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Subband-based Independent Component Analysis

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Subband-based Independent Component Analysis Y. Qi, P.S. Krishnaprasad, and S.A. Shamma ECE Department University of Maryland, College Park – PowerPoint PPT presentation

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Title: Subband-based Independent Component Analysis


1
Subband-based Independent Component Analysis
  • Y. Qi, P.S. Krishnaprasad, and S.A. Shamma
  • ECE Department
  • University of Maryland, College Park

2
Subband-based ICA
  • Classical ICA and Applications
  • Subband-based ICA
  • Experimental Results
  • Conclusions and Future Directions

3
Classical ICA Applications
  • How to make an appropriate representation for
    multivariate data? Based on a linear model,
    Independent Component Analysis offers a method
    to represent the data as independent components
    using higher order statistics.
  • Problems addressed by ICA blind source
    separation (BSS), blind deconvolution, and
    feature extraction.
  • Applications speech enhancement and
    recognition, telecommunication, biomedical
    signal analysis, image denoising and
    recognition, and data mining.

4
Classical ICA Model (1)
  • Mixture Model
  • x As w,
  • Where s is the source signal vector, x is the
    observation signal vector, A is the mixing
    matrix, and w is noise vector.
  • Assumption s s1, s2, , snT comes from n
    independent sources.

5
Classical ICA Model (2)
  • Separation Model
  • y Wx,
  • where y y1, y2, , ynT is the estimate
    source signal vector and W is the unmixing matrix
    s.t. Y Wx WAu Du where D is a permutation
    matrix.

6
Criterion for Statistic Independence
  • Kullback-Leibler Divergence D(f(Y)f(Yi) )
    between pdf f(y) of m1 vector Y and the product
    of its marginal pdf f(Yi) of Yi.
  • Minimizing D(f(Y)f(Yi) ) results in Statistic
    Independence

7
Classical ICA Learning Rules
  • Optimization by Gradient Method
  • Natural Gradient by Amari
  • Estimation of pdf
  • Gram-Charlies Series
  • The Learning Rule
  • W(n1) W(n) g( I - q(Y(n))Y))W(n)
  • Where q(.) is a nonlinear function, f.g.,
  • q(y) 2tanh(y).

8
Motivation for Subband-based ICA
  • Shortcoming of Classical ICA for BBS
  • Not robust in the presence of noise or when
    performed online.
  • Inspiration of Suband-based ICA
  • The psychoacoustic discovery on auditory
    perception
  • Wavelets theory and T-F analysis

9
De-noising
Filtering
Subband-Based ICA
X1
H1
ICA1
X2
X1
Grouping And Competitive Learning
A
H2
ICA2
X2
S1
X1
S2
. . . .
. . . .
X2
X1
HN
ICAN
X2
Early Auditory Models
S1
S2
Hair cell
Cochlea
Lateral Inhibition
10
Subband-based ICA Alogrithm
  1. The observation signal, x, is decomposed into
    subband signals using adaptive basis selection
    algorithm in Wavelet or DCT packet.
  2. The classical ICA learning rule is applied to
    separate signals in those bands which include the
    strongest signal power.
  3. Noise is removed using Donohos soft threshold
    method in subband signals.
  4. Competitive learning is applied to cluster the
    unmixing matrices obtained from different
    subbands. The unmixing matrix W is estimated from
    the cluster peaks.
  5. Finally, y is computed as y Wx.

11
Three Advantages of Subband-based ICA
  • The virtually increased signal-to-noise rate on
    those frequency bands.
  • The fact that subband signals, i.e., wavelet
    coefficients, are more peaky and heavy-tailed
    distributed than the original signals.
  • And the adaptation to the properties of the
    signal and noise by the incorporation of best
    basis selection algorithm.

