Dec 3, 1981 - LINEAR, CHEVRON, AND CROW'S FOOT DETECTOR ARRAYS. 12. IV. BRICKWALL DETECTOR PATTERNS. 31. S,,. 41. V. DISCUSSION ...
.'10
1AD.TR41-32
Technical Report
589 K-P. Dunn
1
Accuracy of Parameter Estimates for Closely Spaced Optical Targets
DTIC DELECTE 1 DEC
3 1981
B
Using Multiple Detectors
23 October 1981 under Electronk Synens Divisi
Conbwc F1g624-C4..• by
Lincoln Laboratory f
MASSACHUSETTS INSTITUTE OF TECHNOLOGY LEXINGTON, MASSwCHUSETTS
Approveed for public velesue; distuibution unlimited.
#,11231 014 L4'
u I
F ~ "••-'
_.z,
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_____________-_--
The work reported in this document was performnd at Lincoln Laboratory, a center for research opevated by Massachusetts Institute of Technology. This program is sponsored by the Ballistic Missile Defense Program Office, Department of the Army; it is supported by the Ballistic Missile Defense Advanced Technology Center under Air Force Contract F1962880-C-0002. This report may be reproduced to satisfy needs of U.S. Government agencies.
The views and conclusions contained in this document are those of the contractor and should not be interpreted as necessarily representing the official policies, either expressed ot implied, of the United States Govwinment.
The Public Affairs Office has reviewed this report, and it is releasable to the National Technical Information Service, where it will be available to the general public, including foreign nationals.
This technical report has been reviewed and is approved for publication. FOR THE COMMANDER
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Non-Lincoln Recipients PLEASE DO NOT REIrRIII Permission is given to destroy this document when it is no longer needed.
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MASSACHUSETTS INSTITUTE OF TECHNOLOGY
LINCOLN LABORATORY
ACCURACY OF PARAMETER ESTIMATES FOR CLOSELY SPACED OPTICAL TARGETS USING MULTIPLE DETECTORS
K-P. DUNN
Group 32
TECHNICAL REPORT 589
23 OCTOBER 1981
Approved for public release; distribution unlimited.
LEXINGTON
jiMASSACHUSETTS
I71
ABSTRACT In order to obtain the cross-scan position of an optical targe., more than one scanning detectors are used.
As expected,
the cro.s-scan position estimation performance degrades when two nearby optical targets interfere with each other. I
on the two-dimensional parameter estimation performance
.bounds
for two closely spaced optical targets are found. classes of scanning detector arrays, the brickwall (or mosaic) patterns,
are considered.
.", For
d':"
-
/'
'/
t_,!iii/I
Two particular
namely, the crcw's foot and
- I
•-. ,.
Theoretical
1
--
C U .,[
CONTENTS
r•
iii
Abstract INTRODUCTION
I. IT.
LINEAR,
III. IV.
7
ANALYSIS
S,,
V.
CHEVRON,
AND CROW'S FOOT DETECTOR ARRAYS
BRICKWALL DETECTOR PATTERNS
12
31
41
DISCUSSION AND SU4MARY
Matrix for Linear, APPENDIX A - The Fisher Information Chevron, and Crow's Fopt Detector Arrays
APPENDIX B
-
4
43
The Fisher Information Matrix For Brickwall4 Detector Patterns
Acknowledgments
51 5
References
iv
•........................V
I.
INTRODUCTION Recently, considerable attention has been focused upon the
S.
accuracy of parameter estimates of an optical systemn for closely spaced objects (CSOs) [1]
r
-
[7].
The intensity and location in-A
~formation of an optical image on the sensor focal plane is collected by a scanning detector as shown in Fig. 1. The detector output,
F
a time function, is a convolution of the optical image (which contains intensity and location information) and the detector response.
There are various types of noise sources at the output
of the detector, therefore estimating the image position and intensity from the noisy output can not be errorless.
