Source localization using a spatial kernel based covariance model andsupervised complex nonnegative matrix factorization
A.J. Muñoz-Montoro(1) , V. Montiel-Zafra(1) , J.J. Carabias-Orti(1) , J. Torre-Cruz(1) , F.J. Canadas-Quesada(1) , P. Vera-Candeas(1)(1) University of Jaen, Spain

Abstract
This paper presents an algorithm for source localization using a beamforming-inspired spatial covariance model(SCM) and complex non-negative matrix factorization (CNMF). The spatial properties are modeled as theweighted sum of spatial kernels which encode the phase an the amplitude differences between microphones
for every possible source location in a grid. The actual localization for each individual source in the multichannel mixture is estimated using complex-valued non-negative matrix factorization (CNMF) where each source
spectrogram is modeled using a dictionary of spectral patterns learned a priori from training material.
Localization performance of the proposed system is evaluated using a multi-channel dataset with configurations(number of simultaneous sources, reverberation time, microphones spacing, source types and spatial locationsof the sources). Finally, a comparison to other state-of-the-art localization methods is performed, showing
competitive localization performance.
Keywords: Localization, DOA estimation, spectral patterns, SCM, CNMF

1

INTRODUCTION

The task of sound source localization (SSL) consists of estimating the spatial positions of the sources in the
acoustic scene and has been applied to multiple applications like audio surveillance, teleconferencing, speechenhancement for hearing-aids, or camera pointing systems [14].
Algorithms for SSL can be classified as indirect and direct approaches [11]. Indirect approaches usually follow
a two-step procedure: 1) estimate the Time Difference Of Arrival (TDOA) [6] between microphone pairs and, 2)
estimate the source position based on the geometry of the array and the estimated delays. Alternatively, direct
approaches perform TDOA estimation and source localization in a single step by analyzing a set of candidate
locations and selecting the optimal spatial position as an estimate of the source location [7].
In fact, SSL performance is closely related to the quality of the TDOAs estimation. According to [3], there arethree general approaches for TDOA estimation: 1) locally for each time-frequency bin and then select the higher
peaks in the resulting histogram [8], 2) using an iterative estimation of the time-frequency bins associated toeach source and the corresponding TDOAs by clustering method using maximum likelihood [18], 3) estimating
an angular spectrum and selecting the high peaks in this function [15]. This angular spectrum can be constructedusing different measures, such as correlation between channels [10] or subspace decomposition of the spatial
covariance matrix (SCM) [9].
Sound field decomposition aims to represent a sound field as a linear combination of fundamental solutionsof the wave equation (or Helmholtz equation) from the results of the pressure measurements. This technique
has been applied for several acoustic signal processing applications, such as sound field analysis, reconstruction
and visualization. Using this technique, the entire sound field can be estimated from the signals received bymultiple microphones which in fact involves source localization or direction of arrival (DOA) estimation because
the power distribution of the pressure field indicates the locations or directions of the sound sources.
Several source localization and DOA estimation methods [12, 2, 17] also assume that the soundfield can be
represented as a spatially sparse distribution of sound sources over an overcomplete linear equation of theobservations. However, in those methods no assumptions are imposed on the structure of the source signals
in the time-frequency domain. Alternatively, NMF-based signal decomposition allows further constraints on the3321

source signal model. For instance, a recent approach in [13] combines supervised CNMF and sparse sound fielddecomposition, obtaining superior results for DOA estimation than other methods using sparse representations.
In this work we propose a supervised SSL estimation method using a beamforming-inspired SCM model and
CNMF. In particular, our method takes the frequency-spectrum structures (spectral patterns) of the source signals into consideration. By exploiting spectral structures trained in advance, it is possible to improve the decomposition accuracy even when the source signals are highly correlated and the sources are in a highly noisy
environment. Moreover, this prior information allows to relate each sources and spatial locations without any
post-processing stage. The mixing filter for the multichannel input signal is modeled as a weighted combinationof the spatial kernels scanning all possible locations in a grid. For the task of DOA estimation, this spatial kernels can be configured to scan all possible directions of arrivals across the 360 degrees [16]. In this work, thespatial kernels are decomposed into two direction dependent SCMs to represent both time and level differences
between microphones caused by a single spatial location and its analytic TDOA for a given array geometry [4].
The level differences are represented using a panning-inspired frequency-independent covariance matrix. On thecontrary, time delays are modeled using a frequency-dependent phase difference covariance matrix. The source
localization is estimated from the frequency-independent directional weights (here denoted as spatial weights)
which represent the spatial distribution of the sources across the spatial grid. As a result, the effect of thespatial aliasing is mitigated since the model accounts for phase difference evidence across frequency by single
frequency-independent time delays of individual spatial kernels. To estimate the model parameter, namely theactivations of the learned spectral patterns for each source, the relative amplitude between microphones and thespatial weights, a CNMF algorithm has been developed using the Itakura Saito optimization function [19].
The estimated spatial location is obtained from the highest peak of the spatial weights associated to each source.
Evaluation is performed using the two datasets with different setups in [3] and [5]. Moreover, the proposed
method has been compared with other localization methods in the literature in terms of SSL.
The rest of the paper is organized as follows. Section 2 presents the multichannel signal mixing model, section
3 introduces the proposed spatial kernel based SCM localization model. The proposed algorithm for source
localization is explained in section 4. Evaluation of the proposed method and comparison with other methods
from the literature is performed in section 5. Finally, conclusions are presented in section 6.

