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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




|X (t)− Pt0 (t)| < C|t − t0|α


α{α : X ∈ Cα(t0)}


α= 0.2

t0




|X (t)− Pt0 (t)| < C|t − t0|α


α{α : X ∈ Cα(t0)}


α= 0.2 0.4

t0




|X (t)− Pt0 (t)| < C|t − t0|α


α{α : X ∈ Cα(t0)}


α= 0.2 0.4 0.6

t0




|X (t)− Pt0 (t)| < C|t − t0|α


α{α : X ∈ Cα(t0)}


h(t0) = 0.6

t0




|X (t)− Pt0 (t)| < C|t − t0|α


α{α : X ∈ Cα(t0)}


h(t0) = 0.6

t0


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






ti

h(ti) = 0.2

0.2

D(h)

h0

d






ti

h(ti) = 0.4

0.4

D(h)

h0

d






ti

h(ti) = 0.6

0.6

D(h)

h0

d






ti

X(t)

t

D(h)

h0

d


Legendre spectrum and log-cumulant

Dyadic wavelet transform (DWT)→ {d (m)(j , ·, ·)}m=1,2,3

Wavelet leaders {`(j ·, ·)} [Jaffard04]

`(j , k1, k2) = supm, λ′⊂3λj,k1,k2

|d (m)(λ′)|


Legendre spectrum and log-cumulant

Dyadic wavelet transform (DWT)→ {d (m)(j , ·, ·)}m=1,2,3

Wavelet leaders {`(j ·, ·)} [Jaffard04]

`(j , k1, k2) = supm, λ′⊂3λj,k1,k2

|d (m)(λ′)|Polynomial expansion [Castaing93]

DL(h) = 2 +c2

2!

(h − c1

c2

)2

− c3

3!

(h − c1

c2

)3

+ · · · ≈ D(h)

Multifractality parameter c2

- width of the spectrum- self-similar processes↔ multiplicative cascades- tied to the variance of log-leaders

Var [ ln `(j , ·, ·) ] = c02 + c2 ln 2j

.

c1

c2

D(h)

h0

d


Estimation of the multifractality parameter

Synthetic multiplicative cascades with different c2c

2=-0.09 c

2=-0.07 c

2=-0.05 c

2=-0.03 c

2=-0.01

Estimation of c2

- benchmark linear regression-based estimation [Castaing93]

X estimation performance

- recently proposed Bayesian estimation [TIP,Combrexelle15]√ estimation performanceX computational cost: Metropolis-Hasting moves

Goal: elimination of Metropolis-Hasting moves in the inference


Space-domain statistical model of log-leaders

1. Marginal distribution of log-leaders approximated by Gaussian

l(j , ·, ·) = ln L(j , ·, ·) ∼ N(·, c0

2 + c2 ln 2j )

2. Intra-scale dependence captured by a radial covariance model

Cov[l(j ,k), l(j ,k + ∆k)]∆r=|∆k |≈ %j (∆r ;θ), θ = (c2, c0

2)

Likelihood of centered log-leaders lj stacked in l = [lTj1 , ..., lTj2 ]T

→ scale-wise product of Gaussian likelihoods

p(l|θ) ∝j2∏

j=j1

|Σj,θ|−12 exp

(−1

2lTj Σ−1

j,θ lj

), with Σj,θ induced by %j (∆r ;θ)

X evaluation of p(l|θ) numerically instable




l(j , ·, ·) = ln L(j , ·, ·) ∼ N(·, c0

2 + c2 ln 2j )2. Intra-scale dependence captured by a radial covariance model


2)



p(l|θ) ∝j2∏

j=j1

|Σj,θ|−12 exp

(−1

2lTj Σ−1

j,θ lj






l(j , ·, ·) = ln L(j , ·, ·) ∼ N(·, c0

2 + c2 ln 2j )2. Intra-scale dependence captured by a radial covariance model


2)



p(l|θ) ∝j2∏

j=j1

|Σj,θ|−12 exp

(−1

2lTj Σ−1

j,θ lj




Whittle approximation

Evaluation of the Gaussian likelihood in the spectral domain

pW (l|θ) ∝j2∏

j=j1

|Γj,θ|−1 exp(−yH

j Γ−1j,θ y j

)- y j vectorized Fourier coefficients of lj- Γj,θ parametric spectral density associated with %j (∆r ;θ)

