Wideangle SAR image formation with migratory scattering centers and regularization in Hough space

Stochastic Systems Group
Wide-Angle SAR Image Formation with
Migratory Scattering Centers and
Regularization in Hough Space
Kush R. Varshney, Müjdat Çetin, John W. Fisher III, and Alan S. Willsky
June 6, 2006
Massachusetts Institute of Technology
Stochastic Systems Group
Outline
• Spotlight-Mode SAR Imaging
• Joint Image Formation and Anisotropy Characterization
• Overcomplete Basis Formulation
• Sparsifying Regularization
• Moving From Pixels to Objects
ƒ
Migratory Scattering Centers
ƒ
Regularization in Hough Space
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Stochastic Systems Group
2-D Spotlight-Mode SAR
s ( x, y )
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Stochastic Systems Group
Observation Model
• Phase history domain:
− j 4 πc f ( x cosθ + y sin θ )
ƒ r ( f , θ ) = ∫∫ s ( x, y )e
dxdy
x 2 + y 2 ≤ L2
• Range profile domain:
ˆ (ρ , θ ) =
ƒ R
s( x, y )δ (ρ − x cos θ − y sin θ )dxdy
2
∫∫
x + y 2 ≤ L2
ƒ
Projection of scattering function in direction specified by θ
• Related by 1-D Fourier transform
L
ƒ r ( f ,θ ) = R
ˆ (ρ ,θ ) exp − j 4πf ρ dρ
c
∫
−L
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{
}
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Stochastic Systems Group
Point Scatterers
• At typical SAR frequencies, scattering appears as though
coming from a set of discrete points
• s( x, y ) = δ (x0 , y0 )
{
}
• Phase history: r ( f ,θ ) = exp − j 4πc f ( x0 cos θ + y0 sin θ )
• Range profile: Rˆ (ρ ,θ ) = δ (ρ − x cos θ − y sin θ )
0
0
• Sinusoid in range profile: ρ = x0 cos θ + y0 sin θ
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Stochastic Systems Group
Wide-Angle Imaging and Anisotropy
• In principle, wide-angle apertures (long flight
path) allow formation of images with high
cross-range resolution
• Isotropic scattering
assumption violated
• Scattering function
depends on aspect
angle θ
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s ( x, y , θ )
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Stochastic Systems Group
Point-Scatterer Model with Anisotropy
• With P point-scattering centers and with anisotropy, phase
history measurements modeled as:
⎧ 4πf
⎫
(
)
(
)
r ( f , θ ) = ∑ s x p , y p , θ exp ⎨− j
x p cos θ + y p sin θ ⎬
c
⎩
⎭
p =1
P
• Joint image formation and anisotropy characterization:
Recover s(x,y,θ) from r(f,θ)
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Approach
• Spatial locations (xp,yp) can be pixels or points of interest
• Expand s(xp,yp,θ) into an overcomplete basis
• Ill-posed inverse problem
• Solve for coefficients with sparsifying regularization
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Stochastic Systems Group
Overcomplete Basis Expansion
• Expand s(xp,yp,θ) into {b1(θ), b2(θ), …, bM(θ)}
⎧ 4πf
⎫
(x p cos θ + y p sin θ )⎬
r ( f , θ ) = ∑∑ a p ,mbm (θ ) exp ⎨− j
c
⎩
⎭
p =1 m =1
P
M
• Problem: determine unknown coefficients ap,m
• f, θ discrete variables
f1, f2, …, fK
ƒ θ1, θ2, …, θN
ƒ
• Overcomplete basis
ƒ
M>N
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Matrix-Vector Form
• Expressible in matrix-vector form:
⎡ r ( f1 , θ ) ⎤ ⎡ b1 (θ )e − 4 jπf1 ( x1 cos θ + y1 sin θ ) c
⎢ r( f , θ)⎥ ⎢
− 4 jπf 2 ( x1 cos θ + y1 sin θ ) c
⎢ 2 ⎥ = ⎢ b1 (θ )e
⎢ # ⎥ ⎢
#
⎥ ⎢
⎢
− 4 jπf K ( x1 cos θ + y1 sin θ ) c
⎣r ( f K , θ )⎦ ⎢⎣b1 (θ )e
" b M (θ )e − 4 jπf1 ( x1 cos θ + y1 sin θ ) c
b1 (θ )e − 4 jπf1 ( x2 cos θ + y2 sin θ ) c
"
" b M (θ )e − 4 jπf 2 ( x1 cos θ + y1 sin θ ) c
%
#
b1 (θ )e − 4 jπf 2 ( x2 cos θ + y2 sin θ ) c
#
" b M (θ )e − 4 jπf K ( x1 cos θ + y1 sin θ ) c
b1 (θ )e − 4 jπf K ( x2 cos θ + y2 sin θ ) c "
"
%
⎡ a1,1 ⎤
⎥
⎢
b M (θ )e − 4 jπf1 ( xP cos θ + y P sin θ ) c ⎤ ⎢ # ⎥
⎥
b M (θ )e − 4 jπf 2 ( xP cos θ + y P sin θ ) c ⎥ ⎢ a1, M ⎥
⎥
⎢
⎥ ⎢ a2,1 ⎥
#
⎥
b M (θ )e − 4 jπf K ( xP cos θ + y P sin θ ) c ⎥⎦ ⎢ # ⎥
⎥
⎢
⎢⎣aP , M ⎥⎦
• r = Φa
• (r = Φa + n)
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Specific Choice of Basis
• What are the bm(θ)?
