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.