第29 回信号処理シンポジウム
2014年11月11日〜14日（京都）チャネル別事前分布推定と両耳共通スペクトルゲインを用いた定位保持型バイノーラル音源分離
Localization-Preserved Binaural Source Separation Using
Channel-Wise Target Prior Estimation and Equi-Binaural Spectral
Gain1

室田勇騎 1 , 北村大地 2 , 小山翔一 3 , 猿渡洋 3 , 中村哲 1
奈良先端科学技術大学院大学, 2 総合研究大学院大学, 3 東京大学

Yuki MUROTA 1 , Daichi KITAMURA2 , Shoichi KOYAMA3 ,Hiroshi SARUWATARI3 , Satoshi NAKAMURA1
1

2

Nara Institute of Science and Technology, Nara, JapanThe Graduate University for Advanced Studies (SOKENDAI), Kanagawa, Japan
3
The University of Tokyo, Tokyo, Japan

Abstruct This paper addresses a new statistical model ofbinaural signals and its application to eﬃcient binaural source
separation. Binaural source separation is always required toretain a spatial cue of the separated sound, such as a headrelated transfer function (HRTF). However, the direct use of
an HRTF is not realistic because this information is normallynot known in advance. To cope with this problem, first, we
focus on the diﬀerence between signal probability density
functions at both ears, which can be blindly estimated by using our previous work on higher-order statistics. Next, wederive a sound-localization-preserved generalized minimum
mean-square error short-time spectral amplitude estimator.
Objective and subjective experiments show the eﬃcacy of the
proposed method in terms of spatial quality.
1 Introduction
Audio signal separation has received much attention insignal-processing research, and many studies have been published in the last decade. Techniques for signal separation
have been developed for many audio applications, including
target speech enhancement for hearing-aid systems [1] andfor controlling each source in a music tune in interactive 3D
audio systems [2]–[5]. In this paper, we also address suchan audio signal separation problem, especially focusing on asignal provided in a binaural format [6].
Compared with simple multichannel signal processing,binaural signal separation includes a relatively diﬃcult task,
namely, extraction of a specific sound while maintaining its
spatial properties. This is because deterioration of the spatial quality of the separated sound has an adverse eﬀect onhuman’s 3D audio perception. Several methods have been
proposed for binaural signal separation, mainly for blindspeech separation and enhancement. To preserve a sound-

localization cue such as the interaural level diﬀerence, thesemethods [7]–[10] apply an equi-binaural spectral gain to boththe left and right ears of the listener, which can be calculatedvia, e.g., Wiener filtering (WF) [11] and the minimum meansquare error short-time spectral amplitude (MMSE-STSA)
estimator [12], [13]. These methods have a drawback thatthey do not take account of binaural information. Ideally, thebest way to enhance it is to explicitly use an important spatial cue, such as a head-related transfer function (HRTF) [6].
We have proposed an algorithm [14] to introduce a user’s
HRTF into a multichannel MMSE-STSA estimator but thismethod was not a blind system; the accurate measurementof the HRTF was required in advance, which is sometimes
impossible in practice.
In another context of signal separation such as informed
source separation, the authors have proposed a combination[15] of supervised nonnegative matrix factorization (SNMF)[16]–[19] and a prior-model-adapted generalized MMSESTSA estimator [20] to deal with music signal separation.
This method requires a certain supervision such as a music scale of the target instrument, but it eﬃciently extracts
a target sound composed with an arbitrary melody from theobserved monaural mixture. In this method, thanks to thehigher-order statistics analysis, hidden parameters of the target statistical model can be estimated in each frequency subband.
Motivated by the above-mentioned prior work, in this paper, we propose a new spatial-cue-aware binaural signal separation algorithm without knowing the user’s HRTF. The key
idea and advantages of the proposed method are summarized
as follows: (I) Instead of using an HRTF, we introduce a target statistical model to express the diﬀerence between user’s
left- and right-ear signals. For instance, if the target sound‑ 486 ‑

