Parametric estimation methods of multivariate data for multi-component image segmentation Stéphane Derrode Institut Fresnel (UR 633) and EGI, arseille stephane.derrode@fresnel.fr Nicolas Brunel and Wojciech Pieczynski Institut National des Télécommunications, CITI Dpt, Evry Journée "Analyse d'images multispectrales" de l'observatoire de Strasbourg
ulti-spectral images (galaxy) + g λ r False color composite image 3 spectral bands of the NGC 303 galaxy (Thuan-Gunn system : g, r, and i). i From the Galaxy Catalog : http://www.astro.princeton.edu/~frei/galaxy_catalog.html 2
ulti-temporal images 4 SAR-ERS images of a rice plantation in Indonesia, 994. 23/0 time 0/02 6/02 06/03 ERS: European radar satellite SAR: Synthetic Aperture Radar 3
ulti-sensor images sensor Nyiragongo volcano (Congo, Goma), January 2002. Radar data Optical data : false colors composite image 4
ulti-scale images ultiscale decomposition Low-pass coef. High-pass horizontal coef. Excerpt from an ERS image showing an oil slick in the editerranean sea High-pass vertical coef. 5
Outline ultivariate parametric p.d.f. ulti-band image classification Statistical segmentation Examples of multivariate parametric p.d.f. ultivariate data analysis viewpoint Independence, PCA, ICA Copulas: a general class of multivariate models Definition Examples: Product, Gaussian and Student copulas Segmentation results multispectral CASI image 6
ulti-band image classification =3 K = Ω = 2 { ω, ω } 2 y y 2 y = { y, y, y } 2 3 x ω ω2 y 3 Real observations y = y y y {,, } 2 3 Classification map x 7
Statistical segmentation x y y 2 y 3 Statistical framework : 2 y 3 One important feature of the statistical modeling of images for segmentation is the choice for laws that represent the randomness within each class. y y = y R 3 ω ( y = ω ) = ( y ) p x x ω 2 f ω 8
ultivariate parametric pdf Gaussian assumption: D exponential law: 2D exponential law: - oran and Downtoon - Arnold and Strauss - Gumbel (, ) y f( y) = e µ µ θ y + θ y θθ2 2 ρθθ2yy2 ρ 2 0 f y y = I e ρ ρ 2 2 ( y 2y2 2 3yy2 ) (, ) = ( β ) β β + + f y y C e β β β β β 2 3 2 ( y ) (, ) = ( ( + θ )( + θ ) θ ) f y y y y e 2 2 t ( y Γ y ) y + y + θ y y ρ 2 2 From Kotz et al, Continuous multivariate distributions, Wiley series in proba. and stat., 2000 f = (2 π) 3 Γ e 2 3 different shapes!!! 9
Outline ultivariate parametric p.d.f. ulti-band image classification Statistical segmentation Examples of multivariate parametric p.d.f. ultivariate data analysis viewpoint Independence, PCA, ICA Copulas: a general class of multivariate models Definition Examples: Product, Gaussian and Student copulas Segmentation results multispectral CASI image 0
ultivariate analysis viewpoint Independence between bands f ( y ) f ( y ) = m= m m f ( y ) f2 ( y2 ) f3 ( y3 ) y y 2 y 3 Supposed to belong to an a priori parametric model such as Beta or Gamma families of distributions
ultivariate analysis viewpoint Principal Component Analysis (PCA) t W z = Wy ( C( ) ) = Γ y z y z 2 y 2 z 3 y 3 f ( z ) f ( z ) f ( y ) W f ( z ) = m= m m = m= m m 2
ultivariate analysis viewpoint Independent Component Analysis (ICA) Find W such that projected data becomes independent (i.e. «decorrelated at all statistical orders»). Linear mixture of observations such that projected bands have the least Gaussian distribution. «Non-gaussianity» criteria: kurtosis or neguentropy. Difficulties: t = W' y Very time consuming (iterative process), even if there exists some Fast ICA algorithms. Often gives data with multimodal histogram (not very interesting for classification purposes!) f ( y ) W' f ( t ) = m= m m 3
ultivariate analysis viewpoint Independence f PCA f ICA f ( y ) f ( y ) = m= ( y ) W f ( z ) = m m m= ( y ) W' f ( t ) = m= m m m m + m m are not necessary Gaussian (Ex: Gamma or Beta laws) - f y are not the margins of f y ( ) ( ) m m R 2 ( ) f ( y ) f ( y) f y, y, y dy dy 2 3 2 3 So, it is impossible to include some physical knowledge about one band. Example: optical (Gaussian) and radar (Gamma) sensors. 4
