Blind Source Separation from source separation to pixel
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 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 1
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/WFPC 2 images – Source separation • Experiment on Multispectral Earth images – Pixel classification • Conclusion 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 2
The Cocktail Party Model • The mixing hypotheses – Linearity – Stationarity – Source independence • The equation: • Xi images - Sj unknown sources - Ni noise • A= [aij] mixing matrix 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 3
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 i. Astro / IDHA Worshop - Strasbourg Observatory 4
Mutual Information • Mutual Information between l variables • Case of Gaussian distributions – R is the matrix of correlation coefficients – In this case : Uncorrelated = Independent 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 5
Independent Component Analysis • Contrast Function : – Mutual information of the sources • Contrast: • Minimum Mutual information = Maximum contrast • How to compute the source entropy ? 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 6
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 i. Astro / IDHA Worshop - Strasbourg Observatory 7
Infomax (Bell & Sejnowski) • ANN output • Minimisation rule of the output entropy • Choice of the activation function • Natural gradient (Amari) 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 8
Fast. ICA • Helsinki : Oja, Karhunen, Hyvärinen • Negentropy – Negentropy = Entropy Gaussian rv – Entropy rv • Negentropy approximation • Choice of the function G - Cumulant order 4, Sigmoid, Gaussian 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 9
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 i. Astro / IDHA Worshop - Strasbourg Observatory 10
The reduced HST images 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 11
KL Expansion of 3 C 120 images 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 12
Best visual Selection : f-SOBI 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 13
CASI Images 9 filters 394 -907 nm Images from GSTB (Groupement Scientifique de Télédétection de Bretagne) with the courtesy of the Pr. Kacem Chehdi ENSSAT Lannion (France) 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 14
Fast. ICA sources after denoising 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 15
Ground analysis 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 16
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 i. Astro / IDHA Worshop - Strasbourg Observatory 17
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, . . 28 November 2002 i. Astro / IDHA Worshop - Strasbourg Observatory 18
- Slides: 18