K-Centroids-Based Supervised Classification of Texture Images using the SIRV modeling

28/08/2013
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K-Centroids-Based Supervised Classification of Texture Images using the SIRV modeling

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Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives K-Centroids-Based Supervised Classification of Texture Images using the SIRV modeling Aurélien Schutz Lionel Bombrun Yannick Berthoumieu IMS Laboratory - CNRS UMR5218, Groupe Signal 28-30 august 2013 Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 1 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Database classification Database classification Musical genres Images databases Textured images databases Video databases Propositions Information geometry Centroid ¯θi [Choy2007] [Fisher1925], [Burbea1982], [Pennec1999], [Banerjee2005], [Amari2007], [Nielsen2009] Bayesian framework of classification Intrinsic prior p(θ | Hi ) [Bayes1763], [Whittaker1915], [Robert1996], [Bernardo2003] Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 2 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Prior capability of handling the intra-class diversity Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 3 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 4 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 5 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Bayesian decision Data space X where x ∼ P Parameter space Θ PΘ a prior parametric model on Θ Riemannian manifold G the Fisher information matrix Nc classes, D = {Hi } Nc i=1 decision space Prerequisites : likelihood p(x | θ, Hi ), prior p(θ | Hi ), 0-1 loss L Decision rule on X : high computational complexity Xi = x | ˆHi = arg min Hj ∈D − log Θj p(x | θ, Hj )p(θ | Hj ).dθ Decision rule on Θ : minimizing conditional risk Duda, Bayesian decision theory Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 6 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Intra-class parametric p(θ | Hi) Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 7 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Intra-class parametric p(θ | Hi) ¯θi centroid of the class i = 1, . . . , Nc Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 7 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Definition of intrinsic prior based on Jeffrey divergence Prior that follow a Gaussian distribution on manifold Θ p(θ | Hi ) = Zi exp − 1 2 (γ¯θi ,θ(1))T Ci γ¯θi ,θ(1) Pennec, Xavier, "Probabilities and statistics on Riemannian manifolds : basic tools for geometric measurements," NSIP, 1999 Proposition Intrinsic prior as Gaussian distribution on manifold Θ, with λi = (¯θi , σ2 i ) p(θ | λi , Hi ) |G(¯θi )|1/2 (σi √ 2π)d exp − 1 2σ2 i J(p(· | θ), p(· | ¯θi )) Jeffrey divergence J(p(· | θ), p(· | ¯θi )) = X (p(x | θ) − p(x | ¯θi )) log p(x | θ) p(x | ¯θi ) .dx Fisher, R.A., "Theory of statistical estimation," Proc. Cambridge Phil. Soc., 22, pp. 700-–725, 1925 Burbea, Jacob et Rao, C.Radhakrishna, "Entropy differential metric, distance and divergence measures in probability spaces : A unified approach ," Journal of Multivariate Analysis, 4, pp. 575—596, 1982 Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 8 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Optimal decision on Θ Decision on X based on Empirical Bayes, Xi = x Hi = arg min Hj ∈D − log Θi p(Xi | θ, Hi )p(θ | Hi ).dθ Kass, R. E. and Steffey, D., "Approximate Bayesian Inference in Conditionally Independent Hierarchical Models (Parametric Empirical Bayes Models)," 1989 Miyata, Y., "Fully Exponential Laplace Approximations Using Asymptotic Modes," Journal of the American Statistical Association, 2004 Proposition Decision on X, Laplace approximation Xi x ˆλi = arg min λj ∈D d 2 log{2σ2 j + 1} + 1 2σ2 j J(p(· | ˆθ(x)), p(· | ¯θj )) ˆθ could be maximum likelihood estimator for p(x | θ, Hi ) [Miyata2004] Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 9 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 10 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Space/scale decomposition Reel/complex wave- lets Gabor Steerable filters Bandelets Grouplets Dual-Tree Mallat, S. A, "Theory for multiresolution signal decomposition : The wavelet representation," IEEE PAMI, 1989 Do, M. and Vetterli, M., "Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance," IEEE IP, 2002 Choy, S.