A Bayesian framework for the multifractal analysis of images using data augmentation and a Whittle approximation S. COMBREXELLE 1,* , H. WENDT 1 , Y. ALTMANN 2 , J.-Y. TOURNERET 1 , S. MCLAUGHLIN 2 ,P.ABRY 3 1 IRIT - ENSEEIHT, Toulouse, France 2 Heriot-Watt University, Edinburgh, Scotland 3 Ecole Normale Supérieure, Lyon, France * Supported by the Direction Générale de l’Armement (DGA) ICASSP 2016, Shanghai
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A Bayesian framework for the multifractalanalysis of images using data augmentation
and a Whittle approximation
S. COMBREXELLE1,∗, H. WENDT1, Y. ALTMANN2, J.-Y.TOURNERET1, S. MCLAUGHLIN2, P. ABRY3
1 IRIT - ENSEEIHT, Toulouse, France2 Heriot-Watt University, Edinburgh, Scotland3 Ecole Normale Supérieure, Lyon, France∗ Supported by the Direction Générale de l’Armement (DGA)
ICASSP 2016, Shanghai
Introduction
Multifractal analysis (MFA)
- a widely used tool in signal/image processing- applications in various fields (texture analysis, . . . )- challenging estimation for images of small sizes
Recent work
- statistical estimation procedure based on a Bayesian framework
Contribution
- an acceptance/reject free estimation algorithm- a framework suitable for extension to multivariate data
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 2/18
Characterization by local regularity
Local regularity of X (t) at t0X ∈ Cα(t0) if ∃C, α > 0; Pt0 (t); deg(Pt0 ) < α :
|X (t)− Pt0 (t)| < C|t − t0|α
Hölder exponent:h(t0) = sup
α{α : X ∈ Cα(t0)}
h(t0)→ 1⇒ smooth, very regular,h(t0)→ 0⇒ rough, very irregular
X(t)
tt0
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 3/18
Characterization by local regularity
Local regularity of X (t) at t0X ∈ Cα(t0) if ∃C, α > 0; Pt0 (t); deg(Pt0 ) < α :
|X (t)− Pt0 (t)| < C|t − t0|α
Hölder exponent:h(t0) = sup
α{α : X ∈ Cα(t0)}
h(t0)→ 1⇒ smooth, very regular,h(t0)→ 0⇒ rough, very irregular
α= 0.2
t0
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 3/18
Characterization by local regularity
Local regularity of X (t) at t0X ∈ Cα(t0) if ∃C, α > 0; Pt0 (t); deg(Pt0 ) < α :
|X (t)− Pt0 (t)| < C|t − t0|α
Hölder exponent:h(t0) = sup
α{α : X ∈ Cα(t0)}
h(t0)→ 1⇒ smooth, very regular,h(t0)→ 0⇒ rough, very irregular
α= 0.2 0.4
t0
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 3/18
Characterization by local regularity
Local regularity of X (t) at t0X ∈ Cα(t0) if ∃C, α > 0; Pt0 (t); deg(Pt0 ) < α :
|X (t)− Pt0 (t)| < C|t − t0|α
Hölder exponent:h(t0) = sup
α{α : X ∈ Cα(t0)}
h(t0)→ 1⇒ smooth, very regular,h(t0)→ 0⇒ rough, very irregular
α= 0.2 0.4 0.6
t0
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 3/18
Characterization by local regularity
Local regularity of X (t) at t0X ∈ Cα(t0) if ∃C, α > 0; Pt0 (t); deg(Pt0 ) < α :
|X (t)− Pt0 (t)| < C|t − t0|α
Hölder exponent:h(t0) = sup
α{α : X ∈ Cα(t0)}
h(t0)→ 1⇒ smooth, very regular,h(t0)→ 0⇒ rough, very irregular
h(t0) = 0.6
t0
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 3/18
Characterization by local regularity
Local regularity of X (t) at t0X ∈ Cα(t0) if ∃C, α > 0; Pt0 (t); deg(Pt0 ) < α :
|X (t)− Pt0 (t)| < C|t − t0|α
Hölder exponent:h(t0) = sup
α{α : X ∈ Cα(t0)}
h(t0)→ 1⇒ smooth, very regular,h(t0)→ 0⇒ rough, very irregular
h(t0) = 0.6
t0
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 3/18
Multifractal spectrum
Multifractal spectrum D(h):- fluctuation of the local regularity- Haussdorf dimension of the sets {ti |h(ti ) = h}
D(h) = dimH{t : h(t) = h}
In practice → multifractal formalism [Parisi85]
ti
X(t)
t
D(h)
h0
d
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 4/18
Multifractal spectrum
Multifractal spectrum D(h):- fluctuation of the local regularity- Haussdorf dimension of the sets {ti |h(ti ) = h}
D(h) = dimH{t : h(t) = h}
In practice → multifractal formalism [Parisi85]
ti
h(ti) = 0.2
0.2
D(h)
h0
d
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 4/18
