a random process which depends upon the portion of the vibration cycle in which the exposure takes place. The focus here is on the modulation transfer function (MTF) because the reduction with image intensity is more critical than that of the phase for human vision. The reduction with resolution in the image plane as result of image motion or vibrations affects the ability of an observer or automatic target recognition (ATR) system to extract the target from its local background. The performance on an ATR system under acceleration is presented elsewhereJ~ For an IR noise-limited imaging system, the critical factor that determines the performance of the FLIR system is the minimum resolvable temperature (MRT). The influence of constant velocity on FLIR systems has been examined in Ref. (5) where the basis for this analysis is the relation between the MRT and the image motion MTF. This dependence of the MRT is investigated here for the case of low frequency vibrations. This latter situation is more complicated because there is no unique MTF that represents the degradation process. Therefore the analysis consists of the all possible MTFs existing during the searching time for the target. The assumption here is that, for an average observer, the search time for the target will include all possible MTFs of low frequency vibrations. This situation is further complicated by the fact that while image restoration tPart of this paper was presented at the SPIE Conference on Acquisition, Tracking and Pointing V, April 1994, Orlando, FL, U.S.A. 691
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techniques can be very effective in deblurring contrast-limited images degraded by image motion and vibration, 16~their utility in correcting noise-limited images is extremely limited at best. The procedure here is similar to the analysis for target detection in the presence of obscurants) 71 However instead of investigating the statistics of the extinction coefficient of the atmosphere fl~,m, the system effect of image motion and vibration on the extinction coefficient flsy,is considered. The first step in the analysis is modeling the behavior of the static probability of detection P:c Ir'sJ which is the critical parameter that determines the dynamic probability of acquiring the target. The dynamic probability of detection is based on the statistical behavior of the static probability P~ as a result of low frequency vibrations. Probability of detection as a function of time is presented for different target ranges and different exposure times. Although emphasis here is on target "detection", similar analyses can be performed for "recognition" and "identification". II. LOW F R E Q U E N C Y VIBRATIONS The effects of sinusoidal image motion, which often results from mechanical vibrations, can be divided into high and low vibration frequencies32'3) In the first case, the time exposure is longer than the vibrational period and the image blur is therefore the entire peak-to-peak translation of the image. The image motion is given by 2~
x (t) = D cos -~--; xo
(I)
the MTF for the high vibration frequency case is (2"3) (2)
M , ( f ) = Jo(2gfD )
The low vibration frequency situation involves exposure time t< shorter than the vibration period To. In this case, the blur radius d and the MTF are random processes that depend upon when the exposure time takes place as shown in Fig. 1. This type of blur is often more demanding than the high vibration frequency blur because in real-life situations d for low vibration frequencies is often much greater than 2D for high vibration frequencies39) The random process of the MTF is described elsewhere°) in detail. The main analysis in Ref. (3) was to calculate the probability of the MTF being higher than a given threshold contrast as a function of spatial frequency. This calculation was obtained for different relative time exposures tflTo and several threshold contrasts. D te-
w
8
l"'"
I
. . . . . .
\ To \ "7"
J"o,_.
I "to T
I ' I
~' / , /3/,To i
I To
. . . . .
Fig. 1. Blur radius d as a function of exposure time t, for low frequencyvibration.
Thermal image target acquisition probabilities in the presence of vibrations
693
The conclusions of that analysis were that the probability function is reciprocal to the threshold contrast and to the parameter t,/T°. This statistical result can be used for defining the probability of detecting a target in an image that was taken during a vibration period by a human visual system and accounting for the integration time of the eye. This analysis was obtained for a contrast-limited imaging system in the visible spectrum. ~4)In a noise-limited thermal imaging system, the parameter that can characterize FLIR performance is the MRT; this parameter is defined as the value of the minimum resolvable temperature difference between a pattern of four identical bars and the three spaces between them. The statistical analysis for IR images can be obtained in the same way as for visible images; however, instead of contrast, the crucial parameter is the difference temperature, AT. The relation between the imaging MTF and the MRT will be presented in the next section. III. T H E R M A L IMAGE P E R F O R M A N C E MODEL The method for calculating the probability of target acquisition from the knowledge of all the MTF possibilities of low frequency vibration is presented here. The goal of this analysis is to obtain the probability of detection as a function of time. Any FLIR performance is calculated as a function of the spatial frequency of the bar pattern. Assume, as in Ref. (5), that the MRT dependence on angular spatial frequency v (in cycles/millirad) is given by MRTo MRT = ~ . exp(fl,,.~v),
(3)
where MRTo is the limiting MRT as v approaches 0, flsysis the system coefficient and E is the ratio of the long to short dimensions of the target, v can be found from the target characteristics by ns0 s
v = - - . R,
(4)
where R is the range, s is the minor dimension of the target, and nso is the task difficulty function from the Johnson chart (1 for detection, 4 for recognition and 6.4 for identification). The relation between the angular spatial frequency v in cycles/mrad to spatial frequency f i n cycles/mm is given by
f = v/F,
(5)
where F is the focal length in mm. The dynamic search model appears in Ref. (8), where the final result for the probability of acquiring the target as a function of time is given by the following equation -t
f -,;.o t '~ -]
P(t,=P=El-exp(-~.~.)]=P=[l-exp~--~ff where t/~ 0.3 s is the time to take a single glimpse and 3,, = small 20 and P=, 2o/ti equals P~/3.4 and hence P(t)=P~
[
l-exp~
)j,
ti/2 o is the mean .