Theoretical
lower bounds on the accuracy of parameter estimates have been derived in [1)
-
[4].
The Cramer-Rao lower bound technique is used in [21
-
[4] to
calculate the best achievable performance of any unbaised estimator
of the unknown parameters.
The results presented in [1] are
obtained using a different error analysis technique.
Other than
somne numerical problems pointed out in [41, which occurred at
small target separations, the approach presented in [1] should have results identical to those obtained by the Cramer-Rao lower bound approach.
There are algorithms developed to perform the
parameter estimation for closely spaced objects from noisy measurements, for example [5)
-
[7].
results are presented in [5] and [7].
Monte Carlo simulation In [5], the well known
j
PSF CONVOLVES WITH DETECTOR RESPONSE FUNCTION
FOCAL PLANE S1st Ring of Optical Image
ScanI
,J
to
O
I
Direction
t
O
0~
t= to 0
t
I
t E
W ,,
to
TI T2 0o÷t Trt
eahtagt amli TIME
-• DETECTION: Identification of the
POINT SPREAD FUNCTION kPSF)prsneoCSs
.,and
Fig 1.Cloelyspaed
0 ESTIMATION: (for each target) Target amplitude position bjet (SO)reslutonproblnem of or determination
1. closely spaced object (CSO) resolution problem for optical system with scanning detector. Fig.
I-
i>I
-
-
--
~-~~-
-
-
.- .-
,•-.
I
n
h
kie
nomto
rtein(]i
mloe
oetmt
maximum-likelihood estimator is implemented for parameter estimation the number of targets in the CSO cluster. ~
All analytical and simulation results presented in the past pertain only to the in-scan direction and assume a vertical
Ii
[
~rectangular detector as shown in Fig. 1. Many detector array] arrangements have been considered by sensor designers to oib'ain position information of a single optical target in the cross-scan direction.
some typical detector patterns considered are shown in
Fig. 2. The time between detector crossings for the chevron and crows's foot patterns can be used to determine the cross-scan ~
I
position of the target.
I
more difficult.
'With two targets, this problem becomes
Figure 3 shows three individual detector outputs
of a crow's foot pattern for different orientations of target locations.
It is obvious that with one linear detector (detector
M) output alone, one can not obtain cross-scan position information. From the results we have presented in (2]
-
[5], one can obtain in-
scan position information of two CSOs no matter how close they are L
if the detector output is noise free. It is, however, not true for a noisy output; the estimation errors increase drastically as the CSO separation decreases.
For the chevron pattern (detectors L
anid R), the cross-scan position information can be obtained without error from noise free outputs for all target locations.
For]
noisy outputs, there are target locations, for example, case II of
3
Chevron
Linear Array
0l Brickwall(Mosaic)
Crow's Foot Fig. 2.
Typical detector patterns have been considered.
4
...........
II
i
LlO11414"t-#-
U
JA
S"-
SCAN DIRECTION
1
2
1
1 2
2
1
2
Cl)
z 0
2 1
o
112
2
A112 1
2
e
~2
1
TIME "'
Individual detector outputs of a crow's foot pattern Fig. 3. for different orientations of target locations.
..
"Fig. 3, where the target position estimation errors may be very large no matter how large the actual target separation is. is because there is a CSO situation for the detector R,
This A more
complicated detector array arrangement may solve this problem, -for erample the crow's foot pattern shown in Fig. 3.
The
achievable estimation performance of the cross-scan position of two CSOs for these particular detector patterns has not been studied analytically previously.
It
is the purpose of this report
to derive the Cramer-Rao lower bounds on the variance of parameter estimates as a function of both scan and cross-scan saparations for CSOs measured by detector patterns shown in Fig. 2.
