-
Acoustic imaging of sparse sources withOrthogonal Matching Pursuit and clustering of
basis vectors
T.F. Bergh1,2 I. Hafizovic2 S. Holm11Department of Informatics, University of Oslo
2Squarehead Technology
ICASSP 2017
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Presentation outline
1. Brief description of deconvolution (DAMAS)2. Proposed deconvolution algorithm3. Simluations & experimental results
ICASSP 2017 - Bergh, Hafizovic, Holm Title 2 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Single-frequency acoustic image
-4 -3 -2 -1 0 1 2 3 4
x [meter]
-3
-2
-1
0
1
2
3
y [m
eter
]
ICASSP 2017 - Bergh, Hafizovic, Holm Introduction 3 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Deconvolved source map
-4 -3 -2 -1 0 1 2 3 4
x [meter]
-3
-2
-1
0
1
2
3
y [m
eter
]
ICASSP 2017 - Bergh, Hafizovic, Holm Introduction 4 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Sound field model
m l
x1(ω)...
xm(ω)...
xM (ω)
=
| | || | || | |
G1(ω) · · · Gl (ω) · · · GL(ω)| | || | || | |
s1(ω)...
sl (ω)...
sL(ω)
+
η1...ηl...ηL
x(ω) = G(ω)s(ω) + η (1)
M: Number of array elements, L: Number of model sources
ICASSP 2017 - Bergh, Hafizovic, Holm Introduction 5 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Acoustic imaging model
n
Yn =1
M2wHn Rx wn =
1
M2wHn E
{xxH
}wn =
1
M2
L∑i=1
L∑j=1
[wHn E
{(Gi si + ηi )(Gj sj + ηj )
∗}
wn]
(2)
0 ≤ n ≤ N, N: Number of image points
ICASSP 2017 - Bergh, Hafizovic, Holm Introduction 6 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Deconvolution Approach for theMapping of Acoustic Sources(DAMAS)Uncorrelated sources and i.i.d. Gaussian noise:
Yn =1
M2
L∑l=1
[∣∣∣wHn Gl ∣∣∣2 E {|sl |2}]+ 1M2 E {|η|2} (3)Y = AQ
(+σ21N
)(4)
Point spread function (PSF): A[n,l] =1
M2
∣∣∣wHn Gl ∣∣∣2 ∈ R+ (5)Source map: Ql = E
{|sl |2
}∈ R+ (6)
Reference: Brooks, Thomas F., and William M. Humphreys. “A Deconvolution Approach for the Mapping of Acoustic Sources (DAMAS)
Determined from Phased Microphone Arrays.” Journal of Sound and Vibration 294, no. 4 (2006): 856–879.
ICASSP 2017 - Bergh, Hafizovic, Holm Introduction 7 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
A as a frame (redundant basis) / dictionary
Y =∑
l AlQlAl : Column l of A, basis vector (or dictionary atom)Ql : Expansion coefficient
Assumptions
1. Ns, number of sources is known2. Number of sources is small, i.e. Q is sparse:
Ns = |Γ| = |supp (Q)|
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Orthogonal Matching PursuitDAMAS (OMP-DAMAS) [1]
for s = 1 to Ns doΓ← Γ ∪ arg maxk
∣∣∣(AHρ)k ∣∣∣Q ← arg minq ||Y − AΓq||2 =
(AHΓ AΓ
)−1AHΓ Y
ρ← Y − AΓQ =[1− AΓ
(AHΓ AΓ
)−1AHΓ
]Y
if stopping criterion fulfilled thenexit loop
end ifend for
Problems with direct application of OMP
I Due to noise and basis mismatch: Y /∈ colsp{A}I Greedy pursuit is inherently suboptimal
[1] Padois, Thomas, and Alain Berry. “Orthogonal Matching Pursuit Applied to the Deconvolution Approach for the Mapping of Acoustic Sources Inverse Problem.”
The Journal of the Acoustical Society of America 138, no. 6 (December 1, 2015): 3678–85. doi:10.1121/1.4937609.
ICASSP 2017 - Bergh, Hafizovic, Holm Introduction 9 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Proposed reconstruction algorithm:Clustered Orthogonal MatchingPursuit - DAMAS (COMP-DAMAS)
Two stages:
1. Exhaustive greedy pursuit.2. Spatial clustering of basis vectors such that Nc ≥ Ns, and
least-squares fitting of each cluster to a single-sourcemodel.
ICASSP 2017 - Bergh, Hafizovic, Holm Algorithm 10 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
COMP-DAMAS Stage 1
Exhaustive greedy pursuitfor s = 1 to L do
Γ← Γ ∪ arg maxk∣∣∣(A†ρ)k ∣∣∣
Q ← (AHΓ AΓ)−1AHΓ Y
ρ← Y − AΓQif(∣∣∣∣∣∣ρ(s−1)∣∣∣∣∣∣
2−∣∣∣∣∣∣ρ(s)∣∣∣∣∣∣
2
)/∣∣∣∣∣∣ρ(s−1)∣∣∣∣∣∣
2≤ δ
thenExit loop
end ifend for
Index selection ruleMaximal least-squares coefficient(Enhanced-OMP [1]):
A†ρ ≈ arg minq||ρ− Aq||2
A† =(
AH A + λσmax(A)1L)−1
AH
λ: Regularization parameter
Terminationρ is dominated by noise
[1] Osamy, Walid, Ahmed Salim, and Ahmed Aziz. “Sparse Signals Reconstruction via Adaptive Iterative Greedy Algorithm.”
