[PPT] - Sparse modal estimation of 2-D NMR signals Souleymen Sahnoun, PowerPoint Presentation

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Sparse modal estimation of 2-D NMR signals

Souleymen Sahnoun, El-Hadi Djermoune and David Brie CRAN UMR 7039 - Universit´ e de Lorraine - CNRS ICASSP 2013 May 31 2013, Vancouver

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Problem statement

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2-D modal signal model

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Simultaneous sparse approximation for modal estimation

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Results

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Conclusions

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Problem statement

Modal estimation of 2-D NMR signals

fx fy −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 0.2 0.3 0.4

– 2-D NMR spectroscopy ⇒ detection of complex chemical interactions ⇒ study of macromolecules – Superposition of 2-D damped complex si- nusoids (2-D case) – Amplitude spectrum ⇒ superposition of 2-D Lorentzian peaks – Modal estimation : determination of the 2-D frequencies and dampings – TLS-Prony, MEMP, 2-D Esprit, IMDF

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Problem statement

Sparse approximation approach for modal estimation

– Sparse approximations for 1-D modal estimation [Goodwin1999, Malioutov 2005, Stoica 2011] – (Very) high resolution ⇒ (Very) large dictionary – [Sahnoun 2012] : multigrid sparse approximation + R-D extension 1-D case : very effective approach (accuracy, computation cost) R ≥ 2 : computational burden is untractable for large signals (dictionary size) How to process large size signals ? – Same idea as TLS-Prony, MEMP – 2-D estimation = 2 × 1-D estimation + mode pairing

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2-D modal signal model

Superposition of 2-D exponentially decaying sinusoids in noise y(m1, m2) =

F

i=1

ciam1−1

i

bm2−1

i

+ e(m1, m2) with : – Number of sample m1 = 1, . . . , M1 (1st dimension), m2 = 1, . . . , M2 (2nd dimension) – Modes : ai = e−αa,i+j2πfa,i (1st dimension), bi = e−αb,i+j2πfb,i (2nd dimension) – Complex amplitudes {ci}F

i=1

– Noise e(m1, m2)

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2-D modal signal model

Matrix form

Y noise-free data matrix containing the samples y(m1, m2) Y = [y1, y2, · · · , yM2] = F

i=1

ciai

F

i=1

cibiai · · ·

F

i=1

cibM2−1

i

ai

with ai = [1, ai, . . . , aM1−1

i

]T . Defining hi = [ci, cibi, · · · , cibM2−1

i

] Y = [a1 · · · aF ][h1 · · · hF ]T = AHb Noisy data : Y = AHb + E Important remark : YT = BHa + ET A last writing : Y = A diag(c) BT + E

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Simultaneous sparse approximation for modal estimation

Sparse approximation

For each column of the data matrix ym2, m2 = 1, . . . , M2, Dictionnary Qa – a(α, f) = [1, e(−α+j2πf), . . . , e(−α+j2πf)(M1−1)]T – q(α, f) = a(α, f)/||a(α, f)||2 – Discretization of the (α, f) plane Sparse modal estimation xm2 = min

x x0

subject to ym2 − Qax2

2 ≤ ǫ

b b b

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Simultaneous sparse approximation for modal estimation

Simultaneous sparse approximation

Each column of the data matrix is a 1-D signal generated by the same modes but with different amplitudes ⇒ Simultaneous sparse approximation

b b b b b b b b b

min

X X2,0

subject to Y − QaX2

f ≤ ǫ

where Y − QaX2

f = vec(Y − QaX)2 2

X2,0 =

X[1, :]2

· · · X[N, :]2 T

and X[n, :] stands for the nth row of X.

Algorithm : Simultaneous OMP [Tropp 2006]

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Simultaneous sparse approximation for modal estimation

Multigrid dictionary

High-resolution modal estimation – high resolution dictionary ⇒ prohibitive computational burden. – multi-grid scheme [Sahnoun2012] – signal dependent dictionary

Add & remove Q(l) : level l Q(l+1) : level l + 1 f f α

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Simultaneous sparse approximation for modal estimation

Mode pairing

Simultaneous sparse approximations Y ⇒ ˆ Qa = [ˆ a1, ˆ a2, . . . , ˆ aFa] YT ⇒ ˆ Qb = [ˆ b1, ˆ b2, . . . , ˆ bFb] Low dimension dictionary ˆ Q = ˆ Qa ⊗ ˆ Qb.

bc bc bc bc bcb

b b b b b b b b b b b

ˆ Qa ˆ Qb

Selection of the pairs of 2-D modes ⇒ sparse approximation min

x x0

subjet to y − ˆ Qx2 ≤ ǫ. Greedy algorithm : OMP, SBR

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Simultaneous sparse approximation for modal estimation

Algorithm summary

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Perform the SVD of Y and take its low rank approximation

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Apply the multi-grid algorithm combined with S-OMP on matrix Y to obtain the modes ai (first dimension)

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Repeat step 2 using YT to estimate the modes bi (second dimension)

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Form the 2-D modes using the pairing procedure

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Results

Numerical simulations

Comparison with 2-D ESPRIT and TLS-Prony – Signal size 30 × 30 – 3 modes – Initial dictionary : 40 frequency points uniformly distributed over the interval [0, 1[, and 4 damping factors α ∈ {0, 0.025, 0.05, 1} – 30 levels of resolution

−10 10 20 30 10

−5

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−4

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−3

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−2

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−1

10 Proposed 2−D ESPRIT 2−D TLS Prony CRB

SNR [dB] RMSE (f1,1)

Accuracy similar to 2D-ESPRIT Lower computational burden than TLS-Prony ⇒ processing of large signals possible

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Results

2-D NMR signal analysis

– Signal size 64 × 2024 – Sub-bands decomposition [Djermoune 2008] – Same setting for Multigrid S-OMP

1 16 18 10 8 16 7 10 8 9 18 6 16 18 19 18 17 15 14 17 5 6 3 14 7 −0.2 −0.1 0.1 0.2 −0.25 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 0.25

f1 f2

0.095 0.1 0.105 0.11 0.115 0.12 0.14 0.16 0.18 0.2 0.22 0.24

f1 f2

−0.185 −0.18 −0.175 −0.17 −0.165 −0.16 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02

f1 f2

−0.185 −0.18 −0.175 −0.17 −0.165 −0.16 0.02 0.04 0.06 0.08 0.1 0.12

f1 f2

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Conclusions

– Sparse modal estimation adapted to large signals – Performances similar to 2D-ESPRIT – Computation time lower than TLS-Prony – Application to other NMR modalities – Extension to the R > 2 case

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