(6) acquisition time. For
(7)
We assume in the following analysis that ti is less than or equal to t,, the exposure time of the camera. Equation (7) is known as the standard military search model for infrared target detection developed in the Center for Night Vision and Electro-Optics (CNVEO). The term P~ in equation (7) represents the fraction of the normal observer ensemble that can successfully find the target in infinite time. P~ is the static probability of detection at infinite time when the position of the target is known. It is given by the target transfer function (TTF) based upon the Johnson chart which describes the number of line pairs (n~) across the target image critical dimension for 50%
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probability of detection, i.e. P~ = 0.50. The TTF curve allows probability calculations for other values of P~ as a function of the number of line pairs (n) across the target image critical dimension, and is given by "'7) (n/nso) E P': = 1 + (n/nso) e'
(8)
where E = 2.7 + 0.7. (n/nso). IV. SENSOR V I B R A T I O N ANALYSIS IV. 1. Basic model The purpose of this section is to introduce a motion-dependent component into the above model. The image motion will alter the value of fl~y,to a new extinction coefficient defined as fl ,*,, according to the statistical behavior of the low frequency vibration MTF. Since M R T is exponentially related to fl,y, (Eq. 3), the MRT can be related to a new fl*.s~5) MRTo MRT = ~ . e x p ( f l * . ~ v ) , where
1(,)
fl*s = fl~.,,+ - In v M T F 0 c)
'
(9)
f
(10)
The second term represents the effect of the blurring wheref~a.~ is the maximum spatial frequency as defined in Refs (2, 3). Equation (10) can be implemented for any kind of motion, including linear motion, iS) f l ~ , = f l , , , + - In • v
sin(nfd) '
f<
1
d'
(11)
The MTF in equation (1 1) equals sin(rcfd)/nfd w h e r e f i s the spatial frequency and d is the blur radius. The maximum spatial frequency is determined to be I/d, which means that theoretically the MRT at this frequency approaches infinity. The MRT curve gives the minimum temperature difference between the target and the background AT that can be resolved by the thermal imaging system at any arbitrary spatial frequency. From equations (9) and (10), it is clear that the M R T is reciprocally related to the motion MTF, so that as the MTF decreases, the MRT increases and image quality become poorer. Image motion or vibration MTF increases fl*.s and therefore increases the MRT, monotonically decreasing the probability of detecting the target. IV.2. Evaluation of the static probability of detection To find the effect of mechanical vibrations on different sensor-target combinations, consider the situation in Fig. 2. In this case, certain statistics for the target and the sensor are assumed (see Tables 1-3). The assumption here is that the image vibration is the factor most limiting image quality. In moving vehicles, planes, rpvs, and ships this is very often the case. Image motion MTF reduces the overall MTF and therefore the maximum spatial frequency of the imaging system. The reduction of the spatial frequency bandwidth causes a reduction in the target acquisition probability for a given range and target size.
Thermal image target acquisition probabilities in the presence of vibrations
Taqlut/Baekil'oun d
Atmosphere
Background,II ATinhsr.nt- TT-T B
ATappat,~t = ATinhurcnt uxP(13atmR)
695
ll>
Critical Tarzet D/mension
('
L-
~?
VibratedPlatform x(t) - A cos( ~ t ) IO
Fig. 2. Description of a real scenario situation.
The minor dimension of the target is assumed here to be s = 3 m and the simulation was done for three ranges, 2, 4 and 8 km. The assumption here is that the IR camera platform is vibrated such that both target and background are smeared in the image plane. This situation is unlike the case where only the target is in motion. (t°) The vibration amplitude in this simulation is determined to be A = 2.5 cm in the object plane. Three values of the relative exposure time te/To are assumed here, 0.05, 0.1 and 0.15, where the presumption is that the integration time of the eye is equal or less to the exposure time duration of the camera. This short exposure time is similar to the conditions postulated in Ref. (1 I) for situations of searching a target in a dynamic scenario. Let's consider now the simulation procedure which is depicted schematically in Fig. 3. 1. Calculate the image vibration MTF according to the method described in Ref. (3) from t = 0 to To/4 over very small intervals. It is enough to comb only a quarter of the vibration period because the M T F repeats itself four times in one cycle. The number of M T F functions for this analysis is chosen to be N = 125. The different M T F curves belong to different points on the time axis, i.e. for each initial exposure time there is an M T F associated with it. The M T F graphs were plotted only for frequencies below the false-resolution criteria, i.e. f
Table I. Characteristics of the sensor MRTo 0.0254°C Bt,-, 0.996 mrad/cyc 1974 FLIR (8-10/zm).
Table 2. Characteristics of the target s E AT R
3m 2 2° 2, 4, 8kin
Table 3. Characteristics of the search process 7 ~a,m t,/T o A CF
1 0.15 km -1 0.05, 0.1, 0.15 2.5 cm 1
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O. Hadar et al. MTF
x(t) - A cos( T~ot)
x(t) ~'
l
~ - 0.05
M'rFt' (best)
> 2 tlt2 .....
ti .....
~
r
¢
>t
)r Spatial frequency [cycles/m]
MRT
II' ~
(worst)
(l~.~t)
v - f. R [cycles/tad]
15sy * s (v) :
13sys +
MP.T = ~
7z1 n ( M T I ~
(v,ti))
A Tappm~.nt
. . . . . . . . . . . . . . . . . .
exp( v IBsys )
(t/7) -
MRT o
ATappm.lm t . ATi~eren t exp(- 13atm R )
L n - v ~
= [hi,
,
,
,
vI
'vi
vN
Spatialfrequency[cycles/millirad]
n 2 , •.........
ni,
-.........
nN]
TTF
;:-7 111111 P ( t ) - Pw[1-exp(- tP-~ft)]
.....
P(t) Without Vibrations .
With Vibrations
> t Fig. 3. Transformation of image vibration MTF to probability of detection performance.
2. Translate the M T F into an equivalent value of MRT. This transform is obtained from equations (9) and (10) where the constants are given in Tables 1-3. The MTF behavior determines the M R T performance. The worst M T F leads to the worst M R T (MRTwo,t) while the best M T F leads to the best M R T (MRTb,,). All MRTs function were plotted f o r f
Thermal image target acquisition probabilities in the presence of vibrations
697
3. Determine the target critical dimension, range, and inherent ATY z~Using knowledge of the atmospheric attenuation as a function of range, calculate the apparent AT of the target according to Beer's Law"'~
A T~,~,,., = A T~.he,,.,exp(-/~,,,. R).
(12)
Here we assume weak turbulence and small scene dynamic range so that Beer's Law approximates atmospheric MTFY 3J For these examples we assume an atmosphere characterized by /~,,,, = 0.15 km -z. 4. Determine the maximum resolvable spatial frequency of the sensor, v~, in cycles per milliradian at this apparent AT by using the obtained value of MRT. The index i indicates the specific spatial frequency that belongs to the specific MRT~. 5. Using the angular subtense of the target critical dimension, s/R, calculate the maximum number of resolvable cycles across the target, N~, using s
N~ = v,-~.