International Journal of Computer Applications 90, no. 17 (March 26, 2014): 5–11. doi:10.5120/15810-4715.
ICASSP 2017 - Bergh, Hafizovic, Holm Algorithm 11 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Stage 2: Clustering & mergingDue to exhaustive pursuit, real sources may
be represented by several basis vectors and
expansion coefficients
3.0 1.5
-0.7
Bottom-up spatial clusteringDistance function:
di,j = 1−1
2
∣∣∣wH (~xi )Gj ∣∣∣2∣∣∣wH (~xj )Gj ∣∣∣2 +
∣∣∣wH (~xj )Gi ∣∣∣2∣∣wH (~xi )Gi ∣∣2
= 1−1
2
(A[i,j]A[j,j]
+A[j,i]A[i,i]
), 0 ≤ di,j ≤ 1
(7)
Continue while:
mini 6=j
di,j < γ = 0.85 AND Nc ≥ Ns
ICASSP 2017 - Bergh, Hafizovic, Holm Algorithm 12 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Stage 2: Clustering & mergingFitting cluster to single source model with
index k̂ and power q̂
3.9
Fitting cluster to a singlesource model
(k̂, q̂
)Γ
= arg min(k,q)
||Ak q − AΓQΓ||2 , (8)
k ∈ Z, q ∈ R+
k̂ = arg mink
∣∣∣∣∣∣∣∣∣∣∣∣Ak
∑j∈Γ
Qj
− AΓQΓ∣∣∣∣∣∣∣∣∣∣∣∣2
(9)
q̂ = (AHk̂ Ak̂ )−1AHk̂ AΓQΓ (10)
ICASSP 2017 - Bergh, Hafizovic, Holm Algorithm 13 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Simulations & Experiments
Benchmark methods
1. Orthogonal Matching Pursuit DAMAS (OMP-DAMAS)2. CLEAN based on Source Coherence (CLEAN-SC)3. Robust DAMAS with Sparse Constraint (SC-RDAMAS)4. Generalized Inverse Beamforming (GIB)
References:
1. Padois, Thomas, and Alain Berry. “Orthogonal Matching Pursuit Applied to the Deconvolution Approach for the Mapping of Acoustic Sources Inverse Problem.”The Journal of the Acoustical Society of America 138, no. 6 (December 1, 2015): 3678–85. doi:10.1121/1.4937609.
2. Sijtsma, Pieter. “CLEAN Based on Spatial Source Coherence.” International Journal of Aeroacoustics 6, no. 4 (December 1, 2007): 357–74.doi:10.1260/147547207783359459.
3. Chu, Ning, José Picheral, Ali Mohammad-djafari, and Nicolas Gac. “A Robust Super-Resolution Approach with Sparsity Constraint in Acoustic Imaging.” AppliedAcoustics 76 (February 2014): 197–208. doi:10.1016/j.apacoust.2013.08.007.
4. Suzuki, Takao. “L1 Generalized Inverse Beam-Forming Algorithm Resolving Coherent/Incoherent, Distributed and Multipole Sources.” Journal of Sound andVibration 330, no. 24 (November 21, 2011): 5835–51. doi:10.1016/j.jsv.2011.05.021.