0.15(b) Right ear(a) Left ear
Amplitude

0.3
Amplitude

is located on the left-hand side, the extracted signal in theleft ear will obey a spiky probability density function (p.d.f.)but the right-ear signal will have a smooth p.d.f. because ofmany diﬀracted waves and the weak direct wave. (II) The
statistical models for each ear can be accurately determinedusing only observable data [15]. Thus, the proposed strategy
is HRTF-blind and a new attempt at establishing a statistical
HRTF approach. (III) Using the same idea as in [10], an equibinaural spectral gain is derived on the basis of statisticalHRTF-adapted generalized MMSE-STSA estimators. Thisavoids the marked deterioration of spatial quality.

0.2

0.1

0.05

0.1

0

1

2
Time [s]

3

4

0

1

2
Time [s]34Fig. 1: Diﬀerence in signals at left and right ears.
teriori SNRs, respectively, which are defined as

2 Conventional Methodξ̃( f, τ) = αγ̃( f, τ − 1)G2 ( f, τ) + (1 − α)max[γ( f, τ) − 1, 0],(5)

2.1 Single-Channel Music Signal Enhancementγ̃( f, τ) =|x( f, τ)|2 /Pñ ( f ),In our previous work, we proposed single-channel musicsignal enhancement based on the generalized Bayesian estimator with automatic target prior adaptation [15]. Here, we
use the SNMF-based dynamic interference spectrogram estimator and closed-form parameter estimation for the statistical model of the target signal based on higher-order statistics.
The details are described below.

where Pñ ( f ) is the estimated interference power spectral density and α is the forgetting factor.
In the generalized MMSE-STSA estimator, the a priori statistical model of the target signal amplitude spectrum is set tothe chi distribution

2.1.1 Music signal separation by generalized
MMSE-STSA estimator with automatic target
prior adaptation
We apply short-time Fourier analysis to the observed signal, which is a mixture of target and interference signals, toobtain the time-frequency-domain complex-valued signalx( f, τ) = s( f, τ)+n( f, τ),(1)where x( f, τ) is the observed signal, s( f, τ) is the target signal, n( f, τ) is the interference signal, f is the frequency binnumber, and τ is the time-frame index.
For the generalized MMSE-STSA estimator, the amplitude
spectrum of the target signal is estimated on the basis of the
MMSE criterion under a certain target prior. The processedsignal s̃( f, τ) via the generalized MMSE-STSA estimator isgiven bys̃( f, τ) = G( f, τ)x( f, τ),
(2)
√
(
)1/β
ν( f, τ) Γ(ρ+0.5) Φ(0.5−ρ, 1, −ν( f, τ))G( f, τ) =··,γ( f, τ)Γ(ρ)Φ(1−ρ, 1, −ν( f, τ))(3)where Γ(·) is the gamma function, Φ(a, b; k) = F 1 (a, b; k)is the confluent hypergeometric function, β is the amplitude
compression parameter, and()−1
ν( f, τ) = γ̃( f, τ)ξ̃( f, τ) 1 + ξ̃( f, τ) .
(4)
Here, ξ̃( f, τ) and γ̃( f, τ) are the estimated a priori and a pos-(6)

p(x) = 2ϕρ Γ(ρ)−1 x2ρ−1 exp(−ϕx2 ),(7)

where p(x) is the p.d.f. of signal x in the amplitude domain,ϕ = ρ/E{|x|2 }, and ρ is the shape parameter. Here, ρ = 1
gives a Rayleigh distribution that corresponds to a Gaussian
distribution in the time domain, and a smaller value of ρ corresponds to a super-Gaussian distribution signal.
In the generalized MMSE-STSA estimator, to calculateγ̃( f, τ), dynamic estimation is required if the interference signal is nonstationary, and estimation of the shape parameter ρ,which depends on the type of target signal, is also required.
2.1.2