(13)
6. Determine the probability, P~, of performing a task (7) from the TTF curve. The value of 7 is obtained from the Johnson chart and it is equal to Nsoo/.. The TTF curve transforms the N vector to P~ vector which varies from P . . . . . ~ to P~_~,. Each value in this vector represents the static probability of detection P~_~which belong to a specific initial exposure time between zero and To~4. 7. Determine the dynamic probability of detection the target as function of time. One possibility for the dynamic analysis is to assign P~_~ in equation (6) and find the probability of detection as function of time for each initial exposure time separately. However this analysis is not helpful for measuring the average performance of the observer. Therefore the analysis must consider the distribution function of the P~ vector. This analysis is discussed in the next section.
1I/.3. Evaluation of the dynamic probability of detection The purpose of the following method is to find the dynamic average probability of detection of a target during image vibration. The basic method for this analysis is described elsewherd7~for the purpose of target acquisition in the presence of obscurants. However, here, we assume more a priori knowledge about the image vibration than was available in the case of smoke in Ref. (7). For this reason the probability of detection that is based on the probability density function (PDF) of P~ is more precise than that in Ref. (7). Let us define another important constant M that is similar to the meaning of P~ from an observer point of view. Observers with a low value of M can detect more difficult targets while those with a high value can detect only easy targets3 7~ For example, if there is a target with a value of P~ observers with value of M less than P~ can detect the target while those with values of M greater Table 4. Results of static probabilities of detection R = 2 (km) No-vibrations t_~,= 0.05 To
R = 4 (kin)
P~ = I P~ ~,t = 0.981 -
P~_.,~,,, = 0.153
P~ ~ 0.944 P~ ~,, -- 0.766 -
P,~_,.o,t ~" 0.148
R - - 8 (kin) P~ = 0.204 P,= b,,t = 0.196 -
P~_,o,,, = 0.127
1¢
To
0.10
t., = 0.15 To
P,:_~,,, = 0.224
P,:_~,,, = 0.223
P~_t*,, = 0.172
P-~_,,.,,,,, = 0.118
P ~ _ , . ~ -- O. 113
P ,~_,o,,, = 0.108
P,: ~,, = 0.132 -
P~.,,.~.~, = 0.108
P~ ~ , = 0.132 -
P~_,,.,,,, = 0.108
P~ ~,,, = 0.132 -
P,%,,,,~, = 0.102
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O. Hadar
et al.
than P~ cannot. The collection of observers evenly distributed in their values of M between 0 and 1 represent the normal observer ensemble. Since the assumption here is that the exposure time is on the order of one glimpse, the analysis must include the statistical behavior of the MRT. The single glimpse probability :-oM for observers (with characteristic M ) of a target (with static probability P=) istT~ for M ~< P~
2oM(t/P~) = P~/3.4,
=0,
(14)
for M > P~,
where M is a random variable evenly distributed between 0 and 1; P~ is a random variable with a PDF fp~(P~) between P~,.o,,, and P~_~,,. The PDF can be found by applying a numerical histogram on the P:~ vector obtained in the previous section.
(a) Without Vibrations 0.9 - /°" ................................................................................
0.8 -/
. . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B . . . .e. . s. . .t. _ . . .~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . .
/
0.7 ;" 0.6 :
~
v
e
r
a
g
e
.
Vibrations
0.5 1i
o., f
~
1~-,~ DT
0.3
0 5488
terI'o - 0.05
0.2
Worst_~
/,.' ........
0.1
°o
,~o
2~o 3~o ~
5~o ~
7~o s~o
~o
~ooo
[s]
Time
(b) 0.3
0.25
''"
0.2
J
0.15
~ i
.......................................... ..W...o. ;~L..c....a~... .............................................................................
r#" If/ .......
lI
=fro - o.o5
0.05
0
0
IO0
200
300
400
500
600
700
800
900
Time [s] Fig. 4. Dynamic probability results for
t~lT. =
0.05: (a) R = 4 km; (b) R -- 8 kin.
1000
699
Thermal image target acquisition probabilities in the presence of vibrations
The value of P (t) is given by P (t) =
fOI (Px best{l -
e x p [ - f p ~ (P~),~.oM (t
"/P,~ )]}dP~
(15)
dM.
The above integral can be resolved numerically.
v. RESULTS The results from the simulation described above are shown in Table 4. This table contains two different parameters for each of the 12 situations that were examined. Three cases are for the "no vibration" case (three different ranges); the other nine are situations with vibrations (three values of R and three values of t,/To). The parameters are considered as static probability
(a) 0.9
....... .wt.~.qu.t v.L~_~._o~._. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.8 0.7 R-- 4[km] DT - 0.5488 re/To - 0.1
0.6
0.5 0.4 0.3
Best...case 0.2
v/,..--"'A"~brations
O.l 0
.............W~/'si.~c~ ..............................................................................................................................
5~0
0
1~
t500
20bo
25'00
3~
35b0
4000
3500
4090
Time [s]
(b) 0.25
°"I/..... . . . . . . oo tt
0
D,-o. o 2
500
1000
1500
2000
2500
3000
Time [s] Fig. 5. Dynamic probability results for
t,/To =
0 . h (a) R ---4 kin; (b) R = 8 kin.
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O. Hadar et al.
constants, P ~ , , and P~_,o,,. The resulting P (t) curves can be seen in Figs 4-6 for three different relative time exposures, t,/To= 0.05, 0.1 and 0.15 respectively. Each figure contains two ranges, (a) 4 km and (b) 8 km. Four curves are presented: P (t) with no vibrations, and three different P (t) functions for vibrations: best case, worst case and average vibrations. Figures 7(a) and (b) demonstrate the influence of the two parameters, te/To and R, respectively. In Fig. 7(a) two groups of M R T are plotted schematically for te/To = 0.05 and 0.15 and the two spatial frequency vectors are obtained for the same apparent AT according to Fig. 3. The MRTs for the case of t,/To = 0.05 vary more than the case of t,/To = 0.15. Therefore the P:~ vector for t,/To = 0.05 is characterized by larger dynamic range and higher values than that for t,/To = 0.15. The reason for these results is that the MTF variance increases as t,/To decreases33~ Also the mean blur radius d increases as t,/To increases. In Fig. 7(b), the influence of the range is presented. The range is determined by the apparent AT that reaches the sensor [equation (12)]; increasing the range causes ATopm,e,, to decrease. The meaning of this result is that R is reciprocal to P~. For high values of R or small values of AT the influence of t,/To is less severe because the values of M R T vary less. (a) It--..