2

MULTICHANNEL SIGNAL MIXING MODEL USING SPATIAL COVARIANCE MATRICESIn this work, we use the spatial covariance matrix (SCM) signal representation used in [4, 16, 19]. Rather
than absolute phase values, a SCM represents the phase and the amplitude difference between every pair of
microphones in the multichannel mixture. Firstly, the magnitude square-rooted matrix x̃ f t for each frequency
bin f and time frame t of the captured signal at each sensor x̃ f t = [x̃ f t1 , ..., x̃ f tM ]T is computed asx̃ f t = [|x̃ f t1 |1/2 sgn(x̃ f t1 ), ..., |x̃ f tM |1/2 sgn(x̃ f tM )]T ,(1)

where sgn(z) = z/|z| is the signum function for complex numbers. Then, the SCM X f t for a single timefrequency point ( f ,t) is defined from the multichannel captured signal x̃ f t as the outer product|x̃ f t1 |..
H
X f t = x̂ f t x̂ f t = .
x̃ f tM x̃∗f t1


···
..
.
···


x̃ f t1 x̃∗f tM

..
,
.
|x̃ f tM |(2)

where H stands for Hermitian transpose. Note that the main diagonal of X f t encodes the magnitude spectrum ofeach channel and the spatial properties of the mixture are represented by its off-diagonal values, which encode
the phase difference between each microphone pair.

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The source mixing model can be defined in terms of the SCM representation asS

X f t ≈ X̂ f t = ∑ H f s |ŷ f ts |

(3)

s=1

where |ŷ f ts | denotes the magnitude spectrogram for each source s ∈ [1, ..., S] and H f s ∈ CM×M is the SCM
representation of the spatial frequency response h̃ f s . As explained in [16], the SCM model in Eq. 3 can beapproximated to be purely additive in anechoic conditions assuming that the sources are uncorrelated and sparse(i.e. only a single source is active at each time frequency ( f ,t) point). However, this supposition does notalways hold in reverberant environments as depicted in [5].

3PROPOSED SCM MODEL FOR SOURCE LOCALIZATIONIn this paper, we present an algorithm for source localization based on SCM and CNMF. In particular, we
propose to incorporate within the signal model a dictionary of spectral patterns for all the sources in the mixture.
Unlike the method proposed in [5], where the localization is carrying out using the magnitude spectrum for eachsource, here a previous NMF stage is performed to learn a set of basis for each source and, then, the estimationof the spectrogram parameters in the STFT domain is computed using CNMF.
The proposed signal model for SCM observation is presented in Eq. 4 as
Wfo
S

O z

}| {
X f t ≈ X̂ f t = ∑ ∑ P f o ◦ A∗o z∗so
s=1 o=1|!