Interference estimation by SNMF

The following equation represents the decomposition
model of SNMF using the trained supervision componentsF( f, k):∑∑A( f, τ) = |x( f, τ)| ≈F( f, k)V(k, τ) +H( f, n)U(n, τ),
k

n

(8)
where F( f, k) is a nonnegative element of the supervised basis matrix trained in advance, which comprises spectral patterns of the target signal as column vectors, V( f, k) is a nonnegative element of an activation matrix that corresponds toF( f, k), H( f, n) represents a nonnegative element of the otherbasis matrix, which comprises residual spectral patterns that∑cannot be expressed by k F( f, k)V(k, τ), and U(n, τ) is anonnegative element of the activation matrix that correspondsto H( f, n). Moreover, k is the basis index of F( f, k) and n is
the basis index of H( f, n). The supervised basis matrix can betrained using sample sounds of the target signal in the training∑
process. Hence, ideally, k F( f, k)V(k, τ) represents the tar∑
get signal components and n H( f, n)U(n, τ) represents the‑ 487 ‑

we call this gain the equi-binaural optimal spectral gain.
3.2 Signal Mixture Model
We consider a mixing model with two inputs, i.e., twoears, and assume that the observed signal contains the target signal and an interference signal. Hereafter, the observed signal vector in the time-frequency domain, x( f, τ) =[xL ( f, τ), xR ( f, τ)]T , is given byx( f, τ) = h( f )s( f, τ) + n( f, τ),

Fig. 2: Conventional and proposed binaural models.
other components diﬀerent from the target signals after the∑decomposition. Thus, ( n H( f, n)U(n, τ))2 is regarded as agood estimate of PÑ ( f ) in (6) in the time-frequency grids.
3 Proposed Method
3.1 Motivation and Strategy
In binaural source separation, it is desirable to use binaural cues such as an HRTF to improve the separation performance. However, this is diﬃcult because we cannot obtainthe HRTF of an unknown user blindly. Therefore, considering the HRTF from a statistical viewpoint, we eﬃciently
express binaural cues using a statistical model based on thechi distribution. First, in the binaural signal, the p.d.fs. of thesignals of each ear are assumed to be diﬀerent by the influence of the diﬀerence in the signals arriving at each ear, andwe can obtain a binaural cue based on the diﬀerence in thesep.d.fs. For example, Fig. 1 shows amplitude of left- and rightear signals of cello in 6 kHz subband, where the sound comesfrom the left-hand side. As shown in this figure, the left-ear
signal contains several dominant components, which yields
spiky p.d.f., but nothing in the right ear. Next, we introduce
the chi distribution to represent the p.d.f. of each ear. By applying the target prior adaption algorithm [15] to each ear, it
is possible to obtain suitable parameters of the p.d.fs. automatically. This means that we can convert the conventional
deterministic HRTF estimation problem into a parameter estimation problem of the corresponding statistical model (seeFig. 2). This can also enable adaptation to unknown users.
However, there is a problem when we apply this strategy
to a binaural source separation system. Generally speaking,a statistical-model-based source separation method (e.g., thegeneralized MMSE-STSA estimator) only provides the statistically fluctuating spectral gains for each of ears independently. However, the fluctuation of the gain function in interaural level diﬀerences at the left and right ears causes the deterioration in sound localization. To resolve this problem, wederive a new optimal spectral gain that minimizes the residualinterference power in terms of the MMSE under the condition
that the spectral gains are equivalent in both ears. Hereafter,(9)