12
The Sound of the Original Mixed Music Signals
  • Music Signal 1
  • Music Signal 2
  • Mixed Signal 1
  • Mixed Signal 2

Example 1 Separation of Mixed Song Signals in
Online Mode
13
The Sound of the Separated Music Signals by
Applying two ICA Algorithms Directly on the
Mixtures
  • Recovered Signal 1 by the Extended Infomax
    algorithm
  • Recovered Signal 2 by the Extended Infomax
    algorithm
  • Recovered Signal 1 by the Nonholonomic ICA
    algorithm
  • Recovered Signal 2 by the Nonholonomic ICA
    algorithm

Example 1 Separation of Mixed Song Signals in
Online Mode
14
The Sound of the Separated Music Signals by
Applying The Subband-Based ICA
  • Recovered signal 1 by the Subband-based ICA
  • Recovered signal 2 by the Subband-based ICA

Example 1 Separation of Mixed Song Signals in
Online Mode
15
Performance Curve Comparison for Online
Separation (1)
16
Time Comparison for Online Separation (1)
Approaches Separation Time (Sec.)
Modified Extended Infomax 1582.62
Nonholonomic ICA 563.24
Subband-based ICA 101.78
Example 1 Separation of Mixed Song Signals in
Online Mode (Run on Sun Ultra10 with 500M memory)
17
Performance Curve Comparison for Online
Separation (2)
18
Time Comparison for Online Separation (2)
Approaches Separation Time (Sec.)
Modified Extended Infomax 61.72
Nonholonomic ICA 86.68
Subband-based ICA 18.05
Example 2 Separation of Mixed Violin and Pop
Music Signals in Online Mode (Run on Sun Ultra10
with 500M memory)
19
Example 3 Separation of Noisy Speech Mixture in
Batch Mode
20
The Sound of the Original Speech Sentences
  • The First Sentence
  • The Second Sentence
  • The Third Sentence
  • The Fourth Sentence

Example 3 Separation of Noisy Speech Mixture in
Batch Mode
21
Example 3 Separation of Noisy Speech Mixture in
Batch Mode
22
The Sound of the Mixtures with Low SNR
  • The First Mixture
  • The Second Mixture
  • The Third Mixture
  • The Fourth Mixture

Example 3 Separation of Noisy Speech Mixture in
Batch Mode
23
Example 3 Separation of Noisy Speech Mixture in
Batch Mode
24
The Sound of the Separated Sentences by
Subband-based ICA
  • The Recovered First Sentence
  • The Recovered Second Sentence
  • The Recovered Third Sentence
  • The Recovered Fourth Sentence

Example 3 Separation of Noisy Speech Mixture in
Batch Mode
25
Example 3 Separation of Noisy Speech Mixture in
Batch Mode
26
The Separation Results by applying a classical
ICA algorithm, Extended Infomax Algorithm (Lee,
Girolami and Sejnowski), directly to the Sound
Mixture
  • The First Output
  • The Second Output
  • The Third Output
  • The Fourth Output

Example 3 Separation of Noisy Speech Mixture in
Batch Mode
27
Quantitative Comparison for Batch Separation
Approaches Performance Index E Average SNR of the Separated Signals
Subband-based ICA 0.051 4.31 dB
Fast ICA 0.1124 -1.63 dB
Extended Infomax 0.118 -1.38 dB
Example 4 Separation of Noisy Mixture in Batch
Mode ( Note The codes of Fast ICA and the
Entended Infomax are downloaded from the authors
websites and the date from ICA99 website.)
28
Conclusions
  • Subband-based ICA is robust to noise.
  • Efficient online learning when other ICA
    algorithms fail.
  • Fast in computation.
  • Possible to address the incomplete mixture
    problem.

29
Future Direction
  • Nonliear ICA by replacing the subband
    decomposition with some appropriate nonlinear
    projection.
  • Kernel ICA. Using the Kernel trick as in Support
    Vector Machines.
  • Using signal cues, for example, pitch of
    acoustic signals, and available prior knowledge,
    to guide separation.

30
Appendix Parameters for Online Music Separation
Experiment 1
  • Data length120,001 Sampling rate 8,000 Hz.
  • Two Source Signals One from Male Singer, anther
    from Female Singer
  • Parameters in Subband ICA
  • Block length 80, Daubechies 10 wavelet
    filter
  • Infomax Algorithm downloaded from
    http//www.cnl.salk.edu/tewon
  • Block length 30 (30 is better than 80
    in the experiment)
  • Modification (A) Maximal number of
    sweeping data, max_sweeps 1, (B) No random
    permutation and PCA processing before applying
    ICA
  • Nonholonomic ICA Block length 30
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