0"9 f 0.8
Without Vibrations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
, '
o'f/ 0.6 ,
0.5 j . . . . Best case 0.4 Average vibrations
.... Worst case 0.3
R- 4
0.2 0.1 ~ 0
0
[km]
DT - 0.5488 re/To - O. 15 iiZT.'ZT-~7..7.~:.'.~7.'.~."i"7.'.7~.~:Z7.77~7.'.'."17~7.7Z7.~.".7ZZ'5.'.7:.. 50
100
150
200
250
300
350
400
450
500
Time [s]
(b)
o.~':t/".... w,'~o~;'~:,i~o~ .............................................. '...........
!i!!I 23
oO; y.-,-o- ............................................................................................................................. 0.06
R - S [km]
0.04
DT - 0.3012 re/To - 0.15
0.02 0
0
58o
l;oo
35'oo Time [s]
Fig. 6. Dynamic probability results for t¢lT~ =0.15: (a) R = 4 km; (b) R = 8 km.
4ooo
Thermal image target acquisition probabilities in the presence of vibrations
701
Ca) t~ ,. 0.15 To
t m - 0.05 To
MR•Tl•ql.kr• }....
~ r~ ftmtt~ MRT .... lk~i~;~ "
MRT (worse) MRTN ....
(better) MRTi
MRTi . . . . . . . .
I /y I1/ ; .
~ T o ~
....
.
.
.
.
.
.
;
.
.
.
.
.
~ , o ~
...........
(gCB-VA) < (VD-VC) } Spatialfrexluency[cycles/tad]
:
Spatialfrequency[cyclesJrad]
(b) te To
-
0.05
MRT (worse) (better)
[R ,, 4 kin]
ATapparent1
J R - 8 l~n]
ATappamzt2
-
-
-
%
I
,
3/A VC
,VB_.VA
i
I
VB
~'D
>v
Spatial frequency [cycles/rad] Fig. 7. MRT as function of spatial frequency: (a) for the same range, R = 4 km, and different relative exposure times, t r / T o = 0.05 and 0.15; (b) for the same relative exposure time, t,/To = 0.05, and different ranges. R = 4 and 8 km.
VI. C O N C L U S I O N S I m a g e vibrations are a d o m i n a n t p a r a m e t e r for unstabilized imaging systems used for target acquisition purposes. The reduction of the probability of detection is especially noticeable for long distance targets and for large relative exposure times. In all cases, the average probability o f detection asymptotically goes to P~:_~,,. The explanation for this result is that by increasing the observer time for acquiring the target, the best case repeats itself m a n y times so that the target will eventually be acquired. It is clear that image motion and vibration can significantly affect target acquisition times and ranges.
702
O. Hadar et al.
F r o m Table 4 several important conclusions can be learned. I. The static probability is not a linear function o f R. F o r example in the case o f no vibration the reduction o f P~ (0.941 ~ 0.204) for range change from 4 to 8 km is more noticeable when R is changing from 2 to 4 km (P:~ : 1 --. 0.941). This can be explained by the exponential nature o f equation (12). II. As te/To increases, the reduction o f P~ is less noticeable for increasing target range. This is because for large value o f relative exposure time the degradation process is more d o m i n a n t so that different ranges become less important. This also can be explained from the graph o f the T T F curve. The difference between P ~ , s , and P ....... , is more prominent when these two values are lying on the linear portion o f the T T F curve. For small values o f P:o (approaching to zero) or for large values o f P:~ (approaching to one) the target transform probability function does not change a great deal. IIl. As it appears from the dynamic probability results the rise time o f the average probability o f detection is quite longer than for the " n o vibration" situation. Vibration lowers both the rate o f acquiring the target and the eventual value o f P ~ .
Acknowledgements--The authors are grateful to the Ministry of Science and Arts in Jerusalem and to the Paul Ivanier Center for Robotics and Production Management for generously supporting part of this research. REFERENCES 1. J. A. Ratches, W. R. Lawson, L. P. Obert, R. J. Bergemann, T. W. Casidy and J. M. Swenson, U.S. Army Electronics Command, Fort Monmouth, NJ, Research and Development Technical Report, ECOM-7043 (April 1975). 2. O. Hadar, M. Fisher and N. S. Kopeika, Optical Eng. 31, 581-589 (1992). 3. O. Hadar, I. Dror and N. S. Kopeika, Optical Eng. 33, 566-578 (1994). 4. O. Hadar, S. R. Rotman and N, S. Kopeika, Optical Eng. 33, 3106-3117 (1994). 5. S. R. Rotman, E. S. Gordon, O. Hadar and N. S. Kopeika, IEEE Trans. Aerospace Electronic. (in press). 6. O. Hadar, Z. Adar, A. Cotter and N. S. Kopeika, Submitted for publication. 7. S. R. Rotman, E. S. Gordon and M. L. Kowalczyk, Optical Eng. 30, 824-829 (1991). 8. S. R. Rotman, Opt. Eng. 28, 1223-1226 (1989). 9. D. Wulich and N. S. Kopeika, Opt. Eng. 2, 529-533 (1987). 10. A. K. Davies, K. J. Cooke and P. A. Stanly, "Modeling the effects of target motion on aquisition performance". U.S. Army Tank-Automotive Command, Research, Development & Engineering Center, Warren, Michigan 483975000, pp. 26-35 (August 1993). II. D. C. Burr, Proc. R. Soc. Lond. 211B, 321-339 (1981). 12. M. C. Dudzik, Electro-Optical Systems Design, Analysis, and Testing, Vol. 4. In The Infrared and Electro-Optical System Handbook (J. S. Accetta and D. L. Shumaker, executive editors). SPIE Press, Bellingham, WA (1993). 13. D. Sadot, N. S. Kopeika and S. R. Rotman, Inf. Phys. Technol. 36, 551-564 (1995).