Outline ultivariate parametric p.d.f. ulti-band image classification Statistical segmentation Examples of multivariate parametric p.d.f. ultivariate data analysis viewpoint Independence, PCA, ICA Copulas: a general class of multivariate models Definition Examples: Product, Gaussian and Student copulas Segmentation results multispectral CASI image 5
Copula: definition The conditional density y regarding class represents our knowledge of the underlying phenomenon. A class is characterized by. the behavior of each component, and 2. the way this components are linked. Copula: Sklar s theorem (959) asserts that any -Dim p.d.f. can be written: ( y ) ( ) ( ( ), L, ( ) ) f = fm ym c F y F y m= f ω ( ) ω. Independent behavior 2. Statistical links F ( ) ( ) with m. the associated c.d.f. of f m., and c( L ) is a p.d.f. on the unit hypercube [ 0,] 6
Property ( y ) ( ) ( ( ), L, ( ) ) f = fm ym c F y F y m=. Independant behaviour 2. Statistical links argins: = y ( ) ( ) f y f dy2... dy R PCA, ICA We can construct multivariate p.d.f. with given margins 7
Product and Gaussian copulas Example #: Product copula (, L, ) ( L ) C u u = u L u c u,, u = Example #2: Gaussian copula ( L ) c2 u,, u = ρ t ξ ρ ξ 2 2 ( u L u ). ξ = Φ ( ),, Φ ( ) Φ m () e t ( ( I ) ) Inverse c.d.f. of the normalized Gaussian density ρ Correlation matrix ( y ) ( ) ( ( ), L, ( ) ) f = fm ym c F y F y m= f ( y ) f ( y ) = m= m m Can be viewed as a multi-dim Gaussian p.d.f. without Gaussian margins! 8
margins ( ) c u, u = ρ 2 2 ξ ρ ξ 2 2 e t ( ( I ) ) 9
Isoprobability levels for a bivariate normal copula with different margins 20
Student copula Example #3: Student copula ν + ν + ν t - 2 Γ ξ ρ ξ 2 Γ 2 + ν 3 (, L, ) = ρ 2 ν + ν + 2 2 Γ 2 ξ + m m= ν c u u t ( y ) ( ) ( ( ), L, ( ) ) ( T u T u ) ξ = ( ), L, ( ) Tm (). f = fm ym c F y F y m=. Independant behaviour 2. Statistical link Inverse c.d.f. of a Student law with deg. of freedom ρ Correlation matrix ν 2
2D copulas with the same Gamma margins ρ =0.5 Product copula Gaussian copula ρ = 0.5, ν = 0 Student copula 22
Outline ultivariate parametric p.d.f. ulti-band image classification Statistical segmentation Examples of multivariate parametric p.d.f. ultivariate data analysis viewpoint Independence, PCA, ICA Copulas: a general class of multivariate models Definition Examples: Product, Gaussian and Student copulas Segmentation results multispectral CASI image 23
Example: CASI image segmentation Airborne hyperspectral CASI image, reduced to 4 bands. Original image contains 7 spectral bands from 450 to 950 nm, with 2 meters ground resolution. Segmentation with 4 classes: forests, fields, roads and wastelands -> 4 4D p.d.f. 24
Segmentation results Independence PCA ICA All results with Gamma laws. Gaussian copula Student copula 25
Conclusion Parametric multivariate modeling -Dim Gaussian and beyond? PCA, ICA Copulas : Product, Gaussian, Student, (Gumbel, Frank, ) Copulas: Can be used to model dependence between random variables in a very general way. Applied to the segmentation of multi-component images, in a vectorial HC model context. Copulas can be used in all situations where multidimensional p.d.f. estimation is required. 26
Some references for copulas [] Joe, H. [997], ultivariate odels and Dependence Concepts, onographs on Statistics and Applied Probability, 73, Chapmann & Hall, London. [2] Hutchinson, T. P. et C.D. Lai [990], Continuous Bivariate Distributions, Emphasising Applications, Rumbsy Scientific Publishing, Adelaide. [3] Nelsen, R.B. [999], An Introduction to Copulas, Lectures Notes in Statistics, 39, Springer Verlag, New-York. [4] Brunel N., Pieczynski W. and Derrode S. [2005], Copulas in HC for multicomponent image segmentation, IEEE ICASSP; arch 8-23 2005, Philadelphia (PA, USA). 27
ulti-spectral images (Earth) 4 Spot images of fields in Brittany, France, 2002. red f green near IF iddle IF Spot : French satellite for Earth observation 28