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).

2

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

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

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

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.....

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

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)

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

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

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

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

• 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.

16

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 ....

!

!

!

"#$%&'!()!!*+,--#.$!/'.0'&!#.!123%4%'&4%'5!67)!

!"#$%&#'&(!)*&+#,%&-./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!