The
performance bounds presented in this zeport are derived with the assumption that all detector outputs are utilized optimally for unbiased parameter estimation; no particular signal processing procedure has been adopted in the analysis. The general formulation of this problem and its associated Cramer-Rao bounds are presented in the next section. Sb F
[
III, results for linear, chevron,
In Section
and crow's foot arrays using a
particular optical point spread function are presented. point spread functions and detector widths,
For other
one can utilize
the formulas provided in Appendix A to calculate the associated estimation performance bounds.
In Section IV,
mosaic) detector patterns are considered.
the brickwall (or
Some typical results fot
the same point spread function considered in Section III are pre-
6
sented.
Formulas that can be used to calculate the estimation
performance bounds are provided in Appendix B.
A summary and
conclusions are given in Section V. II.
ANALYSIS Let us aaume there is an array of M detectors in the focal
plane of a scanning optical sensor; some typical patterns of these detectors are shown in Fig. 2.
Let (xk,yk) be the center and 0
be the orientation of the kth detector with the center of the focal plane scanning along the x-axis as shown in Fig. 4.
Let
pk(ttx,y) be the output of the kth detector to a unit strength point source located at (x,y) at t=01 A point source is called a unit strength point source,
if
for (xy)=(xkyk) and ek =90,
the
output signal Pkltxy) satisfies: f p2(t,x,y)dt = 1. it
(1)
is important to note that, Pk(t,x,y) .As a convolution of the
point source blurred image (e.g., point spread function, PSF,
ior
a monochromatic source) on the focal plane and the response of the kth detector.
In general,
it
is a complicated function of the
detector location (xk' yk) shape, orientation
(6k) , and the
instantaneous target location (x-t,y) on the focal plane at time t. A single point source located at (x,y) with amplitude a will tIn the remainder of this note, a point source (or, target) location is always referced to its location on the focal plane at t=O.
7
,II
.
•!..
y
'O
kth detector
•i ...
I.'/
(Xk
I'
i
(x, Yk)
ui,,er of the
Focal Plane
A l I
/
I Scan Direction
(0,0)
Fig. 4. Individual detector location with respect to thc center of focal plane and its orientation with respect to the scanning direction.
Ia
!I have a noiseless output sk(t) = aPk(t,x,y)
(2)
at the kth detector at time t. The problem treated here concerns the measurements of a pair of point sources from the output of an M detector array. SI'.
(al, (xl' Y()) and the point souzces.
Let
(a 2 , ( Y 2 ) be the amplitude and location of The noiseless output of the kth detector can
be written as the following:
sk(t) We wish to determine
=
+ a 2 Pk(t,x 2 ,Y 2 )o
alPk(t,xl,YI)
(ai,
(iYi)),
i=1,2 from noisy measurements,
taken at the output of each detector, Yk(t)
i.e.,
Sk(t) + nk(t),
=
(3)
k=l,2,...,M,
(4)
where nk(t) is
a white Gaussian noise with two-sided power spectrum
density N /2.
Furthermore,
0
these noises are mutually independent,
that is E[ni(t)nj(t)]
=
0,
for all
t and T, when iyj.
(5)
The Cramer-Rao lower bound on the variance of any unbaised estimator of the unknown parameters: amplitude ratio, R = a 2 /a,, tion,
r,
amplitude of object 1, al,
location of object 1, (xlY 1 ) , separa-
and orientation of objects,
e (the angle between the
detector scan direction and the vector (x 2 ,y
9
2
)
-
(xlyI),
as shown
I'I da2 k Oh
a2
kthadetector
(xk, Y Scan Direction
R =Amplitude Fig.
5.
