Maximum likelihood SAR tomography based on the polarimetric multi-baseline RVoG model:

Transcription

1 Maximum likelihood SAR tomography based on the polarimetric multi-baseline RVoG model: Optimal estimation of a covariance matrix structured as the sum of two Kronecker products. L. Ferro-Famil 1,2, S. Tebaldini 3 1 University of Rennes 1, IETR lab., France 2 University of Tromsø, Physics and Technology Dpt., Norway 3 Politecnico Milano, Italy Laurent.Ferro-Famil@univ-rennes.fr

2 L. Ferro-Famil, S. Tebaldini MB-RVOG, ML SKP-2, PolinSAR 2013, Frascati

3 Outline PolTomSAR using the MB-RVOG model and Kronecker products Maximum Likelihood estimation of SKP-2 covariance matrices Performance of the proposed ML estimator

4 Single-pol MB-RVOG model SB case Uncorrelated responses y i(l) = s gi (l)+s vi (l), I i = I g + I v Homogeneous media E(s x1 s x 2 ) = I xγ x, γ x = f x(z)exp(jk zz)dz Global InSAR response y(l) = [s 1(l), s 2(l)] T R SB = I gr g + I v R v, with R x = [ 1 ] γx γx 1

5 Single-pol MB-RVOG model MB case Point-like scatterer MB-response : s(l) = A c(l)a(z) a = [ 1, exp(jk z2 z),..., exp(jk zns z) ] Global MB-InSAR response y(l) == s g(l)+s v(l) 1 γ x(k z2 )... γ x(k zns ) 1. R MB = I gr g + I vr v with R x =

6 Single-pol MB-RVOG model MB case Point-like scatterer MB-response : s(l) = A c(l)a(z) a = [ 1, exp(jk z2 z),..., exp(jk zns z) ] Global MB-InSAR response y(l) == s g(l)+s v(l) 1 γ x(k z2 )... γ x(k zns ) 1. R MB = I gr g + I vr v with R x =

7 Full-pol MB-RVOG model Point-like scatterer MB-response : y(l) = A c(l) [ ] khh a(z) k hv a(z) = A c(l)k a(z) k vva(z)

8 Full-pol MB-RVOG model Point-like scatterer MB-response : y(l) = A c(l) Uncorrelated P-S processes : E(y iy i ) = IiCi Ri [ ] khh a(z) k hv a(z) = A c(l)k a(z) k vva(z)

9 Full-pol MB-RVOG model Point-like scatterer MB-response : y(l) = A c(l) Uncorrelated P-S processes : E(y iy i ) = IiCi Ri Global MB-POLinSAR response : y(l) = y g(l)+y v(l) R PS = C g R g + C v R v [ ] khh a(z) k hv a(z) = A c(l)k a(z) k vva(z)

10 Full-pol MB-RVOG model Point-like scatterer MB-response : y(l) = A c(l) [ ] khh a(z) k hv a(z) = A c(l)k a(z) k vva(z) Uncorrelated P-S processes : E(y iy i ) = IiCi Ri Global MB-POLinSAR response : y(l) = y g(l)+y v(l) R PS = C g R g + C v R v SB-case (N s = 2) [ ] Cg + C v γ gc g + γ vc v R SP = R g C g + R v C v = γg C g + γv C v C g + C [ ] v C C12 R SP = C 12 C γ(w) = w C 12 w w Cw = γv+µ(w)γg 1+µ(w) MB-RVOG : exactly modeled by an SKP-2 structured covariance matrix

11 MB-RVOG parameter estimation SB case Coherence line estimation GV separation : strong assumptions γ g = 1, rank(c g) < 3 f v(z) known MB case SVD SKP-2 decomposition GV separation : several solutions accuracy increases with N s

12 MB-RVOG parameter estimation SB case Coherence line estimation GV separation : strong assumptions γ g = 1, rank(c g) < 3 f v(z) known MB case SVD SKP-2 decomposition GV separation : several solutions accuracy increases with N s See also : physical interpretation of MB-RVOG JD & SKP decomp. B. El Hajj-Chehade, L. Ferro-Famil, Poster session Wednesday aft.

