Microlocal Structure of Inverse Synthetic Aperture Radar
Data
Margaret Cheney
and Brett Borden
Research Department, Naval Air Warfare Center Weapons Division, China Lake, CA
93555-6100
USA
permanent address: Department of Mathematical Sciences, Rensselaer Polytechnic
Institute,
Troy, NY 12180 USA
permanent address: Physics Department, Naval Postgraduate School, Monterey, CA 93943
USA
Abstract. We consider the problem of all-weather identification of airborne targets. We
show that structural elements of the target correspond to identifiable features of the radar data.
Our approach is based on high-frequency scattering methods but is not limited to the standard
weak-scatterer approximation: we also analyze multiple scattering and structural dispersion
(situations normally interpreted in terms of poorly-behaved image “artifacts”). This work
suggests a method for target identification that circumvents the need to create an intermediate
radar image from which the object’s characteristics are to be extracted. As such, this scheme
may be applicable to efficient machine-based radar identification programs.
PACS numbers: 41.20.Jb, 42.30.Wb
1. Introduction
Object identification from reflected radio waves is an inverse problem with a long history.
This challenging problem is still mostly unsolved but the impetus for the work is high
because, if perfected, such methods would allow for reliable recognition of non-cooperating
targets in all types of weather and at great distances. Radar-based target recognition efforts
share a great deal in common with other problems of remote sensing and current practice
attempts to perform target identification/classification from fully formed radar images. Of
course, constructing an image of a target from radar data is a very difficult task all by itself
since the reflected field data are noisy and are usually collected from a very limited set of
(generally unknown) target orientations [19, 29]. Additional complications arise in the realtime implementation of imaging algorithms in realizable radar systems.
Automatic classification systems, however, should be able to skip this imaging step
because a fully-formed image is probably not required for machine-based target recognition.
This observation, of course, begs the question of “what components of the raw data set are
relevant to target identification?” In this paper we examine a systematic method for extracting
structure-relevant information directly from measured radar data without the need to first
construct an image of the target.
Our approach relates the singular structure (such as edges) of the target to the singular
structure of the data set. Restricting our attention to the singular structure—specifically, to a
certain set in phase space called the wavefront set—allows us to use the tools of microlocal
analysis [9, 13, 32]. This strategy was first applied to imaging problems in [1]; its uses in
Microlocal Structure of ISAR Data
2
seismic prospecting [2, 6, 10], X-ray tomography [11, 16], and Synthetic-Aperture Radar
[23] are active areas of research. An approach similar to the one we pursue here, in which we
use microlocal analysis not to do imaging but instead to study the connection between features
of the target and the data, was considered for the X-ray tomography problem by Quinto [25].
We begin in section 2 by examining the general properties of radar scattering and
developing mathematical models for the measured data. These models involve Fourier
Integral Operators with kernels that are oscillatory integrals; consequently these models can
be studied with the techniques of microlocal analysis. Next we present an overview of the
microlocal concepts and theorems that are relevant to our investigation (section 3.1). These
two sections serve to introduce our notation and define our terminology. Section 3 contains
our main results and calculates the wavefront sets associated with several important radar
scattering situations: weak scatterers; multiple scatterers; and structural dispersion (all cases
are limited to targets whose rotational acceleration is negligible). We conclude our discussion
in section 4 by considering several connections between our results and existing practices and
problems of radar imaging systems. In particular, we briefly discuss targets whose behavior
is not well modeled by our assumptions and suggest potentially fruitful paths for further
research.
2. Radar data
Traditional radar systems transmit an electromagnetic waveform (a pulse) and measure the
time delay and frequency shift of the corresponding waveform reflected from a target so as
to estimate that target’s range and speed. When very short-duration pulses are used, it is
possible to accurately determine the range to individual target substructures. Such high range
resolution (HRR) radar systems can be used to obtain a target’s local integrated scattering
strength as a function of its range.
These one-dimensional “images” are known as range profiles and are used by many
all-weather target recognition systems. But target-identification procedures based on range
profiles suffer from a lack of target information in dimensions orthogonal to range since
range-only radar data maps the reflected energy from all equidistant target substructures to
the same point. Such ambiguity can be partially removed by considering multiple pulses that
interrogate the target from different directions. The different target views, which are also
known as target aspects, collectively define a synthetic aperture and more complete target
images can be recovered from multi-aspect data by, for example, backprojection methods.
In principle, there are two basic schemes for creating synthetic apertures: either the radar
measurement system can move relative to the target (a configuration known as syntheticaperture radar, or SAR); or the target can rotate and sequentially present different aspects
to the radar (a situation known as inverse synthetic-aperture radar, or ISAR). In practice,
of course, one usually sees a combination of these idealized cases and the terminology is
somewhat artificial.
Evidently, cross-range resolution depends on the size of the synthetic aperture. In ISAR
systems, this means that cross-range resolution will also be related to the length of the time
interval over which these data are collected because the observer must wait for an aperture
to be established by the rotating target. For a well-behaved target (i.e., one rotating at
constant rate), cross-range resolution therefore depends on the duration of the measurements.
Rotational target motion also induces a differential Doppler shift in the target’s cross-range
dimension. This observation is the reason why ISAR imaging is sometimes called “rangeDoppler imaging” (although, in HRR systems the Doppler shift associated with target rotation
is usually too small to be directly measured on a pulse-by-pulse basis).
Microlocal Structure of ISAR Data
3
Ultimately, the behavior of radar data is determined by scattered-field solutions to the
wave equation. Since radar systems transmit and receive radio waves, we should generally
examine the electromagnetic (vector) wave equation. For simplicity, however, we will
examine the scalar wave equation and assume that the components of the electromagnetic
field each satisfy "!
#$&%'
(1)
We write the total field as a sum of the incident and scattered fields !
/+ is resulting equation for
!
0 213
8 0 ! /+
170
#4 5!
$
#6
!)( *,+-.!/+ ; the
$
(2)
$
is the target scattering density at time and position :9<;>= .
where
