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					Texture modeling, validation and
   synthesis - The HOS way


          Srikrishna Bhashyam
          Mohammad J Borran
           Mahsa Memarzadeh
              Dinesh Rajan
                  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.
        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...
            What are these Monsters?
• Moments m3 t1, t 2   EXk Xk  t1 Xk  t 2 
           x



       Xt
                           k+t1
                    k                   k+t2
• Cumulants
    – cumulant = central moment (order <= 3)
    – Gaussian processes, all cumulants are zero (order > 2)
• Cumulant Spectra
    – bispectrum = FT { order 3 cumulant }
                      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
  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)
               2-D ARMA Model

        w(m, n)              H(z)             x(m, n)

• Bispectrum
               S3x u, v   c3w H(u) H( v) H(u  v)
• Bicoherence

             B3x u, v  
                                        S3x (u, v)
                             S2x (u) S2x ( v) S2x (u  v)
                                                           1
                                                           2


  – Constant for linear processes
  – Zero for Gaussian processes
             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/
             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 ?
       Statistical Test Results
             Brodatz Textures
http://www.ux.his.no/~tranden/brodatz.html




Linear, non-Gaussian, spatially irreversible
                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
        Texture Synthesis

                             x11       w11
1          M-1     M                   w12
                             x12

M   1
                                   =


2           M      1
                             xMM       wMM


          Image size M x M
      Texture Synthesis

                           x’11       w’11
               0           x’12       w’12



                                  =

0
                          x’MM        w’MM

    M systems of M Linear equations
         Texture Synthesis




Causal AR model    Non-causal AR model
            Parameter Estimation

• Try to match more than the power spectrum
• Cumulants instead of correlations


          a(i) c
         iN
                      3x   (t 2  i , t 1 )  0


• C a = c instead of R a = r
• Calculate only the cumulants that are needed
              Parameter Estimation


• AR parameter estimate with 64 x 64 texture

          Actual a             Estimated a


     -0.9686 0.9704         -0.9662 0.9540

     0.9735                 1.0112
                         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
                        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.

				
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posted:10/5/2011
language:English
pages:18