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 (ordergt2)
  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 lt 3)
• Gaussian processes, all cumulants are zero (order
  gt 2)
• Cumulant Spectra
• bispectrum FT order 3 cumulant

Xt
kt1
k
kt2
5
Challenges

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

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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)

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2-D ARMA Model
H(z)
w(m, n)
x(m, n)

• Bispectrum
• Bicoherence
• Constant for linear processes
• Zero for Gaussian processes

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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/m
iwa/probcalc/chisq/
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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 ?

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Statistical Test Results
Brodatz Textures http//www.ux.his.no/tranden/
brodatz.html

• Linear, non-Gaussian, spatially irreversible

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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

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Texture Synthesis
w11
x11
1
M
M-1
w12
x12
M
1

M
1
2
wMM
xMM
Image size M x M
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Texture Synthesis
w11
x11
w12
x12
0

0
wMM
xMM
M systems of M Linear equations
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Texture Synthesis

• Causal AR model

Non-causal AR model
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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

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Parameter Estimation

• AR parameter estimate with 64 x 64 texture

Actual a Estimated a
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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.