Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering … · 2019. 12....

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


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


Deconvolved source map

-4 -3 -2 -1 0 1 2 3 4

x [meter]

-3

-2

-1

0

1

2

3

y [m

eter

]



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



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



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.



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)|


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.



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


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.



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



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)



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


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



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




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





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




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

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

5

10

15

20

25

30

35

40



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



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)



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



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

Acoustic imaging of sparse sources with Orthogonal Matching Pursuit and clustering … · 2019. 12....

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