Ratio
=a
77I iin
',
I
1
-i'
.---- -
a,
Unknown parameters of a pair of CSOs.
bI
*
2/
A
-*1
i.1
5,
in Fig.
can be obtained by inverting the Fisher information [9]
element is
(i,j)th
matrix, F, whose
(6)
DAlnj
A
= EIln !F.1i
Lij where a. denotes the ith ,nknown parameter, namely, 1, 3= 1t
ratio
a 4 =Yl, [9,
• 5 =r and a6 =G,
al=al,
a 2 =R,
and where in A is the log likelihood
p'. 274]
in A =
yf12(t)s(t)dt
2
(7)
(t)s(t)dt},
-f
and s (t) S2 (t)
s(t)=
s
yl(t) y(t)= Y2 (t)
YM(t)
(t)
and using the statistical model
into (6),
and (8)
Substituting (7)
(8)
•
of the problem described in F
(4)
and (5),
C (a
skt)
we have s(t dt .9)
jF
Having an explicit expression for the output function Pi i=l,2 and k=l,2,...,M,
one can compute F..
1,)
according to (9).
In
the following sections we will consider two simple cases where the partial derivatives,
-i
. sk(t)/D
-.- A~___
i
i=l,..,6, can be obtained explicitly.
Ii
.,
III. LINEAR, CHEVRON, In this section, considered. Swith
AND CROW'S FOOT DETECTOR ARRAYS a special class of detector patterns is
The detector array consists of M parallogram detectors
the shorter width along the scan direction as shown in Fig. 2. The detector pattern is
"a
FI.
chevron array if one less than 900,
6k
called a linear array if
takes only two values, one greater than and
and crow's foot if
ek takes three values (i.e.,
a combination of linear and chevron arrays). the analysis, 1)
faxis,
L
2)
all 0 -90°,
For simplicity in
let us make the following assumptions:
the center of each detector is located along the scan i.e., Yk= 0 for all k. the output of each detector to a unit strength point source located at (x,y) has the same functional form:t Pk(tx,y)
where p(.) more,
is a symmetric, non-negative-valued function.
Further-
let p(t) be the autocorrelation function of p(t),i.e.,
L
p(t) = It
(10)
p(t-(x-xk) + y cotok),
is easy to see that p(0)
J
p(T)p (T-t)dT.
(11)
=1 from the definition of Pk(t,x,y) in
).
Rewriting
(3) in terms of the parameters considered in Section
tThis implicitly assumes that the detector has infinite length or no edge effects on the detector output.
12
I+ j
II with the above assumptions, sk (t)
a P t-(Xl
we obtain +
) + YlCOtek
aaRp t- (xl+rcos0-xk) + (Yl+rsin0)cotO k).
Substituting (12)
into (9),
(12)
we have Fij in terms of the following
functions: p(T)
=J
p(t) p(t-T)dt
(13)
-00
S)
p(t)lt-T)dt
and
"p(T) =-f
(14)
p(t)p(t-T) dt(1)
The expression of each element of F is given in Appendix A. The lower bounds on the variances of the estimates for a,, R, x', yI, r,
and 0 are: CRB (a 1 ) = (F-lll
(16a)
CRB (R)
(16b)
=
(F-I22
CRB (xi) = (F-)33
-1
CRB (yl) = (F-)44
L
CRB (r)
-1 = (F-)55
(16c) (16d) (16e)
51 CRB (0)
r
"13
= (F-I)66
(16f)
Tflhe results in this report are presented in the followlng normalized forms:
CRB(a )/a
(17a)
ER _;CRIB(R)/R
(17b)
E
(/d CRBx
(17c)
CRB(Y)/dx
(17d)
Er =~ EA
CRB (r)/r
(17e)
EA
CRB(0)
(17f)
Ex=A
E
where dx is
the scan direction.
the detector w.idth in
All results
where
will be presented at SNR =10 for each detector,
1I
(18a)
SNR 1 A al/ca 1/2
and
A[
b~
It
is
(18b)
P(t)dt
easy to convert these results to any signal-to-noise
ratio,
SNR, by the following relationship:
E
at SNR = (E
where w = a, R, xl, Yl' r,
at SNR 1 =10)
x (10/SNR)
(19)
and 0.
