Synthetic Aperture Radar Imaging - Purdue University
Transcript of Synthetic Aperture Radar Imaging - Purdue University
Synthetic Aperture Radar Imaging
Margaret CheneyRensselaer Polytechnic Institute
Colorado State University
with thanks to various web authors for images
SAR• developed by engineering community
(for good reasons)
• open problems are mathematical ones
• key technology is mathematics: mathematical synthesis of a large aperture
• mathematically rich: involves PDE, scattering theory, microlocal analysis, integral geometry, harmonic analysis, group theory, statistics, ....
Thumbnail history• 1951: Carl Wiley, Goodyear Aircraft Corp.
• mid-’50s: first operational systems, built by universities & industry
• late 1960s: NASA sponsorship, first digital SAR processors
• 1978: SEASAT-A
• 1981: beginning of SIR series
• since then: satellites sent up by many countries, sent to other planets
SIR-C (1994) image of Weddell Seablue = L band VV, green = L band VH, red = C-band VV
http://southport.jpl.nasa.gov/polar/sarimages.html
JERS (Japan)
Radarsat(Canada)
ERS-1 (Europe)
Envisat (Europe)
TerraSAR-X &Tandem-X
(public-privatepartnership in
Germany)
TerraSAR-X: Copper Mine in Chile
http://www.astrium-geo.com/en/23-sample-imagery
deforestation
internal waves atGibraltar
southern California
topography
Venus
radar penetrates cloud cover
Venus topography
AirSAR (NASA)
CARABAS
Lynx SAR
Airborne Systems
Outline
• Mathematical model for radar data
• Image reconstruction
• The state of the art
• Where mathematical work is needed
3D Mathematical Model
• We should use Maxwell’s equations;but instead we use
�⇥2 � 1
c2(x)�2
t
⇥E(t, x) = j(t, x)⇧ ⌅⇤ ⌃
source
• Scattering is due to a perturbation in the wave speed c:
1c2(x)
=1c20
� V (x)⇧ ⌅⇤ ⌃
reflectivity function
• For a moving target, use V (x, t).
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Basic facts about the wave equation
• fundamental solution g�⇥2 � c�2
0 ⌅2t
⇥g(t, x) = ��(t)�(x)
g(t, x) =�(t� |x|/c0)
4⇥|x| =⇤
e�i�(t�|x|/c0)
8⇥2|x| d⇤
• g(t, x) = field at (t, x) due to a source at the origin at time 0
• Solution of �⇥2 � c�2
0 ⌅2t
⇥u(t, x) = j(t, x),
is
u(t, x) = �⇤
g(t� t⇥,x� y)j(t⇥,y)dt⇥dy
• frequency domain: k = ⇤/c0
(⇥2 + k2)G = �� G(⇤, x) =eik|x|
4⇥|x|
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Introduction to scattering theory
�⇤2 � c�2(x)⇥2
t
⇥E(t, x) = j(t, x)
(⇤2 � c�20 ⇥2
t )E in(t, x) = j(t, x)
write E = E in + Esc, c�2(x) = c�20 � V (x), subtract:
�⇤2 � c�2
0 ⇥2t
⇥Esc(t, x) = �V (x)⇥2
t E(t, x)
use fundamental solution ⇥
Esc(t, x) =⇤
g(t� �,x� z)V (z)⇥2�E(�,z)d�dz.
Lippman-Schwinger integral equation
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frequency domain Lippman-Schwinger equation:
Esc(�, x) = ��
G(�, x� z)V (z)�2E(�, z)dz
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single-scattering or Born approximation
Esc(t, x) ⇥ EscB :=
�g(t� ⇤,x� z)V (z)⇧2
�E in(⇤,z)d⇤dz
useful: makes inverse problem linear
not necessarily a good approximation!
In the frequency domain,
EscB (⌅, x) = �
�eik|x�z|
4⇥|x� z|V (z)⌅2 Ein(⌅, z)⌅ ⇤⇥ ⇧(⇤2+k2)Ein=J
dz
For small far-away target, take J(⌅, x) = P (⌅)�(x� x0) ⇤
Ein(⌅, x) = ��
G(⌅, x� y)P (⌅)�(y � x0)dt⇥dy = �P (⌅)eik|x�x0|
4⇥|x� x0|
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The Incident Wave
The field from the antenna is E in, which satisfies
(⌃2 � c�2⌅2t )E in(t, x) = j(t, x)
⇤
E in(t, x) =�
antenna
�e�i�(t�t��|x�y|/c)
8�2|x� y| j(t⇥,y) d⇥dt⇥dy
=�
antenna
�e�i�(t�|x�y|/c)
8�2|x� y| J(⇥, y) d⇥dy
where j = Fourier transform of J .This model allows for:
• arbitrary waveforms, spatially distributed antennas
• array antennas in which di�erent elements are activated withdi�erent waveforms
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• many wavelengths: narrow beam
• few wavelengths: broad beam
real-aperture imaging versus synthetic-aperture imaging
Plug expression for incident field into Born approximation.....
putting it all together ...
