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R1
Σ
L
i )
B
l/ L
⇒
Cl l ing Antenna:
Synthetic Aperture Radar (SAR)
Consider fixed array
v
Bθ
Assumes phase coherence in oscillator during each image
Lec20.4-1 2/26/01
v (t
Angular reso ution θ ≅ λ
aim Equiva ent to Mov
Basic Concept of SAR:
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R2
v x= t
i
− φjE e
x
si f(aziσ =
Synthetic Aperture Radar (SAR) Ideal point reflector
Maintain coherence
If scat f( ):≠
2π
0
) f(
φ = λ
(t)φ
sf(= σ
B oRθ •
E
0
Lec20.4-2 2/26/01
Process a "w ndow" of data, e.g.
We seek 2-D source character zation: muth, range)
Typical CW pulse signal and echo,
angle
x, t
Phase (t, point source range)
Received phase:
Magnitude E point source , range)
x, t Received magnitude:
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R3
SARθ L λ
0 xψ ψ ∗ ψ = ψx ˆW( ) E( ) E( )
↔
Resolution of “Unfocused” SAR
)
0x L
x
W(x) E(x)•
Reconstruction of image: first select R, xo of interest
SAR angular resolution: θ ≅ λ θ =SAR SAR BL R
≅ λ D
Lec20.4-3 2/26/01
W(x observed
≥ λ D R
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R4
SAR
B
l l R D ( l l , l
l / 2
= θ • ≅ θ
≅
BSAR SARl / L / R D / Rθ ≅ λ θ =
≅ λ / D
Resolution of “Unfocused” SAR
0l
R
R
Yi iR∆
∆ ≅ cTR 2
x x D∆ >
0
R
L, the lateral resolution can be < D.
Lec20.4-4 2/26/01
Lateral spatia reso ution want smal D, arge arge L)
Range reso ution cT
SAR angular reso ution: ≥ λ
Then repeat for al R, x
max
min
elds mage:
Note: If antenna is steered toward the target, increasing
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S1
( )< λ =
> λ
λ
δ
2 2 2 SAR
i/ / D / D .
, R /
Phase-Focused SAR
B /D (x)ψ
/ 2Dλ∼
x 0xD
L
R
and focusing is unnecessary
⎧ ⎛ ⎞+ = ≅⎪ ⎜ ⎟⎝ ⎠⎪
⎪λ λδ < ⇒ ≤⎨
⎪ ⎪ ≥⎪ λ⎩
2 2 2 2
2
2
LR (R ) R 2
L 16 16
2L i
Lec20.4-5 2/26/01
λ = = λ θ
Phase focusing is required if 16, so the target s in the S i.e., if R 2L 2 R where L
AR near fi eld
θ ≅ λ
+ δ + δ
δ ≅
2 R
8R
therefore R "farf eld"
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Phase-Focused SAR →W(x) (x) wi i
(t) l
φ ↑
2π
(t)φ Pulse
xo
spatial resolution in x ca be less than D, using beamsteering (e.g. a phased array)
Also: point sources exist in scene.
S2 Lec20.4-6
2/26/01
Solution: W ' th phase correct on
PRF must be higher; to reconstruct we need at least 2 samp es per phase wrap.
x, t
Note: With phase focusing, L increases and
L.O. phase drift can be corrected if bright
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S3
′= = ′ ′∆ψ = λ
>> x
v / L ( ) / L / L
v/D
L′
L
(x) )Ι
( )
• Ι •
ψ ∗ ψ = ψx x x x
ˆW(x) (x) E(x)
ˆW( ) ( ) E( ) E
( ) ( )Ι xx
xψ
/ Lλ ′ 0
x∆ψ
Required Pulse-Repetition Frequency (PRF)
Lec20.4-7 2/26/01
>> λ >>
PRF e.g., v aircraft velocityWant D, or D Therefore want PRF
W(x
∗ Ι ψ
Observed
↔ Ι ψ
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S4
l i
x
Di iv
and yi l l
x
R
R
y
yBθ xv
( )min x x
1 DR / c L / v v
⎛ ⎞ ′− < =⎜ ⎟⎝ ⎠
PRF Impacts SAR Swath Width
Lec20.4-8 2/26/01
Don't want echoes to over ap for success ve pulses.
Impl es that large swath w dths require large
eld poorer spatia reso ution.
Swath being imaged max
min
max Therefore let 2 R PRF
<<
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S5
0R
B 0Rθ xv
Envelope Delay Focusing
[ ] B 0
i l li l R (l∆ θ
0x (
xo2R c
Envelope delay
Envelope-delay focusing required when R R 3δ > ∆
∆R δR
θB 2
Lec20.4-9 2/26/01
Range Box
Compensation for both envelope delay and phase delay s needed for very high spatia reso ution .e., for smal arge B) and large R .