13 Estimation problem statement and objectives L independent observations {y(l)} L l=1 Estimate RVOG parameters or its SKP-2 covariance matrix SB case Pol-Sampling γ(w i), line LS estimate [C & P ][Flynn & Tabb ] LS estimation (matrix formatting) [LFF et al, ] [L-M 2012] ML estimation [Flynn et al. 2002] and CRLB [Roueff et al. 2011]. MB case LS estimation using joint diagonalization [Ferro-Famil et al. 2010] SVD based SKP-2 decomposition [Tebaldini et al. 2009, 2010, 2012]

14 Estimation problem statement and objectives L independent observations {y(l)} L l=1 Estimate RVOG parameters or its SKP-2 covariance matrix SB case Pol-Sampling γ(w i), line LS estimate [C & P ][Flynn & Tabb ] LS estimation (matrix formatting) [LFF et al, ] [L-M 2012] ML estimation [Flynn et al. 2002] and CRLB [Roueff et al. 2011]. MB case LS estimation using joint diagonalization [Ferro-Famil et al. 2010] SVD based SKP-2 decomposition [Tebaldini et al. 2009, 2010, 2012] LS vs ML estimation Estimate that best fits the observations (LS) vs Estimate most likely to have produced observations (ML). LS : generally fast, often has analytical solutions, no pdf assumption ML : if UMV estimator exists ML. Can be complex and time consuming

15 ML estimation of SKP2 covariance matrices Objectives Find ˆR PS ML = arg max f({y(l)} L l=1 R PS ) under the SKP-2 constraint. ˆR PS ML (Wishart) concentrated criterion : J(R PS ) = tr(r 1 PS ˆR yy)+log R PS ) with ˆR yy = 1 L L y PS (l)y PS (l) l=1

16 ML estimation of SKP2 covariance matrices Objectives Find ˆR PS ML = arg max f({y(l)} L l=1 R PS ) under the SKP-2 constraint. ˆR PS ML (Wishart) concentrated criterion : J(R PS ) = tr(r 1 PS ˆR yy)+log R PS ) with ˆR yy = 1 L Estimation objective : determine L y PS (l)y PS (l) ˆR PS ML = arg min J(R PS ) with and R PS = C g R g + C v R v Find a fast and reliable technique l=1

17 ML estimation of SKP2 covariance matrices Objectives Find ˆR PS ML = arg max f({y(l)} L l=1 R PS ) under the SKP-2 constraint. ˆR PS ML (Wishart) concentrated criterion : J(R PS ) = tr(r 1 PS ˆR yy)+log R PS ) with ˆR yy = 1 L Estimation objective : determine L y PS (l)y PS (l) ˆR PS ML = arg min J(R PS ) with and R PS = C g R g + C v R v Find a fast and reliable technique l=1 Case of a single KP No analytical solution. Fast and reliable iterative alternate (Flip-Flop) optimization [Lu 2004] Asymptotic analytical expression : Weighted LS solution [Werner 2008]

18 ML estimation of SKP2 covariance matrices SKP-2 case In general A B+C D E F No existing solution.

19 ML estimation of SKP2 covariance matrices SKP-2 case In general A B+C D E F No existing solution. Exact joint diagonalization of two matrices A, B > 0 and Hermitian can be exactly jointly diagonalized [Yeredor 2005] : W : WAW = D A, WBW = D B, in general WW I

20 ML estimation of SKP2 covariance matrices SKP-2 case In general A B+C D E F No existing solution. Exact joint diagonalization of two matrices A, B > 0 and Hermitian can be exactly jointly diagonalized [Yeredor 2005] : W : WAW = D A, WBW = D B, in general WW I In the SKP-2 case, R PS = C g R g + C v R v W = W C W R : WR PS W = D Cg D Rg + D Cv D Rv = D

21 ML estimation of SKP2 covariance matrices SKP-2 case In general A B+C D E F No existing solution. Exact joint diagonalization of two matrices A, B > 0 and Hermitian can be exactly jointly diagonalized [Yeredor 2005] : W : WAW = D A, WBW = D B, in general WW I In the SKP-2 case, R PS = C g R g + C v R v W = W C W R : WR PS W = D Cg D Rg + D Cv D Rv = D ML criterion minimization over D Inserting D, W into J(R PS ) J(D, W) = tr(d 1 WˆR yyw )+log D log WW Minimum obtained for : ˆD = diag(wˆr yyw )

22 ML estimation of SKP2 covariance matrices SKP-2 case In general A B+C D E F No existing solution. Exact joint diagonalization of two matrices A, B > 0 and Hermitian can be exactly jointly diagonalized [Yeredor 2005] : W : WAW = D A, WBW = D B, in general WW I In the SKP-2 case, R PS = C g R g + C v R v W = W C W R : WR PS W = D Cg D Rg + D Cv D Rv = D ML criterion minimization over D Inserting D, W into J(R PS ) J(D, W) = tr(d 1 WˆR yyw )+log D log WW Minimum obtained for : ˆD = diag(wˆr yyw ) Resulting concentrated ML criterion J(W) = log diag(wˆr yyw ) log WW

