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Relaxation of rank-1 spatial constraint in overdetermined blind source separation

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Presented at The 2015 European Signal Processing Conference (EUSIPCO 2015, international conference)
Daichi Kitamura, Nobutaka Ono, Hiroshi Sawada, Hirokazu Kameoka, Hiroshi Saruwatari, "Relaxation of rank-1 spatial constraint in overdetermined blind source separation," Proceedings of The 2015 European Signal Processing Conference (EUSIPCO 2015), pp.1271-1275, Nice, France, September 2015 (Invited Special Session).

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Relaxation of rank-1 spatial constraint in overdetermined blind source separation

  1. 1. Daichi Kitamura Nobutaka Ono Hiroshi Sawada Hirokazu Kameoka Hiroshi Saruwatari Relaxation of Rank-1 Spatial Constraint in Overdetermined Blind Source Separation (SOKENDAI) (NII/SOKENDAI) (NTT) (The Univ. of Tokyo/NTT) (The Univ. of Tokyo) EUSIPCO 2015, 2 Sept.,14:30 - 16:10, SS30 Acoustic scene analysis using microphone array
  2. 2. Research Background • Blind source separation (BSS) – Estimation of original sources from the mixture signal – We only focus on overdetermined situations • Number of sources Number of microphones • Ex) Independent component analysis, independent vector analysis • Applications of BSS – Acoustic scene analysis, speech enhancement, music analysis, reproduction of sound field, etc. 2/21 Original sources Observation (mixture) Estimated sources Mixing system BSS Unknown
  3. 3. Problems and Motivations • For reverberant signals – ICA-based methods cannot separate sources well because Linear time-invariant mixing system is assumed – When the number of microphones is grater than the number of sources, PCA is often applied before BSS • Reverberation is also important information to analyze acoustic scenes – We should separate the sources with their own reverberations. 3/21 Original sources Observed signals Mixing Estimated sources BSS Dimension- reduced signals PCA Instantaneous mixing in time-frequency domain To remove weak (reverberant) components of all the sources
  4. 4. • Independent vector analysis (IVA) [Hiroe, 2006], [Kim, 2006] – assumes independence between source vectors – assumes linear time-invariant mixing system • The mixing system can be represented by mixing matrix in each frequency bin. – can efficiently be optimized [Ono, 2011] Conventional Methods (1/4) 4/21 … … Original sources Mixing matrices … … … Observed signals Demixing matrices Estimated sources
  5. 5. Conventional Methods (2/4) • Nonnegative matrix factorization (NMF) [Lee, 2001] – decomposes spectrogram into spectral bases – Decomposed bases should be clustered into each source. • Very difficult problem – Multichannel extension of NMF has been proposed. 5/21 Amplitude Amplitude Observed matrix (power spectrogram) Basis matrix (spectral patterns) Activation matrix (Time-varying gain) Time : Number of frequency bins : Number of time frames : Number of bases Time Frequency Frequency Basis
  6. 6. • Multichannel NMF (MNMF) [Ozerov, 2010], [Sawada, 2013] Conventional Methods (3/4) 6/21 Time-frequency-wise channel correlations Multichannel observation Multichannel vector Instantaneous covariance Source-frequency-wise spatial covariances Basis matrix Activation matrix Spatial model Source model Cluster- indicator Spectral patterns Gains
  7. 7. • MNMF with rank-1 spatial model (Rank-1 MNMF) – Spatial model can be optimized by IVA – Source model and can be optimized by simple NMF We can optimize all the variables using update rules of IVA and simple NMF Time-frequency-wise channel correlations Source-frequency-wise spatial covariances Basis matrix Activation matrix Spectral patterns Gains Conventional Methods (4/4) 7/21 [Kitamura, ICASSP 2015] = Linear mixing assumption as well as IVA Modeled by rank-1 matrices (constraint) Cluster- indicator
  8. 8. • Rank-1 spatial constraint Linear mixing assumption – Instantaneous mixture in a time-frequency domain – Mixing system can be represented by mixing matrix Rank-1 Spatial Constraint 8/21 1. Sources can be modeled as point sources 2. Reverberation time is shorter than FFT length Frequency Time Observed spectrogram Time-invariant mixing matrix Observed signal Source signal
  9. 9. • When reverberation time is longer than FFT length, – the impulse response becomes long – reverberant components leak into the next time frame Problem of Rank-1 Spatial Model 9/21 Mixing system cannot be represented by using only . The separation performance markedly degrades. Frequency Time Observed spectrogram Observed signal Source signal Leaked components
  10. 10. Summary of Conventional methods • MNMF [Ozerov, 2010], [Sawada, 2013] – Full-rank spatial model • does not use rank-1 spatial constraint – much computational costs – strong dependence on initial values • IVA [Hiroe, 2006], [Kim, 2006] & Rank-1 MNMF [Kitamura, 2015] – Rank-1 spatial constraint (linear mixing assumption) • Separation performance degrades for the reverberant signals – Faster and more stable optimization 10/21 Relax the rank-1 spatial constraint while maintaining efficient optimization To achieve good and stable separation even for the reverberant signals,
  11. 11. • Dimensionality reduction with principal component analysis (PCA) – remove reverberant components of all the sources by PCA – But the reverberant components are important! • Utilize extra observations to model direct and reverberant components simultaneously. – microphones for sources, where Proposed Approach 11/21 Original sources Observed signals Mixing Estimated sources BSS Dimension-reduced signals PCA Ex. sources, microphones ( )