• Choose to allow parsimonious representation
|s(θ)|
• Contiguous intervals of anisotropy – all widths and all shifts
Aspect Angle
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Stochastic Systems Group
Specific Choice of Basis
• Rectangular pulses, for example
ƒ
Can seamlessly insert any other shape
ƒ
Hamming, Triangle, Raised Triangle
• Incorporates some prior information
b1
b2
bM
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
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Stochastic Systems Group
Sparsifying Regularization
2
k
• Find a that minimizes J (a) = r − Φa + α a , k < 1
2
k
ƒ •
k
denotes ℓk-norm
• First term: data fidelity
• Second term: regularization term favoring sparsity
• J(a) can be minimized effectively
ƒ
Quasi-Newton method [Çetin & Karl 2001]
ƒ
Greedy graph-structured algorithm [Varshney et al. 2006]
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Stochastic Systems Group
Quick Example
least-squares solution (α = 0)
0.5
0
0
-0.5
0
0.5
0
0
-0.5
0.8
0
3
3
4
2
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θ
θ
0.8
2
4
4
2
2
0.8
0
θ
0
θ
4
0.8
0
0
θ
| s( x , y , θ)|
0.8
0.8
1
3
3
| s( x , y , θ)|
1
0
θ
| s( x , y , θ)|
s(θ)
500 1000
a
0.8
1
| s( x , y , θ)|
0.8
0
| s( x , y , θ)|
500 1000
a
| s( x , y , θ)|
-0.5
0
500 1000
a3
1
0.5
-0.5
500 1000
a1
| s( x , y , θ)|
0
| s( x , y , θ)|
coefficients
0.5
sparse solution (α = 3)
0
θ
0
θ
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Stochastic Systems Group
Moving from Pixels to Objects
• Scattering centers have more meaning than just pixels
• More information is available to allow scene interpretation
• 2 phenomena
ƒ
Anisotropy width and spatial extent are related
ƒ
Certain scattering mechanisms migrate as a function of θ
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Stochastic Systems Group
Glint Anisotropy
• Glint or flash – like a mirror
• Very thin response in θ
• Corresponds to long flat plate in space
• Thinner anisotropy, longer spatial extent
• Would like to explain glint anisotropy parsimoniously
ƒ
Additional sparsity along a line
ƒ
Can infer properties about objects in scene such as orientation
and spatial extent
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Stochastic Systems Group
Range profile
• ρ parameterizes direction of θ
in ground plane geometry
• For fixed θ, ρ = xcosθ + ysinθ
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Hough Transform Ideas
• Points on a line in image space
x
• Intersecting sinusoids in range
profile space
ρ
θ*
ρ*
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y
ρ*
θ*
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θ
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Sparsity Along A Line
• For sparsity along a line, use Hough transform ideas
• Sparsity among scatterers in ρ-θ cells
2
k
• J (a) = r − Φa + α 0 a + α1
2
k
ƒ
Li,j = Si,jFΦ’
Š
Š
Š
K
(ρi,θj)
N
∑∑
i =1 j =1
Li, ja
k
k
Φ’ takes coefficients to phase history domain (not exactly Φ)
F, DFT-like operator, converts to range profile domain
Si,j selects the (i,j)th cell in the range profile domain
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Stochastic Systems Group
Scene with Glint Anisotropy
Conventional image
(formed with polar format algorithm)
True anisotropy at a point
-0.5
magnitude
range (m)
-1
0
0.5
1
1.5
-1
0
cross-range (m)
1
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θ
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Stochastic Systems Group
Example: α0 = 30, α1 = 0
Magnitude
parsimonious in
basis vectors per
pixel
1 211 421 631 8411051126114711681189121012311252127312941315133613571378139914201441146214831
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Stochastic Systems Group