→ Computation of Γj,θ via FFT [TIP,Combrexelle15]

→ Closed-form expression via Hankel transform [E,Combrexelle15]

Γj,θ = c2 F1,j + c02 F2,j , Fi,j = diag(fi,j )

Estimation of θ embedded in a Bayesian framework

- Space-domain likelihood (approximated) + common priors

X non-standard posterior distribution→ acceptance/reject moves



Evaluation of the Gaussian likelihood in the spectral domain

pW (l|θ) ∝j2∏

j=j1


j Γ−1j,θ y j

)- y j vectorized Fourier coefficients of lj- Γj,θ parametric spectral density associated with %j (∆r ;θ)

→ Computation of Γj,θ via FFT [TIP,Combrexelle15]

→ Closed-form expression via Hankel transform [E,Combrexelle15]

Γj,θ = c2 F1,j + c02 F2,j , Fi,j = diag(fi,j )

Estimation of θ embedded in a Bayesian framework

- Space-domain likelihood (approximated) + common priors

X non-standard posterior distribution→ acceptance/reject moves


Fourier-domain statistical model


pW (l|θ) ∝j2∏

j=j1


j Γ−1j,θ y j

)- y j vectorized Fourier coefficients of lj- Γj,θ = c2 F1,j + c0

2 F2,j parametric spectral density

mGenerative model for y = [yT

j1 , ...,yTj2 ]T

p(y |θ) ∝ |Γθ|−1 exp(−yHΓ−1

θ y)

- Complex Gaussian model y ∼ CN (0,Γθ)

- Γθ = c2F 1 + c02F 2 and F i = block(F i,j1 , . . . ,F i,j2 )

X model non-separable in (c2, c02)


Fourier-domain statistical model


pW (l|θ) ∝j2∏

j=j1


j Γ−1j,θ y j

)- y j vectorized Fourier coefficients of lj- Γj,θ = c2 F1,j + c0

2 F2,j parametric spectral density

mGenerative model for y = [yT

j1 , ...,yTj2 ]T

p(y |θ) ∝ |Γθ|−1 exp(−yHΓ−1

θ y)

- Complex Gaussian model y ∼ CN (0,Γθ)

- Γθ = c2F 1 + c02F 2 and F i = block(F i,j1 , . . . ,F i,j2 )

X model non-separable in (c2, c02)


Reparametrization

Non-separable constraints on (c2, c02)

θ ∈ A = {(c2, c02) ∈ R−? × R+

? |Γθ = c2F1 + c02F2 positive-definite}

Design of a linear diffeomorphism ψ

1 mapping joint constraints into independent positivity constraints

ψ : A → R+2?

: θ 7→ ψ(θ) , θ

2 yielding more suitable likelihood

p(y |θ) ∝ |Γθ|−1 exp

(−yHΓ−1

θy)

with

for θ ∈ R+2?

Γθ = θ1F 1 + θ2F 2 positive-definite

θi F i positive-definite

→ separabality of the likelihood via data augmentation


Reparametrization


θ ∈ A = {(c2, c02) ∈ R−? × R+




ψ : A → R+2?

: θ 7→ ψ(θ) , θ



(−yHΓ−1

θy)

with

for θ ∈ R+2?





Reparametrization


θ ∈ A = {(c2, c02) ∈ R−? × R+




ψ : A → R+2?

: θ 7→ ψ(θ) , θ



(−yHΓ−1

θy)

with

for θ ∈ R+2?





Reparametrization


θ ∈ A = {(c2, c02) ∈ R−? × R+




ψ : A → R+2?

