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音源到来方向分布とアクティベーション共有型非負値行列因子分解を用いた音像深度推定∗☆宮内 智, 北村大地, 猿渡 洋, 中村 哲 (奈良先端大)1 Introduction

2 Source Separation Method
In the proposed method, we use the source separation
method based on directional clustering [4] for extracting
the specific source included in the mixed signal. Figure 1
shows the configuration of directional clustering. First,
the time-frequency components of the stereo mixed signal X(ω, τ) = [X (L) (ω, τ), X (R) (ω, τ)]T are represented
into the two-dimensional space that has |X (L) (ω, τ)| and|X (R) (ω, τ)| as coordinate axes, where |X (L) (ω, τ)| and|X (R) (ω, τ)| are the amplitude of each channel. Next,
these components are normalized and separated by kmeans clustering. Obtained components of the individual cluster are used for reproduction by WFS.

3 Proposed Method
3.1 Depth Estimation Based on DOA Distribution
In sound fields, when a sound source is far from thelistener, sound waves arrive from various directions ow∗：Centroid vector

L-ch input signal

L-ch input signal

L-ch input signal：Source component

Wave field synthesis (WFS) [1] is a technique for realizing high quality sound reproduction. WFS allows
us to create sound images at the front of loudspeakers.
However, WFS requires specific localization information of the objective sound sources in a mixed signal
to reproduce the individual sound images. In existing
media contents such as a CD, the localization of eachsound image has been lost because they are generally
provided as a stereo format. Therefore, a method to estimate the localization of each sound image even in themixed source requires urgent attention. In our previousresearch, the estimation method of sound directions has
been proposed [2]. However, as far as we know, there is
no eﬀective method and studies that cope with depth information tracking for sound images. To solve this problem, in this paper, we propose a new depth estimation
method, which utilizes the variation of the direction ofarrival (DOA) of the individual sound source. In thispaper, first, we describe the source separation method
based on directional clustering for the mixed source into
the individual source. Next, we propose a new depth estimation method, which utilizes the variation of DOA. Inaddition, we propose the feature extraction method using
nonnegative matrix factorization (NMF) [3]. Finally, weshow the eﬃcacy of the proposed method by the objective experiment.

R-ch input signal

R-ch input signal

Normalization

Fig. 1R-ch input signal

Clustering

Configuration of directional clustering.

ing to sound diﬀusion. Therefore, as shown in Fig. 2,
the shape of an observed DOA distribution of the target
source can be used as a cue for depth estimation. Process flow of the proposed method are shown in Fig. 3.
In this method, first, we calculate a weighted DOA histogram of a mixed sound source. In this process, DOAs
are calculated as θ = 2 arctan (|X (R) (ω, τ)| / |X (L) (ω, τ)|).
Simultaneously, DOAs are weighted by the magnitude of22each vector w = (|X (L) (ω, τ)| + |X (R) (ω, τ)| )1/2 to avoid
the problem of normalization process. Next, we separate the mixed source into the individual sources by using directional source separation method. In addition,
we propose to extract features of the target components
to eliminate the eﬀect of background noise and artificial
distortion generated by signal processing (windowing inshort-time DFT, etc.). As the feature extraction method,
we propose activation-shared multichannel NMF, whichreduces dimensionality of input stereo data while maintaining directional information. This property is advantageous to polish up the DOA distribution, whereas theconventional NMFs applied to stereo signals in parallel
generate an artificial fluctuation in DOA. Finally, we canestimate the depth of the target source by modeling theresultant DOA distribution. In the following sections, wewill describe activation-shared multichannel NMF andthe modeling method for DOA distribution.
3.2 Activation-Shared Multichannel NMF
NMF is a sparse representation method. The aim ofsparse representations is to reveal certain structures ofa signal, and to represent these structures in a compact.
Also, this representations provide high performance fornoise reduction, compression and feature extraction. Using this property, we eliminate background noise andartificial distortion, which are the problem to evaluate
the shape of DOA distribution. However, if the conventional NMFs are applied to stereo signals in paral-“Depth estimation of sound images using direction of arrival distribution and activation-shared nonnegative matrix
factorization," by Tomo Miyauchi, Daichi Kitamura, Hiroshi Saruwatari, and Satoshi Nakamura (Nara Institute ofScience and Technology).

