Transcript Texture modeling, validation and synthesis

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 m3x t1, t 2   EXkXk  t1 Xk  t 2 
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)
* (-u, -v)
S3x(u, v) = S3x
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)
– Constant for linear processes
– Zero for Gaussian processes
1
2
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
1
M
M-1
M
w11
x11
x12
w12
1
=
2
M
1
xMM
Image size M x M
wMM
Texture Synthesis
w’11
x’11
x’12
w’12
0
=
0
x’MM
M systems of M Linear equations
w’MM
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.