For the numerical results presented here,
14
let us consider that
r the function p(.)
is generated by convolving an infinite length
rectangular detector of width, dx, along the scan direction with point spread function of the form [101
Sa
E sf
•(xxo2
2 (xy)exp
+ (y-yo)2 2c2
[
(20)
where E is the image irradiance at the center of the diffraction pattern and =
.3040
(21)
dx.
The value of 0 is selected such that 90% of the total energy is collected by a detector centered at (xoyo). Figures 6a and 6b present the Cramer-Rao bouinds on the estimation errors for a linear array detector (detector M in Fig. 2)
for the amplitude, a 1 , and the in-scan location, x1, respectively.
The upper half of each figure is the Cramer-Rao bound computed for the 1-D problem, in our previous reports (2] targets are located along the scan axis.
[41,
where
The lower half of each
figure shows the equal performance contours plotted on the sensor focal plane for the 2-D problem with one target at the origin and the other at position (x,y). Because the detector pattern is scanning along the x-axis, the equal performance contours plotted in the lower half of each figure are symmetric about the crossscan axis (y-axis).
This symnetric
figures shown in this report.
property is ii. common to all
Notice that the equal performance
15
Li• .II•
z.. ...
' - . . . .
... -. .
-
.
.
.
..
-_ L _ .. . . .
. . . . T•-- - -M
D '-
40 I
9n r8
.2
!.
SCAN AXIS 34
IP
I
SCNAI
1.0
-
S-
° /
.i
E
,,•7I
B
A
D
/
\
-
-
Si~0.4
/ 0.0 -2 0
"4
-1.0-
1.0
0.0
2.0
Target Cross-Scan Location In Detector Widths Fig. 10. The 5-detector brickwall pattern and the normalized peak of each individual detector as a function of target cross-scan locaticn.
-'
"I
I
I
I
I
_
".
I
center of the detector as indicated in Eq. (26).
The normalized
peak amplitdue of each detector output is plotted as a function of
r
the cross-scan target location in Fig. 10 for the detector pattern
shown in the figure.
The brickwall pattern differs from the pre-
vious chevron and crow's foot patterns in using amplitude rather than separation estimates from individual detectors to obtain information on target separation in the cross-scan direction.
In
spite of this difference, the resulting performance is quite similar to that achieved by the crow's foot. The brickwall pattern occupies much less area on the focal nlane than does the crow's foot pattern.
However to achieve the
performance presented in Fig. 9, optimal signal processing of the outputs of at least 5 detectors was assumed.
Simpler suboptimal
processing may be carried out by combining adjacent detectors as shown by the dotted lines in Fig. 10.
In this way, a crow's foot
pattern may be synthesized from the mosaic pattern.
40
V.
DISCUSSION AND SUMMARY In this report, we have derived the Cramer-Rao lower bounds
on the variances of parameter estimates for two CSOs separated in both the in-scan and the cross-scan direction as measured by two classes of detector patterns.
Numerical results are presented for
a particular point spread function and a fixed detector width in the scan direction for both detector patterns.
The trends of the equi-
performance contours of each parameter of interest for both patterns For the cases considered in this report, the
are quite similar.
achievable parameter estimation performance of the 5-detector brickwall (mosaic) pattern is,
in general, poorer than the 3-de-
tector crow's foot patterns for targets with SNR of 10 per detector.
This is because the overall received SNR of the particular
brickwall (mosaic) pattern considered is crow's foot pattern.
lower than that of the
To achieve the same performance using the
mosaic pattern would require more detector columns (>2) a greater signal processing load. *
pattern may still
However,
as well as
the brickwall
(mosaic)
require a smaller focal plane area to achieve
the same performance as the simple crow's foot pattern. We have not implemented an algorithm to do 2-dimensional CSO detection and parameter estimation.
L
No simulation performance re-
sults have been obtained to compare with the theoretical bounds The signal processing load for the 2-D problem presented here. increases as a power of the number of detectors in the pattern,
41
if
an optimal performance algorithm is implemented.
For a sub-
optimal algorithm, the predicted performance presented here may
I:'
not be easy to achieve.