where h( f ) = [hL ( f ), hR ( f )]T is the column vector of
the transfer functions between the target source and eachear, s( f, τ) is the target signal component, and n( f, τ) =[nL ( f, τ), nR ( f, τ)]T is the column vector of the interference
signal. Throughout this paper, the subscripts ∗ (∗ = {L, R})represent the signals obtained at the left and right ears, respectively.
3.3 Derivation of Equi-Binaural Optimal Spectral Gain
The derivation of the equi-binaural optimal spectral gain is
described below. This is the extended version of [10] for ageneralized cost function, and can be formulated as the minimization problem of the following error e:[{}e =E |hL ( f )s( f, τ)|β − (G( f, τ)|xL ( f, τ)|)β 2
}]
{
(10)
+ |hR ( f )s( f, τ)|β − (G( f, τ)|xR ( f, τ)|)β 2 ,where G( f, τ) is the equi-binaural spectral gain, which is considered as a variable. The optimization problem based on (10)is given by[
Gopt ( f, τ) = argmin E {|hL ( f )s( f, τ)|β − (GL ( f, τ)|xL ( f, τ)|)β }2G( f,τ)+ {|hR ( f )s( f, τ)|β − (GR ( f, τ)|xR ( f, τ)|)β }2+ {(Gβ ( f, τ) − GβL ( f, τ))|xL ( f, τ)|β }2]+ {(Gβ ( f, τ) − GβR ( f, τ))|xR ( f, τ)|β }2 + 2C , (11)where Gopt ( f, τ) is the equi-binaural optimal spectral gain tobe estimated, and GL ( f, τ) and GR ( f, τ) are individual spectral gains for the L and R ears, respectively, which are auxiliary parameters for calculating an approximate solution of
Gopt ( f, τ) because the direct Bayesian estimation of Gopt ( f, τ)
is diﬃcult. In addition, C is related to the correlation between
the estimation error and the observed signal in each channel
when we estimate the target speech signals in the L and R ears
using the parameters GL ( f, τ) and GR ( f, τ), and is defined by
C ={Gβ ( f, τ) − GβL ( f, τ)}· {(GL ( f, τ)|xL ( f, τ)|)β − |hL ( f, τ)s( f, τ)|β }|xL ( f, τ)|β
+ {Gβ ( f, τ) − GβR ( f, τ)}· {(GR ( f, τ)|xR ( f, τ)|)β − |hR ( f, τ)s( f, τ)|β }|xR ( f, τ)|β .
(12)
We discuss the minimization of (11). First, the 1st and 2nd
terms on the right-hand side correspond to the problem oftarget signal estimation in each ear. These terms can be mini-‑ 488 ‑

mized if we obtain the optimal values of GL ( f, τ) and GR ( f, τ)using the generalized MMSE-STSA estimator described in
Sect. 2. Next, C in the 5th term on the right-hand side canbe disregarded if the parameters GL ( f, τ) and GR ( f, τ) provide an accurate estimate of the target signals by approximately considering C to be negligible. Hence, the residual
3rd and 4th terms, i.e., {(Gβ ( f, τ) − GβL ( f, τ))|xL ( f, τ)|β }2 +{(Gβ ( f, τ) − GβR ( f, τ))|xR ( f, τ)|β }2 , should be minimized. This
problem can be formulated as

Therefore, the shape parameter ρ is given byρ = (kurt − 1)−1 .

From this relation, the shape parameter of the target signal
can be estimated by obtaining its amplitude-domain kurtosis
value. In general, however, it is diﬃcult to directly estimate
the kurtosis of a target signal because of its contamination byadditive interference signals.
3.5

Gopt ( f, τ)[≃ argmin E {(Gβ ( f, τ) − GβLopt ( f, τ))|xL ( f, τ)|β }2G( f,τ)]+{(Gβ ( f, τ) − GβRopt ( f, τ))|xR ( f, τ)|β }2 ,(13)

subject to[]
GLopt ( f, τ) = argmin E {|hL ( f )s( f, τ)|β −(GL ( f, τ)|xL ( f, τ)|)β }2 ,GL ( f,τ)(14)[]
GRopt ( f, τ) = argmin E {|hR ( f )s( f, τ)|β −(GR ( f, τ)|xR ( f, τ)|)β }2 ,(20)

Estimation of Hidden Target Kurtosis and Gain
Function

In our previous work, we proposed an algorithm for target
kurtosis estimation in additive signals, which can be derived
from the closed-form relation in higher-order statistics. In
this algorithm, the resultant kurtosis of the target amplitude
spectrum is estimated as