: θ 7→ ψ(θ) , θ



(−yHΓ−1

θy)

with

for θ ∈ R+2?





Data augmentation

Definition of an augmented model y |µ, θ2 ∼ CN (µ, θ2F 2) observed data

µ|θ1 ∼ CN (0, θ1F 1) hidden mean

withp(y |θ) =

∫p(y ,µ|θ)dµ

Virtues of the augmented likelihood p(y ,µ|θ)

p(y ,µ|θ) ∝ θ2−NY exp

(− 1θ2

(y−µ)H F−12 (y−µ)

)×θ1

−NY exp(−1θ1µH F

−11 µ)

√ separable in (θ1, θ2)

√ conjugate to inverse-gamma priors


Data augmentation

Definition of an augmented model y |µ, θ2 ∼ CN (µ, θ2F 2) observed data

µ|θ1 ∼ CN (0, θ1F 1) hidden mean

withp(y |θ) =

∫p(y ,µ|θ)dµ

Virtues of the augmented likelihood p(y ,µ|θ)

p(y ,µ|θ) ∝ θ2−NY exp

(− 1θ2

(y−µ)H F−12 (y−µ)

)×θ1

−NY exp(−1θ1µH F

−11 µ)

√ separable in (θ1, θ2)

√ conjugate to inverse-gamma priors


Bayesian model

Bayesian model

- augmented likelihood p(y ,µ|θ)

- conjugate inverse-gamma priors

π(θi ) ∼ IG(α, β), α, β � 1

- Bayes’ theorem

p(θ,µ|y) ∝ p(y ,µ|θ)×∏

i

π(θi )

Bayesian estimator

- minimum mean squared error

θMMSE

= E[θ|y ]

- untractable integration over the posterior

→ computation via a Markov chain Monte Carlo algorithm


Bayesian model

Bayesian model

- augmented likelihood p(y ,µ|θ)

- conjugate inverse-gamma priors

π(θi ) ∼ IG(α, β), α, β � 1

- Bayes’ theorem

p(θ,µ|y) ∝ p(y ,µ|θ)×∏

i

π(θi )

Bayesian estimator

- minimum mean squared error

θMMSE

= E[θ|y ]

- untractable integration over the posterior

→ computation via a Markov chain Monte Carlo algorithm


Markov chain Monte Carlo algorithm

Generation of a collection of samples (θ(k),µ(k))Nmc

k=1 asymptoticallydistributed according to the posterior p(θ,µ|y)

Approximation of the MMSE

θMMSE

≈ 1Nmc − Nbi

Nmc∑k=Nbi+1

θ(k)

Gibbs sampler

- iterative sampling according to conditional distributions

θ1|y ,µ, θ2 ∼ IG(NY ,µ

H F−11 µ

)θ2|y ,µ, θ1 ∼ IG

(NY , (y − µ)H F

−12 (y − µ)

)µ|y , θ ∼ CN

((θ1F 1 Γ−1

θ

)y ,((θ1F 1

)−1+(θ2F 2

)−1)−1)

√ standard distributions→ no Metropolis-Hasting (MH) moves


Numerical simulations

Synthetic multiplicative cascades: 2D Multifractal Random Walk- non-Gaussian process- multifractal properties ∼ Mandelbrot’s multiplicative cascades

c2=-0.09 c

2=-0.07 c

2=-0.05 c

2=-0.03 c

2=-0.01

Estimation setup

- Daubechies’ mother wavelet- j1 ∈ {1, 2}, j2 = ln2 N − 4

Performance assessment

b= E[c2]− c2, s=

√Var[c2], rms=

√b2 + s2

→ computed over 100 independent realizations


Numerical simulations

Synthetic multiplicative cascades: 2D Multifractal Random Walk- non-Gaussian process- multifractal properties ∼ Mandelbrot’s multiplicative cascades