This is an area which requires further
investigation. General trends in estimation performance were summarized at the ends of Section III and IV.
1
42
F L rAPPENDIX
A.
IJ
THE FISHER INFORMATION MATRIX FOR LINEAR# CHEVRON, AND CROW' S FOT DETECTOR ARRAYS
S
S A0 sk(t) = alp(t-Tk) + alRp(t-•TkAk), EI
where (A.2)
Xk •l-Xk-Ylctk and = rcos8 -
rsinecote
(A.3)
.
k~ke Then,
the derivatives of s sS(t)
___k__
S•~
= p(t-Tk)
are:
+ Rp(trkAk),
(A. 4a)
-A
(A.4b)
a1 Sk(t) a _aP(t-Tkk
D? a R •~~
s k(t)
+Rp(t-Tk-Ak]
=-a[P(t-T)
(A.4c)
xI
[
k) ] , _T k _A~
(A o4 d )
sin6 cotek)Pt- k-Ak),
(A.4e)
k) l c ot e k [p ( t _T a
= t - - --- ---
+ Rp (t
Sosk(t)
kr
-
R(cose
-
43
s of the following functions: P
f
S/
-f
p(t)p(t-r)dt,
(A.s5a)
a(t)p(t-T)dt,
(A. 5a) (A.5b)
-00
and
C
we have
M for i,j=1,2,..,6
(f..)k
F..
(A.6)
where (1+R )p(O)
(f
2 Rp(
(A.7a) (A.7b)
alp(Ak) + a1 R P(O)
f (12)k
a1 R(cosO
(f15)k we
(f
-R
f 22)k S23)k
d
-
hav Mr(sinc
sinecotek);(Ak) + cosecotk)P(A)
(.c (A.7d)J
2,(O
(A.7e)
al P(Atk)
(A.7f)
44.
€24)k = a1 2cotek ( =_a 12
(
R
(A.7g)
Ak
A.h
)*p*_2alRpA
-cote(f) 34 k = tk 33 k (f35)k =--a2 R(cosO - sinecotek) Mp(Ak)
(A.7i)
(f)
(f44)k =-cotek(f (f45 k
3 4 )k
(A.71)
Cotk (f35)k
(A.7m)
wcotok (f
(f46)k
(Ao7k)
+Rp*(0)]
1 R(sinO+cosecotek) [4pO
(36)k
(A.7J)
+ Rp(0)]
(f55)k =-a
1
(A.7n)
3 6 )k
22 R (cose-sinecotek)2p(O)
(A.7o)
( f6
R r sricos-sincote (s in@+cos)cotOk)ý*(0) 222 ~ kk 2*p.10) =a2R 2 R r21 (sin6+cos~cotek
"(fl31k
=
(fR56)k
a
(A.7p) (A1.7q)
and (l4)k
=
(f25)k = (26)k
=0
where p and p are the 1st and 2nd order derivatives of p, respectively.
Inverting this 6 x 6 symmetric matrix, F,
one should
be able to obtain the Cramer-Rao lower bound on the estimates as shown in
(16a)
-
(16f) of Section III.
45 "'•, --.
,.
.-
4.5 _
0..
r
PATTERNS
r
Let Sk
II
ARXFRBIKALDTCO
THE FISHER INOMTO
APPENDIX B.
=
+alRpx(t-T k rcos)pY (yk-71l-rsinf) kt lXtT)Y(Yk-Y1) (B.1)
T
1k-
where (B.2)
Then, the derivatives of skare:
Sk~t
)p (cs1) + pY(Yk
=ap(t-T
_____
1
1yinO -T
) (B.3bO
(B.3d)
~ k~
a
_
-
_
csj (t-krcs Tk-r(yk6) psine),(.