Target location)
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U1
SAR Intensity Estimates: Speckle
( )σ = ∑ 2
iji i, j
l
ˆ K w x EK = range-dependent constant i = pulse number j = subscattering index
Many subscatters j per pixel
( )
= =
σ = ε = =∑ ∑ k
2 2N N j t 2
kk k 1 k 1
ˆ a e rSimplifying:
random variable uniformly over 2π typically
lP
j j 1+
ε =
Ι ε =
∑
∑ k
k
Re
m
mΙ ε
Re ε
v
Scattering centers
Lec20.4-10 2/26/01
For each SAR pixe , the estimated cross-section is:
where
ω +φ
pixe g.r.v.z.m.
g.r.v.z.m.
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U2
r0 σ
⎡ ⎤ = σ ∝ ε⎣ ⎦ 22 2
] / 2
E r (2 / 2)
2
l
E r r 2 / 3 f(N)!⎡ ⎤− ≠⎣ ⎦
σ≅ = ⇒ σ π
i l l li 2 / 3Q i
i / 2
− σε = σ
∑ 2 2
k 2 r p e
SAR Intensity Estimates: Speckle
mΙ ε
Re ε
r
( )− σ> = 2
or / / 2 op r r e
Lec20.4-11 2/26/01
pr
= σ π
− π
E[r
Because we are adding (averaging) phasors, not sca ars
≅ σ
Thus raw SAR mages, ful spatia reso ution, STD dev ation have 0.53 very grainy mages mean intens ty
r / 2
"r "
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U3
Al l l i iti2 ls Q / M
( si i l an be r
smoo d) e
the ⇒ =
τ
Reduction of SAR Speckle
i l i
L Q
N =
N l
i
=
x
Lec20.4-12 2/26/01
ternate ways to reduce "speck e" by averaging pixe ntens es: by averaging M pixe 0.53
using large B gnals y elds high reso ution, c 1) Blu imag
2) Average us ng spatia divers ty
0.53
3 or more ooks possible. Trade against potential resolut on in x.
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U4
f
Q N
=
Reduction of SAR Speckle
, where each band yields an independent image. Note that the same total bandwidth could alternatively yield more range resolution and pixels for averaging.
Lec20.4-13 2/26/01
0.53
N Bands
Image 1 Image 3 Image 2
3) Average using frequency diversity
Alternate ways to reduce “speckle” by average pixel intensities:
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Reduction of SAR Speckle
U5
works only if source varies. For example a narrow swath permits high PRF and an increase in N, but unless the antenna translates more than ~λR/D between pulses [R = range, D = pixel width (m)], then adjacent pulse returns are correlated.
j
D p Dθ ≅ λ
R
~ Q¢ = S.D./M ≅ (1/3)/4 levels = 1/12, versus Q ≅ 1/2. To reduce standard deviation by 6, need N > 62 = 36 looks.
Lec20.4-14 2/26/01
4) Time averaging
In general, reasonably smooth images need > 8 levels, so
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U6
Antenna
v
( )
v
Alternative SAR Geometries and Applications
Velocity dispersion within inverse SAR (ISAR) source (e.g., a moving car) can displace its apparent position laterally.
Antenna
spacecraft)
Antenna
Lec20.4-15 2/26/01
Moves
Source Stationary Case A.
e.g., geology
Case C.
Stationary
(e.g. moon or
Source rotates
Stationary
Case B. Source Moves
(e.g. satellite)
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ESTIMATION
Professor David H. Staelin Massachusetts Institute of Technology
Examples taken from Remote Sensing Principles are Nonetheless Widely Applicable
Linear and Non-Linear Problems Gaussian and Non-Gaussian Statistics
A1 Lec20.4-16
2/26/01
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A2
Linear Estimation: Smoothing and Sharpening
( ) ( ) ( )A B sA A s s4
1T G T d4 π
ψ = ψ Ωπ ∫
Consider antenna response example; we observe:
Smoothing reduces image speckle and other noise; sharpening or “deconvolution” increases it.
Sharpening can “de-blur” images, compensating for diffraction or motion induced blurring. Audio smoothing and sharpening are similar.
Lec20.4-17 2/26/01
ψ − ψ
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A3
i
( ) ( ) ( )A B sA A s s4
1T G T d4 π
ψ = ψ Ωπ ∫
( ) ( ) ( )A BA 1T G T ( ll li )
4ψ ≅ ψ
π
( ) ( ) ( )A B 1T f T f
4ψ ψ ψ = • π
l↑
A↑
( ) ( ) ( )x yx yj2 f f
x y x yx y i G , e d dψ ψ+ψ
ψ ψ
⎡ ⎤ = ψ ψ⎢ ⎥⎣ ⎦∫∫
Linear Estimation: Smoothing and Sharpening
Lec20.4-18 2/26/01
Cons der antenna response example; we observe:
ψ − ψ
sma so d anglesψ ∗
G f
Cyc es/Radian ntenna Spectral Response
.e., G f , f − π ψ
ψ ψ
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A4
( ) ( ) ( )A B 1T f T f
4ψ ψ ψ = • π
l↓
A↓
( ) ( ) ( )A BA 1T G T ( ll li )
4 ψ
π
Example: Square Uniformly-Illuminated Aperture
Dλ λ ψτx x
, f0
D
( ) ( )λ ψτER ( )G ψ ψy
ψx
/ D iλ↔y y, f
λ ψτ D
Aperture
e.g.