For small far-away target, take J(⇤, x) = P (⇤)�(x� x0) ⇥
Ein(⇤, x) = ��
G(⇤, x� y)P (⇤)�(y � x0)dt⇥dy = �P (⇤)eik|x�x0|
4⇥|x� x0|
Then the scattered field back at x0 is
EscB (⇤, x0) = P (⇤) ⇤2
�e2ik|x0�z|
(4⇥)2|x0 � z|2 V (z)dz
In the time domain this is
EscB (t, x0) =
�e�i�(t�2|x0�z|/c)
2⇥(4⇥|x0 � z|)2 k2P (⇤)V (z)d⇤dz
=�
p(t� 2|x0 � z|/c)2⇥(4⇥|x0 � z|)2 V (z)dz
Superposition of scaled, time-shifted versions of transmitted waveform
Note 1/R2 geometrical decay ⇥ power decays like 1/R4
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Antenna moves on path
Fourier transform into frequency domain:
D(�, s) =�
e2ik|Rs,x|A(�, s,x)d�V (x)dx
Choose origin of coordinates in antenna footprint,use far-field approximation|�(s)| >> |x| ⇤ Rs,x = |�(s)� x| ⇥ |�(s)|� ⇥�(s) · x + · · ·
D(�, s) ⇥ e2ik|�(s)|�
e2ik d�(s)·x A(�, s,x)⇧ ⌅⇤ ⌃ V (x)dx
approximate by (function of �, s) (function of x)
same as ISAR! use PFA
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Fourier transform into frequency domain:
D(�, s) =�
e2ik|Rs,x|A(�, s,x)d�V (x)dx
Choose origin of coordinates in antenna footprint,use far-field approximation|�(s)| >> |x| ⇤ Rs,x = |�(s)� x| ⇥ |�(s)|� ⇥�(s) · x + · · ·
D(�, s) ⇥ e2ik|�(s)|�
e2ik d�(s)·x A(�, s,x)⇧ ⌅⇤ ⌃ V (x)dx
approximate by (function of �, s) (function of x)
same as ISAR! use PFA
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data is of the form
d(t, s) =��
e�i�(t�2|Rs,x|/c)A(⇥, s,x)d⇥V (x)dx =: F [V ](t, s)
Cannot use far-field expansion as beforeFrom d, want to reconstruct V .
• d is an oscillatory integral, to which techniques of microlocalanalysis apply (F is a Fourier Integral Operator )
• similar to seismic inversion problem (with constant backgroundbut more limited data)
• d(t, s) depends on two variables.Assume V (x) = V (x1, x2)⌅ ⇤⇥ ⇧ �(x3 � h(x1, x2)).
ground reflectivity function
• if A(⇥, s,x) = 1, want to reconstruct V from its integrals overspheres or circles (integral geometry problem)
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Write
This is a Fourier Integral Operator! (observation of Nolan & Cheney)
Apply matched filter
output of correlation receiver is of the form
d(t, s) =��
e�i�(t�2|Rs,x|/c)A(�, s,x)d�V (x)dx
A includes factors for:1. geometrical spreading2. antenna beam patterns3. waveform sent to antenna
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data is of the form
d(t, s) =��
e�i�(t�2|Rs,x|/c)A(⇥, s,x)d⇥V (x)dx =: F [V ](t, s)
Cannot use far-field expansion as beforeFrom d, want to reconstruct V .
• d is an oscillatory integral, to which techniques of microlocalanalysis apply (F is a Fourier Integral Operator )
• similar to seismic inversion problem (with constant backgroundbut more limited data)
• d(t, s) depends on two variables.Assume V (x) = V (x1, x2)⌅ ⇤⇥ ⇧ �(x3 � h(x1, x2)).
ground reflectivity function
• if A(⇥, s,x) = 1, want to reconstruct V from its integrals overspheres or circles (integral geometry problem)
11
data is of the form
d(t, s) =��
e�i�(t�2|Rs,x|/c)A(⇥, s,x)d⇥V (x)dx =: F [V ](t, s)
Cannot use far-field expansion as beforeFrom d, want to reconstruct V .
• d is an oscillatory integral, to which techniques of microlocalanalysis apply (F is a Fourier Integral Operator )
• similar to seismic inversion problem (with constant backgroundbut more limited data)
• d(t, s) depends on two variables.Assume V (x) = V (x1, x2)⌅ ⇤⇥ ⇧ �(x3 � h(x1, x2)).
ground reflectivity function
• if A(⇥, s,x) = 1, want to reconstruct V from its integrals overspheres or circles (integral geometry problem)
11
Reconstruct a function from its integrals over circles or lines
2
1
x
xspotlight SAR
stripmap SAR
How to invert the radar FIOdata is of the form
d(t, s) =��
e�i�(t�2|Rs,x|/c)A(⇥, s,x)d⇥V (x)dx =: F [V ](t, s)
Cannot use far-field expansion as beforeFrom d, want to reconstruct V .
