# hos

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```					Texture modeling, validation and
synthesis - The HOS way

Srikrishna Bhashyam
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

 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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 views: 2 posted: 10/5/2011 language: English pages: 18