)
-)-
_
y
(B.3c)
k-
(
-T
(
-
)-,
p~-T-
rcosO--)---
rsin6)
a
rsine)
akl)_aRrsinxltT k~-rcos'lp y'k-.a(t-R rCoSPx(t-Tk-rcosy
i
(yk-y-rsin), (B.3f)
[
where 6x(.)
respectively,
are the 1st order derivatives of px and p
y(.)
and
,
For notational simplicity, we will use p, and P2
in the rest of this Appendix to denote py(YkYl
and p yk-il-rsinS),
respectively. Substituting (B.3a)
-
(B.3f) into (9)
and rewriting it
terms of p, the autocorrelation function of p and its
in
defined by (11),
and op,we have
derivatives p,
M F.i
=
2
Eoo = (
NJ ok=1
) ,
for i,j
ij k
= l,2,.,.,6
(B.4)
where (f)
=
(p
2 +p
2
2 2
)p(0)
+
2Plp 2 Rp~rcose)
(B.,5a)
2 (f12)k = a 1 PIP 2 P(rcose)
+
alRP2
(fl4)k =-al(DlPl+R2 p 2 p2 )p(0)
(f15)k = alRplP2 cosOp(rcose)
-
(0) alR(plP 2 +plP 2 )P(rcosO)
6 )k=-alRrplp 2
sineý(rcose)
(B.5c)
- alRplP 2 sinep(rcosO)
-a R2 p 2 P2 sin p(O) (f1
(B.5b)
(B.5d)
- alRrplP2 cos6p (rcose)
-alR2 rP 2 P2 cosep(O)
(B.5e)
47
*1
7--
•,-" a.. . ...• ... .,...•:•:.•',-• :• •.
,•.•.
7
• :..•, . •,•... .......... . .• , ,:.[:,• • • -. -• a .•,"• ,;,:..4%• ,,''• : •• -.• ,•,.. ..vv, • ,
1'
(f
2
I 3)2
(B.5f)
22p
(B. 5g)
.a1 2p p ( o )
f24)k -- al plp2 p(rcos8) f253 k -- a 1 lRp 2
p
2
(B,5j)
ak2 Rrp 2 p•2 C seo) -'al =-26 f33) k -- a1 2 (pI 2 +p
2 2R)°(0)
2al 2 RPlp 2 P(rCOsB)
-
2 (f34)k = a 1 2 a(Pi• 2
I
(B
222
-~2
(B,51)
jiP 2 )•(rCose) . aep(0
(B.5j)
1
o'rcs
(f35)k =-lRpp1
a1~2R2cosep°(0)
-
2 RPlsin S p(rcos)
e*(B. c+aoR
2a "(0
(B, 5n)
+ a 1 2 R2 rP 2 2 sin62'(0) +1
(a 1(
• 2 2 )p()
2
RP l P
+2a1
2
p(rcose)
aI 2 RPlP2 sin)p(rc(se)
_ 2(45)k Pr
+ a1 2Rp 2
(f (f4)
5k)
+ a12 RrPPC°SP(rcose)
(f36)k =-aRrplp 2 sinl°(r
(f
(B 5k)
s22 (O
aa = a 2 Rr lP 2 SinP(rcos0)
2r
(B.5p) lP2cosp"(rs
R
+
22o2
2 =-a
1
2
2 P2
(f~~~~..
2
2
'(0) . +
+a55,k R 1 CSP c..-4-asnO~coO
48
(B.5q)
()
Rr coso a 1 (
2
2ac
p(0(B5r)
(B.5r)
[R
2ri
co
*1R~hncO~
2
o 2,a2 P(O) ' ~()
(f6 ) =-a 2R2r 2[P 22 in2 oop(O) P2i 66 k
F4 II3
2
-
2c0.2ep(o)
(B.5t)
ACKNOWLEDGMENTS The author would like to thank Dr. S. D. Weiner for his
The author also wi hes' .top
critical commients and careful reviews.
th~ank Frances Chen for her programming assistance#, and personally acknowledge C. A. Tisdale for patiently typing this report an
E
~all
its revisions.