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Fig. 2

Direction of arrival

Cluster L

Cluster C

Cluster R

Activation- Activation- Activationshared NMF shared NMF shared NMF
Direction of arrival

DepthDepth
Depthestimation estimation estimationlel, DOA information is disturbed. Therefore, we propose activation-shared multichannel NMF, which provides sparse representation of the signal while maintaining the directional information. In the following section, we describe the update rules of the proposed NMF,
which have been derived by the authors.

Fig. 3(1)where Y (m) (∈ RΩ×T
≥ 0 ) is a spectrogram of the mth channelsignal, F (m) (∈ RΩ×K
≥ 0 ) is a matrix that involves bases ofeach channel signal, and G(∈ R≥K×T0 ) is an activation matrix that is shared in all channels of the signals. In adittion, Ω is the number of frequency bins, K is the numberof bases, and T is the number of frames of the signal．
Since the activation matrix is shared through all channels, we can reduce dimensionality of the input signal
while maintaining directional information that is amplitude ratio between each channel.
Next, we describe the cost function of proposed NMF.
β-divergence is a generalized divergence of variable x
corresponding to y [5]. β-divergence is defined as
yβxβ yxβ−1+−(β ∈ R\ {0, 1} )β (β − 1) β β − 1 ()Dβ (y∥x) = y log y − log x + x − y (β → 1) . (2)


yy

 − log − 1(β → 0)xx
Using β-divergence, the cost function of activationshared multichannel NMF is defined as∑(3)J(Θ) = m Dβ (Y (m) ∥F (m) G),{}
where Θ = F (m) , G is a set of observed variables. Using (2), we can redefine (3) as (∑)β(∑ (m) )β−1 (m)
y(m)k fω,k gk,t∑ω,t k fω,k gk,tJ(Θ) = m,ω,t − ,ββ−1(4)

Direction of arrival

Signal flow of proposed method.

∑() φ(β)
 y(m) gk,t ∑ ′ f (m)′ gk′,t β−2 tkω,tω,k


(m)
(m) 

(5)fω,k← fω,k(∑ (m) )β−1  , ∑t gk,tk′ fω,k′ gk′,t∑()β−2φ(β)(m) (m) ∑(m)
m,k′ fω,k′ gk′,t m,ω yω,t fω,k
gk,t ←gk,t  ∑()β−1  , (6)(m) ∑(m)m,ω fω,km,k′ fω,k′ gk′,tWhen M-channel signals are observed, the trainedbases are constructed as(m = 1, 2, · · · , M),

Direction of arrival(m)(m)
where yω,t， fω,k，and gk,t are entries of matrices Y (m) ，(m)F , and G, respectively. Also, the constant term isabbreviated．Then, the update rules are obtained by theauxiliary function approach as

3.2.1 Cost Function and Update Rules

日本音響学会講演論文集

Frequency ofsource components

Weighted DOA histogram

DOA of far sound source

Example of DOA histogram.

Y (m) ≃ F (m) G

Frequency ofsource components

Direction of arrival

sourcecomponents Noise

Direction of arrival

DOA histogram

Frequency ofsource components

Farsoundsource

Input stereo signal
L-ch
R-ch
STFT

Frequency ofsource components

Frequency ofsource components
Frequency ofsource components

Closesoundsource

DOA of close sound source

where φ (β) is defined as
1/(2 − β)φ (β) = 11/(β − 1)

(β < 1)(1 ≤ β ≤ 2) .