We can write (2) as an integral equation
0FE G
0CDFE
HG 170FE G
FE G
! /+
where [31]
A
#$@?BA
0C&M
M NO
M M
#$LK
P Q
0 $
I !
?SR
4J
J
>V WV XZY
[
( TU M M
J ]
\ Q
(3)
(4)
A
#$
.
satisfies
K
K
In section 2.1, we develop a mathematical model for radar data and explain the
fundamental role played by the weak scattering approximation. We examine the multiplescattering case in section 2.2, where we construct an exact scattering solution for two isotropic
point scatterers. In section 2.3, we consider a model for scattering from a reentrant structure
such as a duct or engine inlet.
2.1. Weak scattering
There are a variety of situations when the approximation known as the Born approximation,
single-scattering approximation, or weak-scattering approximation is appropriate [15, 17].
Under this approximation, we replace the full field ! on the right side of (2) and (3) by the
incident field !( *+ , which
converts 0(3)
CDinto
FE
HG 170FE G
FE G
FE G
! /+
#^@?BA
I ! ( *,+
4J
J
'
(5)
The value of this approximation is that it1 removes the nonlinearity in the1 inverse problem: it
replaces the product of two unknowns ( and ! ) by a single unknown ( ) multiplied by the
known incident field.
For radar measurement systems, the single scattering approximation is the basis for a
crucial method for estimating the scattered field in the presence of system noise. This is a
`
serious issue, because
the energy of the scattered field at the receiver will be reduced by at
least a factor of _
(where _ denotes the distance between the radar and the target and
typically ranges from ten to one hundred kilometers). Thus the signal measured by the
radar will typically be small in comparison with the thermal noise voltage. This difficulty
can be overcome by correlating the received signal with a model of the expected reflection
signal; by this means, radar systems can significantly reduce the effects of system noise and
extend the effective range _ without having to increase the energy of the transmitted signal
to impossible levels [4, 8]. The signal model generally used for such measurements is based
on the single scattering approximation: the scattered field is presumed to be a time- and
Microlocal Structure of ISAR Data
4
frequency-shifted replica of !)( *,+ . The term “radar data” usually refers to these correlation
receiver measurements.
We assume that the incident field is a series of pulses, beginning at times $badc6feg$
h
Zi4','' from an isotropic point radiator at position
6 mF,noso
>V Wpthat
qpV XZY
[
E G
! (c *+
j$
?lk ( *+ ]
where
E
I I
(T U M
\ Q
R
i
J4]
0 E
h
k ( *,+ ]s$&tvuw ( *+x ]s$
E
DGrM
Q
?
w ( *,+
R
(T
I
(6)
E
J
(7)
is the Fourier transform of the signal used to establish the interrogating field transmitted to
E
1z E G
} ~ 0 E G
E
E
the target.
We also
assume that the
target is translating
with }z
velocity
y and rotating, so that at
{$B|
denotes a rotation operator
time , we have
y f , where
(an orthogonal matrix).
0
We consider the monostatic case, in which the transmitter and
receiver are co-located.
)
!
0
/
+
. This field induces a
At the radar, the field due to the e th transmitted pulse is thus c
CV W))qpV XZY
[
system signal whose Born-approximated
value we denote by w /+ €e :
„} "~ E 0G
I
( TU M DGrM
\ Q
‡ E E
E
y f
"m n CV W))q)V X#Y
[
E
FE G
I I
G€M
(T U M H
†
] k ( *,+ ] R
J ] J ]jJ J
'
(8)
\ Q
EE
E } ~ 0 E E
a c , and make the
In (8), we neglect the
overall targetG velocity (set y $ˆ% ), let $
mFnd[Ž V approximation
XZY
[
converts (8) into
change of variables ‰ $ "mFno>- V Wpa ‹ c . This
II
I IŒ
M
( TU M :}70 E E U
| ‚ ‰ w /+ €fe $ ?ŠR
\ Q
- a c ‰ CV W)"‹
mFnd[Ž4V XZY
[
‡ E E
EE
E
I II
I IŒ
M ‘}z UE E
M
(
T
U
†
] k ( *,+ ] R \ Q
J ] J4]jJ
J ‰ '
(9)
M “’H M
M M”–- • a ’ c ‰
M M "~
•
$ {
— • -‘˜
}›šr“
(with the hat denoting
We use the far-field approximationM M
™
$ ,
j$
to rewrite (9) as
unit vector)
and
the
notation
_
"mFn mFnd[¢¡£Žd[0XZY¢¤
‡ E E h
Œ“Ÿ ž I I Œ
II
( Tœ
U
U
w /+ €fe ^
? | ‚ ‰ ] k ( *,+ ] \ Q
R mFn5[¢¡ Žd[0XZY¢¤
_ E FE E
I II
Œ“Ÿ ž I I Œ
†
(T œ
U
U
¢¥s0 E ˆ (10)
J ]jJ4] J
J ‰ '
R
EE
f‡ E E ¥s0 E D
EE
&$
Next we correlate the scattered signal with a signal of the form w ( *,+
¦
]
f4J4]
k ] to
obtain
the
output
of
the
correlation
receiver:
0¥r E H
E
R”¥ §4¨
E
© ‚ adc6 j$ ? w
4J
/+ €e w ( *+
6mfno
mfnO[¢¡ Žd[0XZY¢¤
E
E
0
‡
EE h
Œ“Ÿ ž I I«Œ
I II
( Tœ
U
U
$
?ª|ƒ‚ ‰ ] k ( *,+ ] k ( *+ ] \ Q
_ mFnd[„¡ Žd[0XZY¢¤
R[
E
E E FE FE E
I II
Œ“Ÿ ž I I Œ
I I ¬ I †
(T œ
U
U
(T
U
J ]jJ ] J ] J J
J ‰ (11)
R
R
E
]
w /+ €fe $.?
R
|ƒ‚
where the bar denotes complex conjugation. In (11) we carry out the integrations over
to obtain
mFn5[¢¡ Žd[0XZY¢¤
¥
0‡ E E
EE h
© ‚
adc6 j$
­
i
Q
= _
| ‚
?
†
R
I II
Œ Ÿ ž I IŒ
“
(T œ
U
U
k ( *,+ ] k ( *+ ] 6mfno
mFn5[¢¡ R Žd[0XZY¢¤
E
E E FE E
I I¬ Œ“Ÿ ž I I Œ
II
(T
U
U
U
J ] J ] J
J ‰ '
‰ and
]
(12)
Microlocal Structure of ISAR Data
5
}v
E E
EE
For}zthe
remainder of this section we assume
that ® adc is sufficiently small that we can
0
- ac4 as a function linear in . We use this approximation
expand
in section 3 to explore
the “usual” imaging radar situation corresponding to a signal w ( *,+ made up of a series of
short pulses. In section 4 we also briefly consider the limiting case of a monochromatic signal
}70 E E
}z
made up of a single long-duration pulse.
EE
}z E E
7
}
7
}
E
E
- a c in
When rotational acceleration is negligible ( ® a c 8$ª% ) we can expand
- a c °$
$¯% , so that
a c -²± a c . We introduce the
a Taylor series about
•
• •
•
{} š notation
™
™
•
} a c €µ´ c ‰ } š ³ _ - • ™ c — ‰ v
a c $} š and
¶ c
±
c—
adc ± a c) ‰ $
adc H— ‰ '
‰ ³
(13)
• E E
}
¶
The quantity is the down-range component of velocity at the point ‰ due to rotation. With
• 0 E E
•
E E write
the term involving ™
- a c in
the notation (13) we can use
the orthogonality of to
™
™
•
¶
0
E
E
- ac— ‰ $
c— ‰ c ‰ .
the exponent of (12) as
™
- a c — ‰ in (12), we find
Using
this
expansion
for
6mFno
n
n [0XZY¢¤
¥
0‡ E E
EE h
I I ¬ Œ¹¸ I I
II
© ‚ adc6 j$
(
T
œ
U
·
? | ‚ ‰ ] k ( *,+ ] k ( *+ ] ­ Q
n [0XZY¢¤ R E
n
E E FE E
i = _
I II
Œ¹¸ I I
†
(T œ
U ·
J ] J ] J
J ‰ '
EE
(14)
EE
R
0‡ E in (14)
E yields„¥ E ¥ "~ E
Performing the and ] integrations
¥
h
¥
] k ( *+ ] k ( *,+ Ndc ‰ ] © ‚ adc6 j$
? | ‚ ‰ PQ
h
¶
E
_ n Žd[ 6- mfnd[
c ~ ‰ n Žd[[ n Žd[0XZY¢¤
I ¬5I
Œ¹¬I
†
(T œ U U
U
U
· U
J ] J ‰ (15)
R
NO
where
¥CE h
¶ c NO
¶
i c ‰ ‰
h
$
'
c ‰ ³ h
(16)
¶
- ¶ c ‰ ¥ E
c ‰ ³
c7
™
We interpret c as the Doppler scale factor. Equation (15) is our model for the radar data
in the weak-scattering case. We note that (15) expresses the data © ‚ as a Fourier Integral
E
Operator acting on | ‚ .
remark also that the‡ ] E integral
of (15), ¥ E ¥ "~ E ¥ We
[
E CH E ¥ E ¥ "~
E
E
º
c
$.?
c
]
k ( *+ ]
k ( *+
c
]
R
n
I ¬5I (T U
I
J ]
(17)
is an imaging kernel related to the radar ambiguity function and, in the presence of
measurement noise, determines the resolution to which |ƒ‚ ‰ can be estimated [8].
2.2. Multiple scattering
Multiple scattering does not fit into the weak-scattering model. In the case where there are
only two isotropic point scatterers, we use the exact solution derived in Appendix A for the
}z E
scattered field due to the incident
wave (6). We consider the case of a rotating target; i.e., we
‰» :
replace ‰p» of (A.8) by
E
0
! /0+
$
¼
h
CD E M
~ \ Q
?
»½ V W)"‹
I
M :}70U † Á R (T
Â
I
A
M
E
‰ »
-
M
N4
~ ¾ » k ( * + I«]¿ ‰ » h
P QÀ
5
( T
V W)"‹ ¾ [Ž¾ à V XZR Y
I
"
m n [
E E
I¿
I
I
MÆÅ
I I
(T
( M T :}zU E
I R P5QÀ
(T U
J ] J
R
I
¾ »
‰ »
R
[ŽÄÃ,V X#Y
:}70 E
(18)
Microlocal Structure of ISAR Data
E
6
E
h
h
where Ç $
if Ç $.i and Ç $Èi if Ç $
. Equation (18) is simplified as in section 2: we use
EE
