. Bayesian approach with sparse enforcing prior for acoustic sources localization and imaging Ali MOHAMMAD-DJAFARI Ning CHU, Nicolas GAC and Jos´e PICHERAL Laboratoire des Signaux et Syst`emes (L2S), UMR8506 CNRS-SUPELEC-UNIV PARIS SUD 11 and ´ Dept. Signaux et Syst`emes Electroniques (SSE) SUPELEC, 91192 Gif-sur-Yvette, France http://lss.supelec.free.fr Email:
[email protected] http://djafari.free.fr A. Mohammad-Djafari,
Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
16 Octobre 2014
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Content
◮
Acoustic sources localization and imaging applications
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Forward models
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Beam forming and Deconvolution based models
◮
Regularization methods
◮
Proposed Bayesian inference method
◮
Results on simulated and real data
◮
Conclusions
A. Mohammad-Djafari,
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Acoustic sources localization and imaging
Flyover meausrements at airport
Acoustic imaging of noise sources (dB)
Courtesy of National Aerospace Laboratory (NLA) Holland [Vander 2009] A. Mohammad-Djafari,
Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
16 Octobre 2014
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Acoustic sources localization and imaging
Previous work developed by Renault France [Adam 2010]. Motivation: ◮
Higher spatial resolution for low frequencies.
◮
Robust to measurement errors.
◮
Wide dynamic range of source powers.
◮
Fast acoustic imaging for industry application.
A. Mohammad-Djafari,
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Propagation forward models
◮
Assumptions: Ponctual sources, ideal sensors, no reverberation
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Measured data: z
◮
Unknowns: sources number K, positions P ∗ & amplitudes s∗
A. Mohammad-Djafari,
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Propagation forward model
zm (f ) =
K X
am,k (p∗k , f ) s∗k (f ) + em (f )
k=1
⇓
z(f ) = A(P∗ , f ) s∗ (f ) + e(f ) ◮
am,k (p∗k , f ) =
1 rm,k
exp (−2π f τm,k ): signal propagation;
◮ A(P∗ , f ) ◮ ◮
= [am,k (p∗k , f )]M ×K : signal propagation matrix; p∗k ∈ P∗ : kth source position. Non-linear for P∗ ; Hard to jointly solve P∗ and s∗ .
A. Mohammad-Djafari,
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Imaging forward model
Assumption: s∗ ⊂ s N grids s; P∗ ⊂ P discrete positions P. z(f ) = A(P∗ , f ) s∗ (f ) + e(f ) ⇓ Discretization z(f ) = A(P , f ) s (f ) + e(f ) 1 rm,n
◮
am,n (pn , f ) =
◮
A(P, f ) = [am,n (pn , f )]M×N : discrete propagation matrix;
◮
s = [0, s∗1 , 0, · · · , s∗K , 0, · · · ]T : Spatially K-sparsity;
◮
Linear for s; but under-determined due to M ≤ N.
A. Mohammad-Djafari,
exp (−2π f τm,n )
Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
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Beamforming and Deconvolution based methods
y = C x + σe2 1a
˜ † A|.2 ) C = (|A ◮
˜ = {˜ ˜ : M × N; A an }N : Beamforming steering matrix, A
◮
˜n = a
an ||an ||2 :
Beamforming steering vector of A : M × N .
σe2 : Power of measurement errors e (i.i.d white noise). Linear and determined equations (C : N × N ) for source powers x. A. Mohammad-Djafari,
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Beamforming: low spatial resolution ˜ † z|.2 ] −→ y = Cx + σ 2 1a z = As −→ y = E[|A ǫ ◮
Low spatial resolution (30cm) at low frequency (2500Hz).
