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

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 1

Problem statement 1 2-D modal signal model 2 Simultaneous sparse approximation for modal estimation 3 Results 4 Conclusions 5 2

Problem statement Modal estimation of 2-D NMR signals – 2 -D NMR spectroscopy ⇒ detection of complex chemical interactions 0.4 ⇒ study of macromolecules 0.3 – Superposition of 2 -D damped complex si- 0.2 nusoids ( 2 -D case) 0.1 fy 0 – Amplitude spectrum ⇒ superposition of −0.1 2 -D Lorentzian peaks −0.2 −0.3 – Modal estimation : determination of the −0.4 2 -D frequencies and dampings −0.5 −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 fx – TLS-Prony, MEMP, 2 -D Esprit, IMDF 3

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 4

2-D modal signal model 2-D modal signal model Superposition of 2-D exponentially decaying sinusoids in noise F � c i a m 1 − 1 b m 2 − 1 y ( m 1 , m 2 ) = + e ( m 1 , m 2 ) i i i =1 with : – Number of sample m 1 = 1 , . . . , M 1 (1st dimension), m 2 = 1 , . . . , M 2 (2nd dimension) – Modes : a i = e − α a,i + j 2 πf a,i (1st dimension), b i = e − α b,i + j 2 πf b,i (2nd dimension) – Complex amplitudes { c i } F i =1 – Noise e ( m 1 , m 2 ) 5

2-D modal signal model Matrix form Y noise-free data matrix containing the samples y ( m 1 , m 2 ) � F F F � c i b M 2 − 1 Y = [ y 1 , y 2 , · · · , y M 2 ] = � c i a i � c i b i a i · · · � a i i i =1 i =1 i =1 ] T . Defining h i = [ c i , c i b i , · · · , c i b M 2 − 1 with a i = [1 , a i , . . . , a M 1 − 1 ] i i Y = [ a 1 · · · a F ][ h 1 · · · h F ] T = AH b Noisy data : Y = AH b + E Important remark : Y T = BH a + E T A last writing : Y = A diag ( c ) B T + E 6

b b b Simultaneous sparse approximation for modal estimation Sparse approximation For each column of the data matrix y m 2 , m 2 = 1 , . . . , M 2 , Dictionnary Q a – a ( α, f ) = [1 , e ( − α + j 2 πf ) , . . . , e ( − α + j 2 πf )( M 1 − 1) ] T – q ( α, f ) = a ( α, f ) / || a ( α, f ) || 2 – Discretization of the ( α, f ) plane Sparse modal estimation x m 2 = min x � x � 0 � y m 2 − Q a x � 2 subject to 2 ≤ ǫ 7

b b b b b b b b b 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 � Y − Q a X � 2 min X � X � 2 , 0 subject to f ≤ ǫ where � Y − Q a X � 2 f = � vec( Y − Q a X ) � 2 2 � � T � � � X � 2 , 0 = � X [1 , :] � 2 · · · � X [ N, :] � 2 � � � � 0 and X [ n, :] stands for the n th row of X . Algorithm : Simultaneous OMP [Tropp 2006] 8

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 Q ( l ) : level l Q ( l +1) : level l + 1 Add & α remove f f 9

bc b bc b b b b b b b b b b b bc bc bc Simultaneous sparse approximation for modal estimation Mode pairing ˆ Q b Simultaneous sparse approximations ⇒ ˆ Q a = [ˆ a 1 , ˆ a 2 , . . . , ˆ Y a F a ] Y T ⇒ ˆ Q b = [ˆ b 1 , ˆ b 2 , . . . , ˆ b F b ] Low dimension dictionary ˆ Q a Q = ˆ ˆ Q a ⊗ ˆ Q b . Selection of the pairs of 2-D modes ⇒ sparse approximation Qx � 2 ≤ ǫ. � y − ˆ min x � x � 0 subjet to Greedy algorithm : OMP, SBR 10

Simultaneous sparse approximation for modal estimation Algorithm summary Perform the SVD of Y and take its low rank approximation 1 Apply the multi-grid algorithm combined with S-OMP on matrix Y to obtain the 2 modes a i (first dimension) Repeat step 2 using Y T to estimate the modes b i (second dimension) 3 Form the 2-D modes using the pairing procedure 4 11

Results Numerical simulations Comparison with 2-D ESPRIT and TLS-Prony 0 10 Proposed 2−D ESPRIT −1 2−D TLS Prony – Signal size 30 × 30 10 CRB RMSE ( f 1 , 1 ) – 3 modes −2 10 – Initial dictionary : 40 frequency points uniformly distributed over −3 10 the interval [0 , 1[ , and 4 damping factors α ∈ { 0 , 0 . 025 , 0 . 05 , 1 } −4 10 – 30 levels of resolution −5 10 −10 0 10 20 30 SNR [dB] Accuracy similar to 2D-ESPRIT Lower computational burden than TLS-Prony ⇒ processing of large signals possible 12

Results 2-D NMR signal analysis 0.25 0.2 18 18 17 5 3 8 7 0.15 – Signal size 64 × 2024 0.1 1 7 8 18 18 14 0.05 – Sub-bands decomposition f 1 0 [Djermoune 2008] −0.05 16 10 16 9 16 15 −0.1 – Same setting for Multigrid S-OMP −0.15 6 19 17 6 10 14 −0.2 −0.25 −0.2 −0.1 0 0.1 0.2 f 2 0.12 0.24 −0.02 0.1 0.22 −0.04 0.08 0.2 f 1 f 1 f 1 −0.06 0.06 0.18 −0.08 0.04 0.16 −0.1 0.02 0.14 −0.12 0 0.095 0.1 0.105 0.11 0.115 0.12 −0.185 −0.18 −0.175 −0.17 −0.165 −0.16 −0.185 −0.18 −0.175 −0.17 −0.165 −0.16 f 2 f 2 f 2 13

Conclusions 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 14