Texture modeling, validation and synthesis - The HOS way

1. Texture modeling, validation and synthesis - The HOS way Srikrishna Bhashyam Mohammad J Borran Mahsa Memarzadeh Dinesh Rajan

2. Key Results • Textures can be modeled as linear, non-Gaussian, stationary random field - validated using HOS. • Textures can be synthesized using causal / non-causal AR models. • AR model parameters can be estimated accurately using HOS.

3. Why Higher Order Statistics? • Deviations from Gaussianity • for Gaussian, all higher order spectra (order>2) = 0 • Non-minimum phase extraction • unlike power spectrum, true phase is preserved • Detect and characterize non-linearity • Applications • array processing, pattern/signal classification...

4. What are these Monsters? • Moments • Cumulants • cumulant = central moment (order <= 3) • Gaussian processes, all cumulants are zero (order > 2) • Cumulant Spectra • bispectrum = FT { order 3 cumulant } Xt k+t1 k k+t2

5. Challenges • Storage and computation of bispectrum • 128x128 image • 4D matrix with 268,435,456 elements (1.07 GB) • Symmetry => redundant elements • factor of 12 reduction

6. Non-redundant Region of Bispectrum • 6-fold symmetry S3x(u, v) = S3x(v, u) = S3x(u, -u-v) = S3x(-u-v, u) = S3x(v,-u-v) = S3x(-u-v, v) • If x is real (12-fold symmetry) S3x(u, v) = S3x(-u, -v) *

7. 2-D ARMA Model H(z) x(m, n) w(m, n) • Bispectrum • Bicoherence • Constant for linear processes • Zero for Gaussian processes

8. Model Validation Tests • Gaussianity test • Statistical test to check if the bicoherence is zero • Test statistic is chi-squared distributed National Institute of Agro-Environmental Sciences, Japan http://ss.niaes.affrc.go.jp/pub/miwa/probcalc/chisq/

9. Model Validation Tests • Linearity test • Statistical test to check if the bicoherence is constant • Is the variability of the bicoherence small enough? • Spatial reversibility test • Does the texture have any spatial symmetry ? • Is the imaginary part of bicoherence zero ?

10. Statistical Test Results Brodatz Textures http://www.ux.his.no/~tranden/brodatz.html Linear, non-Gaussian, spatially irreversible

11. Texture Synthesis • 2-D, non-causal, non-Gaussian, AR model • Causal AR • Direct IIR filtering: recursive equation • Non-causal AR • No recursive equation • Calculate truncated impulse response • Solve input-output system of linear equations

12. Texture Synthesis w11 x11 1 M-1 M w12 x12 M 1 = M 1 2 wMM xMM Image size M x M

13. Texture Synthesis w’11 x’11 w’12 x’12 0 = 0 w’MM x’MM M systems of M Linear equations

14. Texture Synthesis Causal AR model Non-causal AR model

15. Parameter Estimation • Try to match more than the power spectrum • Cumulants instead of correlations • C a = c instead of R a = r • Calculate only the cumulants that are needed

16. -0.9686 0.9704 -0.9662 0.9540 1.0112 0.9735 Parameter Estimation • AR parameter estimate with 64 x 64 texture Actual a Estimated a

17. Summary • Higher-order spectrum basics • Linearity, Gaussianity and spatial reversibility • Texture model validation • 2-D Causal and Non-causal AR models • Texture synthesis • Cumulant based causal AR parameter estimation • Modeling of real textures • Useful for texture classification and segmentation • HOS useful but too complex

18. References • T. E. Hall and G. B. Giannakis, “Bispectral Analysis and Model Validation of Texture Images”, Trans. SP, 1995. • S. Das, “Design of Computationally Efficient Multiuser Detectors for CDMA Systems”, M. S. Thesis, Rice University, 1997. • R. Chellappa and R. L. Kashyap, “ Texture Synthesis using 2-D Noncausal Autoregressive Models”, Trans. ASSP, 1985. • A. T. Erdem, “ A Nonredundant set for the Bispectrum of 2-D Signals”, ICASSP, 1993. • C. L. Nikias and A. P. Petropulu, Higher-order Spectra Analysis, 1993.