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:
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:
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
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
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"
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
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
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
<<
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)
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.
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 "
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.
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:
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
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)
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
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
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
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
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
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!
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
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
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.