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

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 ICA/BSS Scenario Signal Processor x1x1 x2x2 xDxD S 1, S 2,…,S L s1s1 s2s2 s L-1 sLsL n L sources D microphones

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

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Sparseness in TF DUET assumption: the maximum number of sources active at any time - frequency point in a mixture of signals is one

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Example Voice Signal and TF Representation Siemens Corporate ResearchJ.Rosca et al. – Scalable BSS under Noise – DAGA, Aachen 2003

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 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) s1s1 s2s2 s3s3

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 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) s1s1 s2s2 s3s3 N=2, L=3

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 The Independence Assumption Assume 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: 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

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Signal Model (1) Assumptions: –L sources, D sensors –Far-field –Direct-path –Noises iid, Gaussian (0,σ 2 ) 1 2 … D

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Signal Model (2) –Mixing model: –Source sparseness in TF –Let those be:

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Example Let N=2, two sources active at any time-freq. point

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 BSS Problem Given measurements {x(t)} 1<=t<=T, D sensors Determine estimate of parameters  : Note: L>D, degenerate BSS problem Mixing parameters Mapping and Source signals

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

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Solution Sketch (Ad-Hoc) Employ principle of coherence (e.g. N=2) –Given a pair of sources S a and S b active at some time-freq. point, then what we know what we should measure at all microphones pairs! –S a and S b 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) i j

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Solution Sketch (ML-1) Maximize likelihood function L( ,R)=p(X| ,R)

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Solution Sketch (ML-2) max L( ,R), after taking log

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Solution Sketch (ML-3) After substituting R:

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Interpretation of Solution Criterion: –projection of X onto the span of columns of M Solution  (“coherent” measurements) –N-dim subspace of C D 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 CDCD

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

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Example Sources L=4, Mics D=2, N=2 Mixing Sources Estimates

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Discussion: L=4 sources, D=2 mics is a case too simple? In some simulations, the N=2 assumption helps Conjecture: approach is useful when N is a small fraction of L

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

Siemens Corporate Research Rosca et al. – Generalized Sparse Mixing Model & BSS – ICASSP, Montreal 2004 Thank you! Real speech separation demo for those interested after session!

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