(7)

(2 < β)3.3 Modeling of DOA distribution
There are many studies on the modeling of DOA distribution, and the modeling methods using a Cauchy distribution and a Gaussian distribution have been proposedin conventional works [6]. However, since these distributions do not have parameters that handle a tail of thedistribution accurately, it is not suitable for the "depth"
estimation. Therefore, in our method, we evaluate theshape of DOA by using generalized Gaussian distribution (GGD) [7]. GGD is a flexible family of probability density function (PDF) modeling with some parameters, and GGD can represent various types of wellknown PDFs, e.g., Gaussian and Laplacian distributions.
The definition of GGD isfGG (z; αscale , βshape ) =βshape
12αscale Γ( βshape [] |z − z| βshape  ,exp −αscale)(8)∫∞where Γ(x) = 0 exp(−t)t x−1 dt is gamma function, z is
the mean of variable z, αscale is scale parameter, βshape- 948 -

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5fGG f(z;αGG (z;α,βshape)
scale, β)544αscale = 0.1βshape = 133αscale = 0.25
βshape = 1.5

2αscale = 0.5
βshape = 22

11: Target source

0-2-20-1.5-1.5-1-1-0.5
-0.5

00

0.5
0.5

11atintervals: Interference source

22

1.5
1.5z

Fig. 4GGD with typical parameters αscale and βshape .
Fig. 5is shape parameter of GGD. Figure 4 shows examplesof diﬀerent PDFs in GGD. As can be seen, the shape ofGGD changes depending on βshape ; note that βshape = 2corresponds to Gaussian PDF and that βshape = 1 corresponds to Laplacian PDF. If βshape is low, GGD becomes a spiky shape, and if βshape is high, GGD becomesa smooth shape. Based on this property, we use the shapeparameter βshape of GGD to evaluate the shape of DOA
distribution. In our depth estimation method, depth ofsound images are evaluated by βshape based on the definition that the target source is close when βshape is low,
and the target source is far when βshape is high.
Although the maximum-likelihood based shape parameter estimation has no closed-form solution in GGD,
we propose a closed-form parameter estimation algorithm based on some approximation and kurtosis. Wewill describe this estimation method as below.
3.3.1 Closed-Form Parameter Estimation in GGD
We can introduce kurtosis to estimate βshape . In general, kurtosis of DOA histogram h(θ) is given bykurt(h(θ)) = ⟨h4 (θ)⟩θ ⟨h2 (θ)⟩−2θ − 3.

(9)

The nth order moment of GGD has the following useful
relationship;⟨zn ⟩ = β−nshape Γ() ()−1n+11Γ.
βshapeβshape) () ()−213ΓΓ− 3. (11)kurt(y(θ)) = Γβshape
βshapeβshape(5In order to obtain βshape value from the measured kurtosis, we should calculate the inverse function of (11).
However, it is well known that there is no exact closedform solution of (11) w.r.t. βshape . Therefore, we hereinafter introduce an approximation for the closed-form
derivation of optimal βshape . With modified Stirling’s formula on gamma function,√Γ(z) ∼ 2π · exp(−z) · zz−0.5 · exp

日本音響学会講演論文集(112z

or equallylog Γ(z) ∼)(12)

11log(2π) − z + (z − 0.5) log z +, (13)212zthen we can obtainlog(kurt(h(θ)) + 3)
()
()
()
5
13= log Γ+ log Γ− 2 log Γβshapeβshapeβshape( 5))(1523∼· log 6 + log √ + β · .
(14)βshape4535This results in the following quadratic equation of βshape
to be solved,
β2shape + βshape ·)
(( 5)
4545
35+log √log 6 = 0,2235(kurt(h(θ)) + 3)

(15)
and we can derive the closed-form estimate of shape parameter βshape from kurtosis, as()453log √45(kurt(h(θ)) + 3)√()2( 5)
1 202535−log √− 90 log 6 .
2435(kurt(h(θ)) + 3)
(16)

βshape = −(10)

From (9) and (10) we have the following equation of kurtosis and β,Sound source geometry used in experiment.