E } A
the oscillatory-integral
representation
(4) for
; make the far-field approximation; apply the
$
ac ; expand
change of variables
in a Taylor series; and use the notation defined
above (10). With these substitutions we obtain
E
n ŽÃf[ [¢X#Y¢¤
6m n n ŽÄÃ[
¼
k ( *+ ] ( Tœ
»
?
S
¾
R
h
\ Q
¢Ã É Ê Ë Ì
_ Ã É5Ã
¾ » ¾
I„Í
n Ž Ãf[ n ŽÃf[ 0[ XZY¢¤
I¿
I I II
Œ¹¸
(T
II
†bÎ ( T œ I R
(T œ
U · U
U
P QÀ
¾ »
R
R
M
M
I
À
where ³ ‰ »
‰p» .
! /0+ €fe $
h
II
»
II
I
R
( T
I¿
UN4
· U
PQÀ
¥
©OÑCÒÓ Ô adc6 h
s$
PQ
¥ "~
Î k
( *+Õ
-
» c
»ZÖ
]
¥
I
P ¾ Q» À k ( *+ Õ
5
where
» c
»ZÖ
c ³
h
( T
(19)
E
Nk ] I ¿¾ » P ( *Q+ À
h
I
¶ c
Nd
‰ » R
Û d
N ¶ c d
N ‰
'
¶ c
‰ » h
C
Ñ
Ò
Ó
Ô
, (20) reduces to (15) for |
‰ Æ$
¾ »7Ý
-
(20)
à n no ~
à n [ n ŽÄÃ[
n ¡ ŽÃ )ŽÃf[[¢X#Y¢¤ Ð
E
I
I ¬ ÃÙI Ø Œ¹¬ ÃÙI Ø
Œ ¿ ŒÚŸ ž
(T œ
UU
· U
U
J ] E×
]
h
h
II
U
¾ » ¾ » R
à n n ~
à n [ n ŽÃf[0XZY¢¤
I ¬ ÙÃ Ø ŒÏ¬ ÃÙØ
(T œ
U
· U
R
¥ ~
I
¥ÏÛ
»ZÖ
¼
Ã¢É Ê Ë Ì ?
à É5Ã
IÍ
E× _ ¥
†
¥
n ŽÄà [
n ŽÃ [ [¢X#Y¢¤«Ð
E
EE
I ŒÏ¸
I II
U · U
U
J4] J ]jJ
The output of the correlation receiver is
Œ¹¸
~
~
(21)
IrÜ
(Observe
that
when
‰
‰ ¾ » ¾ »
¾ K
‰
‰ .)
¾ K
Expanding the denominator of (20), retaining only terms cubic and lower in , and
¾ »
simplifying,
we obtain
¥
E
h
© ÑCÒÓ Ô a c s^
¥
PQ
_ ¼
Ã¢É Ê Ë Ì ? h à É5Ã
I„Í
¥
¥ "~ E ×
†<Î k
( *+ Õ
» c
»ZÖ
I
- 5
P ¾ Q» À k ( *+€Õ
-
» ¾À »
¾ PQ
I
]
¥
E ×
]
R
¥ "~
¥
» c
»ZÖ
(22)
à n n ~
à n [ n ŽÃf[0XZY¢¤
I ¬ ÃÙØ ŒÏ¬ ÃÙØ
(T œ
U
· U
R
¥ ~
I
» c
»ZÖ
k ( *+Õ
Nd
k ( *,+ ]
¶ ¾c »
‰ » ]
à n n n ŽÃ [
à n n ŽÃf[
[¢X#Y¢¤
I ¹
I ¬ ÃÙI Ø Œ ¬ ÃFI Ø
Π(T œ
U· U
· U
E×
R
à n n ~
à n [ n ŽÃf[ [0XZY¢¤ Ð
E
I ¬ ÃÙØ Œ¹¬ ÃFØ
Π(T œ
UU
· U
J ] '
Equation (22) is our model for radar data in the multiple-scattering case. We note that (22) is
a sum of oscillatory integrals, to which the techniques of microlocal analysis can be applied.
Our multiple-scattering model (22) differs significantly from that of the weak-scattering
case in that additional bookkeeping must be performed to account for target substructure
‡
Nd
position relative to other scatterers. In addition, the multiple-scattering expression
depends
À
]Þ
on the overall target orientation and involves multiplicative terms of the form
R§4¨
(for some integer Þ ).
Microlocal Structure of ISAR Data
7
2.3. Dispersive scattering by reentrant structures
For reentrant structures with openings that can be associated with the location ‰ , the most
complicated aspect of multiple scattering (i.e., the accounting) can be eliminated. This
simplification is made possible by a model [3] for scattering from such structures that includes
wave propagation within the duct or cavity. Here, the analysis is done by treating the reentrant
ã
structure as a waveguide:
for | in (9), we use
Žd[0YæoÊFç
Ì 6è Ì
|›ß ]àfa c ‰ s$Èá,â
¼
ãåä
a c ‰ (
R
ã
¿
é
U
T
'
(23)
ã
In this equation, Þ indexes the eigen-solutions
(modes) of the waveguide problem, ê
ä
denotes the mode cutoff frequency,
is the strength of the mode, áâ is proportional to
À
the amount of energy that gets coupled into the reentrant feature, and ‰ is the distance
from the mouth of the duct/cavity to a scattering center within. We denote by ë the mouth
À
of the structure and assume that ‰ is constant over ë and zero off ë . We note that this
scattering model includes dependencies on ] and a .
•
•
We take
•
á,â
í
a c ‰ j$@ì
í
—
™
c Fî>â
‰ 6
(24)
where is the (effective) normal to the waveguide opening, ì is a coupling pattern that gives
the angular dependence of the coupling strength, and î â is a function that is supported in a
neighborhood of ë and is further characterized in section 3.4. Equation (23) models only the
contribution to the scattered field from scatterers within the waveguide; scattering from the
edges of the waveguide mouth is handled separately (as in sections 2.1 or 2.2).
In the time domain,
to
ã
Žd[0YæoÊç
è
FE (23) corresponds
áß
Since
À
fa c ‰ $Èá,â
a c ‰ a c ‰ $Èá,â
¼ã
¼
ã
ä
ä
Ì
¿
(
?
ã{ï”ã R FE
U
Ì
é
T
I
R
(T
J ]
6'
(25)
‰ is assumed
ïã 0FE to be constant
YæoÊfon
ç ë Ì ,è Ì
¿
I
é
(
T
(T J ]
$ˆ?
R
R
(26)
is independent of ‰ . This integral can be expressed in terms of the Heaviside function ð
the Bessel functionïã ñ5ò0Fas
ã°ó E E ô
E
FE E
Nd
NO ×
$ˆ?
†
ïã 0
Consequently,
õ
À
i
ð
?
ñ5ò Õ ê
‡ó
]
C
ê
is the convolution of ð
i
ã ÷ 3
0 ‡ "~ö ó
$
t
]
ê
ã
6'
À
NO R
( TU
ñ5ò Õ ê
I
À
i
II
ã ó
[
and
EE
J j
] J
ô
i
'
À
(27)
NO ×
with
(28)
Since the downrange dimension of a typical radar image is actually travel time, we can
see from (27) that the image of scattering centers located within ducts/cavities that obey this
model will not
be localized to a point. Instead, the associated image will be stretched and
ã
extended in the downrange dimension. The “stretching” property follows from the scaling
behavior of ê
in the argument of ñ ò . This general behavior is a consequence of dispersion—
waves reflected from such scattering centers exhibit a frequency-dependent time delay (as in
Microlocal Structure of ISAR Data
8
equation (25)). In practice, such nonlocal image elements can be difficult to map to the local
target structures that created them, and are usually considered to be image artifacts.
To obtain a model
for radar data, we substitute
for | in
(9) the expression
E
E
E
| ß ]
fadc6 ‰ j$
?
á ß
fadc6 ‰ R
II
(T
J
'
(29)
We carry out the computations (10) through (15) as before and, finally, in (15) we substitute
expression (29). We thus obtain
for the output
of the E correlation
receiver
¥
¥
E
E
© ß adc" where
$
¥
ø ß a c ?ªø3ß adc6 FE
‰ $
h
P5Q
_ ‰ Fá ß
‡
¥
E ]
?
†
fadc6 ‰ J
R
k ( *,+
n
I ¬I
(T œ
J ‰ (30)
0¥ E ¥ ~ E
k ( *NO+ c
] h
¶
n - ~ c n [ n ZX Y ¤
E
Œ¹¬I
I
U
·
J ] '
4
E
]
(31)
This equation is our model for radar data from structurally dispersive target elements.
Again, this result involves oscillatory integrals which can be studied with the techniques of
microlocal analysis. Although equation (30) is not quite in the form of a Fourier Integral
Operator (because it involves mutliplication in the adc variable as well as integration), it
could be converted into one by introducing another variable and modifying the phase of (31)
appropriately.
3. Wavefront sets for radar data
We focus on localized scattering centers such as corners, specular “flashes” from smooth
surfaces, and re-entrant structures such as ducts and engine inlets. These target features we
characterize by the singular structure of | , which we describe in terms of its wavefront set.
3.1. Wavefront sets
Mathematically the singular structure of a function can be characterized by its wavefront
set, which involves both the location and corresponding directions ù of singularities
[9, 13, 30, 32].
õ
õ
¢ÿ
of the function if there
Definition. The point ò ù ò is not in the wavefront set úÈû
õ
ÿ
3
þ
$
ý
%
t
, for which the Fourier transform
is a smooth cutoff function ü with ü
ò
ü
ù decays rapidly (i.e., faster than any polynomial) as
for ù in a neighborhood of ù ò .
õ
This definition says that to determine whether ò ù ò is in the wavefront set of ,
one should 1) localize around ò by multiplying
by a smooth function ü supported in the
õ
neighborhood of ò , 2) Fourier transform ü , and 3) examine the decay of the Fourier
transform in the direction ù ò . Rapid
decay of the Fourier transform in direction ù ò corresponds
õ
to smoothness of the function in the direction ù ò [16].
Example: a point scatterer. If |
>$
K
, then úÈû
|8$Âu
ù ù
$ý
x .
Microlocal Structure of ISAR Data
9
¥
| 3$ ¥ ð ô— C , where ð
Example: a specular
flash. Suppose
$ %p $@
ý % x .
function. Then úÈû |8j$ u € > — È
denotes the Heaviside
be specified closed sets ([14], p. 255):
Wavefront sets can
c