Pergamon
THERMAL PROBABILITIES
IMAGE IN THE
TARGET ACQUISITION PRESENCE OF VIBRATIONSt
O. HADAR, S. R. ROTMAN and N. S. KOPEIKA Department of Electrical and Computer Engineering, Ben-Gurion University of the Negev, P.O. Box 653, 84105 Beer-Sheva, Israel
(Received 2 August 1994) Abstract--The effects of low frequency mechanical vibrations on thermal imaging target acquisition are considered. A model is described that takes as input mechanical vibration data such as amplitude and frequency as well as the physical characteristics of the target. The output is the probability of detection as a function of time. Analysis indicates that even low amplitude vibration greatly affects the predicted target detection times and ranges and, hence, the utility of IR systems in realistic scenarios.
I.
INTRODUCTION
There are numerous factors which influence an observer's ability to acquire a target. One of the common problems in real time imaging systems is image motion and vibration. Image motion as a result of vibrations is often the limiting factor in image resolution for moving systems, such in reconnaissance, robotics, computer vision, etc. The reduction of image resolution due to image motion influences the ability of an observer to detect the target. This paper presents a modification to the standard military model for IR target acquisition of the Center for Night Vision and Electro-Optics (CNVEO)J ~ The modification is for dynamic imaging systems where image motion or vibrations are involved in the acquisition process. Motion may derive from vehicular motors, airplanes turbines, or even the refrigeration system cooling the IR receiver. The development is based on a new method for modeling the motion degradation as an optical transfer function (OTF) appropriate to any kind of image motionJ 2"3~This paper deals with low frequency vibrations where the OTF is a random process which depends upon the portion of the vibration cycle in which the exposure takes place. The focus here is on the modulation transfer function (MTF) because the reduction with image intensity is more critical than that of the phase for human vision. The reduction with resolution in the image plane as result of image motion or vibrations affects the ability of an observer or automatic target recognition (ATR) system to extract the target from its local background. The performance on an ATR system under acceleration is presented elsewhereJ~ For an IR noise-limited imaging system, the critical factor that determines the performance of the FLIR system is the minimum resolvable temperature (MRT). The influence of constant velocity on FLIR systems has been examined in Ref. (5) where the basis for this analysis is the relation between the MRT and the image motion MTF. This dependence of the MRT is investigated here for the case of low frequency vibrations. This latter situation is more complicated because there is no unique MTF that represents the degradation process. Therefore the analysis consists of the all possible MTFs existing during the searching time for the target. The assumption here is that, for an average observer, the search time for the target will include all possible MTFs of low frequency vibrations. This situation is further complicated by the fact that while image restoration tPart of this paper was presented at the SPIE Conference on Acquisition, Tracking and Pointing V, April 1994, Orlando, FL, U.S.A. 691
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techniques can be very effective in deblurring contrast-limited images degraded by image motion and vibration, 16~their utility in correcting noise-limited images is extremely limited at best. The procedure here is similar to the analysis for target detection in the presence of obscurants) 71 However instead of investigating the statistics of the extinction coefficient of the atmosphere fl~,m, the system effect of image motion and vibration on the extinction coefficient flsy,is considered. The first step in the analysis is modeling the behavior of the static probability of detection P:c Ir'sJ which is the critical parameter that determines the dynamic probability of acquiring the target. The dynamic probability of detection is based on the statistical behavior of the static probability P~ as a result of low frequency vibrations. Probability of detection as a function of time is presented for different target ranges and different exposure times. Although emphasis here is on target "detection", similar analyses can be performed for "recognition" and "identification". II. LOW F R E Q U E N C Y VIBRATIONS The effects of sinusoidal image motion, which often results from mechanical vibrations, can be divided into high and low vibration frequencies32'3) In the first case, the time exposure is longer than the vibrational period and the image blur is therefore the entire peak-to-peak translation of the image. The image motion is given by 2~
x (t) = D cos -~--; xo
(I)
the MTF for the high vibration frequency case is (2"3) (2)
M , ( f ) = Jo(2gfD )
The low vibration frequency situation involves exposure time t< shorter than the vibration period To. In this case, the blur radius d and the MTF are random processes that depend upon when the exposure time takes place as shown in Fig. 1. This type of blur is often more demanding than the high vibration frequency blur because in real-life situations d for low vibration frequencies is often much greater than 2D for high vibration frequencies39) The random process of the MTF is described elsewhere°) in detail. The main analysis in Ref. (3) was to calculate the probability of the MTF being higher than a given threshold contrast as a function of spatial frequency. This calculation was obtained for different relative time exposures tflTo and several threshold contrasts. D te-
w
8
l"'"
I
. . . . . .
\ To \ "7"
J"o,_.
I "to T
I ' I
~' / , /3/,To i
I To
. . . . .
Fig. 1. Blur radius d as a function of exposure time t, for low frequencyvibration.
