Blind separation of infinitely many sparse sources
1,21131Hirokazu Kameoka , Misa Sato , Takuma Ono , Nobutaka Ono , Shigeki Sagayama
1 The University of Tokyo, 2 NTT Communication Science Laboratories, 3 National Institute of Informatics

3. Generative Model Mixture of Direction-of-Arrivals (DOAs)

 Generative process of observed signal
Gaussian noise assumption:(2)... (1)

1. Introduction

 Mixture of infinitely many sparse sources Blind source separation (BSS)Technique for separating sources only from microphone inputs
Observed 1 Estimate 1Source 1
 Potential applications include
hands-free teleconference
system and automatic meetingtranscription system

Incorporating appropriate constraint into
may help solve
permutation alignment problem through parameter inference.

(1)

Finite case:Estimate 2

It is often difficult to pre-specify the exact number of all possible
sources present in real environments.
 In meeting situations, # of speakers can change during the meetinge.g.)
or loud, unexpected noise such as door slamming can occur.

(3)
Infinite case:... (2)

When # of sources is unknown, we shall always consider
underdetermined case (# of microphones # of sources) One successful approach for underdetermined BSS involves utilizing“*sparseness” of speech [1]* T-F components of speech are near zero across most of T-F bins

To exploit the sparseness of speech, mixing model must berepresented in T-F domain
 Permutation alignment problem needs to be solved

TimeTimeTime

1…
0(Narrowband approximation of
convolutive mixture in time domain)

T-F component observed at
microphone mT-F component of source kSteering vector of source kT-F component of noise
K (# of sources)

M (# of microphones)

 Assuming sparseness of sources in T-F domain
Frequency

Assumption: only one sourceis active in each T-F bin

TimeSuperposition of 3 speech spectrograms

5. Experiment Variational inference Experimental condition

Goal is to compute From (1)-(4) :Index of source active at T-F bin

i=8i=2i=9i=1i = 10... (4)Process by which direction
index is generated(indicates how likely
direction i is to be chosen)

(2)

(3)

(4)

Instead, we consider finding

Tractable choices ofallows us toobtain principled approximate solutions

Kullback-Leibler divergence
betweenand Results
Average Signal-to-Distortion Ratios(SDRs) [6] obtained with Sawada’s andproposed methods with various K andK∗ settings (See upper-right) Proposed method improves withincreasing K* (implying that there isno need to pre-specify # of sources)

Mean-field approximation
 Truncated approximation
(Thus, we can ignorefor

Truncating does not mean that true posterior itself is truncated. As truncationlevel
becomes larger, approximation to true posterior become more accurate.

 Coordinate ascent

Speech data: excerpted from ATR Japanese speech database
 Observed signals: mixed three source signals (two female & onemale speakers) into two mixtures using measured impulse
response (reverberation time 0ms)
 T-F components were obtained by short-time Fourier transform(Frame length 64 ms, Hamming window, overlap length 16ms)
 Three sources were spaced 30 degrees apart Comparison: Sawada’s method [5] (# of sources must be pre-specified)
K*: Truncation level (Proposed)
K : Assumed # of sources (Conventional)
 # of iterations = 100, # of DOA candidates I = 180
 Separated signals =Interchannel phase differences (Correct)
To obtain exact posterior, we must derive
marginal distribution, but this involves
intractable integrals.

 Instantaneous mixture ini=3

Probability distributions

4. Posterior Inference(1)

2. Mixing Modelfrequency domain:a.k.a. constructive definitionof Dirichlet process [4]Produces exponentially decaying weights

Time

Process by which complex array response
is generated given direction

Stick-breaking process [3]Process by which
active source index
is generated(indicates how
likely source index
k is to be chosen)We propose BSS approach thatsimultaneously performs(1) estimation of # of sources,(2) source separation based onsparseness of speech, and(3) permutation alignment.

... (3)Discrete set of 10 Directions(4)

Complex array response associated withdirection i (constant value)...

 Motivation and objective

Observed 2

Source 2

Direction index of source kAssumption: Each source
propagates as a plane wave
and emanates from one of I
directions.

)Interchannel phase differenceIf index k indicates identical source across ,must have certain
structure based on the property of acoustic wave propagation.

Interchannel phase difference

Abstract: This paper presents a novel BSS approach thatsimultaneously performs an estimation of the number of sources,
source separation based on the sparseness of speech, andpermutation alignment, based on Bayesian nonparametric approach.

Frequency (kHz)Frequency (kHz)References[1] O. Yılmaz, S. Rickard, “Blind separation of speech mixtures via time-frequency masking,” IEEE Trans. Signal Process., vol. 52, no. 7, pp. 1830–1847, 2004.
[2] Y. Izumi, N. Ono, S. Sagayama, “Sparseness-based 2ch BSS using the EM algorithm in reverberant environment,” in Proc. WASPAA, pp. 147–150, 2007.
[3] J. Sethuraman, “A constructive definition of Dirichlet priors,” Statistica Sinica, vol. 4, pp. 639–650, 1994.
[4] T. S. Ferguson, “A Bayesian analysis of some nonparametric problems,” Annals of Statistics, vol. 1, no. 2, pp. 209–230, 1973.
[5] H. Sawada, S. Araki, S. Makino, “Underdetermined convolutive blind source separation via frequency bin-wise clustering and permutation alignment,”
IEEE Trans. ASLP, vol. 19, no. 3, pp. 516–527, 2010.
[6] E. Vincent, R. Gribonval, C. F´evotte, “Performance measurement in blind audio source separation,” IEEE Trans. ASLP, pp. 1462–1469, 2006.