Geometrically Constrained Independent Vector Analysis

A Geometrically Constrained Independent Vector Analysis
Algorithm for Online Source Extraction
Affan Khan, Maja Taseska, Emanu¨el Habets
affan.khan@audiolabs-erlangen.de
September 7, 2015

Content
1. Introduction
Motivation
State of the art vs Proposed algorithm
Signal model
Independent Vector Analysis (IVA)
2. Constrained IVA (CIVA)
3. Results
4. Conclusions
© AudioLabs 2015
Slide 1
Constrained Independent Vector Analysis

Introduction
Motivation
Many applications require a handsfree capture of speech using one or more
microphones.
Signal received at the microphones is usually a mixture of desired and
undesired source signals.
Extraction of desired source from the mixture is required.
© AudioLabs 2015
Slide 2

Introduction
State of the art
Beamforming algorithms
Commonly used to extract one source signal.
Beamformers can either be data-independent or data-dependent.
Data-dependent beamformers require an accurate estimate of SOS.
Independent component analysis (ICA)
Goal is to extract all source signals.
Assumption of statistical independence.
Limited performance in reverberant and non-determined scenarios.
Researchers incorporated prior knowledge in ICA to improve performance.
© AudioLabs 2015
Slide 3

Introduction
State of the art vs Proposed algorithm
Geometric prior information can be added to ICA through soft and hard
constrained optimization [1-2].
These algorithms require prior knowledge on the number of sources.
The algorithms assume all samples are available (batch mode).
Permutation ambiguity not completely resolved.
Proposed algorithm
No prior knowledge on the number of sources required.
An online algorithm.
No permutation ambiguity due to the use of IVA instead of ICA.
[1] Parra and Alvino ”Geometric Source Separation: Merging Convolutive Source Separation With Geometric
Beamforming”, IEEE Transactions on Speech and Audio Processing, 2002, pp. 352-362.
[2] Knaak, Araki and Makino ”Geometrically Constrained Independent Component Analysis”, IEEE Transactions on Audio,
Speech and Language Processing, 2007, pp. 715-726.
© AudioLabs 2015
Slide 4

Introduction
Signal model
The mixture signal received at the m − th microphone
Ym(n, k) =
L
l=1
Am,l(k) Sl(n, k) + Vm(n, k)
We consider the problem where only one of the L sources is desired
y(n, k) = a1(k) S1(n, k) +
L
u=2
au(k) Su(n, k) + v(n, k)
Extraction of source signals from the received mixture in a standard BSS
algorithm is achieved by a demixing matrix W(k)
ˆs(n, k) = W(k) y(n, k)
© AudioLabs 2015
Slide 5

Introduction
Signal model
The demixing matrix W(k) can be written out as follows
W(k) = [w1(k) w2(k) w3(k) · · · wM (k)]H
Goal: Given the DOA of the desired source, compute a demixing matrix
W(k) while ensuring that w1(k) extracts the signal originating from the
desired direction.
© AudioLabs 2015
Slide 6

Introduction
Independent Vector Analysis (IVA)
Computes the demixing matrix through minimization of statistical
dependence between the output signals across all frequency bins.
Jiva = KL p ˆS1(n) · · · ˆSM (n) ||
M
m=1
p ˆSm(n)
Jiva = −
M
m=1
E log p ˆSm(n) −
K
k=1
log|det [W(k)] |
An online variant of IVA was proposed in [3].
[3] Kim ”Real-Time Independent Vector Analysis for Convolutive Blind Source Separation”, IEEE Transactions on Circuits
and Systems 1, 2010, pp. 1431-1438.
© AudioLabs 2015
Slide 7

Content
1. Introduction
IVA with geometric constraints
Online CIVA algorithm
3. Results
4. Conclusions
© AudioLabs 2015
Slide 8

