Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M. Nannini and A. Reigber

IGARSS 2011Esteban Aguilera

Compressed Sensing forPolarimetric SAR Tomography

E. Aguilera, M. Nannini and A. Reigber


1. Polarimetric SAR tomography2. Compressive sensing of single signals3. Multiple signals compressive sensing: Exploiting

correlations4. Compressive sensing for volumetric scatterers5. Conclusions

Overview


azimuthground range

M parallel tracks for 3D imaging

Tomographic SAR data acquisition

Side-looking illumination at L-Band


The tomographic data stack

Our dataset is a stack of M two-dimensional SAR images per polarimetric channel

M images

azimuthrange


The tomographic data stack

Projections of the reflectivity in the elevation direction are encoded in M pixels (complex valued)

azimuthrange

1

2

M

bb

B

b


The tomographic signal model: B = AX

11,1 1,2 1,3 1,1

22,1 2,2 2,3 2,2

33,1 3,2 3,3 3,

,1 ,2 ,3 ,

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

N

N

N

MM M M M N N

xa r a r a r a rb

xa r a r a r a rb

xa r a r a r a r

ba r a r a r a r x

,4

,( ) i jj r

i ja r e

height

B : measurementsA : steering matrixX : unknown reflectivity


What’s the problem?

High resolution and low ambiguity require a large number of tracks:

1. Expensive and time consuming2. Sometimes infeasible3. Long temporal baselines affect reconstruction


Where does this work fit?

Beamforming (SAR tomography):1. Beamforming (Reigber, Nannini, Frey)2. Adaptive beamforming (Lombardini, Guillaso)3. Covariance matrix decomposition (Tebaldini)

Physical Models (SAR interferometry):1. PolInSAR (Cloude, Papathanassiou)2. PCT (Cloude)

Compressed sensing (SAR tomography)1. Single signal approach (Zhu, Budillon)2. Multiple signal/channel approach


Elevation profile reconstruction

A

B AX

AMxN : steering matrixXN : unknown reflectivityBM : stack of pixels

height

gnd. rangeazimuth


The compressive sensing approach

We look for the sparsest solution that matches the measurements

minX 1

X

2AX B subject to

Convex optimization problem


How many tracks?

In theory:

take

measurements

frequencies selected at random

In practice:

we can use our knowledge about the signal and sample less:

low frequency components seem to do the job!

0 log( )M C S N

2M S


CS for vegetation mapping ?

The elevation profile can be approximated by a summation of sparse profilesDifferent to conventional models (non-sparse). And probably a bad one…

elevation

amplitude

= + + … +


Tomographic E-SAR CampaignTestsite: Dornstetten, GermanyHorizontal baselines: ~ 20mVertical baselines: ~ 0mAltitude above ground: ~ 3800m# of baselines: 23

3,5 m

2 corner reflectors in layover and ground


CAPON using 23 tracks (13x13 window) = ground truth

40 m

2 corner reflectors in layover

Canopy and groundGround

40 m

Single Channel Compressive Sensing

CS using only 5 tracks


Normalized intensity – 40 m

Beamforming (23 passes, 3x3)

SSCS (5 passes, 3x3)


Multiple Signal Compressive Sensing

Assumption: adjacent azimuth-range positions are likely to have targets at about the same elevation

1 1 1

2 2 2...

M M M

b c db c d

b c d

L columnsazimuthrange

rangeazimuth

M images GHH


Polarimetric correlations

We can further exploit correlations between polarimetric channels

G

3L columns

GHH GHV GVV



A

G AY

AMxN : steering matrixYNx3L : unknown reflectivities HH HV VV Mx3L : stacks of pixelsG


YNx3L : unknown reflectivity

Y

minY

2AY G subject to

2,1Y


We look for a matrix with the least number of non-zero rows that matches the measurements


Mixed-norm minimization

minY

2AY G subject to

0

Number of columns in Y (window size + polarizations)

Probability of recovery failure

(Eldar and Rauhut, 2010)

2,1Y


SSCS (saturated) MSCS (span saturated)

MSCS (polar) MSCS (span)

Layover recovery with CS


Beamforming (23 passes, 3x3)

SSCS (5 passes, 3x3)

MSCS (5 passes, 3x3)

MSCS (pre-denoised) (5 passes, 3x3)

Layover recovery with CS


Volumetric ImagingSingle signal CS (5 tracks)

Multiple signal CS (5 tracks)

40 m


Volumetric ImagingSingle signal CS (5 tracks)


40 m


Volumetric ImagingPolarimetric Capon beamforming (5 tracks)


40 m


Towards a “realistic” sparse vegetation model

elevation

amplitude

Canopy and ground component

Possible sparse description in wavelet domain!


Sparsity in the wavelet domain

Daubechies wavelet example: 4 vanishing moments 3 levels of decomposition

groundcanopy ground

canopy

0.5

1

0

0.5

1

0



minY 1

WY

( )AY D Gs.t.

Additional regularization

1

L1 norm of wavelet expansion (W: transform matrix)

synthetic aperture

2,1Y


Volumetric Imaging in Wavelet DomainFourier beamforming using 23 tracks (23x23 window)

Wavelet-based CS (5 tracks)

40 m


Volumetric Imaging in Wavelet DomainFourier beamforming using 23 tracks (23x23 window)

Wavelet-based CS (5 tracks)

40 m


Conclusions

Single signal CS:

1. High resolution with reduced number of tracks2. Recovers complex reflectivities but polarimetry problematic3. Model mismatch is not catastrophic (CS theory)4. It’s time-consuming (Convex optimization)

Multiple signal CS:

1. Polarimetric extension of CS2. Higher probability of reconstruction, less noise3. More robust for distributed targets4. Vegetation reconstruction in the wavelet domain


Convex optimization solvers

CVX (Disciplined Convex Programming): http://cvxr.com/cvx/

SEDUMI: http://sedumi.ie.lehigh.edu/

http://cvxr.com/cvx/

http://cvxr.com/cvx/

http://sedumi.ie.lehigh.edu/

http://sedumi.ie.lehigh.edu/

Documents

Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M. Nannini and A. Reigber