Thermal image target acquisition probabilities in the presence of vibrations
693
The conclusions of that analysis were that the probability function is reciprocal to the threshold contrast and to the parameter t,/T°. This statistical result can be used for defining the probability of detecting a target in an image that was taken during a vibration period by a human visual system and accounting for the integration time of the eye. This analysis was obtained for a contrast-limited imaging system in the visible spectrum. ~4)In a noise-limited thermal imaging system, the parameter that can characterize FLIR performance is the MRT; this parameter is defined as the value of the minimum resolvable temperature difference between a pattern of four identical bars and the three spaces between them. The statistical analysis for IR images can be obtained in the same way as for visible images; however, instead of contrast, the crucial parameter is the difference temperature, AT. The relation between the imaging MTF and the MRT will be presented in the next section. III. T H E R M A L IMAGE P E R F O R M A N C E MODEL The method for calculating the probability of target acquisition from the knowledge of all the MTF possibilities of low frequency vibration is presented here. The goal of this analysis is to obtain the probability of detection as a function of time. Any FLIR performance is calculated as a function of the spatial frequency of the bar pattern. Assume, as in Ref. (5), that the MRT dependence on angular spatial frequency v (in cycles/millirad) is given by MRTo MRT = ~ . exp(fl,,.~v),
(3)
where MRTo is the limiting MRT as v approaches 0, flsysis the system coefficient and E is the ratio of the long to short dimensions of the target, v can be found from the target characteristics by ns0 s
v = - - . R,
(4)
where R is the range, s is the minor dimension of the target, and nso is the task difficulty function from the Johnson chart (1 for detection, 4 for recognition and 6.4 for identification). The relation between the angular spatial frequency v in cycles/mrad to spatial frequency f i n cycles/mm is given by
f = v/F,
(5)
where F is the focal length in mm. The dynamic search model appears in Ref. (8), where the final result for the probability of acquiring the target as a function of time is given by the following equation -t
f -,;.o t '~ -]
P(t,=P=El-exp(-~.~.)]=P=[l-exp~--~ff where t/~ 0.3 s is the time to take a single glimpse and 3,, = small 20 and P=, 2o/ti equals P~/3.4 and hence P(t)=P~
[
l-exp~
)j,
ti/2 o is the mean .
(6) acquisition time. For
(7)
We assume in the following analysis that ti is less than or equal to t,, the exposure time of the camera. Equation (7) is known as the standard military search model for infrared target detection developed in the Center for Night Vision and Electro-Optics (CNVEO). The term P~ in equation (7) represents the fraction of the normal observer ensemble that can successfully find the target in infinite time. P~ is the static probability of detection at infinite time when the position of the target is known. It is given by the target transfer function (TTF) based upon the Johnson chart which describes the number of line pairs (n~) across the target image critical dimension for 50%
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probability of detection, i.e. P~ = 0.50. The TTF curve allows probability calculations for other values of P~ as a function of the number of line pairs (n) across the target image critical dimension, and is given by "'7) (n/nso) E P': = 1 + (n/nso) e'
(8)
where E = 2.7 + 0.7. (n/nso). IV. SENSOR V I B R A T I O N ANALYSIS IV. 1. Basic model The purpose of this section is to introduce a motion-dependent component into the above model. The image motion will alter the value of fl~y,to a new extinction coefficient defined as fl ,*,, according to the statistical behavior of the low frequency vibration MTF. Since M R T is exponentially related to fl,y, (Eq. 3), the MRT can be related to a new fl*.s~5) MRTo MRT = ~ . e x p ( f l * . ~ v ) , where
1(,)
fl*s = fl~.,,+ - In v M T F 0 c)
'
(9)
f
(10)
The second term represents the effect of the blurring wheref~a.~ is the maximum spatial frequency as defined in Refs (2, 3). Equation (10) can be implemented for any kind of motion, including linear motion, iS) f l ~ , = f l , , , + - In • v
sin(nfd) '
f<
1
d'
(11)
The MTF in equation (1 1) equals sin(rcfd)/nfd w h e r e f i s the spatial frequency and d is the blur radius. The maximum spatial frequency is determined to be I/d, which means that theoretically the MRT at this frequency approaches infinity. The MRT curve gives the minimum temperature difference between the target and the background AT that can be resolved by the thermal imaging system at any arbitrary spatial frequency. From equations (9) and (10), it is clear that the M R T is reciprocally related to the motion MTF, so that as the MTF decreases, the MRT increases and image quality become poorer. Image motion or vibration MTF increases fl*.s and therefore increases the MRT, monotonically decreasing the probability of detecting the target. IV.2. Evaluation of the static probability of detection To find the effect of mechanical vibrations on different sensor-target combinations, consider the situation in Fig. 2. In this case, certain statistics for the target and the sensor are assumed (see Tables 1-3). The assumption here is that the image vibration is the factor most limiting image quality. In moving vehicles, planes, rpvs, and ships this is very often the case. Image motion MTF reduces the overall MTF and therefore the maximum spatial frequency of the imaging system. The reduction of the spatial frequency bandwidth causes a reduction in the target acquisition probability for a given range and target size.
Thermal image target acquisition probabilities in the presence of vibrations
Taqlut/Baekil'oun d
Atmosphere
Background,II ATinhsr.nt- TT-T B
ATappat,~t = ATinhurcnt uxP(13atmR)
695
ll>
Critical Tarzet D/mension
('
L-
~?
VibratedPlatform x(t) - A cos( ~ t ) IO
Fig. 2. Description of a real scenario situation.
The minor dimension of the target is assumed here to be s = 3 m and the simulation was done for three ranges, 2, 4 and 8 km. The assumption here is that the IR camera platform is vibrated such that both target and background are smeared in the image plane. This situation is unlike the case where only the target is in motion. (t°) The vibration amplitude in this simulation is determined to be A = 2.5 cm in the object plane. Three values of the relative exposure time te/To are assumed here, 0.05, 0.1 and 0.15, where the presumption is that the integration time of the eye is equal or less to the exposure time duration of the camera. This short exposure time is similar to the conditions postulated in Ref. (1 I) for situations of searching a target in a dynamic scenario. Let's consider now the simulation procedure which is depicted schematically in Fig. 3. 1. Calculate the image vibration MTF according to the method described in Ref. (3) from t = 0 to To/4 over very small intervals. It is enough to comb only a quarter of the vibration period because the M T F repeats itself four times in one cycle. The number of M T F functions for this analysis is chosen to be N = 125. The different M T F curves belong to different points on the time axis, i.e. for each initial exposure time there is an M T F associated with it. The M T F graphs were plotted only for frequencies below the false-resolution criteria, i.e. f
Table I. Characteristics of the sensor MRTo 0.0254°C Bt,-, 0.996 mrad/cyc 1974 FLIR (8-10/zm).
Table 2. Characteristics of the target s E AT R
3m 2 2° 2, 4, 8kin
Table 3. Characteristics of the search process 7 ~a,m t,/T o A CF
1 0.15 km -1 0.05, 0.1, 0.15 2.5 cm 1
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x(t) - A cos( T~ot)
x(t) ~'
l
~ - 0.05
M'rFt' (best)
> 2 tlt2 .....
ti .....