Constrained IVA (CIVA)
Given the DOA of the desired source, we constrain the look direction of
w1(k) using RTF vector under far ﬁeld assumption
g1(k) = 1 ej(2πf/c)[r2 − r1]T
q1
· · · ej(2πf/c)[rM − r1]T
q1
T
where q1 is a unit vector pointing in the direction of the desired source.
A penalty function to restrict the Euclidean angle between w1(k) and g1(k)
Jp =
K
k=1
[cos Θ(k) − 1]2
cos Θ(k) =
Re wH
1 (k) g1(k)
||w1(k)|| ||g1(k)||
© AudioLabs 2015
Slide 9

The cost function of IVA with geometric constraints can be expressed as
Jciva = Jiva + λ Jp
The gradient of Jciva with respect to the elements of the demixing matrix
Wciva(k) = E{ϕ(k)
(n)y
H
(n, k)} − W
−H
(k)
Wiva(k)
+λ Wp(k)
where
Wp(k) = C · (cos Θ(k) − 1) g1(k) −
w1(k)
||w1(k)||2
Re w
H
1 (k)g1(k)
© AudioLabs 2015
Slide 10

An online CIVA algorithm
An online variant of the CIVA algorithm is obtained by omitting the
expectation operator in the calculation of Wiva(k).
To avoid divergence of the algorithm due to source signal ﬂuctuations, we
normalize the gradient matrix at each frame by its Frobenius norm || · ||F
Wn(k) = Wn−1(k) − η
Wn,civa(k)
|| Wn,civa(k)||F
© AudioLabs 2015
Slide 11

Content
1. Introduction
3. Results
4. Conclusions
© AudioLabs 2015
Slide 12

Results: Constrained IVA
Experimental setup
Dimensions of the simulated room: 7.5m × 5.5m × 3m.
Uniform circular array with a diameter of 2.5 cm.
Sampling rate 16 kHz, STFT frame 1024 samples (50% overlap).
Diffuse noise 30dB SDR, uncorrelated sensor noise 50dB SNR.
Segmental speech distortion index (SegSD) and segmental signal to
interference ratio (SegSIR) are used as performance measures.
© AudioLabs 2015
Slide 13

Results
Fixed number of active sources
Proposed constrained IVA algorithm converges in all scenarios.
Provides low desired speech signal distortion.
T60 = 150 ms T60 = 300 ms
segSD segSIR (dB) segSD segSIR (dB)
Unprocessed mixture 1.9 1.5
IVA (M = 3, L = 3) 0.21 14.9 0.31 4.7
CIVA (M = 3, L = 3) 0.12 11.7 0.24 5.2
Unprocessed mixture 1.9 1.5
IVA (M = 4, L = 3) 0.27 8.2 0.39 3.8
CIVA (M = 4, L = 3) 0.04 17.0 0.07 8.7
Unprocessed mixture 0.7 -0.3
IVA (M = 4, L = 5) 0.54 0.2 0.57 0.1
CIVA (M = 4, L = 5) 0.05 8.7 0.08 3.6
Table: Performance measures for ﬁxed number of sources
© AudioLabs 2015
Slide 14

Results
Time-varying number of active sources
Figure: Top: Activity pattern of speech sources over time; Bottom: Segmental SIR
improvement over time. M = 6, T60 = 150 ms.
© AudioLabs 2015
Slide 15

Conclusions
An geometrically constrained online IVA algorithm was proposed to extract
the desired speech signal.
The applicability of the algorithm to scenarios with ﬁxed and time varying
number of interferers was demonstrated.
Future work includes comparison of the algorithm against beamforming
algorithms and evaluation with recorded data.
© AudioLabs 2015
Slide 17

Conclusions
An geometrically constrained online IVA algorithm was proposed to extract
the desired speech signal.
The applicability of the algorithm to scenarios with ﬁxed and time varying
number of interferers was demonstrated.
Future work includes comparison of the algorithm against beamforming
algorithms and evaluation with recorded data.
Thank you for your attention!
© AudioLabs 2015
Slide 17

Geometrically Constrained Independent Vector Analysis

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