∑



kurt∗ = µ4 (A∗ ) − µ4  (HU)∗ 
n


∑
∑

2
+4µ2 ( (HU)∗ )−4µ2 (A∗ )µ2 ( (HU)∗ )

GR ( f,τ)(15)n

To solve (13), we first obtain GLopt ( f, τ) and GRopt ( f, τ)from the generalized MMSE-STSA estimator in (14) and(15), then by substituting them into (13), we solve the following equation in G( f, τ):∂e= Gβ ( f, τ)|xL ( f, τ)|2β − GβLopt ( f, τ)|xL ( f, τ)|2β∂G( f, τ)+ Gβ ( f, τ)|xR ( f, τ)|2β − GβRopt ( f, τ)|xR ( f, τ)|2β= 0.

nwhere we ignore the indexes f and τ for saving the space. For
the detailed derivation of (21), see Ref. [15].
The shape parameter of the target signal p.d.f. at each ear
can be estimated using the kurtosis and (20). Therefore, theequi-binaural optimal spectral gain estimated by the proposed
method is obtained as follows by substituting (2) into (17):G̃opt ( f, τ){ |x ( f, τ)|2β (ν̃ ( f, τ))β/2 Γ ((kurt − 1)−1 + 0.5)
L
L
L
=
(
)
β
β
2
2β
{|xL ( f, τ)| + |xR ( f, τ)| }γ̂L ( f, τ)Γ (kurtL − 1)−1

The solution of (16) is given by
Gopt ( f, τ) β1/β GLopt ( f, τ)|xL ( f, τ)|2β + GβRopt ( f, τ)|xR ( f, τ)|2β  .
= 
|xL ( f, τ)|2β + |xR ( f, τ)|2βΦ(0.5 − (kurtL − 1)−1 , 1, −ν̃L ( f, τ))Φ(1 − (kurtL − 1)−1 , 1, −ν̃L ( f, τ))()|xR ( f, τ)|2β (ν̃R ( f, τ))β/2 Γ (kurtR − 1)−1 + 0.5+(){|xL ( f, τ)|2β + |xR ( f, τ)|2β }γ̂Rβ ( f, τ)Γ (kurtR − 1)−1
}1/β
Φ(0.5 − (kurtR − 1)−1 , 1, −ν̃R ( f, τ))·.
(22)
Φ(1 − (kurtR − 1)−1 , 1, −ν̃R ( f, τ))·(17)
3.4 Shape Parameter and Kurtosis

Regarding the chi distribution p(x) in (7), the mth-ordermoment can be written as∫ ∞Γ(ρ + m2 ) − mµm (x) =ϕ 2.
xm p(x)dx =
(18)
Γ(ρ)
0
Then, the kurtosis of the chi distribution is calculated as
kurt = µ4 (x)/µ22 (x) = (ρ + 1)/ρ.

(19)

n

(21)

(16)

In (17), we need to calculate GLopt ( f, τ) and GRopt ( f, τ),which include a shape parameter ρ that should represent thea priori distribution of the target signal. In Sects. 3.4 and 3.5,
we describe how to blindly estimate ρ.

n

−1

∑
∑

 2
2
· µ2 (A∗ )+µ2 ( (HU)∗ )−2µ2 (A∗ )µ2 ( (HU)∗ ) ,where GLopt ( f, τ) and GRopt ( f, τ) are the L- and R-ear optimal
spectral gains, respectively.