~
r
¢
>t
)r Spatial frequency [cycles/m]
MRT
II' ~
(worst)
(l~.~t)
v - f. R [cycles/tad]
15sy * s (v) :
13sys +
MP.T = ~
7z1 n ( M T I ~
(v,ti))
A Tappm~.nt
. . . . . . . . . . . . . . . . . .
exp( v IBsys )
(t/7) -
MRT o
ATappm.lm t . ATi~eren t exp(- 13atm R )
L n - v ~
= [hi,
,
,
,
vI
'vi
vN
Spatialfrequency[cycles/millirad]
n 2 , •.........
ni,
-.........
nN]
TTF
;:-7 111111 P ( t ) - Pw[1-exp(- tP-~ft)]
.....
P(t) Without Vibrations .
With Vibrations
> t Fig. 3. Transformation of image vibration MTF to probability of detection performance.
2. Translate the M T F into an equivalent value of MRT. This transform is obtained from equations (9) and (10) where the constants are given in Tables 1-3. The MTF behavior determines the M R T performance. The worst M T F leads to the worst M R T (MRTwo,t) while the best M T F leads to the best M R T (MRTb,,). All MRTs function were plotted f o r f
Thermal image target acquisition probabilities in the presence of vibrations
697
3. Determine the target critical dimension, range, and inherent ATY z~Using knowledge of the atmospheric attenuation as a function of range, calculate the apparent AT of the target according to Beer's Law"'~
A T~,~,,., = A T~.he,,.,exp(-/~,,,. R).
(12)
Here we assume weak turbulence and small scene dynamic range so that Beer's Law approximates atmospheric MTFY 3J For these examples we assume an atmosphere characterized by /~,,,, = 0.15 km -z. 4. Determine the maximum resolvable spatial frequency of the sensor, v~, in cycles per milliradian at this apparent AT by using the obtained value of MRT. The index i indicates the specific spatial frequency that belongs to the specific MRT~. 5. Using the angular subtense of the target critical dimension, s/R, calculate the maximum number of resolvable cycles across the target, N~, using s
N~ = v,-~.
(13)
6. Determine the probability, P~, of performing a task (7) from the TTF curve. The value of 7 is obtained from the Johnson chart and it is equal to Nsoo/.. The TTF curve transforms the N vector to P~ vector which varies from P . . . . . ~ to P~_~,. Each value in this vector represents the static probability of detection P~_~which belong to a specific initial exposure time between zero and To~4. 7. Determine the dynamic probability of detection the target as function of time. One possibility for the dynamic analysis is to assign P~_~ in equation (6) and find the probability of detection as function of time for each initial exposure time separately. However this analysis is not helpful for measuring the average performance of the observer. Therefore the analysis must consider the distribution function of the P~ vector. This analysis is discussed in the next section.
1I/.3. Evaluation of the dynamic probability of detection The purpose of the following method is to find the dynamic average probability of detection of a target during image vibration. The basic method for this analysis is described elsewherd7~for the purpose of target acquisition in the presence of obscurants. However, here, we assume more a priori knowledge about the image vibration than was available in the case of smoke in Ref. (7). For this reason the probability of detection that is based on the probability density function (PDF) of P~ is more precise than that in Ref. (7). Let us define another important constant M that is similar to the meaning of P~ from an observer point of view. Observers with a low value of M can detect more difficult targets while those with a high value can detect only easy targets3 7~ For example, if there is a target with a value of P~ observers with value of M less than P~ can detect the target while those with values of M greater Table 4. Results of static probabilities of detection R = 2 (km) No-vibrations t_~,= 0.05 To
R = 4 (kin)
P~ = I P~ ~,t = 0.981 -
P~_.,~,,, = 0.153
P~ ~ 0.944 P~ ~,, -- 0.766 -
P,~_,.o,t ~" 0.148
R - - 8 (kin) P~ = 0.204 P,= b,,t = 0.196 -
P~_,o,,, = 0.127
1¢
To
0.10
t., = 0.15 To
P,:_~,,, = 0.224
P,:_~,,, = 0.223
P~_t*,, = 0.172
P-~_,,.,,,,, = 0.118
P ~ _ , . ~ -- O. 113
P ,~_,o,,, = 0.108
P,: ~,, = 0.132 -
P~.,,.~.~, = 0.108
P~ ~ , = 0.132 -
P~_,,.,,,, = 0.108
P~ ~,,, = 0.132 -
P,%,,,,~, = 0.102
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O. Hadar
et al.
than P~ cannot. The collection of observers evenly distributed in their values of M between 0 and 1 represent the normal observer ensemble. Since the assumption here is that the exposure time is on the order of one glimpse, the analysis must include the statistical behavior of the MRT. The single glimpse probability :-oM for observers (with characteristic M ) of a target (with static probability P=) istT~ for M ~< P~
2oM(t/P~) = P~/3.4,
=0,
(14)
for M > P~,
where M is a random variable evenly distributed between 0 and 1; P~ is a random variable with a PDF fp~(P~) between P~,.o,,, and P~_~,,. The PDF can be found by applying a numerical histogram on the P:~ vector obtained in the previous section.
(a) Without Vibrations 0.9 - /°" ................................................................................
0.8 -/
. . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .B . . . .e. . s. . .t. _ . . .~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... . . . .
/
0.7 ;" 0.6 :
~
v
e
r
a
g
e
.
Vibrations
0.5 1i
o., f
~
1~-,~ DT
0.3
0 5488
terI'o - 0.05
0.2
Worst_~
/,.' ........
0.1
°o
,~o
2~o 3~o ~
5~o ~
7~o s~o
~o
~ooo
[s]
Time
(b) 0.3
0.25
''"
0.2
J
0.15
~ i
.......................................... ..W...o. ;~L..c....a~... .............................................................................
r#" If/ .......
lI
=fro - o.o5
0.05
0
0
IO0
200
300
400
500
600
700
800
900
Time [s] Fig. 4. Dynamic probability results for
t~lT. =
0.05: (a) R = 4 km; (b) R -- 8 kin.