I50 k
~
..
REFERENCE83 1.0
D, L.'Fried, "Resolution, ,Signal-to-Noise Ratio and Measuremenit Precision,".Opt~ical Science Consultants, Report No. TR-034, (October 1971), also published in J. Opt. Soc. Am.. 69, 399 (1979).
*2.
R. W. '1%iller, ":1Aqcuracy of Parameter Estimates for Unresolved'ýbjects," .Technical.Note 1978-20, Lincoln Laboratory IAD1016 M. I.T.(8J (8 June J9 78,, DDCAD02l8
I. 3.
M J. Tsai.and. K.. P. Dunn, "Perforioancd -Limi-tation on Par'ameter Estimation of .Cloiscly Spaced Optioal'Targets Using Shot-N.oi se, Juetco
oe,1Tefn6a`Nte97-5
Lincoln
5.
n Estimation of eeto e'oTal'iuaio'S~yo CIlosely Spaced Optica~l Targets,h: technical Note 1980-19, Lincoln'Laboratory.. M.I.T.-(18 March 1960) DTIC-,AD-A088090/9 .
6e;
D,, -L. F'ried,. "Distinguishing. Between a SigeObject 'and Closely Spaced Objects., Signal"Processing Analysis," the Optical 'Scier.,e Crmipany,- Report No. T1R-299, (February 1978).
7*
Y. Kima~hi, ".Dua~l Pulse Matcher Performance Analysis," Passive'Optical Al ori~hzn Development for Layered Defense :Program, Nichols Research Corporation, Report No. NRC-LDST-017A Tr October 1978)..
Be8
H. Akaike, "A-New Lcok at the Statistical Model Identification," IEEE. Trans.-Automat. Contr. AC-19, 716 (1974).
9.
He. L. Van Trees , Detection,ý Estimation and Modulation Theory, Part I, (Wiley, New York 1968).?
1.0,
'
K. Seyrafi, Electra-Optical Systems Analysis, (Electro-Optical Research Compnay, Los Angeles, 1973).
4T.I
4A
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I ;5. TYPE OF REPORT & PERIOD COVERED
TITLE (and Subtitle)
4.
3. RECIPIENT'S CATALOG NUMBER
Accuracy of Parameter Estimates for Closely
Spaced
i ,
Optical Targets Using
6. PERFORMING ORG. REPORT NUMBER
•~( ) j71
7. AUTHOR(s)
Keh-Ping Dunn 9.
Technical Report
Multiple Detectors
Technical Report 589
--
S. CONTRACT OR GRANT NUMBER(s)
"
F19628-80-C-0002
PERFORMING ORGANiZATION NAME AND ADDRESS
10.
Lincoln M.I.T. P.O. BoxLaboratory, 73 Lexington, MA 02173"/./
Program Element No. 63308A
/
11.. CONTROLLING OFFICE NAE AND ADDRESS
•Air
12.
Force SystemF Command, USAF Andrews AFB
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NATE
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Electronic Systems Division Bedford, MA 01731 Hanscom AFB
16.
REPORT
23 October 1981
DC• 20331
Washington, S~58
PROGRAM ELEMENT, PROJECT, TASK AREA & WORK UNIT NUMBERS
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KEY WORDS (Continue on reverse side it necessary and identify by block numlber)
Cramer-Rao bound closely spaced objects parameter estimation
passive optical sensors degradation multiple detectors
70. ASiSTRACT {Continue on reverie aide if necessariy and identify by block number.'
(In order to obtain the cross-scan position of an optical target, more than one scanning detectors are iF
used. As expected, the cross-scan position estimation performance degrades when two nearby optical targets interfere with each other. Theoretical bounds on the two-dimensional parameter estimation performance for two closely spaced optical targets are found. Two particular classes of scanning detector arrays, namely, the crow's foot and the brickwall (or mosaic) patterns, are nontidered.
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