synthetic aperture radar imaging - purdue university
TRANSCRIPT
![Page 1: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/1.jpg)
Synthetic Aperture Radar Imaging
Margaret CheneyRensselaer Polytechnic Institute
Colorado State University
with thanks to various web authors for images
![Page 2: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/2.jpg)
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, ....
![Page 3: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/3.jpg)
![Page 4: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/4.jpg)
![Page 5: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/5.jpg)
![Page 6: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/6.jpg)
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
![Page 7: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/7.jpg)
![Page 8: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/8.jpg)
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
![Page 9: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/9.jpg)
JERS (Japan)
Radarsat(Canada)
ERS-1 (Europe)
Envisat (Europe)
TerraSAR-X &Tandem-X
(public-privatepartnership in
Germany)
![Page 10: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/10.jpg)
TerraSAR-X: Copper Mine in Chile
http://www.astrium-geo.com/en/23-sample-imagery
![Page 11: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/11.jpg)
deforestation
![Page 12: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/12.jpg)
internal waves atGibraltar
![Page 13: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/13.jpg)
southern California
topography
![Page 14: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/14.jpg)
![Page 15: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/15.jpg)
Venus
radar penetrates cloud cover
![Page 16: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/16.jpg)
Venus topography
![Page 17: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/17.jpg)
AirSAR (NASA)
CARABAS
Lynx SAR
Airborne Systems
![Page 18: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/18.jpg)
![Page 19: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/19.jpg)
![Page 20: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/20.jpg)
Outline
• Mathematical model for radar data
• Image reconstruction
• The state of the art
• Where mathematical work is needed
![Page 21: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/21.jpg)
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).
2
![Page 22: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/22.jpg)
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|
3
![Page 23: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/23.jpg)
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
4
frequency domain Lippman-Schwinger equation:
Esc(�, x) = ��
G(�, x� z)V (z)�2E(�, z)dz
5
![Page 24: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/24.jpg)
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|
6
![Page 25: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/25.jpg)
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
1
• many wavelengths: narrow beam
• few wavelengths: broad beam
real-aperture imaging versus synthetic-aperture imaging
Plug expression for incident field into Born approximation.....
![Page 26: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/26.jpg)
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
7
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
7
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
7
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)
9
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
6
![Page 27: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/27.jpg)
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
![Page 28: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/28.jpg)
Reconstruct a function from its integrals over circles or lines
2
1
x
xspotlight SAR
stripmap SAR
![Page 29: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/29.jpg)
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)
9
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
9
radar application: Nolan & Cheney
![Page 30: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/30.jpg)
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
13
![Page 31: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/31.jpg)
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
13
![Page 32: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/32.jpg)
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
15
![Page 33: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/33.jpg)
• Do Taylor expansion in exponent
• Change variables
At critical point z = x :
Choose
data manifold
![Page 34: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/34.jpg)
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.
16
length of antenna in along-track direction
Resolution is determined by the region in Fourier space where we have data:
short calculation
![Page 35: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/35.jpg)
State of the Art
• motion compensation
• interferometric SAR
![Page 36: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/36.jpg)
Multi-pass interferometry
Landers earthquake 1992 Hector mine earthquake
http://topex.ucsd.edu/WWW_html/sar.html
![Page 37: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/37.jpg)
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, ....
![Page 38: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/38.jpg)
We want to track vehicles
and pedestrians in
the urban areas.
![Page 39: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/39.jpg)
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?
![Page 40: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/40.jpg)
Wide-angle SAR and 3D imaging
![Page 41: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/41.jpg)
Moving objects cause streaking or ....
!
!
!
"#$%&'!()!!*+,--#.$!/'.0'&!#.!123%4%'&4%'5!67)!
!"#$%&#'&(!)*&+#,%&-./0&&-./00.12
![Page 42: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/42.jpg)
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
![Page 43: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/43.jpg)
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
![Page 44: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/44.jpg)
Radar imaging with multiple antennas
Antennas are few and irregularly spaced
Where should antennas be positioned?What paths should they follow?
![Page 45: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/45.jpg)
Extraction of information from images
image of same scene at two different frequencies
![Page 46: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/46.jpg)
Infer material properties from radar images
![Page 47: Synthetic Aperture Radar Imaging - Purdue University](https://reader031.vdocuments.us/reader031/viewer/2022022519/6217da2aa3c2e57eff50808d/html5/thumbnails/47.jpg)
papers and lectures available athttp://eaton.math.rpi.edu/Faculty/cheney
Radar imaging is a field that is ripe for mathematical attention!