Theorem 1 If k $ u € ù x is a closed subset of ;
c
;
whose wavefront set is k .
†
c
;
%o , then there is a function on
Our strategy is to work out explicitly how the wavefront set of |
to the wavefront set of © . We take
the wavefront set of | to be
D$ ý |8$ˆu ‰ 6
©
úÈû
corresponds (via (12))
x '
(32)
To compute the wavefront set of , we use the following four theorems [9].
Theorem 2 (Wavefront
set of an oscillatory
integral) Suppose ø
W [ $.?
ø
R
( T U
is defined by
€F]sJ ]à
(33)
there is some and ë
where satisfies the following condition:
ã
ã ¾
set , the estimateM Ê Ì
M
M M
!
M 8M
#"
$
holds, with
c c Ê
c Ì
T
&%
€F]s
% )(
c h
-
Ý
]
for ã which, on any compact
~Z [FV CV
Œ
U
(34)
e . Then the wavefront
C set of ø satisfies
»
s$È% x '
úÈû ø @u €
$
'
Theorem
3 (Wavefront
Suppose
set of a product)
(35)
* - ù+ € ù,* r9 úÈû € ù+ €9 úÈû A4 x
õ
contains no points of the
form € . Then the wavefront set of the product
õ
õ
õ
A4-$
- úÈû A4/. úÈû
/. úÈû A6'
úÈû
úÈû
úÈû
õ
-
úÈû
A ³
3(
u € ù
0
3(
1
õ
1
(36)
A satisfies
(37)
Definition. Suppose
is a mapping from to , where and are assumed
to be smooth
õ
õ
manifolds, andõ
is a õ function defined on . Then the pull-back
is a function on
j$
f .
defined by
õ
We can alsoõ extend this notion to apply to distributions defined on , provided the
š
wavefront set of avoids
G
theG “bad” set
02
0
4 6 5
$
0
1
W
7 0 5$ 02
”
1
0
for some 9
98
'
1
(38)
Theorem
4 (Pull-back of a wavefront set) Suppose is a smooth mapping from to , and
õ
isõ a distribution on whose wavefront set avoids the set (38). Then the wavefront set of
š
is contained
in
02
1
W
(39)
0 2 õ $ 4 € ù ù $:7 0 5 for some 5 such that 0 ”65€9 úÈû õ 8 '
õ
(
(
Application to embedding a function in a larger space. If we have a function of the
)(
>(
variable õ , and we want to consider
it to be a function of the variables and ; , then we
õ
can write as the
pull-back 02 for the mapping
0 ; =< . Then, since the Jacobian
õ
h
#% , the
of 0 is 7 0 $
wavefront
f)( set of 02 šCB is )( B
4
õ
õ
3( ; @?AB 7 0
)( B €9 úÈ û
'
8
úÈû 0 2 $
õ
$ u
; D? #%
€9 úÈû
x '
(40)
úÈû
Microlocal Structure of ISAR Data
10
02
02
š
Definition.
If A is a distribution defined on , the push-forward A of A satisfies ADõ $
A , where the superscript T denotes transpose; in other words, for any test function ,
õ
õ
A
$
A6
.
E 00 2 F E 0 2 F
2
0
Theorem 5 (Push-forward of a wavefront set) Suppose is a smooth mapping from
š
‡ 8
and A is a distribution defined onG . ThenG
0 2 A$@u 5>
JIAK
to
1
,
A7 0 5€9 úÈû A4 x '
(41)
Application to calculating wavefront sets of integrals. We use( push-forwards in the
)(
)(
following¦ way.
Suppose we have a distribution A in the variables and
Then we can
; . L(
3(
)A (
(
)
(
)
(
(
; 4J ; as the push-forward 0 A for the mapping 0 ; <
interpret
, because
¦
E A6 02 õF $ E 0 A6 õ2 F $ ¦ 0 A4 õ 4J .
õ
A
; J J ; $
M
2
2
úÈû
HG
0
$
€
3.2. Wavefront set for>the
(traditional) weak scattering case
a c denote the fast time (similarly, a c is the slow time). Then we can
In (15), we let cv³
write (15) as
¥
© ‚
a c where
j$@?
‰ j$
a c ø ‚
¥
adc6 ø3‚
¥
‡
h
P5Q
‰ |ƒ‚
¥
?
_ ‰ 4J ‰ (42)
0¥ E ¥ "
~
N
O
] k ( *,+ ] k ( *+
]
c
h
-n no¶ c ~
[
n
5
Z
X
¢
Y
¤
n
I ¬I ŒÏ¬5I
†
(T œ
U
·
R
©
E E
E
E
J ]
'
(43)
Under the assumptions on k ( *+ of Theorem 2, equation (42) expresses ‚ in terms of
a Fourier Integral Operator applied to |ƒ‚ , and therefore the wavefront set of © ‚ can be
calculated in terms of that of |ƒ‚ by standard means [9, 13, 32]. We illustrate here an
alternative method for calculating the wavefront set that we will use in section 2.3.
This alternative approach considers © ‚ to be a push-forward of the product ø ‚>|Ƃ , and
we use the theorems of section 3.1. More specifically, © ‚ is formed from two operations:
¥
| ‚ ¥ by ø3‚ , and then
first we multiply
push forward the product by means of the projection
adc" ‰ ac . In the process of multiplying | ‚ by ø°‚ , we need to
operator
consider | ‚ to be a function of the same variables as ø°‚ ; for this we consider the pull-back
of | ‚ . Calculation of the wavefront set of © ‚ can therefore be carried out in the following
steps: a) calculate the wavefront set of ø°‚ from theorem 2; b) calculate the wavefront set of
©
ø ‚>|Ƃ from theorems 4 and 3; and c) calculate the wavefront set of ‚ $
ø ‚|ƒ‚> from
theorem 5.
N <
02
‡
E 0¥ E ¥ "~
E
E
]
Step a) We assume that ] k ( *,+ ] k ( *+ c
¥ E C
2. The phase of ø ‚ is E ¥ÏE ô
%
PO
$
and so
úÈû
$
ø ‚
¥ÏE S
¥
a c g
T
‰ c
mFn
“$@
c
]
%
h
‰
XW
c
h
‰ c
?T 6U V 6 p > 6
¥ÏE - E ¥Cc E ‰ f Ù´ô
c ‰ h
$g]
c ‰ c
‰ Ù´ c
satisfies the hypothesis of theorem
Nd
RQ
‰ (44)
(45)
Nd
¥C$@
E % } š •
‰ f ±
a c D— ‰
c
RY
NO
%
Microlocal Structure of ISAR
Data
¥ Uz$& % $ ]
V“$&[¬ Z\% $@% $
z
$
]
E
11
E
‰ ”
c
} š E
O i
h
Î
•
a c NO ±
¶ c
}z ‰
-
c
Nd
´,c
-
‰ h
-
‰ c
']
} š ¥ÏE h
• Ð
adcp %
¥
where (as before) we have taken ® adcp{^ % . Some of the details of the calculations can be
found in Appendix B. (Note that for these short-duration signals, will be independent of
and so “$@% for all cases considered in the remainder of section 3.)
We can simplify the expressions for and in (45) as follows. First, a straightforward
computation shows that
V
T
C^
E ¥CE E
i ¶ c ‰ - ¶ c ‰ T<$&] c ‰ (46)
`_ $g] '
~
¥ E ¥ E ~ $
Then,
to simplify
fact that from the criticality condition we have ´ c
ac O ¥ h E N4
¥
Q , we useNOthe
N
¥ E
NO
c
c E . Substitution of this relation
intoN4 the result
for together with the facts
h
h
h
h
h
- ¶ E i and - ¶
yields
c
c $
c $Èi
•
• ~ Y
W } š } š ±
c
$ - ¶ idc] E ‰ z
a c a c •
•
i] ™Î
c
$
- ¶ c ‰ •
™
The Taylor series expansion for
E
write
• ~
i] ™
z$
¶
c ‰ ™
J
-
c J4adc •
obeys
™
~ Ð
c
~ '
•
™
cƒ^
c -
(47)
•
~ ™
c€J
N
c
J4adc and so we can
c6'
(48)
Step b) By theorem 3,
the wavefront
set
of the product
of ø ‚ and |ƒ‚ obeys $
ø°‚ | ‚ úÈû
ø3‚ -
úÈû
cb
~F
We write the wavefront
set of | ‚ in ;
d
(via ~FTheorem
4)öto
; ¥ it is
| ‚ j$
úÈû
adc6 ~f úÈû
‚ ø 
úÈû
\d $
and E
d
S
¥
| ‚
]
id]
- ¶ c
E ‰ •
~ Ä
™
c ‰ \d
where ‰
are defined by (50). Note that
(since k ( *+ ] has no DC component).
÷
b
úÈû
e \d ?fT e HU e \V de
d
c
d
9“;r ‰ >r9
a c ‰  ¹ Ú$@% ô
¥ E C
N d
h
E ¥ c E ‰ Ù´ c ‰ $È%
‰ Ec
T“e $&] gUze d $
ze $: - z$
/. úÈö>\û d ø3d ‚ /d. úÈû ÷ | ‚ 6' (49)
j$
$ ý ; pulled back
‰ ? )
| ‚ f
b
¥
'
|ƒ‚
¥ÏE c
d
?
as úÈû
‰ %#%#% Oadc6 Consequently, we can write
d
úÈû
| ‚ '
(50)
d
and
d
‰
(51)
$
‰ for some
Te hU"e VCe )e v$ ý d d
r9
‰ E
because ]
]
úÈû
|ƒ‚
$l% is excluded
Microlocal Structure of ISAR Data
ji
N The Jacobian matrix
of the projection
Step c)
7 N
$
\k \= k4= = 4=
¥
-<
adc" ‰ 12
¥
adc6 is
(52)
(assuming ‰ 9Ú; = ).
¥
¥
š © l
E E E E E 5,
E the
‚E E $ E E
From
theorem
wavefront
set of
N
?AT U 6V ? 7 N T 6U V 2
7 N T 6U V
nmo T
om T
^ i
U gqp
7 N
rk _ U gqp
\k
V
V
N 2
l$
E E ø ‚>|ƒ‚ obeys úÈû
ø ‚|ƒ‚f
š
E
E
E
E
E
E
a c a c ‰
f9 úÈû ø ‚>|ƒ‚ .
In order to
9 úÈû ø ‚>|ƒ‚ , we note that E E
€
determine when
EE
EE
EE
š
EE
EE
EE
EE
EE
= =
$
$
'
(53)
= 4=
• ~ from elements of (49) with Ú$
. There are
In other words, contributions
to úÈû © ‚ come
™
c ƒ$ ý
no such elements in úÈû ø ‚> because
in (48). Similarly, there is no contribution
4
8
d
d
z$
\d
id] ¶ c
-
~ ™
‰ e
from úÈû |ƒ‚> because elements with v$
E
(51), on the other hand, yields
•
mss T
o U up tt
V q
do not appear in (50). Comparison of (53) with
c 6'
(54)
This relation describes specular reflection.
S
For a weakly scattering target, the wavefront set of © ‚ is contained in the set
Summary
¥
EE
EE
?T 6 U V
a c U
EE
EE
$
$&%o
\d
d
¥CE d
‰ for which v$
c
g
‰ c
c
E¥ÏE ]
h
-
\d
i] ¶ c
-
‰ fF´ c
c
E
d
¥CE • Ä~ ™
‰ d
NO
‰ cT
$@%
c for some
EE
E
$&]
\d d
r9
‰ )
¥ E ô
¥ E NO
EE EE
In