Multifractal spectrum
Multifractal spectrum D(h):- fluctuation of the local regularity- Haussdorf dimension of the sets {ti |h(ti ) = h}
D(h) = dimH{t : h(t) = h}
In practice → multifractal formalism [Parisi85]
ti
h(ti) = 0.4
0.4
D(h)
h0
d
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 4/18
Multifractal spectrum
Multifractal spectrum D(h):- fluctuation of the local regularity- Haussdorf dimension of the sets {ti |h(ti ) = h}
D(h) = dimH{t : h(t) = h}
In practice → multifractal formalism [Parisi85]
ti
h(ti) = 0.6
0.6
D(h)
h0
d
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 4/18
Multifractal spectrum
Multifractal spectrum D(h):- fluctuation of the local regularity- Haussdorf dimension of the sets {ti |h(ti ) = h}
D(h) = dimH{t : h(t) = h}
In practice → multifractal formalism [Parisi85]
ti
X(t)
t
D(h)
h0
d
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 4/18
G Fourier-domain model+ Data augmentation+ Conjugate prior→ Gibbs
LF vs MHG/G→ significant reduction of the standard deviation→ root mean square error divided by 4
MHG vs G→ slight difference for the bias, vanishing for large sample sizes
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 15/18
Results: convergence and computational cost
Convergence of Markov Chains
MHG long burn-in (tuning of the proposals) ∼ 3000 iterations
G almost immediate convergence ∼ 600 iterations
Computational time T = DWT + estimation algorithm
−0.1
−0.09
−0.08
−0.07
−0.06
500 1500 2500
c(k)2 N = 210
k
Averaged Markov Chains
log2N
log2T
−8
−6
−4
−2
0
2
4
7 8 9 10 11
LF
MHG
G
Computational time
→ MHG 25 ∼ 5 slower than LF
→ G 5 ∼ 2 slower than LF
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 16/18
Results: convergence and computational cost
Convergence of Markov Chains
MHG long burn-in (tuning of the proposals) ∼ 3000 iterations
G almost immediate convergence ∼ 600 iterations
Computational time T = DWT + estimation algorithm
−0.1
−0.09
−0.08
−0.07
−0.06
500 1500 2500
c(k)2 N = 210
k
Averaged Markov Chains
log2N
log2T
−8
−6
−4
−2
0
2
4
7 8 9 10 11
LF
MHG
G
Computational time
→ MHG 25 ∼ 5 slower than LF
→ G 5 ∼ 2 slower than LF
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 16/18
Conclusion and future work
Conclusion- Bayesian estimation of c2 in image texture- novel statistical model generative model for Fourier coefficients (Whittle approximation)
separable model (reparametrization + data augmentation)
- inference via a simplified Gibbs sampler→ efficient estimator with reduced computational cost
Work under investigation- MFA of multivariate data via the design of multivariate priors
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 17/18
Thanks for your attention
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 18/18
References
- [Parisi85] U. Frish, and G. Parisi, On the singularity structure of fullydeveloped turbulence; appendix to Fully developped turbulence andintermittency, by U. Frisch, in Proc. Int. Summer School Phys. Enrico Fermi,North-Holland, 1985, pp. 84-88
- [Jaffard04] S. Jaffard, Wavelet techniques in multifractal analysis, in FractalGeometry and Applications: A Jubilee of Benoît Mandelbrot, Proc. Symp.Pure Math., M. Lapidus and M. van Frankenhuijsen, Eds. 2004, vol. 72(2),pp. 91-152, AMS
- [Castaing93] B. Castaing, Y. Gagne, and M. Marchand, Log-similarity forturbulent flows?, Physica D, vol. 68, no. 34, pp. 387-400, 1993
- [TIP,Combrexelle15] S. Combrexelle, H. Wendt, N. Dobigeon, J.-Y. Tourneret,S. McLaughlin, and P. Abry, Bayesian Estimation of the MultifractalityParameter for Image Texture Using a Whittle Approximation, IEEE T. ImageProces., vol. 24, no. 8, pp. 2540-2551, Aug. 2015
- [E,Combrexelle15] S. Combrexelle et al, Bayesian Estimation of theMultifractality Parameter for Images Via a Closed-Form Whittle Likelihood,Proc. 23rd European Signal Process. Conf. (EUSIPCO), Nice, France, 2015.
