Sparse Adaptive Template Matching and Filtering for 2D Seismic. Images with Dual-Tree Wavelets and Proximal Methods. Mai Quyen Pham1,3, Caroline ...
Sparse Adaptive Template Matching and Filtering for 2D Seismic Images with Dual-Tree Wavelets and Proximal Methods Mai Quyen Pham1,3, Caroline Chaux2, Laurent Duval1, and Jean-Christophe Pesquet3 1
3
IFP Energies nouvelles, 92852 Rueil-Malmaison Cedex, France 2Aix-Marseille Univ. I2M UMR CNRS 7373, 13453 Marseille, France Univ. Paris-Est, LIGM UMR CNRS 8049, 77454 Marne-la-Vall´ee, France
Signal formation model z (n) = y (n)+s(n)+b(n) = y (n)+
p′+Pj −1 J−1 X X j=0
(n)
Principles of seismic acquisition (nt−p,nx)
hj (p)rj
Proximity operator H: Hilbert space, ψ : H → H: lower semi-continuous convex function
Hydrophone
Towed streamer • • • • • • •
+b(n)
p=p′
1 proxψ : H → H : u 7→ argmin kv − uk2 + ψ(v) v∈H 2 C : convex set in H ֒→ proxιC (x) =
♣ n = (nt, nx), nt ∈ {0, . . . , Nt − 1} and nx ∈ {0, . . . , Nx − 1} PJ−1 ′ ♣ Pj tap coefficients, −Pj + 1 ≤ p ≤ 0, P = j=0 Pj
projection onto C proxλ|·|p
(∀x ∈ R)
(n) ♣ J approximate templates rj for the multiple s(n) assumed to be
a) proxλ|·|2 (x) =
known
(n) ♣ Inverse problem reduced to time-varying filters hj (p) estimation
Reflections on different layers (primaries in blue), and reverberated disturbances (multiples in dotted and dashed red).
−λ
λ
x
b) proxλ|·|(x) = sign(x) max{z(|x| − λ, 0)} | shrinkage operator
Dual-tree M-band wavelet
MAP estimation minimize
y∈RNtNx ,h∈RNtNxP
M+L FBF algorithm
kz − Rh − yk2 + ιD (F y) + ιC (h)
♣ R: defined from J templates ♣ F ∈ RK×NtNx : hybrid dual-tree wavelet ֒→ x = F y ∈ RK S2L ♣ D = D1 × . . . × D2L, {1, ( . . . , K} = l=1 Kl , βl ∈ ]0, +∞[ X ∀l ∈ {1, . . . , L}, Dl = (xk )k∈Kl ∪Kl+L : |xk | ≤ βl and k∈Kl
|
{z
}
primal coefficients
♣ C = C1 ∩ C2 ∩ C3
X
k∈Kl+L
|
|xk | ≤ βl {z
}
dual coefficients
♣ Slow variations of the filters: o n (nt,nx) (nt+1,nx) 1. Along time C1 = h | hj (p) − hj (p) ≤ εnj,px o n (n ,n ) (n ,n +1) 2. Along sensors C2 = h | hj t x (p) − hj t x (p) ≤ εnj,pt
[i]
[i]Π ((γ [i])−1s[i]) = v [i] + γ [i]F y [i] and w1[i] = s[i] − γ D 2 2 [i] [i] [i] [i] t2 = u[i] + γ [i]h[i] and w2 = t2 − γ [i]ΠC (γ [i])−1t2 ) Averaging
s2
[i] [i] [i] [i] [i] [i+1] [i] w1 + γ F s1 and v = v − s2 + q 1 [i] [i] [i] [i] [i] [i+1] [i] w2 + γ t1 and u = u − t 2 + q2
= = Update
[i] ∗w [i] Rt[i] + s − z + F 1 1 1 [i] [i] h[i+1] = h[i] − γ [i] R⊤(Rt[i] + s − z ) + w 1 1 2 end for
y [i+1] = y [i] − γ [i]
P 1/2 P Nt−1 (nt,nx) 2 x−1 ♣ ∀d ∈ RNt×Nx , ℓ1,2(d) = N ) nx=0 nt=0 (d
Real data
Synthetic data
)
Set γ [i] ∈ [ǫ, 1−ǫ ϑ ]. Initialization: y [0] ∈ RNtNx , h[0] ∈ RNtNxP , v [0] ∈ RK , u[0] ∈ RNtNxP Iterations: for i = 0, 1, . . . do Gradient computation [i] s1 = y [i] − γ [i] Rh[i] + y [i] − z + F ∗v [i] [i] [i] [i] ⊤ [i] [i] [i] t1 = h − γ R (Rh + y − z ) + u Projection computation
[i] q1 [i] q2
♣ Concentration metrics on the filters (λ ∈ ]0, +∞[) n o C3 = h ∈ RNtNxP | ℓ1,2(Rh) ≤ λ Dual coefficients
1 x |1 +{z2λ }
p=1
p=2
“Wiener” filter
♣ Additional sparsity constraints taken into account.
Primal coefficients
∀x ∈ H
Π | C{z(x)}
Numerical experiments X Filter lengths: P0 = 4, P1 = 4 (J = 2) X Iterations: 10000 (stopping if ky [i+1] − y [i]k < 10−6)
50
X Constraint choice: εnj,px = 0.05 and εnj,pt = 0.0001 ∀(j, p).
100
Primary
150 200 250 300 350 400
100
200
300
400
Observed data z
Observed data z (σ = 0.04)
F \σ orthogonal basis y shift-invariant frame M -band dual-tree orthogonal basis s shift-invariant frame M -band dual-tree
0.04 13.93 15.51 17.17 7.79 7.74 9.82
0.08 11.05 13.21 15.60 7.09 6.64 9.37
0.16 7.45 10.71 12.67 4.65 5.26 7.01
SNR for the estimations of y and s in dB considering different wavelet transforms F and three noise levels.
Primary
50
Key message
100 150
X Adaptive filtering with convex optimization,
200
X Large choice in sparse 2D wavelet transforms, 250 50
100
150
200
250
Estimated yb by 1D method
Estimated yb by 1D method (SNR = 7.08 dB) 50
X Efficiency of the low-redundant M -band dual-tree.
References
Primary [1] M. Q. Pham, L. Duval, C. Chaux, and J.-C. Pesquet, “A primal-dual proximal algorithm for sparse template-based adaptive filtering: Application to seismic multiple removal”, IEEE Trans. Signal Process., vol. 62, no. 16, pp. 4256–4269, Aug. 2014.
100 150
[2] C. Chaux, L. Duval, and J.-C. Pesquet, “Image analysis using a dual-tree M -band wavelet transform”, IEEE Trans. Image Process., vol. 15, no. 8, pp. 2397–2412, Aug. 2006.
200 250 50
Estimated yb by 2D method (SNR = 17.17 dB)
100
150
200
Estimated yb by 2D method
250
[3] P. L. Combettes and J.-C. Pesquet, “Primal-dual splitting algorithm for solving inclusions with mixtures of composite, Lipschitzian, and parallel-sum type monotone operators”, Set-Valued Var. Anal., vol. 20, no. 2, pp. 307–330, Jun. 2012. LATEX Tik Zposter