-K. and Tong, C.-S., "Supervised Texture Classification Using Characteristic Generalized Gaussian Density," Journal of Mathematical Imaging and Vision, 2007 Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 11 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Stochastic models for likelihood p(x | θ, Hi) Spherically Invariant Random Vector (SIRV) x = g √ τ : g multivariate gaussian distribution Σ τ Weibull distribution a Joint distribution y = (τ, g), θ = (Σ, a) p(y | θ) = pG (g | Σ)pw (τ | a) Separability of Jeffrey divergence J(p(· | θ), p(· | θ )) = J(pG (· | Σ), pG (· | Σ ))+J(pw (· | a), pw (· | a )) Bombrun, L., Lasmar, N.-E., Berthoumieu, Y. and Verdoolaege, G., Multivariate texture retrieval using the SIRV representation and the geodesic distance, 2011 Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 12 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Centroid ¯θi computation State of the art : exponential families ; centred multivariate Gaussian ¯ΣR,i = 1 Ni Ni n=1 Σ−1 n −1 and ¯ΣL,i = 1 Ni Ni n=1 Σn Banerjee, A., Merugu, S., Dhillon, I. and Ghosh, J., Clustering with Bregman divergences, 2005 Nielsen, F. and Nock, R. Sided and Symmetrized Bregman Centroids, 2009 Steepest descent algorithm for Weibull centroid Dekker, T. J., Finding a zero by means of successive linear interpolation, 1969 Brent, R. P., An algorithm with guaranteed convergence for finding a zero of a function, 1971 Proposition Separated estimation of each centroid. ¯θi = (1 − i )¯ΣR,i + i ¯ΣL,i , arg min a∈R+ 1 Ni Ni n=1 J(pw (· | an), pw (· | a)) Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 13 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 14 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Unique centroid versus multiple centroids (K-CB) Varma, M. and Zisserman, A., A Statistical Approach to Texture Classification from Single Images, 2005 Several centroids per class (¯θi,k )K k=1, likelihood K-CB with binary weights wk pm(θ | (Hi,k )K k=1) = K k=1 wk Zi,k exp − 1 2σ2 i J(p(· | θ), p(· | ¯θi,k )) Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 15 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Algorithms for K-CB K-means ( Hard C-Means ) Proposition 1. Assignment of parametric vector θ Θi,k = θ | ˆθi,k = arg min θi,l ∈Hi 1 2σ2 i J(p(· | θ), p(· | ¯θi,l )) 2. Update ¯θi,k ¯θi,k = arg min ¯θ∈Θ Θi,k 1 2σ2 i J(p(· | θ), p(· | ¯θ)).dθ Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 16 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Textured image database Vision Texture database (VisTex) Brodatz database Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 17 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Vistex, SIRV Weibull, Jeffrey divergence 2/16 5/16 8/16 11/16 14/16 85 90 95 100 No. of training sample Averagekappaindex(%) 1−CB [1] 3−CB 1−NN Spatial Database K 1-NN 1-CB [1] K-CB neigh. NTr = K NTr = NSa/2 NTr = NSa/2 3 × 3 VisTex 3 83.7 % ±2.0 90.4 % ±1.3 96.8 % ±1.2 Brodatz 10 50.6 % ±2.6 79.9 % ±1.5 96.2 % ±1.2 1 × 1 VisTex 3 78.7 % ±2.3 72.7 % ±2.0 88.9 % ±1.7 Brodatz 10 65.8 % ±2.7 70 % ±1 97 % ±2 [1] Choy, S.K., Tong, C.S. : Supervised texture classification using characteristic generalized Gaussian density. Journal of Mathematical Imaging and Vision 29 (Aug. 2007) 35–47 Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 18 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 19 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Conclusions et Perspectives Conclusion 1. Bayesian classification theory and Information geometry 2. Concentred Gaussian distribution as prior p(θ | Hi ) intrinsic when p(θ) or L depends on Fisher information matrix G(θ) Decision rule done on Θ 3. K-Centroids based (K-CB) classification Diversity intra-class too high : a class, K centroids K-means on each class Numerical application : K-CB performances close to 1-NN performances K-CB give a low computing complexity Perspectives 1. K-CB with Possibilistic Fuzzy C-Means (PFCM) algorithm 2. Adapting the number of centroid needed by class Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 20 / 21 Introduction Bayesian classification and Information geometry Textured images K-CB and results Conclusion and Perspectives Content Brodatz, P., Textures : A Photographic Album for Artists and Designers, 1966. Schutz, A. and Bombrun, L. and Berthoumieu, Y. (Labo IMS) K-CB of images with SIRV 28-30 august 2013 21 / 21