particular,
the wavefront
set corresponding
to a single point scatterer
at ‰
h
ò c
ò
ò
$@% whose normal vector is
c ‰
c ‰ fÙ´Äc ‰ T 6U wv
ò
]
|Ƃ
úÈû
(55)
'
ॠE will
be the curve
h
ò
c ‰ .
3.3. Wavefront sets for multiple scattering
~
In the case of the two isotropic point scatters that we modeled
in section
2.2, the target is
Ï
Ñ
,
Ò
Ó
Ô
‰ s$
‰
‰ ‰
‰ . The corresponding
simply a sum of two delta functions |
K
K
wavefront set is
~
| ÑCÒÓ Ô $
úÈû
4
‰
6
all D$ ý
.
x
u ‰
all D$ ý
8
'
(56)
We see from (22)
that
~
multiple-scattering data can be expressed as a sum of oscillatory
integrals © ÑCÒÓ Ô ^ © - © - © = ; to each we can simply apply Theorem 2. The corresponding
phases are
E ¢¥
g
¥
NO
%
~
%
%
$
=
$
$
h
] E¢¢¥ » c c g
- » c F´ ¥ c ‰ » NO
»ZIÖ
»ZI Ö
I
À
» c ´ c ‰ » - NO ] E¢¢¥ » c c g
f´ c ‰ » ¥ »ZÖ
»ZÖ
À
h
» c c
» c Ù´Äc ‰ » - i 6
]
'
»ZÖ
»ZÖ
We have constructed explicitly the action on
given by (42).
(57)
xyz|{~}/ of the canonical relation for the Fourier Integral Operator
Microlocal Structure of ISAR Data
The wavefront set of¥
~
© ~ a c 4
$€
úÈû
$ €
©
úÈû
isE E theE E same
for
E E as determined
¥
ô
the¥ weak scatterer
Nd case:
?AT U 6 V
4
©
»½
EE
is¥
4
©
EE
The wavefront set of
= is¥
©
úÈû
=
$ €
~ EE
h
c
» c
»ZÖ
¥
-
EE
EE
c
-
EE
ô
» c
»ZÖ
c
EE
T 6 U
h
E„
h
j$ô]
h
I
‰ » -
» c ´ c
»ZÖ
à¥
¥
$&%p
8
» c »ZÖ
¥
I
EE
T 6 U
$@%
´ c
‰ » à¥
j$ô]
ô
» c
»ZÖ
» c Ù ´ c
»ZÖ E EE
T 6 U
¥
-
EE
I
$@%
?AT U 6 V
a c »½
EE
?AT U 6 V
a c ~ -
$@%
The wavefront set of
©
»½
13
~
'
(58)
I
» c »ZÖ
8
Nd
À
‰ » -
$@%
'
(59)
À
» c F´ c ‰ » - i Z
»
Ö
à¥
EE
E
h
» c '
j$ô]
»ZÖ
© ÑCÒÓ Ô
Nd
-
$&%p
8
©
~
(60)
‚.
Finally, the wavefront set of our three-term approximation to
is the union úÈû
© © .
úÈû
úÈû
=
We note that the critical curves in the adc – c plane are somewhat different for the single-,
} š •
double-, and triple-scattering contributions.
In particular, single-scattering curves are
.
‰
} š a c •
_
c3$.i
±
-
‰
a c (61)
double-scattering
• are described
× by ‘} š ~ } šcurves
E c $.i
Õ
h
-
#Y
±
a c ‰ ‰
Õ _
•
a c • } š -
±
adc ‰
‰
E× E
•
} š ~
#Y À
±
a c ‰
-
~
À
(62)
and triple-scattering curves obey‘} š c $.i
_
-
À
} š adc - •
-
±
NO
À
‰
‰
a c •
'
(63)
Multiple scattering from pairs of scattering centers can potentially be recognized in the data
by the occurrence of collections of such curves.
3.4. Wavefront sets for scattering by reentrant structures
The dispersive-scattering model of equation
(23) links the downrange artifacts of equation
(27) to the target image through the î>⠉ factor. We choose î>â by Theorem 1 so that it is
supported in a neighborhood
oföCë and its wavefront
÷
set • is
î â $
úÈû
ï ã
\d d d
‰
‰ )
9
d
ë
ì
í ƒg%
'
(64)
© ß úÈû
We ï compute
in several steps: a) compute úÈû á â from Theorem 3; b)
ã
compute úÈû
; c) compute úÈû ø3ß from Theorem 2; d) compute úÈû ø3ß | ß .$
¥
¥
©
á â fromE Theorem 3; and finally e) consider ß as the push-forward
ø7ß | ß úÈû ø3ß
© ß
a
6
c
d
a
"
c
‰
for
, and compute the wavefront set of from Theorem 5.
N „<
02
Microlocal Structure of ISAR Data
O
set of á â
Step a) The wavefront
$
á â úÈû
•
í
ad c" ‰ j$Èì ìƒ -
úÈû
™
—
Q
•
cfî ⠉ is obtained
from Theorem 3:
.
î â úÈû
14
/.
ì{
úÈû
î â 6'
úÈû
(65)
The coupling pattern ì , however, is assumed to be smooth; its wavefront set is therefore
empty. Consequently, the wavefront set of á â is simply the pull-back of úÈû î â to ; by
•
÷
•
} š •
Theorem
4: öC
d d d d d
a) ‰ ? T ) ‰
á â $
úÈû
Step b)
9
ë
T
d
d
í ƒô%
$È%Ïì
ï ã
and ì
d
í
—
~
†
ao ô%
' (66)
ã
The wavefrontï set
of õ can be calculated by cutting out a small interval about
ã
in the definition (28) of , and then using standard theory [9, 13, 32] to conclude
is the same as that of ðñ ò . Alternatively, one can apply theorems
that the wavefront set of
ï ã and 5 to
FE draw
E the
FE same Ndconclusion.
E
2, 3, 4,
Accordingly we have
]Â$
ê
$
úÈû
d d d
U 4
$.i
À
U
d
$
an arbitrary nonzero real number
We note that this wavefront set is independent of Þ .
%
S
and so, by Theorem 2,
úÈû
'$
¥
FE
a c ø ß5
‰ E
?T66UpV“$È%¹6U E
T<$&] c U3$
E ¥ÏE ]
‚
¥CE $
á â ø3ß
úÈû
ø3ß -
-
úÈû
ô
:
(68)
¥CE h
NO€HFE
c ‰ Ù´ c ‰ ‰ c
E
• ~ 0 FE
E
E
i] ™
c f ”
$ ]
g
- ¶ c ‰ ï ã
c
ã before, we use
Step d) ï As
Theorem ï3ã to obtain úÈû
úÈû
(67)
XU
‰ ” z$
c
'
E
ø3ß has a phase
E 0¥ E ô
function
¥ E on Ndthe
rD
E
depending
additional
variable
h
$
]
c c
c Ù´Äc
Step c)
8
ø3ß | ß $
/.
á â f
úÈû
/.
ø3ß úÈû
]
ø3ß ï ã á â :
úÈû
/.
úÈû
$È%p
'
(69)
úÈû
á â 6'
(70)
The sum term is ï ã
úÈû
ø3ß -
úÈû
-
S
$
úÈû á â ¥CE g
$
‰
where
E
T…e $ô]
Uze $
e $
v
E
Ue
h
‰ c
c
E
$g]
ï ã
FE
adc6 ¥CE d dc
-
‰ 9
d %
ë
c E ‰ - • % ~ - %p
i] ™
c ¶
c ‰ E
-
]
%
E
?HTe HU"e V“e $È% e U e NO€HFE
‰ fF´ c E ‰ ‰ ‰
- E«¥C% E ]
¥
$
À
i $@% •
í
and ì
} š —
d
E
-
% -
Ï
(71)
•
†
a g%
where ì
d
d
•
í ƒô%
ï ã
where ] and are arbitrary nonzero real numbers.
~Ù`
á â by (70), where úÈ û 3
is given
ø ß is given by (69), úÈû
We compute úÈû ø7~Fß `
by (67) (pulled back to ;
by Theorem 4), and úÈû á â is given by (66) (pulled back to ;
).
Microlocal Structure of ISAR Data
ï ã
Step e)
N 2
ø7ß
N 2
15
¥
¥
ø7ß | ß The radar data (30) can
be E written
as
the push-forward © ß $
á â where
a dc6 ‰ ac . The Jacobian of this projection is
7 N
N ji
$
\k \= k
= 4=
`
„<
ƒ'
(72)
©
ï ã therefore
The wavefront set
ö be calculated
E E from
E E E E Theorem 5:
of ß can
¥
© ß $
á â f
adc6 ø3ß
úÈû
úÈû
ï”ã
š EE EE EE
¥
FE N 2
$
$
?AT U 6V T U 6V 9
(73)
÷
@? 7 N á,⛠úÈû ø ß
E E E E ø E ßE
š EE EE EE
á,â
where
the wavefront set
of
is given by (70).
From (72) we see ï”that
ã
E
7 N T U 6V °$ T 6U V %o ; in other words,
we consider elements of (70) for
E
which $
and U $ % . From
these requirements, we see that the elements
from úÈû
e
$b
ý % . Similarly, úÈû ø ßo and úÈû á,âÆ do not contribute
do not contribute because U $
$ý .
because D
Consequently, úÈû © ß is contained in the set
S ¥ EE EE EE
¥ E ô
¥ E Nd
À
h
© ßo$
a c ?AT U 6V
$@%
$È%p
c ‰ c
c ‰ fF´ c ‰ - i úÈû
• ~ • ×
•
} š d •
Nd
í
—
ao ô
† % and ì Õ ™ c - À í ô
ƒ %
where ‰ 9 ë Cì
]
E
EE EE
ॠE T 6U $g] h c ‰ f '
(74)
N
a c ‰ ïã
We see that the critical curve in the a c – c plane is the same as (63), and is associated with
À
scattering centers lying within the duct/cavity at distance from the mouth. The point ‰ in
the critical set corresponds to a point at the mouth of the reentrant structure. In addition, the
critical curve is present in the data only at angles for which energy couples into the dispersive
structure, and for times after which the wave has reached the scattering center within.
4. Examples and interpretation
4.1. Ordinary ISAR
“Ordinary ISAR” [3, 4, 28] considers the target to be composed entirely of weak scatterers
and is described by the discussion of section 2.1. In addition, ISAR traditionally considers
the target to be rotating at a constant
rate about a fixed axis, and that this rate is sufficiently
¶ c
Ü
slow to ensure that
for all ‰ in the target’s support. (More accurately, traditional
‰ ISAR is usually constrained to use data collected over a sufficiently small time interval that
a constant rotation rate can be considered a good approximation.) In practice, data collection
times are on the order of seconds, so that typical apertures are several degrees.
To illustrate the ideas, we assume that the target is rotating at a constant rate about the
h
‡
axis %p% , so that
}z
s$
mo Dˆ ‰h‡ŒŠ ‹ ‡
Š ‡
%
Œ‹
Š ‡
ˆu‰Š ‡
%
‡
%
%
h
pq
'
(75)
•
h
š
3 
• • %# %~ direction.
Further, we assume that
the radar
is located
in the µ$
In this
‡
a c , where ‰
$
and
a c g
$
case, we can write ´ c ‰ &$ _
‰
a c a c . Substituting (75) into (15), performing the integration over the rotation
ˆu‰Š ‡
Œ‹
Š ‡
, Ž ‡
,
Ž ‡

