1. IGARSS 2011 Esteban Aguilera Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M. Nannini and A. Reigber

2. IGARSS 2011 Esteban Aguilera 1. Polarimetric SAR tomography 2. Compressive sensing of single signals 3. Multiple signals compressive sensing: Exploiting correlations 4. Compressive sensing for volumetric scatterers 5. Conclusions Overview

3. IGARSS 2011 Esteban Aguilera azimuth ground range M parallel tracks for 3D imaging Tomographic SAR data acquisition Side-looking illumination at L-Band

4. IGARSS 2011 Esteban Aguilera The tomographic data stack Our dataset is a stack of M two-dimensional SAR images per polarimetric channel M images azimuth range

5. IGARSS 2011 Esteban Aguilera The tomographic data stack Projections of the reflectivity in the elevation direction are encoded in M pixels (complex valued) azimuth range

6. IGARSS 2011 Esteban Aguilera The tomographic signal model: B = AX height B : measurements A : steering matrix X : unknown reflectivity

7. IGARSS 2011 Esteban Aguilera What’s the problem? High resolution and low ambiguity require a large number of tracks: 1. Expensive and time consuming 2. Sometimes infeasible 3. Long temporal baselines affect reconstruction

8. IGARSS 2011 Esteban Aguilera 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

9. IGARSS 2011 Esteban Aguilera Elevation profile reconstruction A A MxN : steering matrix X N : unknown reflectivity B M : stack of pixels height gnd. rangeazimuth

10. IGARSS 2011 Esteban Aguilera The compressive sensing approach We look for the sparsest solution that matches the measurements subject to Convex optimization problem

11. IGARSS 2011 Esteban Aguilera 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!

12. IGARSS 2011 Esteban Aguilera CS for vegetation mapping ? The elevation profile can be approximated by a summation of sparse profiles Different to conventional models (non- sparse). And probably a bad one… elevation amplitude =++ … +

13. IGARSS 2011 Esteban Aguilera Tomographic E-SAR Campaign Testsite:Dornstetten, Germany Horizontal baselines:~ 20m Vertical baselines:~ 0m Altitude above ground:~ 3800m # of baselines:23 3,5 m 2 corner reflectors in layover and ground

14. IGARSS 2011 Esteban Aguilera CAPON using 23 tracks (13x13 window) = ground truth 40 m 2 corner reflectors in layover Canopy and ground Ground 40 m Single Channel Compressive Sensing CS using only 5 tracks

15. IGARSS 2011 Esteban Aguilera Normalized intensity – 40 m Beamforming (23 passes, 3x3) SSCS (5 passes, 3x3)

16. IGARSS 2011 Esteban Aguilera Multiple Signal Compressive Sensing Assumption: adjacent azimuth-range positions are likely to have targets at about the same elevation L columns azimuth range azimuth M images G HH

17. IGARSS 2011 Esteban Aguilera Polarimetric correlations We can further exploit correlations between polarimetric channels 3L columns G HH G HV G VV

18. IGARSS 2011 Esteban Aguilera Elevation profile reconstruction A MxN : steering matrix Y Nx3L : unknown reflectivities HH HV VV Mx3L : stacks of pixels

19. IGARSS 2011 Esteban Aguilera Y Nx3L : unknown reflectivity subject to Elevation profile reconstruction We look for a matrix with the least number of non-zero rows that matches the measurements

20. IGARSS 2011 Esteban Aguilera Mixed-norm minimization subject to Number of columns in Y (window size + polarizations) Probability of recovery failure (Eldar and Rauhut, 2010)

21. IGARSS 2011 Esteban Aguilera SSCS (saturated)MSCS (span saturated) MSCS (polar)MSCS (span) Layover recovery with CS

22. IGARSS 2011 Esteban Aguilera Beamforming (23 passes, 3x3) SSCS (5 passes, 3x3) MSCS (5 passes, 3x3) MSCS (pre-denoised) (5 passes, 3x3) Layover recovery with CS

23. IGARSS 2011 Esteban Aguilera Volumetric Imaging Single signal CS (5 tracks) Multiple signal CS (5 tracks) 40 m

24. IGARSS 2011 Esteban Aguilera Volumetric Imaging Single signal CS (5 tracks) Multiple signal CS (5 tracks) 40 m

25. IGARSS 2011 Esteban Aguilera Volumetric Imaging Polarimetric Capon beamforming (5 tracks) Multiple signal CS (5 tracks) 40 m

26. IGARSS 2011 Esteban Aguilera Towards a “realistic” sparse vegetation model elevation amplitude Canopy and ground component Possible sparse description in wavelet domain!

27. IGARSS 2011 Esteban Aguilera Sparsity in the wavelet domain Daubechies wavelet example: 4 vanishing moments 3 levels of decomposition ground canopyground canopy 0.5 1 0 1 0

28. IGARSS 2011 Esteban Aguilera Elevation profile reconstruction s.t. Additional regularization L1 norm of wavelet expansion (W: transform matrix) synthetic aperture

29. IGARSS 2011 Esteban Aguilera Volumetric Imaging in Wavelet Domain Fourier beamforming using 23 tracks (23x23 window) Wavelet-based CS (5 tracks) 40 m

30. IGARSS 2011 Esteban Aguilera Volumetric Imaging in Wavelet Domain Fourier beamforming using 23 tracks (23x23 window) Wavelet-based CS (5 tracks) 40 m

31. IGARSS 2011 Esteban Aguilera Conclusions Single signal CS: 1. High resolution with reduced number of tracks 2. Recovers complex reflectivities but polarimetry problematic 3. Model mismatch is not catastrophic (CS theory) 4. It’s time-consuming (Convex optimization) Multiple signal CS: 1. Polarimetric extension of CS 2. Higher probability of reconstruction, less noise 3. More robust for distributed targets 4. Vegetation reconstruction in the wavelet domain

32. IGARSS 2011 Esteban Aguilera Convex optimization solvers CVX (Disciplined Convex Programming): http://cvxr.com/cvx/http://cvxr.com/cvx/ SEDUMI: http://sedumi.ie.lehigh.edu/http://sedumi.ie.lehigh.edu/