• d is an oscillatory integral, to which techniques of microlocalanalysis apply (F is a Fourier Integral Operator )
• similar to seismic inversion problem (with constant backgroundbut more limited data)
• d(t, s) depends on two variables.Assume V (x) = V (x1, x2)⌅ ⇤⇥ ⇧ �(x3 � h(x1, x2)).
ground reflectivity function
• if A(⇥, s,x) = 1, want to reconstruct V from its integrals overspheres or circles (integral geometry problem)
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Strategy for inversion scheme
G. Beylkin (JMP ’85)
Construct approximate inverse to F
Want B (relative parametrix) so that BF = I+(smoother terms)Then image = Bd � BF [V ] = V +(smooth error).
microlocal analysis ⌅a) method for constructing relative parametrixb) theory ⌅ BF preserves singularities
“local” ⇥⇤ location of singularities“micro” ⇥⇤ orientation of singularitiessingularities ⇥⇤ high frequenciesbasic tool is method of stationary phase
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radar application: Nolan & Cheney
Construction of imaging operator
recall
d(s, t) =� �
e�i�(t�2|Rs,x|/c)A(�, s,x)d�V (x)dx
image= Bd where
Bd(z) =� �
ei�(t�2|Rs,z|/c)Q(z, s, �)d� d(s, t)dsdt
where Q is to be determined.
• B has phase of F ⇥ (L2 adjoint)
• Compare:
– inverse Fourier transform
– inverse Radon transform
• This approach often results in exact inversion formula
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Analysis of approximate inverse of F
I(z) =�
ei�(t�2|Rs,z|/c)Q(z, s, ⇥)d⇥ d(s, t)dsdt
where Q is to be determined below.
• Plug in expression for the data and do the t integration:
I(z) =� �
ei2k(|Rs,z|�|Rs,x|)QA(. . .) d⇥ds⌅ ⇤⇥ ⇧
K(z,x)
V (x)d2x
point spread function
• Want K to look like a delta function
�(z � x) =�
ei(z�x)·�d�
• Analyze K by the method of stationary phase
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K(z,x) =⇥
ei2k(|Rs,z|�|Rs,x|)QA(. . .)d⇥ds
main contribution comes fromcritical points
|Rs,z| = |Rs,x|⇤Rs,z · �(s) = ⇤Rs,x · �(s)
If K is to look like�(z � x) =
�ei(z�x)·�d2⇥,
we want critical points only when z = x.
Antenna beamshould illuminate only one of the criticalpoints ⇥ use side-looking antenna
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• Do Taylor expansion in exponent
• Change variables
At critical point z = x :
Choose
data manifold
Resolution
• independent of range!
• independent of wavelength!
• better for small antennas!
Along-track resolution is L/2.
This is ...
• independent of range!
• independent of λ!
• better for small antennas!
These are all explained by noting that when a point
z stays in the beam longer, the effective aperture
for that point is larger.
In range direction, want broad frequency band ⇒
get largest coverage in ξ.
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length of antenna in along-track direction
Resolution is determined by the region in Fourier space where we have data:
short calculation
State of the Art
• motion compensation
• interferometric SAR
Multi-pass interferometry
Landers earthquake 1992 Hector mine earthquake
http://topex.ucsd.edu/WWW_html/sar.html
Where mathematical work is neededDealing with complex wave propagation
Incorporate more scattering physics: multiple scattering (avoid Born approx.), shadowing, geometrical effects,
resonances, wave propagation through random media, ....
We want to track vehicles
and pedestrians in
the urban areas.
We would like to identifyobjects under foliage
The shadow sometimesseems to show the object
more clearly than the directscattering. How can we exploit
the shadow?
Wide-angle SAR and 3D imaging
Moving objects cause streaking or ....
!
!
!
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!"#$%&#'&(!)*&+#,%&-./0&&-./00.12
Incorrect positioning:
A train off its trackA ship off its wake
Incorrect positioning:
A train off its trackA ship off its wake
incorrect positioning.
a train off its track a ship off its wake
Waveform designEach antenna element
can transmit a different waveform. What waveforms
should we transmit?
Want to suppressscattering fromuninteresting
objects (leaves, etc.)
coding theory, number theory,group theory+ statistics +
physics
Can we transmit different signals in different directions?
Antenna modeling & design
spectrum congestion
Radar imaging with multiple antennas
Antennas are few and irregularly spaced
Where should antennas be positioned?What paths should they follow?
Extraction of information from images
image of same scene at two different frequencies
Infer material properties from radar images
papers and lectures available athttp://eaton.math.rpi.edu/Faculty/cheney
Radar imaging is a field that is ripe for mathematical attention!