Within the Resolution Cell_Super-resolution in Tomographic SAR Imaging.pdf

Within h R
Wi hi the Resolution Cell:
l i C ll
Super-resolution in Tomographic SAR Imaging

Xiao Xiang Zhu, Richard Bamler

Remote Sensing Technology Institute, DLR/TUM
g gy ,

IGARSS 2011, 24-29 July 2011, Vancouver, Canada

Very high resolution acquisition (1.1×0.6m2)
Medium resolution acquisition (5×25m2)
TerraSAR-X
ERS

Very high resolution (VHR)
opens up for the first time the opportunity to use
SAR for urban infrastructure monitoring
from space ...

Problems arise ...
P bl i

SAR Side-looking Imaging Geometry

elevation
angle 

s

r
x

SAR Geometry in Range-Elevation Plane

s
z 3-D reflectivity
distribution
x
y
  x, r , s 

r

Within the Resolution Cell_Super-resolution in Tomographic SAR Imaging.pdf

Complex 3-D Structures as Imaged by SAR

Bellagio hotel, Las Vegas
interpretation difficult  real 3 D imaging required  TomoSAR
3-D

Optical image, © Google Earth TerraSAR-X spotlight mode

TomoSAR, a Spectral Estimation Problem
Complex pixel value in acquisition n (after some phase corrections):
b
g n     s  exp   j 2 n s  ds
 d
s
b
 FT [ ( s )] |n , n  1,..., N
2b
n  n
r
TomoSAR = spectral estimation

elevation • irregular sampling
aperture • small N
• motion must be considered

s s
z 3-D reflectivity
distribution
x
y
  x, r , s 

r

TomoSAR System Model

 Discrete system model:
 
     
g     
 R γ  ε
     
  N 1
  
  N L
  
  L1
 
Measurements Irregular FT Elevation profile Noise

 SVD-Wiener, a MAP estimator with white noise and prior (reference)

γ   R T Cεε1R  C1  R T Cεε1g
 1 
ˆ γγ

noise covariance prior covariance

Single + Double
Topography
[m]

Sparsity of the Signal in Elevation

Elevation resolution ca. 50 times worse than in azimuth and range
 Super resolution is crucial
Super-resolution
 Signal sparse in elevation

The SL1MMER Algorithm

Scale-down by

L1 norm  Compressive sensing theory

Minimization –  Candès 2006, Donoho 2006 Baraniuk 2007
2006
Candès &Wakin 2008
2006, 2007,

Model selection –
Estimation
Reconstruction

Proposed by Zhu et al in IGARSS 2010, offers an aesthetic non parametric realization of NLS
al. 2010 non-parametric

X. Zhu, R. Bamler, Tomographic SAR Inversion by L1 Norm Regularization – The Compressive Sensing Approach,
IEEE Transactions on Geoscience and Remote Sensing, 48(10), pp. 3839-3846.
X. Zhu R Bamler Super-Resolution
X Zhu, R. Bamler, Super Resolution Power and Robustness of Compressive Sensing for Spectral Estimation with
Application to Spaceborne Tomographic SAR, IEEE Transactions on Geoscience and Remote Sensing, in press.

Scale-down by L1 Norm Minimization
 
 
      under determined
under-determined system
g    R  γ   infinitely many solutions
     
  N1
  
  NL
  
  L1
 
Make use of special prior ─ sparsity

 Nonlinear least squares (NLS) ─ maximum likelihood estimator (MLE)
(Theoretically the best but NP hard)
best, NP-hard)

 2
γ  arg min g  Rγ 2  MS γ
ˆ
γ 0 
 Compressive sensing (CS) based sparse reconstruction algorithms
(Convex optimization, solved b li
(C ti i ti l d by linear programming)
i )
 2
γ  arg min g  Rγ 2  K γ
ˆ
γ 1 

Scale-down by L1 Norm Minimization

 
 
     
g    R  γ 
     
  N1
  
  NL
  
  L1
 

Model Selection

 
 
       
g    R  γ  R ( s)  
ˆ 
       
  N1
  
  NL
   
  N K
 ˆ
  L1
 

Estimation (De-biasing)

 
 
       
g    R  γ  R ( s)  
ˆ 
       
  N1
  
  NL
   
  N K
 ˆ
  L1
 

   
g    R ( s) 
ˆ  γ ( s) 
ˆ γ ( s)   
ˆ ˆ
    
  K1
ˆ   K1
ˆ
  N 1
  
  N K
 ˆ

Super-resolution of SL1MMER — Simulation
Two scatterers inside one SAR pixel
δs: Distance between two scatterers



z  40m
z

x y
y  69m

Super-resolution of SL1MMER — Simulation

Key Questions

1) Can we separate two close scatterers?

2) If yes: How accurately can we estimate their
) y y
▫ positions, 
▫ amplitudes,   SL1MMER is an efficient estimator
▫ phases?  I.e. it
I its estimation accuracy approaches th
ti ti h the
Cramér–Rao lower bound

… as a function of SNR, N, distance, phase difference, …

X. Zhu, R. Bamler, Super-Resolution Power and Robustness of Compressive Sensing for Spectral Estimation with
Application to Spaceborne Tomographic SAR, IEEE Transactions on Geoscience and Remote Sensing, in press.

Key Questions

1) Can we separate two close scatterers?

2) If yes: How accurately can we estimate their
) y y
▫ positions, 
▫ amplitudes,   SL1MMER is an efficient estimator
▫ phases?  I.e. it
I its estimation accuracy approaches th
ti ti h the
Cramér–Rao lower bound

… as a function of SNR, N, distance, phase difference, …

Super-Resolution, a Detection Problem

s
s  :
s

a1 a2
s
s

PD  SNR  N , , a1 a2 ,  

H0: single scatterer SNR  0 ... 10dB H1: double scatterers
(not resolved) N  10 ... 100 (resolved)
(resol ed)

Super-Resolution Power

PD = 50%

 The definition of resolution:
The minimum distance between two scatterers that are separable at a
prespecified probability of detection PD

Super-resolution Factor
s 1
 50%  
 50% 50%

PD = 50%
0%

50%  0 4
0.4   50%  2.5
 25

Fundamental Bounds for SR Factors
Δφ uniformly distributed in [-π, π]

 . d 
SR
factors:

1.5-25

Super-resolution of SL1MMER — Real Data

Bellagio hotel, Las Vegas

Optical image, © Google Earth TerraSAR-X spotlight mode

Number of Scatterers

 SVD - Wiener 13% double  SL1MMER 30% double

Blue: Null scatterers per pixel; Green: Single; Red: Double

Super-resolution of SL1MMER — Final Proof

1) SL1MMER detects
much more double
scatterers
3 dB point response
width 2) Mainly contributed by
the SR power

s
Normalized elevation distance  :
s

Conclusions

 VHR tomographic SAR inversion is able to reconstruct the shape and
motion of individual buildings and city areas
areas.

 Super-resolution is crucial and possible.

 The achievable super-resolution factors with the newly proposed
SL1MMER algorithm in the typical parameter range of tomographic SAR
are found to be promising and are on the order 1.5~25.

 SL1MMER algorithm is an efficient estimator, and offers an aesthetic
non-parametric realization of NLS

City Tour
Cit T

Thanks to Y. Wang & G.Hochleitner for visualization

Within the Resolution Cell_Super-resolution in Tomographic SAR Imaging.pdf

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