Blind Source Separation : from source separation to pixel classication

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Blind Source Separation from source separation
to pixel classication

• Albert Bijaoui1, Danielle Nuzillard2
• Frédéric Falzon3
• 1 Observatoire de la Côte d'Azur (Nice)
• 2 Université de Reims Champagne Ardenne
• 3 Alcatel Space Cannes-la-Bocca

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Outlines

• What is Blind Source Separation (BSS)?
• Different BSS tools
• Karhunen-Loève expansion (KL/PCA)
• Independent Component Analysis (ICA)
• Use of spatial correlations (SOBI, ..)
• Experiment on HST/WFPC2 images
• Source separation
• Experiment on Multispectral Earth images
• Pixel classification
• Conclusion

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The Cocktail Party Model

• The mixing hypotheses
• Linearity
• Stationarity
• Source independence
• The equation

• Xi images - Sj unknown sources - Ni noise
• A aij mixing matrix

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KL and PCA

• Search of uncorrelated images
• The Principal Component Analysis
• Iterative extraction of the linear combinations
  having the greatest variance
• PCA application to images ? KL
• KL limitations
• If Gaussian Probability Density Functions (PDF)
• uncorrelated independent
• If not
• It may exist more independent sources than the
  ones resulting from the KL expansion

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Mutual Information

• Mutual Information between l variables
• Case of Gaussian distributions
• R is the matrix of correlation coefficients
• In this case Uncorrelated Independent

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Independent Component Analysis

• Contrast Function
• Mutual information of the sources
• Contrast
• Minimum Mutual information Maximum contrast
• How to compute the source entropy ?

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JADE

• Comons approach
• PDF Edgeworth Approximation
• Cumulants use
• JADE (Cardoso Souloumiac)
• Based on order 4 cumulants
• Rotation of KL separation matrix
• Jacobi decomposition (2 à 2)
• Joint Diagonalisation

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Infomax (Bell Sejnowski)

• ANN output
• Minimisation rule of the output entropy
• Choice of the activation function
• Natural gradient (Amari)

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FastICA

• Helsinki Oja, Karhunen, Hyvärinen
• Negentropy
• Negentropy Entropy Gaussian rv Entropy rv
• Negentropy approximation
• Choice of the function G
• Cumulant order 4, Sigmoid, Gaussian

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BSS from spatial correlations

• SOBI (Belouchrani et al.)
• Cross-correlations between sources and shifted
  sources
• Number p of cross correlation matrices
• Jacobi / Givens decomposition
• Joint diagonalization
• F-SOBI (Nuzillard)
• Cross-correlations are made in the Fourier space

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The reduced HST images
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KL Expansion of 3C120 images
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Best visual Selection f-SOBI
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CASIImages 9 filters394-907nmImages from
GSTB (Groupement Scientifique de Télédétection de
Bretagne) with the courtesy of the Pr. Kacem
Chehdi ENSSAT Lannion (France)
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FastICAsources after denoising
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Ground analysis
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Classification

• A source is not a pure element
• Pixel classification is easily deduced by
  comparison to the ground analysis
• BSS allows one to facilitate classification
• New classes are probed by BSS analysis

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Conclusion

• Used BSS methods were based on the cocktail party
  model.
• Typical tools for Data Mining
• Adapted to multi-wavelengths observations or data
  from spectroimagers
• Many applications source identification, pixel
  classification, denoising, compression, ..