Siemens Corporate Research Rosca et al. Generalized Sparse Mixing Model BSS ICASSP, Montreal 2004...

Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004

Siemens Corporate Research

Generalized Sparse Signal Mixing Model and Application to Noisy Blind Source Separation

Justinian Rosca

Christian Borss #

Radu Balan

Siemens Corporate Research, Princeton, USA

# Presently: University of Brawnschweig



ICA/BSS Scenario

Signal Processor

x1 x2 xD

S1, S2 ,…,SL

s1

s2

sL-1

sL

n

L sourcesD microphones

ji n s i

jij ax

NSAX

DL ,"fat" isA



Motivation

• Solve ICA problem in realistic scenarios• In the presence of noise. Is this really feasible?• When A is “fat” (degenerate)

• Successful DUET/Time-frequency masking - approach and implementation

• Can we do better if we relax the DUET assumption about number of sources “active” at any time-frequency point? [Rickard et al. 2000,2001]



Sparseness in TF

• DUET assumption: the maximum number of sources active at any time - frequency point in a mixture of signals is one

ji 0),(),( :ityorthogonaldisjoint -W tStS ji



Example Voice Signal and TF Representation

Siemens Corporate ResearchJ.Rosca et al. – Scalable BSS under Noise – DAGA, Aachen 2003



Sparseness in TF

• Sources hop from one set of frequencies to another over time, with no collisions (at most one source active at any time-freq. point)

s1

s2

s3

1

2

3



Generalized Sparseness in TF

• Sources hop from one set of frequencies to another over time, with collisions (at most N sources active at any time-freq. point)

s1

s2

s3

N=2, L=3



The Independence AssumptionAssume the TF coefficient S(k,) is modeled as a product of a Bernoulli (0/1) r.v., V, and a continuous r.v. G:

The p.d.f. of S becomes:

For L independent signals the joint source pdf becomes:

),...,,(Rest)()()1()()1(),...,,( 212

1 ,1

1

121 L

L

l

L

ljjjl

LL

ll

LL SSSqSSpqqSqSSSp

),(),(),( kGkVkS

)()1()()( SqSqpSpS

•W-Disjoint Orthogonality (DUET): q very small→ retain first two terms; at most one source is active at any time-freq. point

•Generalized W-Disj.Orth.: q very small→ retain first N+1 terms; at most N sources are active at any time-freq. point



Signal Model (1)• Assumptions:

– L sources, D sensors– Far-field– Direct-path– Noises iid, Gaussian (0,σ2)

lklk

L

lkllk

L

ll

kDktntstx

tntstx

1 ;2 ),()()(

)()()(

1

11

1

bS

1 2 … D

aS



jkS

NmkRkS

j

mjm

,0),(

1 ),,(),(

Signal Model (2)

– Mixing model:

– Source sparseness in TF

– Let those be:

NmtS,...L},{},...,j{jN(t,ω

mj

N

1 ,0),( such that 21 indices most at ), 1

L

ldl

did kNkSekX l

1

)1( ),(),(),(



Example

Let N=2, two sources active at any time-freq. point

L

k

tRtRtt

kiftRkiftR

kkiftS

Ltt

...),(),(),(),(

:are problem theof Parameters

),(),(

,0),(

},,...,1{),(),,(

21

21

21

22

11

21

2121

active is source iff ),(

},,...,1{)},{(:

llk

Lk

m

m



BSS Problem

• Given measurements {x(t)}1<=t<=T , D sensors

• Determine estimate of parameters :

• Note: L>D, degenerate BSS problem

L

N kRkRRk

,...,)),(),...,,((

),(

1

1

Mixing parameters

Mapping and Source signals



Approach: Two Steps

1. Estimate mixing parameters, e.g. using the stronger constraint of W-disjoint orthogonality

2. Estimate the source signals under the generalized W-disjoint orthogonality assumption



Solution Sketch (Ad-Hoc)• Employ principle of coherence (e.g. N=2)

– Given a pair of sources Sa and Sb active at some time-freq. point, then what we know what we should measure at all microphones pairs!

– Sa and Sb are the true ones if they result in minimum variance across all microphone pairs, i.e. coherent measurements

– Note: For N=2 and L=4 there are 6 pair of sources to be tested! (1,2),(1,3),(1,4),(2,3),(2,4),(3,4)

),(

),(

j)(i, b),(a, edhypothesizfor known

2221

1211

),(),(),(),(

),( pairs mic allfor and ),(

jib

jia

j

i

SS

baRbaRbaRbaR

XX

jitfixed

bS

i j

aS



Solution Sketch (ML-1)

• Maximize likelihood function L(,R)=p(X| ,R)

),(),(

where

}),(),(1exp{1),(

1

212

1

0 ),(2

L

ll

idd

dd

D

d k

kSekY

kYkXRL

lj

l




• max L(,R), after taking log

10,

)(ˆ

),(),(min

,1

*1*

),(

1

0

21,

DdeMwhere

XMMMR

kYkX

ljidld

k

D

dddR

Njjj

jNjj

jNjj

iDiDiD

iii

iii

eeeeeeeee

M

)1()1()1(

222

111

...

...

...1...11

21

21

21




• After substituting R:

projection orthogonal,

max

})({max

)(min

2*

2

*1**

),(

2*1*

MMMM

M

P

k

PPPP

XP

XMMMMX

XMMMMX

M



Interpretation of Solution• Criterion:

– projection of X onto the span of columns of M

• Solution (“coherent” measurements)– N-dim subspace of CD closest to X among

all L-choose-N subspaces spanned by different combinations of N columns of the matrix M

• Existence iff N≤D-1

CD

1M

2M

NLM



Experimental Results (1)

• Algorithm applied to realistic synthetic mixtures

• From anechoic, low echoic, echoic to strongly echoic

• 16kHz data, 256 sample window, 50% overlap, coherent noise, SIR (-5dB,10dB), 30 gradient steps/iteration (Step 2), 5 iterations

• Evaluation: SIRGain, SegmentalSNR, Distortion



Example Sources L=4, Mics D=2, N=2

Mixing

Sources

Estimates



Discussion: L=4 sources, D=2 mics is a case too simple?

-10123456789

10

1 2

# Sources Simultaneously Active

SIR

Gai

n ML-anechoic

AH-anechoicML-echoic

AH-echoic

• In some simulations, the N=2 assumption helps • Conjecture: approach is useful when N is a small

fraction of L



Conclusion

• Contribution: ML approach to noisy BSS problem under generalized sparseness assumptions, addressing degenerate case D<L

• Estimation problem can be addressed using sparse decomposition techniques: progress is needed



Thank you!

Real speech separation demo for those interested after session!



Outline

• Generalized sparseness assumption• Signal model and assumptions• BSS problem definition• Solution sketch: Ad-hoc and ML estimators• Geometrical interpretation of solution• Experimental results• Conclusion

Siemens Corporate Research Rosca et al. Generalized Sparse Mixing Model BSS ICASSP, Montreal 2004...

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