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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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]
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Example Voice Signal and TF Representation
Siemens Corporate ResearchJ.Rosca et al. – Scalable BSS under Noise – DAGA, Aachen 2003
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
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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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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( ),(),(),(
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Example
Let N=2, two sources active at any time-freq. point
L
k
tRtR
tt
kiftR
kiftR
kkif
tS
Ltt
...
),(),(
),(),(
:are problem theof Parameters
),(
),(
,0
),(
},,...,1{),(),,(
21
21
21
22
11
21
2121
active is source iff
),(
},,...,1{)},{(:
l
lk
Lk
m
m
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
BSS Problem
• Given measurements {x(t)}1<=t<=T , D sensors
• Determine estimate of parameters :
• Note: L>D, degenerate BSS problem
L
N kRkRR
k
,...,
)),(),...,,((
),(
1
1
Mixing parameters
Mapping and Source signals
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
S
S
baRbaR
baRbaR
X
X
jitfixed
bS
i j
aS
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Solution Sketch (ML-1)
• Maximize likelihood function L(,R)=p(X| ,R)
),(),(
where
}),(),(1
exp{1
),(
1
2
12
1
0 ),(2
L
ll
id
d
dd
D
d k
kSekY
kYkXRL
lj
l
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Solution Sketch (ML-2)
• max L(,R), after taking log
10,
)(ˆ
),(),(min
,1
*1*
),(
1
0
2
1,
DdeMwhere
XMMMR
kYkX
ljid
ld
k
D
dddR
Njjj
jNjj
jNjj
iDiDiD
iii
iii
eee
eee
eeeM
)1()1()1(
222
111
...
...
...
1...11
21
21
21
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Solution Sketch (ML-3)
• After substituting R:
projection orthogonal,
max
})({max
)(min
2*
2
*1**
),(
2*1*
MMMM
M
P
k
PPPP
XP
XMMMMX
XMMMMX
M
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
N
LM
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Example Sources L=4, Mics D=2, N=2
Mixing
Sources
Estimates
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Discussion: L=4 sources, D=2 mics is a case too simple?
-1
0
1
2
3
45
6
7
8
9
10
1 2
# Sources Simultaneously Active
SIR
Gai
n
ML-anechoic
AH-anechoic
ML-echoic
AH-echoic
• In some simulations, the N=2 assumption helps • Conjecture: approach is useful when N is a small
fraction of L
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
Thank you!
Real speech separation demo for those interested after session!
Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004
Siemens Corporate Research
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