K∑ B f ks g∗kts

(4)

k=1

{z

Hfs

}|

{z

|ŷ f ts |}where the superscript ∗ refers to a free parameter during the factorization and ◦ stands for the Hadamard
product. The SCM mixing filter H f s is computed as a linear combination of spatial kernels W f o ∈ CM×M
multiplied by the spatial weights matrix zso ∈ RS×O
which relates sources s with spatial elements of the grid o.
+
Moreover, we propose to decompose the spatial kernels W f o into two covariance matrices: the phase differences
covariance matrix (PDCM) P f o and the level differences covariance matrix (LDCM) Ao .
As in [4], the phase difference between microphones is kept fixed during the factorization process while therelative amplitudes are estimated using the LDCM. In this manner, the parameter P f o is obtained a priori forevery spatial position as [P f o ] pm = e jθ pm ( f ,o) , where the function θ pm ( f , o) = 2π fi τ pm (xo ) encodes the phasedifference for each pair of sensors (p, m) at each bin f and spatial position o.
Note that a fixed number of look position o = 1, ..., O are used to cover the grid space. For a pair of microphones (p, m), each spatial position xo can be turned into a TDOA (in seconds) using the following expression:
τ pm (xo ) =||xo − p||2 − ||xo − m||2c(5)

where || · ||2 denotes the L2-norm, xo , p and m are the source spatial position o and the microphone m and p
locations using the Cartesian coordinate system, respectively, and c is the speed of sound.
In this work, the magnitude time-frequency spectrogram |ŷ f ts | for each source s is computed as a linear combination of basis functions B f ks ∈ RF×K×Sand their corresponding time-varying gains gkts . In particular, the+basis functions for each source s consist of a set of spectral patterns learned in advance using a standard NMF.
During the factorization process, these basis function are fixed.

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4

CNMF ALGORITHM FOR SOURCE LOCALIZATIONIn this work, the parameters of the proposed SCM model in Eq. 4 are estimated using CNMF. We have used asimilar approach to [19] to obtain the multiplicative updates via auxiliary functions for the case of the Itakura
Saito (IS) divergence. As demonstrated in [19], MU updates provide faster convergence than the EM algorithms.
4.1 Formulation for Itakura Saito Divergence
The Itakura Saito divergence between the observed X f t and the estimated X̂ f t SCM observations is expressedas:−1
DIS (X f t , X̂ f t ) = tr(X f t X̂−1
(6)f t ) − log(det(X f t X̂ f t )) − M
In pursuit of brevity, we write directly the multiplicative update rules of the IS divergence for each free parameter gkts and zso in Eq. 6 as:v
u−1u ∑ f ,t,k B f ks gkts tr(X̂−1
f t X f t X̂ f t W f o )zso ← zso t∑ f ,t,k B f ks gkts tr(X̂−1
f t W f o)v
u−1u ∑ f ,o zso B f ks tr(X̂−1
f t X f t X̂ f t W f o )gkts ← gkts t∑ f ,o zso B f ks tr(X̂−1
f t W f o)

(7)

Finally, update rules for the level matrix Ao are obtained as in [19] by solving an algebraic Ricatti equation as
Ao CAo = D, where C and D are defined as∗
C = ∑ zso B f ks ∑ gkts X̂−1ft ◦ Pfo
s,k

f ,t

−1
0
D = ∑(W0f o (∑ zso B f ks ∑ gkts X̂−1
f t X f t X̂ f t )W f o ) ◦ P f o
f

s,k

(8)

t

Note that W0f o is the target matrix before the update. The solution of the Riccati equation is obtaining in three
0 −C
step. Firstly, a 2M × 2M matrix E =is defined. Then, the 2M eigenvectors from E is computed as−D 0
e1 , ..., e2M and, finally, the eigenvectors is sorted according to the associated eigenvalues in ascending order andremove the vectors corresponding to the smallest eigenvalues discarding the negative eigenvalues. As a result,
M sorted eigenvectors e01 , ..., e0M are achieved.
Then, the new Ao is obtained as Ao ← IJ−1 , where the M × M matrices I and J are defined from the sorted
eigenvectors e0 as J = [e01,1:M , ..., e0M,1:M ] and I = [e01,M+1:2M , ..., e0M,M+1:2M ]. Finally, to compensate for computerarithmetical error, we ensure Ao is Hermitian by Ao ← 21 (Ao + AH
o ).