Microlocal Structure
of ISAR Data

¦
Nd
axis,
writing | / ÒÑ ‰ ³
¶
in the
modulus
yields
¥
© ‚
a c c h
j^
| ‚

ÒÑ
_ \‘

?ª| /
‰
‡ E †
] k ( *,+E ]
E¥ E
where we have written ]
‰ $ ] PQ
16
Nd ¶
]
in the phase and
= , and dropping terms in
mFn5[[0XZY¢¤
n Ž [ [ n n Ž )[[ )Ž
IŒ I Ø
I«Œ I Ø
ž

(
œ
«
U
T
T
U
U
T
T
U
U

U
¥ "~ E
E
E
R E
‰ fJ
¥
E
r‘

-
]
k ( *+ c ‰
]
Ö
]
•
c J ]
r‘/’
“ Œ”

J ‰ E Nd
(76)
N
d
¶
c ‰ $
, with ]
i c ‰ ]
Ö
denoting the Doppler frequency shift. We note that dropping terms of order ] ¶ and
higher
assumption [3, 8], i.e., k ( *,+ is negligible outside an interval
~
involves a narrowband
~
~
Ü ] .
] F] with ]
]
• 
,
•
Ö


Equation (76) demonstrates that standard ISAR imaging can only recover the axis0
integrated target scattering density function. In high range resolution
(HRR) ISAR imaging,
0¥rg E ¥
g E
equation (76) can º be further reduced since such systems use w ( *,+ that will best approximate
¢
¥
"
~
z^
. For HRR signals, k ( *+ ]s will be a slowly
equation (17) as
K
M
M
M M
] ]s will also be slowly varying when
varying
function
of ] and k ( *,+ ]s k ( *+
K
E
¥ ~ E
E
] 0Ü ‡ E ] . Consequently,
HRR systems
will have poor frequency resolution capabilities
K
] - ]
and ] k ( * + ~ ] k ( *,+
c is often approximated as being constant over
Ö
] f] Ì . In this case, equation (76) reduces to
k ( *+ $
Ž [[ n Ž )[[ )Ž
m n [[0XZY¢¤
E
•
Š6–
¨ ¨
© ‚
wv
T
a c c .?5?
T
Ê
ÒÑ
| /

‰
(œ U«T
R
IŒ
T
I Ø
n
U
\‘
U T
I«Œ
T
n
I Ø
U
r‘
r‘/’
U 
“ ž UŒ”
,
J4]
J ‰
¥
(77)
¥
where we have suppressed the (now formal) -dependence on the left side because the right
side is independent of .
Systems
with coarse frequency resolution are also insensitive to local Doppler shift
N
N
]
c ‰ . These frequency
shifts are actually quite small; typical values for maneuvering
Ö
h
h
%
aircraft targets are ] c ]
and we can make the approximation ] c ] Ü
.
Ö
Ö
In other words, we assume that the target is effectively stationary for the duration of the
fast-time measurement. This simplification is known as the start-stop approximation. Such
measurements are Doppler-free and, in this case, the term “range-Doppler” imaging is
something of a misnomer—the methods used are really closer to tomographic techniques.
When ] c $@% , equation (77) becomes
Ì
•