The final output is given by s̃∗ ( f, τ) = G̃opt ( f, τ)x∗ ( f, τ).
4
4.1

Evaluation Experiments
Experimental Conditions

In this experiment, we used four binaural instrumental signals, namely, an oboe, clarinet, cello, and piano, where thetarget instrument s( f, τ) is the oboe (each melody part is depicted in [15]). These signals were artificially generated bya MIDI synthesizer and the directions of arrival of these signals were set from −90◦ to 90◦ with 15◦ intervals by using the‑ 489 ‑

Equi-gain WF

SNMF
Equi-gain MMSE-STSA
Generalized MMSE-STSA

low spatial distortion.
Figure 3 shows the average SDRs for each method andeach direction. From these results, we can confirm that the
separation performance of the proposed method is better thanthose of the other methods. This result indicates the efficacy of introducing the flexible a priori statistical modelof the target signal and equi-binaural spectral gain. The
simple MMSE-STSA estimators (Equi-gain MMSE-STSA
and Gain-min MMSE-STSA) also assume the fixed a priori
model of the Gaussian distribution but the assumption is not
appropriate for representing a music signal and a spatial difference between both ears. In contrast, the proposed method
automatically chooses a spikier p.d.f. (ρ ≪ 1) for the ear
closer to the source location and a smoother p.d.f. (ρ ≃ 1) for
the opposite ear. These p.d.fs. match the binaural target.

Gain-min MMSE-STSA
Proposed method

8

SDR [dB]

7

6

5

4
-90

-75

-60

-45

-30

-15

0

15

30

45

60

75

90

Angle [deg]Fig. 3: Average SDRs for each method and each direction.
Proposedmethod4.3

Generalized
MMSE-STSA

We next conducted a subjective test to evaluate the performance of the proposed method, focusing on the humanimpression of the separated signal from the viewpoint of spatial quality. In the subjective experiment, we employed the
XAB method and compared two methods, i.e., Generalized
MMSE-STSA and Proposed method. The participants in theexperiment comprised four males and two females.
Figure 4 shows the result of the subjective experiment,
which indicates that the proposed method using the equibinaural spectral gain markedly outperforms Generalized
MMSE-STSA. Therefore, we confirmed the eﬀectiveness ofusing the equi-binaural spectral gain to improve the spatial
quality.

0

10

20

30 40 50 60 70
Preference score [%]8090Fig. 4: Result of XAB test for evaluation of spatial quality.
corresponding HRTF h( f ). The HRTFs were selected fromthe open database of the “Samurai” dummy head [21]. The
observed signals were produced by mixing two sources selected from the target signal and the other three signals withthe same power. In this observed signal, the target and interference signals were located in the same direction. In theestimation of the interference signal using SNMF, we usedartificial clean MIDI sounds of the target instrument without
an HRTF as supervision for the training process. The training sounds contained two octave notes that covered all thenotes of the target signal in the observed signal. The sampling frequency of all signals was 44.1 kHz. Spectrograms
were computed using a 92-ms-long rectangular window withan 11-ms-long overlap shift. Moreover, the number of trainedbases was 100 and the number of other bases was 50. The
forgetting factor α was 0.97, and the amplitude compression
parameter β was 1.0.
4.2 Objective Experiment
In the objective experiment, we compared six methods,i.e., SNMF (SNMF) [19], the equi-binaural spectral gain
via WF (Equi-gain WF) [9], the equi-binaural spectral
gain via the MMSE-STSA estimator (Equi-gain MMSESTSA) [10], the minimum-gain-based MMSE-STSA estimator (Gain-min MMSE-STSA) [8], the generalized MMSESTSA estimators independently applied to each ear (Generalized MMSE-STSA), and the equi-binaural spectral
gain via the generalized MMSE-STSA estimator (Proposed
method). Every method uses the same SNMF as the interference estimator. We used the signal-to-distortion ratio (SDR)
defined in [22] as the evaluation score. The SDR indicatesthe overall quality of the separated target sound, and is highin the case of high separation, low artificial distortion, and5

Subjective Experiment

Conclusion

In this paper, to address the eﬀect of statistical models forboth ears on binaural signal source separation, we applied
the generalized MMSE-STSA estimator with automatic prioradaptation to a binaural signal. The proposed method of binaural signal separation using equi-binaural spectral gain canalso improve the sound-localization properties. From the results of and subjective experiments, it was found that the proposed method outperforms conventional methods from theviewpoint of separation performance and sound-localization
preservation.
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