1000
699
Thermal image target acquisition probabilities in the presence of vibrations
The value of P (t) is given by P (t) =
fOI (Px best{l -
e x p [ - f p ~ (P~),~.oM (t
"/P,~ )]}dP~
(15)
dM.
The above integral can be resolved numerically.
v. RESULTS The results from the simulation described above are shown in Table 4. This table contains two different parameters for each of the 12 situations that were examined. Three cases are for the "no vibration" case (three different ranges); the other nine are situations with vibrations (three values of R and three values of t,/To). The parameters are considered as static probability
(a) 0.9
....... .wt.~.qu.t v.L~_~._o~._. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 0.8 0.7 R-- 4[km] DT - 0.5488 re/To - 0.1
0.6
0.5 0.4 0.3
Best...case 0.2
v/,..--"'A"~brations
O.l 0
.............W~/'si.~c~ ..............................................................................................................................
5~0
0
1~
t500
20bo
25'00
3~
35b0
4000
3500
4090
Time [s]
(b) 0.25
°"I/..... . . . . . . oo tt
0
D,-o. o 2
500
1000
1500
2000
2500
3000
Time [s] Fig. 5. Dynamic probability results for
t,/To =
0 . h (a) R ---4 kin; (b) R = 8 kin.
700
O. Hadar et al.
constants, P ~ , , and P~_,o,,. The resulting P (t) curves can be seen in Figs 4-6 for three different relative time exposures, t,/To= 0.05, 0.1 and 0.15 respectively. Each figure contains two ranges, (a) 4 km and (b) 8 km. Four curves are presented: P (t) with no vibrations, and three different P (t) functions for vibrations: best case, worst case and average vibrations. Figures 7(a) and (b) demonstrate the influence of the two parameters, te/To and R, respectively. In Fig. 7(a) two groups of M R T are plotted schematically for te/To = 0.05 and 0.15 and the two spatial frequency vectors are obtained for the same apparent AT according to Fig. 3. The MRTs for the case of t,/To = 0.05 vary more than the case of t,/To = 0.15. Therefore the P:~ vector for t,/To = 0.05 is characterized by larger dynamic range and higher values than that for t,/To = 0.15. The reason for these results is that the MTF variance increases as t,/To decreases33~ Also the mean blur radius d increases as t,/To increases. In Fig. 7(b), the influence of the range is presented. The range is determined by the apparent AT that reaches the sensor [equation (12)]; increasing the range causes ATopm,e,, to decrease. The meaning of this result is that R is reciprocal to P~. For high values of R or small values of AT the influence of t,/To is less severe because the values of M R T vary less. (a) It--..
0"9 f 0.8
Without Vibrations
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
, '
o'f/ 0.6 ,
0.5 j . . . . Best case 0.4 Average vibrations
.... Worst case 0.3
R- 4
0.2 0.1 ~ 0
0
[km]
DT - 0.5488 re/To - O. 15 iiZT.'ZT-~7..7.~:.'.~7.'.~."i"7.'.7~.~:Z7.77~7.'.'."17~7.7Z7.~.".7ZZ'5.'.7:.. 50
100
150
200
250
300
350
400
450
500
Time [s]
(b)
o.~':t/".... w,'~o~;'~:,i~o~ .............................................. '...........
!i!!I 23
oO; y.-,-o- ............................................................................................................................. 0.06
R - S [km]
0.04
DT - 0.3012 re/To - 0.15
0.02 0
0
58o
l;oo
35'oo Time [s]
Fig. 6. Dynamic probability results for t¢lT~ =0.15: (a) R = 4 km; (b) R = 8 km.
4ooo
Thermal image target acquisition probabilities in the presence of vibrations
701
Ca) t~ ,. 0.15 To
t m - 0.05 To
MR•Tl•ql.kr• }....
~ r~ ftmtt~ MRT .... lk~i~;~ "
MRT (worse) MRTN ....
(better) MRTi
MRTi . . . . . . . .
I /y I1/ ; .
~ T o ~
....
.
.
.
.
.
.
;
.
.
.
.
.
~ , o ~
...........
(gCB-VA) < (VD-VC) } Spatialfrexluency[cycles/tad]
:
Spatialfrequency[cyclesJrad]
(b) te To
-
0.05
MRT (worse) (better)
[R ,, 4 kin]
ATapparent1
J R - 8 l~n]
ATappamzt2
-
-
-
%
I
,
3/A VC
,VB_.VA
i
I
VB
~'D
>v
Spatial frequency [cycles/rad] Fig. 7. MRT as function of spatial frequency: (a) for the same range, R = 4 km, and different relative exposure times, t r / T o = 0.05 and 0.15; (b) for the same relative exposure time, t,/To = 0.05, and different ranges. R = 4 and 8 km.
VI. C O N C L U S I O N S I m a g e vibrations are a d o m i n a n t p a r a m e t e r for unstabilized imaging systems used for target acquisition purposes. The reduction of the probability of detection is especially noticeable for long distance targets and for large relative exposure times. In all cases, the average probability o f detection asymptotically goes to P~:_~,,. The explanation for this result is that by increasing the observer time for acquiring the target, the best case repeats itself m a n y times so that the target will eventually be acquired. It is clear that image motion and vibration can significantly affect target acquisition times and ranges.
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F r o m Table 4 several important conclusions can be learned. I. The static probability is not a linear function o f R. F o r example in the case o f no vibration the reduction o f P~ (0.941 ~ 0.204) for range change from 4 to 8 km is more noticeable when R is changing from 2 to 4 km (P:~ : 1 --. 0.941). This can be explained by the exponential nature o f equation (12). II. As te/To increases, the reduction o f P~ is less noticeable for increasing target range. This is because for large value o f relative exposure time the degradation process is more d o m i n a n t so that different ranges become less important. This also can be explained from the graph o f the T T F curve. The difference between P ~ , s , and P ....... , is more prominent when these two values are lying on the linear portion o f the T T F curve. For small values o f P:o (approaching to zero) or for large values o f P:~ (approaching to one) the target transform probability function does not change a great deal. IIl. As it appears from the dynamic probability results the rise time o f the average probability o f detection is quite longer than for the " n o vibration" situation. Vibration lowers both the rate o f acquiring the target and the eventual value o f P ~ .
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