23 ML estimation of SKP2 covariance matrices ML criterion minimization over W : formulation Explicit ML criterion J(W C, W R ) = log diag( D) log WW with D = WˆR yyw and W = W C W R

24 ML estimation of SKP2 covariance matrices ML criterion minimization over W : formulation Explicit ML criterion J(W C, W R ) = log diag( D) log WW with D = WˆR yyw and W = W C W R Conditional criterion : known spatial diagonalizer N s J(W C W R ) = log diag(w C DCi W C ) log W C W C i=1 Conditional criterion : known polarimetric diagonalizer N p J(W R W C ) = log diag(w R D Ri W R ) log W R W R i=1

25 ML estimation of SKP2 covariance matrices ML criterion minimization over W : formulation Explicit ML criterion J(W C, W R ) = log diag( D) log WW with D = WˆR yyw and W = W C W R Conditional criterion : known spatial diagonalizer N s J(W C W R ) = log diag(w C DCi W C ) log W C W C i=1 Conditional criterion : known polarimetric diagonalizer N p J(W R W C ) = log diag(w R D Ri W R ) log W R W R i=1 ML criterion minimization over W : joint diagonalization J(W C W R ) or J(W R W C ) efficiently minimized modified Pham s technique [Pham 2001] Iterative constrained joint diagonalization of { D Ci } or { D Ri } Convergence is proven, generally very fast (4 iterations)

26 ML estimation of SKP2 covariance matrices : algorithm Step 0 Initialisation L Compute ˆR yy = 1 y L PS (l)y PS (l), set W C(0), W R (0), n = 0 l=1

27 ML estimation of SKP2 covariance matrices : algorithm Step 0 Initialisation L Compute ˆR yy = 1 y L PS (l)y PS (l), set W C(0), W R (0), n = 0 l=1 Step 1 Diagonalization Compute D(n+1) = W(n)ˆR yyw (n), n = n+1

28 ML estimation of SKP2 covariance matrices : algorithm Step 0 Initialisation L Compute ˆR yy = 1 y L PS (l)y PS (l), set W C(0), W R (0), n = 0 l=1 Step 1 Diagonalization Compute D(n+1) = W(n)ˆR yyw (n), n = n+1 Step 2 Minimization using Pham s technique W C (n) = arg min J(W C W R (n 1)), W R (n) = arg min J(W R W C (n 1)) W C W R

29 ML estimation of SKP2 covariance matrices : algorithm Step 0 Initialisation L Compute ˆR yy = 1 y L PS (l)y PS (l), set W C(0), W R (0), n = 0 l=1 Step 1 Diagonalization Compute D(n+1) = W(n)ˆR yyw (n), n = n+1 Step 2 Minimization using Pham s technique W C (n) = arg min J(W C W R (n 1)), W R (n) = arg min J(W R W C (n 1)) W C W R Step 3 Convergence? If no : Go to Step 1 Step 4 Identification D = diag(ŵˆr yy Ŵ ), ˆR PS ML = Ŵ 1 DŴ From ˆR PS ML, identify Ĉg,ˆR g, Ĉv,ˆR v

30 SignificantL. improvement, Ferro-Famil, S. Tebaldini in general MB-RVOG, ML SMP LS SKP-2, PolinSAR 2013, LS ML Frascati Global relative rms error rmse = E R PS ˆR PS 2 F, with ˆR R PS 2 PS = ˆR yy, ˆR PS LS, ˆR PS ML F

31 Coherence locations in the complex plane N p = 3, N s = 5, Ground : rank(c g) = 2, Volume : full rank, minimum phase

32 Coherence wise rms error rmse(γ ij) = E( γ ij ˆγ ij 2 ) Equal performance for low rank (and DoF) R g, well chosen solution.

33 Polarimetric rms error rmse i = E C PS i Ĉ PSi 2 F C PSi 2 F Large LS error over low rank C g

34 MB-RVOG model validity vs estimation numb. unfeasible exp. numb.exp 100

35 Tomography at P band of a TROPISAR profile (ONERA/SETHI) N p = 3, N s = 6 L = 16 < N sn p

36 LS & ML SKP-2 Tomography N p = 3, N s = 6 L = 16 < N sn p

37 Conclusion Numerical ML estimation of the MB-RVOG model Rigorous and fast Adapted to POL-inSAR and TOMO-SAR Could also be used in medical imaging, image processing... Performance Fewer looks needed Significant variance reduction LS ML unformated LS Better estimation of polarimetric features, not only spatial ones