© ‚
•
•
˜—
Ö
wv
adc6 c
?
~
| /
ÒÑ
š™
‰

?
T
T
Ê
R
•
I (T œ
\‘’
n4
)Ž
U
“ ž U›”
mfnO[[0XZY¢¤
E
J4]
J ‰

'
(78)
• NO
We see that if ]
and ]
, then equation (78)
is precisely
a Radon
transform
,
Ò
Ñ
Ò
Ñ
a c . For
‰
of | / : for each a c , one integrates | /
over the line c $Bi _
real systems, (78) is a bandlimited version of the Radon transform and ISAR images are
traditionally produced by Radon inversion methods (such as filtered backprojection [20, 21]).
While ISAR imaging schemes are usually based on equation (78), the analysis of section
3 shows that the wavefront set of the data contains considerable information about the target,
}z
which can be extracted without forming an image. The wavefront analysis
of section 3.2
• ~ • ¥
E
± 2$ % during
simplifies somewhat with the start-stop approximation.
In
particular,
since
™
™
h
•
the expression for of
the fast time interval, c $
and we N can take • c{$ ‡ adcp in
™
ac $
adc6
adc , which implies
(55). Moreover, for (78) we have d J4adc:$
that (55)
reduces
öC to
Nd
• NO ×
úÈû
$
© ‚  Ž ‡
\d
Ž 
š?AT66Upd c3 ^@id´Äc• ‰  NO
™
i]
a c where  $
adc" c
Š
Œ‹
d
ˆD‰hŠ
T66Upr$ d ] d Õ i ‰  Ž  adcp ÷
for some ‰,  r9 úÈû |ƒ‚
h
'
(79)
d
Microlocal Structure of ISAR
Data
U
h
•
Nd
 Ž 
h
Nd
š
17
8
cœ
- i ‰
- i ¶ ‰ j$
]
$
]
] - ] à encodes an
It is easy to see that 7$
inferred Doppler shift across the synthetic aperture collection interval (even though no local
frequency shifts are measured).
• A point scatterer
(whose
wavefront
set contains all directions ù ) located at ‰ corresponds
i ‰
ac in the data domain. The coordinates of this scatterer are
to the curve c7$Èi _
• usually estimated from the intersection of the
backprojections constructed from data (i.e., lines
• a c oriented with angle a c and offset i ‰
). But the wavefront-set
analysis suggests
•
O
N
a c from knowledge of c and estimate the
another possibility: find the range i ‰
h
] i ‰
.
cross-range position from the directions 6 ps$
Strictly speaking, of course, bandlimited data are smooth and therefore the wavefront
set is empty. Our analysis, however, views the bandlimited case as an approximation to the
infinite-bandwidth problem.
 Ž ‡
, Ž Ž ‡

‡
T U
‡
d
 Ž 
4.2. High Doppler resolution
E
=
E ]
] ò , the radar
In the situation where the incident signal is chosen so that k ( *+ ] K
¢¥¹
becomes a high Doppler resolution system [26]. Of course, in this situation we cannot actually
~
EE
since such a signal would be infinitely long.
correlate with w ( *,+
Instead, we set a $B%
and consider a “long-duration pulse” approximation to tvu w ( *+ x ] :
EE
where
•
¥
©
£
žŸ)Ÿ
]
k +
¡ ? $
( U«T
II
R
¢
T
[
I
J
E
(80)
is “large.” Substitution into equation (12) yields
h
s$
¤ ? ? 5
Q
i
= _
¥
E [¢¡ Žd[0XZY¢¤
I II
Œ“Ÿ ž I I
( U«T
T
(T œ
U
U
[¢¡ Žd[0XZY¢¤ FE R E
E E F E E
II
]
¢
[
I
¢
¢
§ £
¦
£
¦
‰ ]
] ò R
6mfnoK I I ¬ Œ“Ÿ ž I I
II
†
(T
U
U
U
J J4] J ] J
J ‰
„[ ¡ Žd[0XZY
[
R I I I
Œ“Ÿ ž I I
] ò (T U
U
U
$
?5? |ƒ‚ ‰ P5Q
• 0FE E
_
FR E
¥r>HFE Epô
Nd
FE FE E
™
†
>— 4J J
J ‰
~ [ ~_
[
[¢¡ ŽOXZY
[‰ E E
K
¬
Œ
¬
I
I
¹
Œ
¬
ž
I
I
Ÿ ]ò ( T ”U
U
U
U
$
| ‚ ‰ J
J ‰ '
?
P5Q
(81)
• E E
_
R
0 E E
Nd¥
›þ E E ª
NO
¥
™
EE
$
u
‰ — ‰ _ The set
x in the
• E E
last equation of (81) consists of those values of
and ‰ for ™ which
the argument of the
Ɨ ‰ , we can use the
delta function in the previous line can vanish. When _
Nd¥
j: E E Nd
NO¥
fact thatE E high Doppler resolution
signals yield coarse time resolution to approximate by
E
E E g>
Nd
^ u
_
x . In equation (81) we then make the change of
variables $ _ ~Z [ to X#obtain
Y¢¤
~Z [ ~
[
XZY
[¢¡ ŽOXZY¢¤ FE
¥
¬
(T œ U ¬ I
Œ¹¬ Ÿ ž I Œ 
] ò
©
(
T
œ
U
U
U

s^
R PQ
?O?
|ƒ‚ ‰ J J ‰ '
_
R
­
| ‚
E 0‡
—
E
— £
©¨
— £
¢
•
—
¤ ¦
¢
(82)
We would like to carry out a wavefront-set analysis for (82) as we did in section 3. Here
the time variable should correspond to the frequency variable ] in section 3. Unfortunately,
E
for rotating targets, the
phase of (82) is not generally homogeneous of degree one in the
integration variable , and a more traditional approach is required (e.g., the method of
stationary phase).
Microlocal Structure of ISAR Data
18
We can apply the wavefront-set analysis of
section 3, however, when
the rotation rate
• E
}70
}z E

NO
E • 
InNd this case,
we NOuse
the fact that for
rotations
of the }z
form
(75), weNdcan
write
™
™
r$
, so that
s$
. When
_
_
_
_
E
E

is sufficiently slow that the E small-angle‡ approximation
can E be used over the
whole data
h
^
^
collection interval, i.e.,
and
for all in the interval
, then
(82) becomes
~Z [ X#Y¢¤
>
‡}70is E small.
‡
›‹
Š ‡
ˆD‰Š ‡
¢
O
‡
\‘’
£ £
Q
YæoÊ ~
[Ž
XZY
[[
¬
(T œ U Œ¹¬

ž
,
Ò
Ñ
(T
U
U "U 
s^
?O?
| /
R PQ
‰ (83)
R XZY
[[ ¤ FE
_ ~Z YæoÊ ~
[Ž
¬
ŒÏ¬
I
ž
†
œ ‡ U
U "U 
J J ‰
• ( T
R
E
©
<$
. The wavefront set of
where, as before,
can be calculated in
• 0>
«‡
"~ ¥
the same manner as done in section 3.2, where Nd] is replaced by and |ƒ‚ is replaced by
h
- D ¥ ‰ | / Ò,Ñ ‰ ] ò d"~d
• this
0case
C the
Nd phase is
_ ¥ f . In
R§4¨
h
h
$
] ò
‰
'
_
(84)
©
•
¥
] ò

¢
O
Ž 
%
Š
Œ‹ ”
\‘/’ ‘ 
“ ›”
ˆD‰hŠ
 Ž ‡
‡
O
For the wavefront set we
obtain: ¥
úÈû
©
h¢

Q
“ ›”
•
Q
 Ž  ‡
¥
•
0>
Nd
‡ Ž 
?UV h
% - $ ~ ‰  ¥ ‡ • _ >
h
"- ~
Uz$È % $ FE ] òr‡ O
‰ • Ž 0‡ C
V“$@ ¬ $ ] ~ òFE«Oh - ‡ ‰  >Ž  ‡ NO Q _
$ ~ i¥] ò • - ] • 0C
"
‰ Nd_ š
h

