Parametric estimation methods of multivariate data for multi-component image segmentation

Transcription

1 Parametric estimation methods of multivariate data for multi-component image segmentation Stéphane Derrode Institut Fresnel (UR 633) and EGI, arseille Nicolas Brunel and Wojciech Pieczynski Institut National des Télécommunications, CITI Dpt, Evry Journée "Analyse d'images multispectrales" de l'observatoire de Strasbourg

2 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 : 2

3 ulti-temporal images 4 SAR-ERS images of a rice plantation in Indonesia, /0 time 0/02 6/02 06/03 ERS: European radar satellite SAR: Synthetic Aperture Radar 3

4 ulti-sensor images sensor Nyiragongo volcano (Congo, Goma), January Radar data Optical data : false colors composite image 4

5 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

6 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

7 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

8 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

9 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 ( 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

10 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

11 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

12 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

13 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

14 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 So, it is impossible to include some physical knowledge about one band. Example: optical (Gaussian) and radar (Gamma) sensors. 4

15 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

16 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

17 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

18 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

19 margins ( ) c u, u = ρ 2 2 ξ ρ ξ 2 2 e t ( ( I ) ) 9

20 Isoprobability levels for a bivariate normal copula with different margins 20

21 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 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

22 2D copulas with the same Gamma margins ρ =0.5 Product copula Gaussian copula ρ = 0.5, ν = 0 Student copula 22

23 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

24 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

25 Segmentation results Independence PCA ICA All results with Gamma laws. Gaussian copula Student copula 25

26 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

27 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 , Philadelphia (PA, USA). 27

28 ulti-spectral images (Earth) 4 Spot images of fields in Brittany, France, red f green near IF iddle IF Spot : French satellite for Earth observation 28