Example: α0 = 0, α1 = 20
Magnitude
parsimonious in
pixels
1 211 421 631 8411051126114711681189121012311252127312941315133613571378139914201441146214831
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Stochastic Systems Group
Example: α0 = 30, α1 = 20
Magnitude
solution represents
entire glint with
one basis vector
1 211 421 631 8411051126114711681189121012311252127312941315133613571378139914201441146214831
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Stochastic Systems Group
Other Hough-Type Regularization Terms
• Sparsity along a line is not the only possibility
• Can favor other types of spatial geometry in image formation
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Stochastic Systems Group
Migratory Scattering Centers
• Other effect besides anisotropy
prominent in wide-angle SAR
• Certain types of scattering
centers migrate as function of θ
• Parameterize migration around
a ‘center’
• Scattering center closer to radar
by R(θ)
• If circle, R(θ) = R
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Stochastic Systems Group
Trigonometry
Rcosθ
Rsinθ
θ
(xc, yc)
R
(xc–Rcosθ, yc–Rsinθ)
• s(x, y) = δ(xc–Rcosθ, yc–Rsinθ)
• ρ = (xc–Rcosθ)cosθ + (yc–Rsinθ)sinθ
= xccosθ + ycsinθ – R
• ρ = xccosθ + ycsinθ – R(θ)
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Stochastic Systems Group
Migratory Scattering Center
Observation Model
(x
p
, yp )
invariant spatial location
(when θ = 0)
⎧ 4πf
r ( f , θ ) = ∑ s (x p , y p , θ )exp ⎨− j
c
⎩
p =1
P
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((x
⎫
)
)
(
)
(
)
+
+
−
R
0
cos
θ
y
sin
θ
R
θ
⎬
p
p
p
p
⎭
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Stochastic Systems Group
Procedures for
Circular Migration Characterization
• Restricting to migration in a circle, two methods
• Method 1: even more overcomplete basis
P L M
⎧ 4πf
⎫
ƒ
(
(
)
)
(
)
−
+
+
−
a
b
θ
exp
j
x
R
cos
θ
y
sin
θ
R
⎨
∑∑∑
p ,l , m m
p
p ,l
p
p ,l ⎬
c
⎩
⎭
p =1 l =1 m =1
• Techniques to minimize cost function unchanged
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Stochastic Systems Group
Procedures for
Circular Migration Characterization
• Method 2: optimize over vector of radii R (one entry per
scatterer)
ƒ
R
ƒ
{
min r − Φ(R ) arg min r − Φ(R ) 2 + α a
a
k
k
}
2
2
basis vectors as function of radius:
⎧
⎩
φk , p ,m (R p ) = bm (θ ) ⋅ exp⎨− j
ƒ
2
4πf k
((x p + R p )cosθ + y p sin θ − R p )⎫⎬
c
⎭
minimization by nonlinear least-squares optimization techniques,
where the nonlinear function involves solving the inner
sparsifying regularization minimization
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Stochastic Systems Group
Example
true migratory scattering shape
overlaid on conventional image
migratory scattering solution
overlaid on conventional image
solution gives much more
object-level information than
conventional image
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Non-Circular Migration
• Characterize circular migration over subapertures
-0.4
-1
-0.8
-0.6
-0.2
-0.2
range (m)
range (m)
-0.4
0
0.2
0
0.4
0.6
0.8
1
-1
-0.5
0
cross-range (m)
0.5
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1
0.2
-0.2
-0.1
0
0.1
0.2
cross-range (m)
0.3
0.4
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Stochastic Systems Group
Conclusion
• Novel overcomplete basis and sparse signal representation
formulation for anisotropy characterization in wide-angle SAR
• Exploit available information in phase history measurements to
extract object-level information
ƒ
Hough space regularization for glint anisotropy
ƒ
Migratory basis vectors in overcomplete basis for migratory
scattering
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Questions
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