◮
spatially variant PSF convolution
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Source power x A. Mohammad-Djafari,
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Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
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Deconvolution methods for power propagation model 1.4 2
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y = C x + σe2 1a b = arg min ||y − C x||22 + αF(x) x x s.t. x ≥ 0 and sparse 0.6
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Iterative: CLEAN used by [Stoica 2003], Parameter selection;
Breakthrough: Deconvolution Approach for Mapping Acoustic Sources (DAMAS) proposed by [Brooks 2005]; Sensitive ◮ Robustness: Diagonal Removal DAMAS (DR-DAMAS) [Brooks 2005]; Over suppression ◮ F (x) = ||w x||l with 0 ≤ l ≤ 1, weight vector w: ◮ l = 0: Real sparsity, but hard to solve; ◮ l = 1: Sparsity, well solved by LASSO algorithm; DAMAS with sparsity constraint (SC-DAMAS) [Yardi 2008]; ◮ α: to be tuned carefully; A. Mohammad-Djafari, Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif, 16 Octobre 2014
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Deconvolution and regularization results 1.4
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(f) SC-DAMAS
Deconvolution: High spatial resolution, but sensitive to errors Regularization: High spatial resolution, robust to errors, but α selection. A. Mohammad-Djafari,
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General performance of classical methods
Methods
CBF1
CLEAN
DAMAS
DR-DAMAS
SC-DAMAS
Resolutions
Low
Normal
Normal
Normal
High
Dynamic Range
Narrow
Normal
Normal
Normal
Normal
Noise
Robust
Sensitive
Sensitive
Normal
Normal
Parameter2 number
No
Required
No
No
Required
Computation
Least
Normal
Normal
Normal
High
1 2
Conventional Beamforming Parameter to be tuned
A. Mohammad-Djafari,
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16 Octobre 2014
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Improved power propagation model and sparsity
y = C x + σe2 1a −→ y = C x + σe2 1a + ξ Contribution: improve robustness using model uncertainty ξ caused by unknown effects: multi-path propagation and prior of sparsity information (b x, σˆ2 ) = arg min ||y − C x − σe2 1a ||22 (x,σe2 ) , s.t. kxk1 = β, x 0, σe2 ≥ 0
(b x, σˆ2 ) jointly estimated; Sparse solution on x ˆ sparsity parameter on total source powers kxk1 ; ◮ β: ◮ Sensitive to β A. Mohammad-Djafari, Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif, 16 Octobre 2014 ◮
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Robust DAMAS with sparsity constraint (SC-RDAMAS) 1.4
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Higher resolution; Robust to measurement errors; but Sensitive to sparsity parameter. A. Mohammad-Djafari,
Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
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Bayesian approach with a sparse prior ◮
[Chu et al. 2012 JSVJournal]
Bayesian approach infers (x, θ) from y using p(x, θ|y) p(x, θ|y) ∝ p(y|x, θ 1 ) p(x|θ 2 ) | {z } | {z } Likelihood
θ = [θ 1 , θ 2 ]
◮
P rior
p(θ) |{z}
Hyper−prior
Likelihood
y = C x + σe2 1N + ξ " # 1 p(y|x, θ 1 ) ∝ exp − 2 ky − C x − σe2 1a k2 2σξ θ 1 = [σe2 , σξ2 ] : Hyper-parameters
◮
p(x|θ 2 ) Sparsity enforcing
◮
p(θ) Conjugate priors
A. Mohammad-Djafari,
Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