4 Experiment and Results
A geometry of an experiment is shown in Fig. 5. We
prepared two mixed stereo signals containing three instruments. Mixed signal 1 consists of vocal (Vo.), piano(Pf.), and guitar (Gt.). Also, mixed source 2 consists ofVo., electric guitar (E.Gt.), and synthesizer (Syn.). The
target source was located in the center direction withseven distances. In addition, the interference sourceswere located in the left- and right-hand sides. Test
sources were generated using a room impulse response
recorded at each position. We used the image method[8] as a reference for this experiment, which is a technique of simulating the room impulse response. The signal not processed by NMF was evaluated as conventional- 949 -2014年3月

Image method (oracle)Conventional method 1

Proposed method
Conventional method 2

Image method (oracle)
Conventional method 1

3.5
5.5

4553

3.5
4.5

Proposed method
Conventional method 2

34

Shape parameter

Shape parameter

2.5
4.5

2

1.5

1

0.5

2.5
3.5
2
1.5
1
0.5

00
0

0.5

1

1.5

2

2.5

3

3.5

40

Distance between source and listener [m]Fig. 6
Table 1

Estimated value of mixed signal 1.

Fig. 7

Correlation coeﬃcient of mixed signal 1

Data set
Target source
Interference source (left)
Interference source (right)
Proposed method
Conventional method 1Conventional method 2

1Vo.
Pf.
Gt.
0.903
0.419
0.233

2Vo.
Gt.
Pf.
0.891
0.532
0.165

3
Gt.
Pf.
Vo.
0.777
0.154
0.044

4
Gt.
Vo.
Pf.
0.651
0.277
-0.037

5
Pf.
Vo.
Gt.
0.791
0.602
0.426

6Pf.
Gt.
Vo.
0.816
0.496
0.157

method 1. Also, the signal processed by conventional
NMF, which was applied to each channel independently,
was evaluated as conventional method 2.
The experimental results are shown in Fig. 6 andFig. 7. From these results, the shape parameters of theproposed method are proportional to the distance of thetarget source, whereas the results of the conventional
methods have no agreement with the oracle. In addition,
the correlation coeﬃcient between the reference value ofimage method and the estimated value of other methods
are shown in Table 1 and Table 2. These results indicate strong relation between the estimated value of theproposed method and the distance of the target source.
Thus, the eﬃcacy of the proposed method as the depth
estimation can be confirmed.

5 Conclusion
In this paper, we proposed a new depth estimation
method of sound source in mixed signal using the DOA
distribution. We also proposed a new feature extraction
method for the multichannel signal, activation-shared
NMF. The result of the experiment indicated the eﬃcacyof the proposed method.

日本音響学会講演論文集

0.5

1

1.5

2

2.5

3

3.5

4

Distance between source and listener [m]Table 2Estimated value of mixed signal 2.
Correlation coeﬃcient of mixed signal 2

Data set
Target source
Interference source (left)
Interference source (right)
Proposed method
Conventional method 1
Conventional method 2

1Vo.
Syn.
E.Gt.
0.901
0.353
0.116

2Vo.
E.Gt.
Syn.
0.882
0.298
0.208

3
E.Gt.
Syn.
Vo.
0.797
0.127
-0.0494E.Gt.
Vo.
Syn.
0.899
0.419
0.457

5
Syn.
Vo.
E.Gt.
0.940
0.419
0.239

6
Syn.
E.Gt.
Vo.
0.735
0.042
0.370

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pp.977–995, 1988.
[2] M. Hirata, et al., “Spatial depth perception of focusing source by wave field synthesis for 3D soundreproduction," Proc. 3DSA, pp.137–140, 2011.
[3] D. D. Lee, H. S. Seung, “Algorithms for nonnegative matrix factorization," Neural Info. Process. Syst., vol.13, pp.556–562, 2001.
[4] S. Araki, H. Sawada, R. Mukai, S. Makino, “Underdetermined blind sparse source separation forarbitrarily arranged multiple sensors,” Signal Processing, vol.87, no.8, pp.1833–1847, 2007.
[5] S. Eguchi, Y. Kano, “Robustifying maximum likefood estimation," Technical Report of Institute of
Statistical Mathematics, 2001.
[6] M. Hirakawa, K. Suyama, “Multiple sound source
tracking by two microphones using PSO," Proc. ISPACS, pp.467–479, 2013.
[7] G. Box, et al., “Bayesian Inference in Statistical Analysis,” Addison Wesley, Reading, Massachusetts, 1973.
[8] B. Allen and A. Berkley “Image method foreﬃciently simulating small-room acoustics," J.
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