  12. 12. Proposed Approach 12/21 • Utilize extra observations to model direct and reverberant components simultaneously. – microphones for sources, where Original sources Observed signals Mixing Ex. sources, microphones ( ) Estimated sources Reconstruction Separated components BSS IVA or Rank-1 MNMF
  13. 13. Proposed Approach 13/21 • Utilize extra observations to model direct and reverberant components simultaneously. – microphones for sources, where Original sources Observed signals Mixing Ex. sources, microphones ( ) Direct Reverb. Direct Reverb. Estimated sources Reconstruction Separated components BSS • We assume the independence between not only sources but also the direct and reverberant components of the same sources.
  14. 14. • Permutation problem of separated components – Order of separated components depends on initial values • We propose two methods to cluster the components – 1. Using cross-correlations for IVA – 2. Sharing basis matrices for Rank-1 MNMF Clustering of Separated Components 14/21 Separated components Which separated components belong to which source?
  15. 15. • Permutation problem of separated components – Order of separated components depends on initial values • We propose two methods to cluster the components – 1. Using cross-correlations for IVA – 2. Sharing basis matrices for Rank-1 MNMF Clustering of Separated Components 15/21 Estimated source Reconstruction Separated components Clustered components Direct component of source 1 Clustering Reverb. component of source 1 Direct component of source 2 Reverb. component of source 2
  16. 16. Clustering Using Spectrogram Correlation • Direct and reverberant components of the same source have a strong cross-correlation. • Cross-correlation of two power spectrograms – Calculate for all combination of separated components – Merge the components in a descending order of 16/21 Power spectrogram of Power spectrogram of ・・・
  17. 17. • Direct and reverberant components can be modeled by the same bases (spectral patterns) • Estimate signals with Basis-Shared Rank-1 MNMF – Only for Rank-1 MNMF • because IVA doesn’t have NMF source model – By imposing basis-shared source model, Rank-1 MNMF can automatically cluster the components. Auto-Clustering by Sharing Basis Matrix 17/21 Separated components Source model of Basis- Shared Rank-1 MNMF Shared basis matrix for source 1 Reconstruction Estimated sources Shared basis matrix for source 2 Direct component of source 1 Reverb. component of source 1 Direct component of source 2 Reverb. component of source 2
  18. 18. • Conditions – JR2 impulse response Experiments Original source Professionally-produced music signals from SiSEC database JR2 impulse response in RWCP database is used Two sources and four microphones Sampling frequency Down sampled from 44.1 kHz to 16 kHz FFT length in STFT 8192 points (128 ms, Hamming window) Shift length in STFT 2048 points (64 ms) Number of bases 15 bases for each source (30 bases for all the sources) Number of iterations 200 Number of trials 10 times with various seeds of random initialization Evaluation criterion Average SDR improvement and its deviation 18/21 Reverberation time: 470 ms 2 m Source 1 80 60 Microphone spacing: 2.83 cm Source 2
  19. 19. • Compared methods (7 methods) – PCA + 2ch IVA • Apply PCA before IVA – PCA + 2ch Rank-1 MNMF • Apply PCA before Rank-1 MNMF – 4ch IVA + Clustering • Apply IVA without PCA, and cluster the components – 4ch Basis-Shared Rank-1 MNMF • Apply Basis-Shared Rank-1 MNMF without PCA – 4ch MNMF-based BF (beam forming) • Apply maximum SNR beam forming (time-invariant filtering) using full-rank covariance estimated by 4ch MNMF – 4ch MNMF • Apply conventional MNMF (full-rank model), and apply multichannel Wiener filtering (time-variant filtering) – Ideal time-invariant filtering • The upper limit of time-invariant filtering (supervised) Experiments 19/21 Conventional methods Proposed methods Conventional methods Reference score
  20. 20. • Results (song: ultimate_nz_tour__snip_43_61) – Source 1: Guitar – Source 2: Vocals 16 14 12 10 8 6 4 2 0 SDRimprovement[dB] Experiments 20/21 Rank-1 spatial model Time-invariant filter (1/src) Full-rank model Time-invariant filter (1/src) Full-rank spatial model Time-variant filter (1/src) Upper limit of time-invariant filter (1/src) Rank-1 spatial model Time-invariant filter (2/src) : Source 1 : Source 2 PCA+ 2ch IVA PCA+ 2ch Rank1 MNMF 4ch IVA+ Clustering 4ch MNMF- based BF 4ch MNMF Ideal time- invariant filtering (supervised) 4ch Basis- Shared Rank-1 MNMF
  21. 21. • Results (song: bearlin-roads__snip_85_99) – Source 1: Acoustic guitar – Source 2: Piano 12 10 8 6 4 2 0 -2 -4 SDRimprovement[dB] Experiments 21/21 : Source 1 : Source 2 PCA+ 2ch IVA PCA+ 2ch Rank1 MNMF 4ch IVA+ Clustering 4ch MNMF4ch Basis- Shared Rank-1 MNMF Ideal time- invariant filtering (supervised) 4ch MNMF- based BF
  22. 22. Experiments 22/21 • Comparison of computational times – Conditions • CPU: Intel Core i7-4790 (3.60GHz) • MATLAB 8.3 (64-bit) • Song: ultimate_nz_tour__snip_43_61 (18s, 16kHz sampling) PCA + 2ch IVA PCA + 2ch Rank1MNMF 4ch IVA+ Clustering 4ch Basis- Shared Rank1 MNMF 4ch MNMF 23.4 s 29.4 s 60.1 s 143.9 s 3611.8 s Achieve efficient optimization compared with MNMF (The performance is comparable with MNMF) 1h!2.4m
  23. 23. Conclusion • For the case of reverberant signals – Achieve both good performance and efficient optimization • The proposed method – Can be applied when the number of microphones is grater than twice the number of sources – separately estimates direct and reverberant components utilizing extra observations – can be thought as a relaxation of rank-1 spatial constraint • Experimental results show better performance – The proposed method outperforms the upper limit of time- invariant filtering in some cases 23/21 Thank you for your attention!

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