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Sparsity enforcing prior on source powers ◮
Sparsity enforcing prior in Generalized Gaussian family p(x|θ 2 ) ∝ exp(−γ |xn |β ), θ 2 = [γ, β] 0.06
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p(x) = γβ/(Γ(β)) exp(−γ|x|β)
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β = 1 (fixed) enforces sparsity distribution: sharp peak; 0 ≤ γ ≤ 1 (to be estimated) enlarges dynamic range: long tail. ◮
p(θ): Positive priors using Jeffrey priors: p(γ) ∼ γ1 , p(σξ2 ) ∼ σ12 , p(σe2 ) ∼ σ12 , θ = [γ, σe2 , σξ2 ] ξ
A. Mohammad-Djafari,
e
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Joint Maximum A Posteriori (JMAP)
[Chu et al. 2012 JSVJournal]
ˆ = arg max {p(x, θ|y) ∝ p(y|x, θ 1 ) p(x|θ 2 ) p(θ)} (b x, θ) (x,θ )
= arg min {J (x, θ)} , with J (x, θ) = − ln p(x, θ|y) (x,θ ) N 1 ln σξ2 − N ln γ J (x, θ) = 2 ky − C x − σe2 1a k2 + γ kxkβ=1 + 2 2σ {z } | ξ | {z } {z } Sparse prior Hyper−parameter | prior Likelihood: data f itting s.t. x 0, σe2 ≥ 0, σξ2 ≥ 0, γ ≥ 0; θ = [σe2 , σξ2 , γ] Advantages ◮ Super spatial resolution: sparse prior; ◮ Wide dynamic range: γ estimation; ◮ robustness to errors: σe2 , σ 2 estimations ξ Limitations ◮ Non-quadratic optimization: x and θ; ◮ High computational costs O(N 2 ): C x, N × N dimension. A. Mohammad-Djafari,
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JMAP) optimization
[Chu et al. 2012 JSVJournal]
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Higher resolution, robust, wide dynamics, parameter-independent, but time-consuming A. Mohammad-Djafari,
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Method comparisons
Methods
CBF
DR-DAMAS
SC-DAMAS
SC-RDAMAS
JMAP
Resolutions
Low
Normal
High
Higher
Higher
Dynamics3
Narrow
Normal
Normal
Wide
Wide
Noise
Robust
Normal
Normal
Robust
Robust
Parameter
No
No
Required
Required
No
Cost
Least
Normal
Higher
Higher
Higher
3
Dynamic range
A. Mohammad-Djafari,
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Invariant convolution model of power propagation
[Chu et al.
ICA2013a]
y = C x + σe2 1a + ξ −→ y = H x + ǫ = h ∗ x + η
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C: spatially variant in near-field Operation Matrix multiplication 2D invariant convolution 1D separable convolution
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H: spatially invariant in far-field
Expression4 Cx h∗x h1 ∗ h2 ∗x
Complexity O(N 2 ) O(Nh 2 N ) O(2 Nh N )
Speed gain 1 N/Nh 2 N/(2 Nh )
4 h: Nh × Nh matrix; h1 , h2 : Nh length vector; A. Mohammad-Djafari, Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
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Short summary of the works z(f ) = A(P∗ , f ) s∗ (f ) + e(f ): Given z find (P∗ , , s∗ ) ⇓ Discretization z(f ) = A(P , f ) s (f ) + e(f ): Given (z, A) find s (sparse) . ⇓ Beamforming power y = C x + σe2 1a : Given (y, C) find x (sparse) by deconvolution/regularization. ⇓ Model uncertainty ξ y = C x + σe2 1a + ξ: Given (y, C) find (x, σe2 ) by proposed SC-RDAMAS. Given (y, C) find (x, σe2 , σξ2 , γ ) by proposed Bayesian JMAP. ⇓ Invariant convolution model C x ≈ h ∗ x ≈ h1 ∗ h2 ∗ x y = H x + ǫ: Given (y, C) to find (x, σǫ2 ) by proposed Bayesian JMAP: fast, but low quality ⇓ ǫ: Model errors are spatially variant noises To be continued... A. Mohammad-Djafari,
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Spatialy variant noise model and Students-t priors
[Chu et al.
