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28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
1
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
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
2
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
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
3
The Cocktail Party Model• The mixing hypotheses
– Linearity– Stationarity– Source independence
• The equation:
ijj iji NSaX • Xi images - Sj unknown sources - Ni noise
• A= [aij] mixing matrix
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
4
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
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
5
Mutual Information
• Mutual Information between l variables
• Case of Gaussian distributions
– R is the matrix of correlation coefficients– In this case : Uncorrelated = Independent
li
i
ll
nn
ln
s
nnpnnpSSIp
i
l
,1
121
,...,
1)(),...,(log),...,(),...,(
1
RI detlog21
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
6
Independent Component Analysis• Contrast Function :
– Mutual information of the sources
• Contrast:
• Minimum Mutual information = Maximum contrast
• How to compute the source entropy ?
AXXESESSI n
l
ll detlog),...,()(),...,( 211
ASESSCl
ll detlog)(),...,( 21
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
7
JADE
• Comon’s 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