Texture modeling, validation and synthesis - The HOS way

Srikrishna BhashyamMohammad 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

• Cumulants– cumulant = central moment (order <= 3)

– Gaussian processes, all cumulants are zero (order > 2)

• Cumulant Spectra – bispectrum = FT { order 3 cumulant }

2121x3 tkXtkXkXEt,tm

k

k+t1

k+t2

Xt

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)*

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

2-D ARMA Model

• Bispectrum

• Bicoherence

– Constant for linear processes

– Zero for Gaussian processes

2

1

2x2x2x

3x3x

)(S )(S )(S

),(S ,B

vuvu

vuvu

)H( )H( )H( c ,S 3w3x vuvuvu

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

Linear, non-Gaussian, spatially irreversible

Brodatz Textures

http://www.ux.his.no/~tranden/brodatz.html

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

x12

xMM

w11

w12

wMM

1 M

M2

M

M-1

1

1

Image size M x M

Texture Synthesis

=

x’11

x’12

x’MM

w’11

w’12

w’MM

00

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

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

0)(c )a( N

3x ,

i

12 titi

Parameter Estimation

• AR parameter estimate with 64 x 64 texture

Actual a Estimated a

-0.9662 0.9540

1.0112

-0.9686

0.9735

0.9704

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.