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Blind Source Separation : from source separation to pixel classication. Albert Bijaoui 1 , Danielle Nuzillard 2 & Frédéric Falzon 3 1 Observatoire de la Côte d'Azur (Nice) 2 Université de Reims Champagne Ardenne 3 Alcatel Space – Cannes-la-Bocca. O utlines. - PowerPoint PPT Presentation
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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
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
8
Infomax (Bell & Sejnowski)
• ANN output
• Minimisation rule of the output entropy
• Choice of the activation function
• Natural gradient (Amari)
)(XY g
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
9
FastICA• Helsinki : Oja, Karhunen, Hyvärinen
• Negentropy– Negentropy = Entropy Gaussian rv – Entropy rv
• Negentropy approximation
• Choice of the function G- Cumulant order 4, Sigmoid, Gaussian
pyG GEyGEyJ )()()(
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
10
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
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
11
The reduced HST images
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
12
KL Expansion of 3C120 images
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
13
Best visual Selection : f-SOBI
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
14
CASIImages 9 filters
394-907nmImages from GSTB (Groupement Scientifique
de Télédétection de Bretagne) with the courtesy of the Pr. Kacem Chehdi ENSSAT
Lannion (France)
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
15
FastICAsources
after denoising
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
16
Ground analysis
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
17
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
28 November 2002
iAstro / IDHA Worshop - Strasbourg Observatory
18
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, ..