c2=-0.09 c

2=-0.07 c

2=-0.05 c

2=-0.03 c

2=-0.01

Estimation setup

- Daubechies’ mother wavelet- j1 ∈ {1, 2}, j2 = ln2 N − 4

Performance assessment

b= E[c2]− c2, s=

√Var[c2], rms=

√b2 + s2

→ computed over 100 independent realizations


Results: estimation performance

b

0

0.02

0.04 LF

MHG

G

N = 27

s rms

c20

0.02

0.04

−0.08 −0.04

N = 211

c2

−0.08 −0.04

c2

−0.08 −0.04

LF linear regression

MHG space-domain model+ Whittle approximation+ Uniform prior→ Gibbs/MH

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


Results: estimation performance

b

0

0.02

0.04 LF

MHG

G

N = 27

s rms

c20

0.02

0.04

−0.08 −0.04

N = 211

c2

−0.08 −0.04

c2

−0.08 −0.04

LF linear regression

MHG space-domain model+ Whittle approximation+ Uniform prior→ Gibbs/MH

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


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


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


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


Thanks for your attention


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.


Multifractal formalism

Wavelet coefficients {d (m)(j , ·, ·)}m=1,2,3 → wavelet leaders {L(j , ·, ·)}

Scaling exponents ζL(q) q ∈ [q−,q+]

SL(j ,q) = 1nj

∑L(j , ·, ·)q ' 2jζL(q)

Legendre spectrum

DL(h) = minq 6=0

(d + qh − ζL(q)

)≥ D(h)

Polynomial expansion

ζL(q) = c1q + c2q2/2 + c3q3/6 + · · ·

=2qlog2 S( , ):qa

alog2

ζ( )q

q0

D(h)

h0

d


Multifractal formalism

Wavelet coefficients {d (m)(j , ·, ·)}m=1,2,3 → wavelet leaders {L(j , ·, ·)}

Scaling exponents ζL(q) q ∈ [q−,q+]

SL(j ,q) = 1nj

∑L(j , ·, ·)q ' 2jζL(q)

Legendre spectrum

DL(h) = minq 6=0

(d + qh − ζL(q)

)≥ D(h)

Polynomial expansion

ζL(q) = c1q + c2q2/2 + c3q3/6 + · · ·

=2qlog2 S( , ):qa

alog2

ζ( )q

q0

D(h)

h0

d


Gaussian assumption

CMC-LN c2 = −0.08 QQ-plot (process) QQ-plot (leaders)

QQ-plot (log-wavelet modulus) QQ-plot (log-leaders)


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



2. Covariance model




%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



2. Covariance model




%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


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


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

)


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

φj (|ω|;θ) = c2 fj (|ω|) + c02 gj (|ω|), θ = [c2, c0

2 ]T

fj (|ω|) = 2π(J0(rjρ)−J0(3ρ)

ρ2+

3ln(rj 2j/3)J1(3ρ)

ρ)−

2π ln(rj 2j/3)

ln 4

∫ 3


gj (|ω|) = 6πJ1(3|ω|)

ρ− 2π

∫ 3


√nj/4




φj (ω;θ) =



φj (ω;θ) = φj (|ω|;θ) = 2π∫ ∞




2 ]T

fj (|ω|) = 2π(J0(rjρ)−J0(3ρ)

ρ2+

3ln(rj 2j/3)J1(3ρ)

ρ)−

2π ln(rj 2j/3)

ln 4

∫ 3


gj (|ω|) = 6πJ1(3|ω|)

ρ− 2π

∫ 3


√nj/4




φj (ω;θ) =



φj (ω;θ) = φj (|ω|;θ) = 2π∫ ∞




2 ]T

fj (|ω|) = 2π(J0(rjρ)−J0(3ρ)

ρ2+

3ln(rj 2j/3)J1(3ρ)

ρ)−

2π ln(rj 2j/3)

ln 4

∫ 3


gj (|ω|) = 6πJ1(3|ω|)

ρ− 2π

∫ 3


√nj/4


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