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

Compressed Sensing for Polarimetric SAR Tomography E. Aguilera, M. Nannini and A. Reigber. Polarimetric SAR tomography Compressive sensing of single signals Multiple signals compressive sensing: Exploiting correlations Compressive sensing for volumetric scatterers Conclusions. Overview.

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

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  1. Compressed Sensing forPolarimetric SAR TomographyE. Aguilera, M. Nannini and A. Reigber

  2. Polarimetric SAR tomography Compressive sensing of single signals Multiple signals compressive sensing: Exploiting correlations Compressive sensing for volumetric scatterers Conclusions Overview

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

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

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

  6. The tomographic signal model: B = AX height B : measurementsA : steering matrixX : unknown reflectivity

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

  8. Where does this work fit? Beamforming (SAR tomography): Beamforming (Reigber, Nannini, Frey) Adaptive beamforming (Lombardini, Guillaso) Covariance matrix decomposition (Tebaldini) • Physical Models (SAR interferometry): • PolInSAR (Cloude, Papathanassiou) • PCT (Cloude) • Compressed sensing (SAR tomography) • Single signal approach (Zhu, Budillon) • Multiple signal/channel approach

  9. Elevation profile reconstruction height AMxN: steering matrix XN : unknown reflectivity BM : stack of pixels A azimuth gnd. range

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

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

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

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

  16. Multiple Signal Compressive Sensing Assumption: adjacent azimuth-range positions are likely to have targets at about the same elevation M images azimuth azimuth range range GHH L columns

  17. Polarimetric correlations We can further exploit correlations between polarimetric channels GHH GHV GVV 3L columns

  18. Elevation profile reconstruction Mx3L: stacks of pixels AMxN: steering matrix YNx3L: unknown reflectivities HHHVVV

  19. Elevation profile reconstruction YNx3L: unknown reflectivity subject to We look for a matrix with the least number of non-zero rows that matches the measurements

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

  21. Layover recovery with CS MSCS (polar) MSCS (span) SSCS (saturated) MSCS (span saturated)

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

  23. Volumetric Imaging Single signal CS (5 tracks) 40 m Multiple signal CS (5 tracks)

  24. Volumetric Imaging Single signal CS (5 tracks) 40 m Multiple signal CS (5 tracks)

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

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

  27. Sparsity in the wavelet domain • Daubechies wavelet example: • 4 vanishing moments • 3 levels of decomposition canopy 1 1 0.5 0.5 0 0 canopy ground ground

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

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

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

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

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

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