Upload
others
View
3
Download
0
Embed Size (px)
Citation preview
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
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
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
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
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
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
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
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)|
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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