ICA2013a] 1.3
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y = Hx + ǫ ◮
ǫ: Model errors are spatially variant noises: p(ǫi |νi ) = N (ǫi |0, 1/νi )
◮
p(νi ) = G(νi |aν , bν ) ,
Non-stationary prior p(ǫ): Z p(ǫi ) = p(ǫi |νi )p(νi ) dνi = St (ǫi )
A. Mohammad-Djafari,
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Sparsity enforcing via Students-t priors
[Chu et al. ICA2013a]
y = Hx + ǫ x: Sparsity enforcing prior from Student-t family (Cauchy): 0.5 Normal DE Laplace Students−t
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p(xj ) = A. Mohammad-Djafari,
−2
Z
0
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p(xi |γj )p(γj ) dγj = St (xj )
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Non stationary noise model and sparsity enforcing via Students-t priors [Chu et al. ICA2013a] y =Hx+ǫ ◮
Non stationary noise model via St p(y|x) = N (y|Hx, Σǫ ), Σǫ = diag [ν] , p(ν) =
M Y
G(νm |aγ , bγ )
m=1 ◮
Sparsity enforcing prior via St prior p(x|γ) = N (x|0,
◮
Σ−1 γ )
Σǫ = diag [γ] p(γ) =
N Y
G(γn |aγ , bγ ) ,
n=1
Joint posterior law
N N Y Y −1 N (x|0, G(ν ) |a , b ) Σ ) G(γn |aγ , bγ ) p(x, γ, ν|y) ∝ N (y|H x , Σ−1 n ν ν ν γ {z } n=1 | | {z } n=1 Likelihood | {z } | {z } P rior Hyper−prior
◮
◮
Hyper−prior
JMAP could solve 3N -dimensional variables by alternate optimization JMAP is a point estimator. How to quantify estimation
A. Mohammad-Djafari,
Commision Signaux et Statistiques du Conseil Scientifique, Suplec Gif,
16 Octobre 2014
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Bayesian VBA via Students-t priors
[Chu et al. ICA2013a]
To find x, θ = [γ, ν] from y (Z ) q(x, θ) p(x, θ|y) ≈ qˆ(x, θ) = arg min q(x, θ) ln d(x, θ) p(x, θ|y) q(x,θ ) (x,θ ) {z } | Minimizing K-L divergence qˆ(x, θ) = qˆ1 (x) qˆ2 (γ) qˆ3 (ν), ◮
Analytical solutions: Conjugate priors: ˆ x) µx , Σ qˆ1 (x) = N (x|ˆ N Y qˆ2 (γ) = G(γn |ˆ aγ , ˆbnγ ) n=1 N Y G(νn |ˆ q ˆ (ν) = aν , ˆbnν ) 3
,
n=1
◮
ˆ x , γˆ , νˆ) for quantifying estimation VBA jointly obtains (ˆ µx , Σ uncertainty (confidential interval) ; Outperforms JMAP.
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Simulation in colored (spatially non-stationary) noises 1.4
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−0.2
−1.2
0
(b) Beamforming y
1.4
−1
−0.8
−0.6 x (m)
−0.4
−0.2
0
(c) DR-DAMAS.
1.4
1.4
4
2
2 1.3
1.3
1.3 2
0
0
1.2
1.2
1.2 0
−2
−2
1.1
1.1
1.1 y (m)
y (m)
y (m)
−2 −4 1
1
−4
1
−4 −6
0.9
0.9
−6
0.9 −6
0.8
−8
0.8
0.7
−10
0.7
0.6
−12
0.6
−8
0.8 −8
−10
0.7 −10
−1.2
−1
−0.8
−0.6 x (m)
(d) JMAP A. Mohammad-Djafari,
−0.4
−0.2
0
−12
0.6 −1.2
−1
−0.8
−0.6 x (m)
−0.4
(e) JMAP+Conv
−0.2
0
−1.2
−1
−0.8
−0.6 x (m)
−0.4
−0.2
0
(f) VBA+Conv
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Real data in wind tunnel S2A at 2500Hz 1
0
0
−2
−2
−4
−4
−6
−6
1
0.5
0.5
−8 0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
−10 −2
−8 −1.5
(a) Beamforming
−1
−0.5
0
0.5
1
1.5
2
(b) DR-DAMAS
1
0
0
−2
−2
−4
−4
−6
−6
1
0.5
0.5
−8 0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
0
−10 −2
−8 −1.5
−1
(c) SC-RDAMAS
−0.5
0
0.5
1
1.5
2
−10
(d) JMAP
1
0
0
−2
−2
−4
−4
−6
−6
−8
−8
1
0.5
0
−10
0.5
−2
−1.5
−1
−0.5
0
0.5
(e) JMAP+Conv A. Mohammad-Djafari,
1
1.5
2
0
−10 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−10
(f) VBA+Conv
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Covariance matrix estimation in VBA: an advantage compared with JMAP 2.5 2 1 1.5 1
0.5
0.5 0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
0
2
ˆ x (solution uncertainty) of estimated source powers x b. Estimated covariance Σ 0 −2 1 −4 −6
0.5
−8 0
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
−10
b by VBA+Conv approach Estimated source powers x
A. Mohammad-Djafari,
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Contributions and Conclusions ◮
Improved forward models of acoustic power propagation: ◮
◮
◮
Proposed approaches: ◮
◮ ◮
◮
◮
Robust acoustic power propagation model by model uncertainty; Efficient invariant convolution model by reduced PSF size, separable PSF and GPU acceleration; Robust deconvolution approach with sparsity constraint (SC-RDAMAS) Bayesian approach with sparsity enforcing prior (JMAP) Non stationnarity of the model errors and sparsity enforcing with Student-t priors JMAP and Variational Bayesian Approximation (VBA)
Advantages: ◮ ◮ ◮
◮
Higher spatial resolution: sparsity constraint/ prior; Wide dynamic range: sparsity parameter estimation Robust to non-stationary errors: hyper-parameter estimation of measurement errors and model uncertainty; Adaptive hyper-parameter estimations: Bayesian inference.