( ) ( ) ( )
ψ ψ
ψ
π =
A B
4 T fT f
G f
i
ψ← > λ x
f D / !
Try:
Lec20.4-19 2/26/01
G f
Cyc es/Radian ntenna Spectral Response
sma so d anglesψ ≅ ψ ∗
, G f
Blurr ng Function
Th s restores high-frequency components, but
goes to zero for
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A5
( )ψi i i
"princi l
.
n"
( ) ( ) ( ) ( )ψ
ψ ψ ψ
π =
A B
4 T fˆ f W f
G f
( )ψW f
λψ τxxf ,
ψy f
0 D λ
Example: Square Uniformly-Illuminated Aperture
cycles/radian
( ) ( ) ( )
ψ ψ
ψ
π =
A B
4 T fT f
G f
i
ψ← > λ x
f D / !
Lec20.4-20 2/26/01
The w ndow function W f avoids the s ngular ty
pal so utio The uses a boxcar W (s).
Try: T
Th s restores high-frequency components, but
goes to zero for
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A6
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
ψ ψ ψψ ψ ψ
ψ ψ ψ
ψ
= =
= π
ψ =
B
B B A
A B
B
i i :
ˆf f f W f W f ,
f T f 4
T i i ly illumi
( )B T ψ = xψ
yψ
/λ
( )B ψ <
Example: Square Uniformly-Illuminated Aperture
( )B T ψ ↔ ( )B T fψ
x fψ00
Can set to zero (apriori information)
Can clip, transform, iterate
Lec20.4-21 2/26/01
= δ ψ
= π
Cons der the restored (sharpened) mage of a point source, T
Then T 1, so T 4 T G f
since T G f
2-D s nc funct on for a uniform nated rectangular
antenna aperture.
Zero at 2D Predicts T 0!
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A7
( ) ( ) ( )A A T f T fψ ψ ψ = + ↑
T ↑
∆
( ) ( ) ( ) ( )B A ˆ ˆT f f /ψ ψ ψ ψ
⎡ ⎤= π⎣ ⎦
( )i ψ
l→ ↓
( ) ( ) ifi i
llψ ψ
( )B T fψ
D /− λ D / λ x
fψ 0
Sharpening Noisy Images
0 fψ
( )AT fψ
( )N fψ
e.g.
D/λ
Lec20.4-22 2/26/01
N f
Truth rms
4 T G f W f
Therefore optim ze W f
0! Amp ifier Noise
Ampl es wh te noise N f for sma G f
for point source
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A8
( ) ( ) ( ) ( ) ( )( )0
2
B A
4 W f Minimi T f T f Q
G f
∆ψ ψ ψ ψ
ψ
π − + =
⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦
( )i f :ψ
iel∂ ∂ =
( ) ( ) ( ) ( ) ( ) ( )
( ) ( )
2
2
o o o
o
1 1 * E T f T f T fA A A2 2W f E T fA
∗ ψ ψ ψψ ψ
ψ
ψ ψ
+ +
=
+
⎡ ⎤ ⎢ ⎥⎣ ⎦
⎡ ⎤ ⎢ ⎥⎣ ⎦
( ) ( )
( )o
2
2 A
1 1W f
N1
S1 E T f
ψ
ψ
ψ
= ≅
+ +
⎛ ⎞ ⎜ ⎟⎡ ⎤ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
⎡ ⎤ ⎢ ⎥⎣ ⎦
[ ]A N =
Sharpening Noisy Images
Lec20.4-23 2/26/01
ze E N f
Therefore optim ze W
Q / W 0 y ds:
optimum
N f N f
N f
optimum E N f
If E T 0, then
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A9
1(1 N / S) i i
−+
( ) ( )
( )o
2
2 A
1 1W f
N1
S1 E T f
ψ
ψ
ψ
= ≅
+ +
⎛ ⎞ ⎜ ⎟⎡ ⎤ ⎜ ⎟⎢ ⎥⎣ ⎦ ⎝ ⎠
⎡ ⎤ ⎢ ⎥⎣ ⎦
[ ]A =
( )AA
l i lψ
= fψ0 D / λ 0
N 0≠
( )optW f wi (l l lψ ⇒
( )optW fψ
Sharpening Noisy Images
fψ
( )optW f ψ
Lec20.4-24 2/26/01
The weighting s w dely used
optimum E N f
If E T N 0, then
Can be used for restoration of blurred images of al types: photographs TV, T maps, SAR mages, fi tered speech, etc.
for N 0 :
der beam ower spatia resolution), ower sidelobes.