5

EVALUATION

5.1 Experimental Setup
The proposed method is evaluated using two datasets for two-channels and four-channel mixture signals. For
the task of DOA estimation, the amount of look directions O conforms an angular grid that covers 180 degrees
for the two-channel and 360 degrees for the four-channel dataset, respectively. To alleviate the computational
complexity, all the directions have been scanned in the zero elevation plane. In addition, in the four-channel
dataset and for the task of source localization, the position of the source is estimated as a specific position intoa spatial grid in x, y and z axes, where the grid resolution is set as 50 cm for the three axes.
First, a two-channel dataset is composed of 2964 mixtures of 11 s duration corresponding to the male andfemale speeches belonging to the database used in [3]. The sampling frequency of the mixture signals is 16
kHz. It covers an extensive number of configurations (2 to 6 sources), 4 microphone spacings (5 cm, 15 cm,
30 cm and 1 m), 6 reverberation times RT60 (50 ms, 100 ms, 150 ms, 250 ms, 500 ms and 750 ms), 2 source
types (male and female speeches) and multiple angular positions of the sources. The mixing filters are generated
according to a room of dimensions 4.45 m × 3.55 m × 2.5 m.

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The four-channel dataset is composed of 96 mixtures of 30 s duration that corresponds to male and female
speeches from the original database used in [5]. The sampling frequency of the signals is 16 kHz. This dataset
comprises several number of configurations (2 to 3 sources), 2 microphone spacings (5 cm and 54 cm) for the
angular grid and 3 microphone positions (located at the ends of the room, and located at the center of the roomwith spacings of 5 cm and 54 cm) and 2 source types (male and female speeches). The average reverberation
time RT60 is 350 ms. The room dimensions are 7.95 m × 4.90 m × 3.25 m.
The performance of the proposed algorithm has been compared with five state-of-the-art methods: the generalized cross-correlation with phase transform (GCC-PHAT) [10], the multiple signal classification (MUSIC) [20],the minimum variance distortionless response (MVDR) beamformer [1] and the diffuse noise model (DNM) [3].
Evaluation of the proposed method is performed applying the criteria and evaluation code used in [3]. An
estimated DOA is considered as correct if its difference with the true DOA is less than 5 degrees for the
angular grid whereas for the spatial grid it is correct if the position is correctly estimated for the three axes.
The following metrics are computed: recall R, precision P and F-measure F. These metrics are defined by
R = ISJ , P = IJJ and F = 2 R×P
R+P , where J is the number of estimated DOAs, IJ is the number of correct DOAs
and finally S is the number of sources. For a fair comparison, the number of sources is assumed to be known(J = S). Then, same values for R, P and F are obtained so the latter metric will be evaluated.
5.2 Results for DOA estimation
0.85

Global results for the 2-channel dataset
1

Average F-measure vs N

Average F-measure vs mic spacing Average F-measure vs RT60
0.85

0.85

0.8

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0.75

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F-measure

0.6

F-measureF-measureF-measure

0.7

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Proposed

GCC-PHAT

MUSIC

MVDR

DNM

(a) Average F-measure of the proposed algorithm
and the state-of-the art methods for the 2-channel
dataset.

0.5

2

3

4

5

Number of sources N

6

0.5

0

0.5

Microphone spacing (m)

1

0.5

0

200

400

600

Reverberation time RT60 (ms)(b) Average F-measure of the proposed algorithm as a function of thenumber of sources (left), microphone spacing (center) and reverberation
time RT60 (right) for the 2-channel dataset.

Figure 1. Average results for the 2-channel dataset.
First, the proposed method applying the angular grid is analyzed. Figure 1a shows a comparison with the stateof-the-art methods enumerated in Section 5.1 for the 2-channel dataset. It can be observed that the proposed
method and GCC-PHAT, MVDR and DNM methods obtain similar results. For this dataset, DNM method
slightly over-performs the proposed method (F = 0.76 and F = 0.7, respectively).
The F-measure as a function of the number of sources, microphone spacing and reverberation time RT60 for
the proposed method is presented in Figure 1b. The proposed algorithm obtains higher localization performance
when the number of sources decreases. According to the microphone spacing, optimal values are obtained for
15 and 30 cm. Since the proposed algorithm models both the inter-microphone amplitude and the phase difference, when dealing with short spacing arrays (5 cm) the inter-microphone amplitudes are negligible and the
localization is performed using the phase information. In the case of large spacing arrays (1 m), the localization is carried out using the amplitude information, because the phase information is ambiguous due to spatial
aliasing. Finally, regarding the reverberation time, the localization performance is lower as this time increases.
Figure 2a shows the average F-measure of the proposed and the state-of-the-art methods for the 4-channel

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Results for the 4-channel dataset (Number of sources and Microphone spacing (m))
1

Global results for the 4-channel dataset
1
0.9

0.8

0.8

F-measureF-measure

0.7
0.6
0.5
0.4

Proposed
GCC-PHAT
MUSIC
MVDR
DNM

0.6

0.4

0.3
0.2

0.2

0.1
0

Proposed

GCC-PHAT

MUSIC

MVDR

DNM

0

(a) Average F-measure of the proposed algorithm
and the state-of-the art methods for the 4-channel
dataset.