A. Mohammad-Djafari,
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Perspectives ◮
In short term: ◮
◮
In middle term: ◮
◮
◮
Real-time realization of 3D acoustic imaging by GPU: programming proposed approaches directly on GPU in order to utilize 25% and more of peak computational performance; More sophisticated prior models : Group sparsity prior for correlated sources; G or χ2 distribution for positive source powers... In middle term, forward models of full-wave propagation for correlated sources (directivity pattern);
In long term: ◮
◮ ◮ ◮
Inversion methods based on signal models to jointly solve signal amplitude (power), phase, characteristic frequency; De-reverberation in non-anechoic chamber; Acoustic separation; ......
A. Mohammad-Djafari,
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List of publication Published journals (2) ◮
◮
N. CHU, A. Mohammad-Djafari and J. Picheral, Robust Bayesian super-resolution approach via sparsity enforcing priors for near-field acoustic source imaging, Journal of Sound and Vibration, Vol. 332, No. 18, pp 4369-4389, Feb. 2013. N. CHU, J. Picheral and A. Mohammad-Djafari, N. Gac, A robust super-resolution approach with sparsity constraint in acoustic imaging, Applied Acoustics, vol.76, pp.197-208, 2014.
To submit: (2) ◮
N. CHU, A. Mohammad-Djafari, N. Gac, and J. Picheral, A 2D invariant convolution model for acoustic imaging, International Journal of Aeroacoustics, 2013.
◮
N. CHU, A. Mohammad-Djafari, N. Gac, and J. Picheral, A hierarchical variational Bayesian approach approach via Student’s-t priors for acoustic imaging with non-stationary noises, Journal of the Acoustical Society of America, 2013.
A. Mohammad-Djafari,
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List of publications Published in conference (5) ◮ N. CHU, A. Mohammad-Djafari, N. Gac, and J. Picheral, An efficient variational Bayesian inference approach via Student’s-t priors for acoustic imaging in colored noises , Journal of the Acoustical Society of America, Vol. 133, No.5. Pt.2, POMA Vol 19, pp. 055031-40, International Conference of Acoustics (ICA2013), Montreal, Canada, 2013. ◮ N. CHU, A. Mohammad-Djafari and J. Picheral, A Bayesian sparse inference approach in near-field wideband aeroacoustic imaging, 2012 IEEE International Conference on Image Processing, Orlando (ICIP2012), USA, Sep. 30-Oct. 04, 2012. (EI) ◮ N. CHU, A. Mohammad-Djafari and J. Picheral, Bayesian sparse regularization in near-field wideband aeroacoustic imaging for wind tunnel test, 2012 IOA annual meeting and 11th Congr` es Fran¸cais d’Acoustique (ACOUSTICS2012), Nantes, France, Apr. 23-27, 2012, pp. 1391-1396. ◮ N. CHU, A. Mohammad-Djafari and J. Picheral, Two robust super-resolution approaches with sparsity constraint and sparse regularization for near-field wideband extended aeroacoustic source imaging, Berlin Beamforming Conference 2012 (BeBeC2012), Berlin, Germany, Feb. 22-23, 2012, pp. 29. ◮ N. CHU, J. Picheral and A.Mohammad-Djafari, A robust super-resolution approach with sparsity constraint for near-field wideband acoustic imaging, IEEE International Symposium on Signal Processing and Information Technology (ISSPIT2011), Bilbao, Spain, Dec. 14-17, 2011, pp. 310-315. (EI) A. Mohammad-Djafari,
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