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 14 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Simulation setup
I Single frequency delay-and-sum at 2kHz, sampled at16kHz
I White noise sources, 100 snapshots of 128 samplesI 8m x 6m imaging area, 20x15=300 grid points, distance
4.3m from arrayI 16x16-element array, 39cm x 39cm
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 15 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Simulation results, scenario 1
DAS
-4 -2 0 2 4
-2
0
2
COMP-DAMAS
-4 -2 0 2 4
-2
0
2
OMP-DAMAS
-4 -2 0 2 4
-2
0
2
CLEAN-SC
-4 -2 0 2 4
-2
0
2
SC-RDAMAS
-4 -2 0 2 4
-2
0
2
GIB
-4 -2 0 2 4
-2
0
2
0
0.2
0.4
0.6
0.8
1
No noise
DAS
-4 -2 0 2 4
-2
0
2
COMP-DAMAS
-4 -2 0 2 4
-2
0
2
OMP-DAMAS
-4 -2 0 2 4
-2
0
2
CLEAN-SC
-4 -2 0 2 4
-2
0
2
SC-RDAMAS
-4 -2 0 2 4
-2
0
2
GIB
-4 -2 0 2 4
-2
0
2
0
0.2
0.4
0.6
0.8
1
Single-channel SNR = -9dB
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 16 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Simulation results, scenario 2
DAS
-4 -2 0 2 4
-2
0
2
COMP-DAMAS
-4 -2 0 2 4
-2
0
2
OMP-DAMAS
-4 -2 0 2 4
-2
0
2
CLEAN-SC
-4 -2 0 2 4
-2
0
2
SC-RDAMAS
-4 -2 0 2 4
-2
0
2
GIB
-4 -2 0 2 4
-2
0
2
0
0.2
0.4
0.6
0.8
1
1.2
No noise
DAS
-4 -2 0 2 4
-2
0
2
COMP-DAMAS
-4 -2 0 2 4
-2
0
2
OMP-DAMAS
-4 -2 0 2 4
-2
0
2
CLEAN-SC
-4 -2 0 2 4
-2
0
2
SC-RDAMAS
-4 -2 0 2 4
-2
0
2
GIB
-4 -2 0 2 4
-2
0
2
0
0.2
0.4
0.6
0.8
1
1.2
Single-channel SNR = -9dB
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 17 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Simulation results, scenario 3
DAS
-4 -2 0 2 4
-2
0
2
COMP-DAMAS
-4 -2 0 2 4
-2
0
2
OMP-DAMAS
-4 -2 0 2 4
-2
0
2
CLEAN-SC
-4 -2 0 2 4
-2
0
2
SC-RDAMAS
-4 -2 0 2 4
-2
0
2
GIB
-4 -2 0 2 4
-2
0
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
No noise
DAS
-4 -2 0 2 4
-2
0
2
COMP-DAMAS
-4 -2 0 2 4
-2
0
2
OMP-DAMAS
-4 -2 0 2 4
-2
0
2
CLEAN-SC
-4 -2 0 2 4
-2
0
2
SC-RDAMAS
-4 -2 0 2 4
-2
0
2
GIB
-4 -2 0 2 4
-2
0
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Single-channel SNR = -9dB
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 18 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
0 -6 -9 -120.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
Aver
age
posit
ion
erro
r [m
]Scenario 1
0 -6 -9 -12Single-channel SNR [dB]
0
5
10
15
20
25
30
35
40
Aver
age
rela
tive
powe
r erro
r [%
]
0 -6 -9 -120.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35Scenario 2
0 -6 -9 -12Single-channel SNR [dB]
0
5
10
15
20
25
30
35
40
0 -6 -9 -120.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35Scenario 3
COMP-DAMASOMP-DAMASCLEAN-SCSC-RDAMASGIB
0 -6 -9 -12Single-channel SNR [dB]
0
5
10
15
20
25
30
35
40
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 19 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Experimental setupI Single frequency delay-and-sum at 2.4kHz, sampled at 44.1kHzI Two white noise sources, 10, 100 & 1000 snapshots of 128 samplesI 4m x 3m imaging area, 40x30=1200 grid points, distance 2.7m from arrayI 16x16-element array, 39cm x 39cm
0.39cm
0.39cm
2.7m
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 20 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Experimental results
DAS
-2 0 2
-1
0
1
COMP-DAMAS
-2 0 2
-1
0
1
OMP-DAMAS
-2 0 2
-1
0
1
CLEAN-SC
-2 0 2
-1
0
1
SC-RDAMAS
-2 0 2
-1
0
1
GIB
-2 0 2
-1
0
1
0
0.5
1
1.5
2
2.5
3
10 -7
30ms (10 snapshots)
DAS
-2 0 2
-1
0
1
COMP-DAMAS
-2 0 2
-1
0
1
OMP-DAMAS
-2 0 2
-1
0
1
CLEAN-SC
-2 0 2
-1
0
1
SC-RDAMAS
-2 0 2
-1
0
1
GIB
-2 0 2
-1
0
1
0
0.5
1
1.5
2
2.5
10 -7
3s (1000 snapshots)
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 21 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Experimental accuracy
Average position error [m]
K = 10 K = 100 K = 1000
COMP-DAMAS 0.03 0.07 0.03
OMP-DAMAS 0.13 0.15 0.15
CLEAN-SC 0.10 0.04 0.04
SC-RDAMAS 0.07 0.07 0.07
GIB 0.28† 0.08† 0.18
Average relative runtimes (300 grid points, deconvolutiononly)
COMP-DAMAS OMP-DAMAS CLEAN-SC SC-RDAMAS GIB (Beamforming)
2.5 1 160 800 1900 20
ICASSP 2017 - Bergh, Hafizovic, Holm Simulations & Experiments 22 / 23
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Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering of basis vectors
Conclusion
I Lower overall position & power error than benchmarkmethods
I Low runtime and short required integration time => suitablefor realtime implementation
Future work
I Parameter-free regularizationI Simultaneous fitting of multiple sources in stage 2
Thank you for listening.
ICASSP 2017 - Bergh, Hafizovic, Holm Conclusion 23 / 23
TitleIntroductionAlgorithmSimulations & ExperimentsConclusion