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 19/18
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 21/18
Statistical model of log-leaders: intra-scale covariance
2. Covariance model
- radial symmetry of the covariance
→ Cov[l(j, k), l(j, k + ∆k)]∆r=|∆k |≈ %j (∆r ;θ)
- piece-wise logarithmic model parametrized by θ = [c2, c02 ]T
%j (∆r ;θ) =
c0
2 + c2 ln 2j ∆r = 0
%(0)j (∆r ;θ) 0 ≤ ∆r ≤ 3
max(0, %(1)j (∆r ;θ)) 3 ≤ ∆r ≤
√2nj
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 22/18
Statistical model of log-leaders: intra-scale covariance
2. Covariance model
- radial symmetry of the covariance
→ Cov[l(j, k), l(j, k + ∆k)]∆r=|∆k |≈ %j (∆r ;θ)
- piece-wise logarithmic model parametrized by θ = [c2, c02 ]T
%j (∆r ;θ) =
c0
2 + c2 ln 2j ∆r = 0
%(0)j (∆r ;θ) 0 ≤ ∆r ≤ 3
max(0, %(1)j (∆r ;θ)) 3 ≤ ∆r ≤
√2nj
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 22/18
Statistical model of log-leaders: intra-scale covariance
2. Covariance model
- radial symmetry of the covariance
→ Cov[l(j, k), l(j, k + ∆k)]∆r=|∆k |≈ %j (∆r ;θ)
- piece-wise logarithmic model parametrized by θ = [c2, c02 ]T
%j (∆r ;θ) =
c0
2 + c2 ln 2j ∆r = 0%
(0)j (∆r ;θ) 0 ≤ ∆r ≤ 3
max(0, %(1)j (∆r ;θ)) 3 ≤ ∆r ≤
√2nj
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 22/18
Spectral density
Closed-form expression of the spectral density
φj (|ω|;θ) = c2 fj (|ω|) + c02 gj (|ω|)
fj (|ω|) = 2π(J0(rjρ)−J0(3ρ)
ρ2+
3ln(rj 2j/3)J1(3ρ)
ρ)−
2π ln(rj 2j/3)
ln 4
∫ 3
0r ln(1 + r)J0(rρ) dr
gj (|ω|) = 6πJ1(3|ω|)
ρ− 2π
∫ 3
0r ln(1 + r)J0(rρ) dr/ ln 4, with rj =
√nj/4
Model fitting
→ spectral grid restricted to low frequencies |ω| ≤ π√η with η = 0.25
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 23/18
MCMC algorithm
Strategy of Gibbs sampler
- iterative sampling according to conditional laws- not standard conditional laws→ Metropolis-within-Gibbs- computation of acceptance ratio at each iteration
rc2 =
√det Σ(θ(t))
det Σ(θ(?))×
j2∏j=j1
exp(−1
2lTj
(Σj,θ(θ(?))−1 −Σj,θ(θ(t))−1
)lj
)
Combrexelle et al. – A Bayesian framework for the multifractal analysis – ICASSP 2016, Shanghai – 24/18
Closed-form Whittle approximationClose-form expression for the spectral density
- Bochner’s theorem→ spectral density
φj (ω;θ) =
∫R2%j (|x|;θ)e−i(xTω)dx
- radial symmetry→ Hankel transform
φj (ω;θ) = φj (|ω|;θ) = 2π∫ ∞
0r%j (r ;θ)J0(rρ) dr
- piece-wise logarithmic covariance model→ analytical expression for Hankel transform