2src-5cm

2src-54cm

3src-5cm

3src-54cm(b) Average F-measure of the proposed algorithm and the state-of-the art
methods for the 4-channel dataset according to the number of sources andthe microphone spacing.

Figure 2. Average results for the 4-channel dataset.
dataset. The proposed method obtains the best results among the compared methods (F = 0.87). Regarding theother methods, best results are obtained by the DNM method (F = 0.79).
A comparison of the proposed method with the state-of-the-art methods according to the number of sources andmicrophone spacing is presented in Figure 2b. The best localization performance is obtained by the proposed
method for the second and fourth setups, where the microphone spacing is 54 cm. For the first and third setups,
the proposed method obtains the second best performance.
5.3 Results for spatial position estimation in a grid
Here, the proposed method is evaluated for sources located into specific positions into a spatial grid in x, y andz axes, where the grid resolution is set as 50 cm for the three axes. Then, a comparison of the proposed method
according to the microphones position is performed. Locating the microphones in the center of the room, theproposed method is not able to correctly estimate the specific position of the source (F = 0) for the case of
short arrays (5 cm). Dealing with large spacing arrays (1 m), the localization performance improves, obtaining
an average of F = 0.45 for the cases of 2 and 3 sources. However, when the microphones are placed on thewalls of the room, the results outperform, obtaining an average of F = 0.9 for the case of 2 sources and F = 0.4
for the case of 3 sources. This performance was expected, because, when the array is placed in the center of theroom, the distance between the sources and the microphones is large enough to consider a far-field problem,
which causes the phase differences between positions with the same DOA to be indistinguishable. However,when the microphones are placed on the walls of the room, a near-field problem is carried out, obtaining better
results due to major phase differences for all grid positions.
Figure 3 shows an example of the spatial localization for two concurrent sources. The sources are located into
the spatial position in the grid o = 283, corresponding to x = 1 m, y = 3 m, z = 1 m and at 0◦ azimuth; ando = 1026, corresponding to x = 4 m, y = 5 m, z = 1 m and at 90◦ azimuth. For the case of DOA estimation, theproposed method allows to identify the actual DOA for each source. However, it can only identify the correct
position in the grid when the microphones are placed on the walls of the room.

6

CONCLUSION

In this work, we proposed a multi-source localization method using a beamforming-inspired SCM model and
CNMF with the IS optimization function. The proposed method uses a dictionary of spectral patterns learnedin advance for each source, what improves the estimation of the mixing parameters and allows to relate each

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0.8

0.08

Source 1
Source 2

Source 2

0.07

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Norm. Salience

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Source 1

Source 1
Source 2

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0
-90

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Azimuth (degrees)(a) Example of DOA estimation for two
concurrent sources.

0
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1051

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1

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Spatial position in the grid

Spatial position in the grid(b) Example of spatial position estimation in a grid for two concurrent sources
with the microphone placed in the centerof the room.

(c) Example of spatial position estimation in a grid for two concurrent sources
with the microphone placed on the wallsof the room.

Figure 3. Example of the spatial weights matrix zso estimation for two concurrent sources from the cases: (a)DOA estimation, (b) spatial position estimation in a grid with the microphone placed in the center of the room,and (c) spatial position estimation in a grid with the microphone placed on the walls of the room. The lines inblue and red represents the estimated localization for sources 1 and 2.
sources and spatial positions without any post-processing stage. The mixing model is obtained as a weighted
combination of spatial kernels that model the amplitude and the phase differences for every possible source
location in a grid.
Localization performance of the proposed system is evaluated using two multichannel datasets with several
setups (number of simultaneous sources, reverberation time, microphones spacing, source types and spatial locations of the sources). Finally, a comparison to other state-of-the-art localization methods is performed, showing
competitive localization performance.

ACKNOWLEDGEMENTS
This work is supported by Pre-doctoral Fellowship Program from the “Ministerio de Ciencia, Innovación y Universidades” of Spain under the reference BES-2016-078512 and by the University of Jaén under the program“Acción 1. Apoyo a las estructuras de investigación de la Universidad de Jaén para incrementar su competitividad atendiendo a sus singularidades”.

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