The document discusses algorithms for 3D reconstruction of urban areas from multi-channel InSAR data. It presents the MCPU algorithm which uses phase data and the MCPAR algorithm which jointly uses phase and amplitude data. MCPAR provides better reconstruction in low coherence areas by exploiting amplitude data to help preserve discontinuities. Both algorithms formulate the reconstruction as a MAP estimation problem solved using Markov random fields and graph cuts optimization. Results on simulated and real data show MCPAR produces more accurate reconstructions than MCPU.

1. Three Dimensional Reconstruction of Urban Areas Using Jointly Phase and Amplitude Multi-channel Images Università di Napoli Parthenope Aymen Shabou Florence Tupin Giampaolo Ferraioli Vito Pascazio IGARSS 2010 – Honolulu July 25-30 2010 TELECOM ParisTech

3. Problem Statement , , Interferometric SAR systems 3D reconstruction starting from two different SAR complex acquisitions A 1 ( R , x )=| A 1 ( R , x )|exp{ j  1 ( R , x )} A 2 ( R , x )=| A 2 ( R , x )|exp{ j  2 ( R , x )} v (·)= u 1 (·) u 2 *(·)=| u 1 (·)|| u 2 (·)|exp{ j  (·)}  (·)=  1 (·)-  2 (·) Phase Unwrapping is needed! SAR 1 SAR 2

5. Three Dimensional Reconstruction - MCPU Multi-Channel Phase Unwrapping (MCPU) algorithm ( Ferraioli et al . [09]) provides the 3D reconstruction in terms of MAP estimation. The problem can be solved using Markov Random Fields (MRF). , , The MRF a posterior Energy is given by :

6. Three Dimensional Reconstruction - MCPU Multi-Channel Phase Unwrapping (MCPU) algorithm ( Ferraioli et al . [09]) provides the 3D reconstruction in terms of MAP estimation. The problem can be solved using Markov Random Fields (MRF). , , f (.) single-channel likelihood p = 1,…, P pixel position m = 1,…, M interferogram (channel) The multi-channel likelihood Energy is given by : The MRF a posterior Energy is given by :

14. Three Dimensional Reconstruction - MCPU The Graph construction used by the MCPU algorithm is the one proposed by Ishikawa This construction ( ordered set of labels ) and the considered a priori model ( convex model ) ensures to reach the global optimal solution ! h 3 h 2 h 1 o p q Red arrows are representative of the likelihood term, the blue of the a priori energy t s

16. Three Dimensional Reconstruction - MCPAR , , MCPAR algorithm provides the 3D reconstruction in terms of MAP estimation. The problem can be solved using MRF The MRF a posterior Energy is given by : where x =[ a h ] is the vector collecting the unknown parameters y =[ A 1 A 2  ] is the vector collecting the multi-channel observed data

17. Three Dimensional Reconstruction - MCPAR The likelihood Energy is given by

18. Three Dimensional Reconstruction - MCPAR The likelihood Energy is given by The a priori Energy is given by This a priori allows us to preserve the edges and encourage their co-location in the restored amplitude and phase images

19. Three Dimensional Reconstruction - MCPAR The MCPAR solution is found by minimizing the a posteriori Energy For the optimization we use Graph Cut theory. It is not possible to use Ishikawa construction: - no ordered set of labels (a vector of two sets of labels) - a priori energy is not convex . The global optimum solution cannot be guaranteed.

20. Three Dimensional Reconstruction - MCPAR   p q Alpha -expansion scalar: proposing to all the pixels to change their label to  or to remain in their current label The graph is constructed iteratively for each possible label until convergence To perform the optimization we use the vectorial alpha -expansion The energy is minimized using the Max-Flow algorithm. Alpha -expansion is able to provide a “good” local minima (i.e. a local minima near the global one) in case the a priori energy is a metric

21. Three Dimensional Reconstruction - MCPAR Vectorial alpha -expansion: finding the minimum energy when a vector of labels  =[  ( a )   ( h ) ] is proposed to both current amplitude and phase The algorithm iterates for all the possible couple of labels until convergence The amplitude and the phase are forced to move together to the label  or are forced to stay together in their current labels. The a priori energy turns to be a metric and thus it ensures us to obtain a solution which is a not far from the optimal one

22. Results – Simulated data b= [60 140 220 300] Bc=1000 2 coherence areas  s  =  s(1-b/Bc) 64x64 pixels Reference profile Coherence map

23. Results – Simulated data b= [60 140 220 300] Bc=1000 2 coherence areas  s  =  s(1-b/Bc) 64x64 pixels Reference profile Coherence map First Interferogram Last Interferogram Noisy Amplitude

24. Results – Simulated data b= [60 140 220 300] Bc=1000 2 coherence areas  s  =  s(1-b/Bc) 64x64 pixels Reconstructed Profile MCPU

25. Results – Simulated data b= [60 140 220 300] Bc=1000 2 coherence areas  s  =  s(1-b/Bc) 64x64 pixels Reconstructed Profile MCPU Reconstructed Profile MCPAR

26. Results – Real data E-SAR L-Band data Dresden b= [9 40] 120x220 pixels First Interferogram Noisy Amplitude Optical Image – Google ©

27. Results – Real data E-SAR L-Band data Dresden b= [9 40] 120x220 pixels MCPU Reconstruction MCPAR Reconstruction Optical Image – Google ©

28. Conclusions A new multi-channel phase unwrapping methodology (MCPAR) for the 3D reconstruction in urban scenario exploiting both phase and amplitude data has been presented. The use of amplitude data helps the 3D reconstruction preserving the discontinuities Compared to MCPU it is able to provide a better reconstruction in low coherence areas (with a low number of interferograms) MCPAR is based on Graph Cuts optimization algorithms: it provides a “good” solution in short time Future works: analyze other a priori models, take into account geometrical distortions and test the MCPAR on other real data sets

29. Special thanks to DLR for providing E-SAR data