Ž
$ ‡ - ‡ _ ¬ '
for some ‰  €9 úÈû | / Ò,Ñ \v
• $§ª
h
¥
•
 Ž 
f6
NO
Nd
_ f"
(85)
 Ž
•
We see that knowledge of the wavefront set again enables us to estimate the location of point
scatterers: the critical curve (or ) gives us ‰ —
, and knowledge of gives us ‰ — .
V
U
4.3. AcceleratingM }z
targets
M
E E
}z
The case where ® }zis
not vanishingly
small }7
can
be E E even more difficult. In this situation,
± a c - a c Ú^
a c may not be valid and, consequently,
the approximation
equation (12) cannot be simplified to the form of equation (14). The manifestation of this
failure is progressive phase error in © and degradation of ISAR images constructed under the
}70
}z
assumption that
is constant—such images are said to be “de-focused.”
When ® is smooth and there are no abrupt changes in ± , then it is possible
Ê
Ìto
~ E E }v
a
fa
consider these data as having been collected over a collection of subintervals
Ê
Ì
}
® adc7^¯% for adc.9
a
a
. Over
whose lengths are sufficiently
small so that
each such subinterval, ± can be considered to be constant in a manner similar to the start-stop
approximation discussed in section 4.1. Within each subinterval the analysis of this paper can
be applied; but over the collection ofn subintervals
the critical points will not form sine curves.
mFn
#
track these non-sinusoidal variations in HRR
The normal directions 6 p7$
data.
E
In the high-Doppler-resolution case, the small-angle expedient
used in section 4.2
(approximating the phase as a degree-one homogeneous function of ) will generally fail to
F
accurately capture the aspect dependence of data collected from accelerating
Ê
Ì targets. The
with a
fa
and choose
usual plan of attack for this problem is to identify
time-domain signals that are short enough for to be considered approximately constant on
‡
\­ r­
T 6U
%
%
‡
£ £
\­ \­
\­ \­
Microlocal
Structure of ISAR Data
\­ Ê fa\­
Ì
19
but are also long enough to enable good estimation of the instantaneous frequency
(for retrieval of adc ). These considerations are the same as those normally encountered in
a
‡
time-frequency signal processing [7].
Alternatively, it is possible that the analysis of this paper could be extended to treat the
accelerating-target case; this we leave for future work.
5. Conclusions and Future Work
Our discussion has not actually been about radar imaging. Instead, it has focused on the
structure imposed upon measured radar data by a class of image features associated with the
singular set of the radar target. Standard radar imaging schemes attempt to estimate precisely
this class of features, however, and so our approach has “imaging” at its heart. In particular,
we have shown that when the weak-scattering approximation is valid, the location of the
target’s scattering centers can be estimated directly from the data wavefront set.
We have also shown that the mapping from target to data for the important cases of
structural dispersion and multiple scattering displays fundamental differences from the weakscatterer case. In particular, we demonstrated that the wavefront set for multiple-scattering
events can be distinguished from single-scattering data. We also showed that the wavefront
set for scattering from ducts and cavities is similar to that of a triply-scattered wave. Both
these observations are potentially significant: they may lead to schemes for eliminating ISAR
image artifacts by first isolating the wavefront set of the measured data and then constructing
an image from this reduced data set (this is an area of future research).
}z
The case of three-dimensional ISAR imaging—which relies on a more general version
than considered in equation (75)—also fits neatly into this framework. For nonof
cooperative targets, a principal problem lies in discovering the various roll/pitch/yaw data
variations due to target maneuvers from the data themselves. These dependencies must be
separately isolated if an accurate image is to be formed, and wavefront-set analysis offers a
systematic approach for investigating the target behavior.
We leave for the future the question of how knowledge of the singular structure of the
radar data can best be exploited for target imaging and identification. There are a number of
issues here. For image formation, the wavefront-set analysis suggests that reconstruction
methods related to local tomography [11, 16] may be useful. In particular, analysis of
wavefront sets can determine whether backprojection will provide an image free of certain
artifacts [22, 24]. In addition, wavefront-set analysis suggests an approach for producing
artifact-free, superresolved images: remove all components of the data set except those that
correspond to well-understood target features, and form an image from those components
only.
Practical implementation of the analysis in this paper requires that we be able to extract
the wavefront set from noisy, band-limited and discretely sampled radar data. The problem
of extracting wavefront sets under such conditions is closely related to image processing
problems such as edge detection, and these are active areas of current research. We explore
one possible approach in [5], where we provide numerical examples of synthetic radar data
and show how the wavefront set analysis enables us to estimate target parameters from very
noisy data.
Our ultimate goal, of course, is to identify targets under all weather conditions. For
target identification, it may not be necessary to form an image. We have shown that wavefront
set analysis distills the data set into a set of primitives that are easily related to a target’s
structural properties. As such, this approach constitutes a promising new tool that deserves
further investigation.
Microlocal Structure of ISAR Data
20
Acknowledgments
We are grateful to the Mathematical Sciences Research Institute and in particular to the
organizers of the fall 2001 program on Inverse Problems for bringing us together; it was there
that we began the discussions that led to this work. We also thank Rensselaer Polytechnic
Institute and the Naval Air Warfare Center Weapons Division for making possible M.C.’s
visit to China Lake during the spring of 2002. M.C. is also grateful to George Papanicolaou,
Air Force Office of Scientific Research grant F49620-01-1-0465, and Stanford University for
hospitality during the spring of 2002, when much of the paper was written. We would like to
thank Rafe Mazzeo for pointing out Theorem 1 to us.
This work was supported by the Office of Naval Research. This work was also supported
in part by CenSSIS, the Center for Subsurface Sensing and Imaging Systems, under the
Engineering Research Centers Program of the National Science Foundation (award number
EEC-9986821), and by the NSF Focused Research Groups in the Mathematical Sciences
program.
Appendix A. Multiple Scattering
®
¯
®
For a time-harmonic incident wave ( *+ , the frequency-domain field /+ scattered from
“point” scatterers can be obtained from the Foldy-Lax [33] equations together with the
assumption that the scattered field from a single “point” scatterer is proportional to the Green’s
function [27]:
°
®
°
­
¼
/+ $
®
°
#M
~
M
‰ » »½
®
¼
( *+ Â$
»
‡
N O
~
P5Q
±²
®
¾ »
#M
°
±M ± ± ±
® ‰ ¾
‰ » »
‰
½»
(A.1)
h
Ç $
Zi4,''' ¯
(A.2)
´O
]´
. Equation (A.1) says that the scattered field is the sum of
where ´Os$
R§4¨
the fields scattered from each scatterer; moreover, the field scattered from the Ç th scatterer is
proportional to the field
that is incident upon the Ç th scatterer. Equations (A.2) say that the
»
Ç th local incident field is the overall incident
~
field plus the field scattered from all the other
scatterers. If the scattering strengths ',''”
are known, the equations (A.2) can be
¾
¾
¾
solved for the ; then the total field can be found from (A.1).
»
~ M (A.2) are
“point” scatterers,
#equations
M In the case of two
~
®
­
®
®
®
^
where
À
$
®
~
$@i if Ç $
®
¾
»
E
where Ç
‰
°
#M ( *,+ - ‰ M
¾ ~
‰
^
¾
®
®
~ ~ ‰
‰ (A.3)
(A.4)
^
~
~
and (A.4) at ‰ gives
rise to the
~ system of equations
À
h
~ M
‰
°
( *,+ -
®
~ $
Evaluating (A.3) at ‰
M ®
$
°
À
¾
°h
_
®
®
‰ ‰ _
®
( *+ ‰ ( *+ ‰ ®
$
_
(A.5)
. These equations
have the
solutions
‰ » $
h
and Ç
/+ $
E
®
°
( *+ ‰p» h
­
¼
»½
¾
h
$
~
I
¾ ~» ¾
°
À ®
À
I
( *,+ ‰p» h
Ç $
°
if Ç $@i . Using (A.6)
in (A.1) yields
#M
M
‰ » ¾ »
®
°
( *+ ‰p» h
¾
I
¾ ~» ¾
°
À ®
À
#i
(A.6)
I
( *,+ ‰p» '
(A.7)
Microlocal Structure of ISAR Data
21
M M
0‡ E r: E j E M M E
E M The time-domain
scattered field due to the incident
field (6) can be found
by taking
\ Q M "~
›
‡ $EM ¦ M NO
0
‡
E
E
]
A
‰»
‰p» J
and ( *+ ‰p» ›$ k ( *+ ] R”§4¨ ] ] adc in (A.7) and Fourier transforming from ] to . The
‰ » ‰ »
R§4¨
R§ ¨
exponentials involving cancel, and we obtain
E
0
0>H E M M
N
¼
~ ¾ » k ( * + I«]¿ ! /0+ $
~ ? A
P5QÀ
‰ » h
( T
V WpŽÃV XZY
V W))ŽÃ V XZY
¾ ¾ R
»½
I
6m n [
E E
I
I¿
I
M M
M0Å
I I
( T
(T M †@Á R ( T I
(T U
R PQÀ
J ] J ' (A.8)
I
\ Q
\R Q
¾ »
R
‰ »
‰ »
°
®
­
Appendix B. Some Formulas
•
a c ¥ Õ
™
} š ×
c— ‰
$
±
•
adc D— ‰
$
5
i NO c
$
h
¶
¶
c ‰ c ‰ •
} š ¶
c ‰ $
a c H— ‰
®
a c } š •
Ž
¶ c
‰ s2$ „} ± š a c •
{}
Ž
Ž
´,c7$
d
a
c
D
—
j
$
‰
¥
}
¥
Ž
Ž
i
O
c
¶ c
c7$
‰ j$
h
¶
c
3
¶ c
‰ (B.1)
(B.2)
(B.3)
š •
(B.4)
±
š ac •
¶ c
(B.5)
a c Nd ‰ (B.6)
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³C´
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