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

Xiao Xiang Zhu, Richard Bamler

Remote Sensing Technology Institute, DLR/TUM

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

g gy ,

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

Very high resolution (VHR)Very high resolution (VHR)opens up for the first time the opportunity to useSAR for urban infrastructure monitoringSAR for urban infrastructure monitoringfrom space ...

P bl iProblems arise ...

SAR Side-looking Imaging Geometry

elevation

angle

s

rx

SAR Geometry in Range-Elevation Plane

z

x x r s

3-D reflectivity distribution

s

yx

r

, ,x r s

Complex 3-D Structures as Imaged by SAR

Bellagio hotel, Las VegasBellagio hotel, Las Vegasinterpretation difficult real 3 D imaging required TomoSARinterpretation difficult real 3-D imaging required TomoSAR

Optical image, © Google Earth TerraSAR-X spotlight mode

TomoSAR, a Spectral Estimation Problem

bComplex pixel value in acquisition n (after some phase corrections):

2j db

exp 2

[ ( )] | , 1,...,n

n ns

g s j s ds

FT s n N

2 nn

br

TomoSAR = spectral estimation

elevationaperture

TomoSAR = spectral estimation• irregular sampling

• small Naperture

s

small N

• motion must be considered

sz

x x r s

3-D reflectivity distribution

s

yx

r

, ,x r s

TomoSAR System Model

Discrete system model:

g R εγ N 1 N L

L 1

Measurements Irregular FT Elevation profile Noise

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

1 1T 1 1 T 1ˆ εε γγ εεR C R C R Cγ g

noise covariance prior covariancenoise covariance prior covariance

SingleTopography

[m]

Single + DoubleTopography

[m]

Sparsity of the Signal in Elevation

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

The SL1MMER Algorithm

Scale-down byScale-down by

L1 norm Compressive sensing theory Candès 2006 Donoho 2006 Baraniuk 2007

Minimization –

Model selection –

Candès 2006, Donoho 2006, Baraniuk 2007, Candès &Wakin 2008

Model selection

Estimation

Reconstruction

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

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 Power and Robustness of Compressive Sensing for Spectral Estimation with

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

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 system

N 1 N L

g R γ under determined system infinitely many solutions

L 1

Make use of special prior ─ sparsity

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

2 0

2ˆ arg min MS γ

g Rγ γγ

Compressive sensing (CS) based sparse reconstruction algorithms (C ti i ti l d b li i )(Convex optimization, solved by linear programming)

1

2

2ˆ arg min K

γγ g γ γR

Scale-down by L1 Norm Minimization

g R γ N 1 N L

L 1

Model Selection

g R γ ˆ( )

R s

N 1 N L

L 1

ˆN K

Estimation (De-biasing)

g R γ ˆ( )

R s

N 1 N L

L 1

ˆN K

K̂ 1ˆN 1 N K

ˆ ˆ( ) ( )

g R s γ s K̂ 1

ˆ ˆ( )

γ s

N 1 N K

Super-resolution of SL1MMER — SimulationTwo scatterers inside one SAR pixelδs: Distance between two scatterers

40z m

z

yx y

69y m

Super-resolution of SL1MMER — Simulation

Super-resolution of SL1MMER — Simulation

Super-resolution of SL1MMER — Simulation

Key Questions

1) Can we separate two close scatterers?1) Can we separate two close scatterers?

2) If yes: How accurately can we estimate their ) y y▫ positions,▫ amplitudes,▫ phases?

SL1MMER is an efficient estimator

I it ti ti h th▫ phases? I.e. its estimation accuracy approaches 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?1) Can we separate two close scatterers?

2) If yes: How accurately can we estimate their ) y y▫ positions,▫ amplitudes,▫ phases?

SL1MMER is an efficient estimator

I it ti ti h th▫ phases? I.e. its estimation accuracy approaches the Cramér–Rao lower bound

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

Super-Resolution, a Detection Problem

s : s s

1a 2a

s

s1 2

s s

1 2, , ,D SNR N aP a

H0: single scatterer

(not resolved)

H1: double scatterers

(resol ed)10 100N 0 ... 10dBSNR

(not resolved) (resolved) 10 ... 100N

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

50%50% 50%

1s

0%PD = 50%

0 4 2 550% 50%0.4 2.5

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

. d

SRfactors:

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

3 dB point response

1) SL1MMER detects much more double scatterers3 dB point response

width 2) Mainly contributed by the SR power

s:s

s

Normalized elevation distance

Bellagio Hotel in 3D

Conclusions

VHR tomographic SAR inversion is able to reconstruct the shape and motion of individual buildings and city areasmotion of individual buildings and city 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 SARSL1MMER 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

Cit TCity Tour

Thanks to Y. Wang & G.Hochleitner for visualization