BAYESIAN JOINT ESTIMATION OF THE MULTIFRACTALITY PARAMETER OFIMAGE PATCHES USING GAMMA MARKOV RANDOM FIELD PRIORS

S. Combrexelle1, H. Wendt1, Y. Altmann2, J.-Y. Tourneret1, S. McLaughlin2 and P. Abry31 IRIT, CNRS, Toulouse Univ., France first.lastname@irit.fr

2 School Engineering Physical Sci., Heriot-Watt Univ., Edinburgh, UK, initial.lastname@hw.ac.uk3 Physics Dept., CNRS, ENS Lyon, France patrice.abry@ens-lyon.fr

SummaryMultifractal (MF) analysis enables homogeneous image texture to be modeled and studied via the regularity fluctuations of

image amplitudes. In this work, we propose a Bayesian approach for the local (patch-wise) MF analysis of images with

heterogeneous MF properties. To this end, a joint Bayesian model for image patches is formulated using spatially smoothinggamma Markov Random Field priors, yielding a fast algorithm and significantly improving the state-of-the-art performance.

Numerical simulations based on synthetic MF images illustrate the benefits of the proposed MF analysis procedure for images.

Multifractal analysis-Holder exponent:

Roughness of image X around t0�! comparison to a local power law behavior

�! ||X(t)�X(t0)|| C||t� t0||↵ C > 0, ↵ > 0

�! largest such ↵: Holder exponent h(t0)

X(t0)

⇢rough h(t0) ⇠ 0

smooth h(t0) ⇠ 1

-Multifractal spectrum:

D(h) = dimHausdor↵{t : h(t) = h}�! geometric structure of iso-Holder sets of points ti

ti

h(ti) = 0.2

0.2

D(h)

h0

d

-Wavelet-leaders:

Local supremum of wavelet coe�cients

d(m)X (j,k) – 2D DWT coe�cients (m = 1, 2, 3)

9�j,k – dyadic cube centred at k2j and its 8 neighbors

l(j,k) = supm2(1,2,3),�0⇢9�j,k

|d(m)X (�0)|

-Multifractal formalism:

-D(h) 1� (h�c1)2/(2|c2|)+ · · ·

Cp(j) = c0p + cp ln 2j

x p-th cumulant of ln l(j,k)- c2 – multifractality parameter c1

c2

D(h)

h0

d

C2(j) = Var [ln l(j,k)] = c02 + c2 ln 2j

�! classical estimation by linear regression (LF)

Statistical model for log-leaders-Time-domain statistical model: [Combrexelle15]

Centered log-leaders `j = [log l(j, ·)] ⇠ Gaussian random field

– radial symmetric covariance model %j(�k;✓)– parametrized by ✓ = [✓1, ✓2]

T = [c2, c02]T

– assuming independence between scales:

p(`|✓) =j2Y

j=j1

p(`j|✓)/j2Y

j=j1

|⌃(✓)|�12 e

⇣�12`

Tj ⌃(✓)�1`j

⌘

, ` = [`Tj1, ..., `Tj2]T

⌃(✓) - covariance matrix induced by %j(�k;✓)

– Whittle approximation of p(`j|✓)

p(`j|✓) / exp⇣�X

m

log �j(m;✓) +y⇤j(m)yj(m)

�j(m;✓)

⌘

yj = DFT (`j) - Fourier transform of centered log-leaders

�j(m;✓) - spectral density associated with %j(�k;✓)

-Data augmented model in the Fourier domain:

1 [Combrexelle16]– statistical interpretation of Whittle approximation

�! �! y = [yTj1, ...,yT

j2]T complex Gaussian random variable

– reparametrization v = (✓) 2 R+2?

y ⇠ CN (v1F + v2G)

independent positivity contraints on viF and G - diagonal, positive-definite, known and fixed

– data augmentation of the model(y|µ, v2 ⇠ CN (µ, v2G) observed data

µ|v1 ⇠ CN (0, v1F ) hidden mean

�! associated with augmented likelihood

p(y,µ|✓) / v2�NY exp

⇣� 1

v2(y � µ)HG

�1(y � µ)

⌘

⇥ v1�NY exp

⇣� 1

v1µHF

�1µ⌘

pinverse-gamma IG priors on vi are conjugate

[Robert05] Monte Carlo Statistical Methods, Springer, New York, USA, 2005

[Dikmen10] Gamma Markov Random Fields for Audio Source Modeling, IEEE T. Audio,

Speech, and Language Proces., vol. 18, no. 3, pp. 589-601, March 2010

[Combrexelle15] Bayesian estimation of the multifractality parameter for image texture using

a Whittle approximation, IEEE T. Image Proces., vol. 24, no. 8, pp. 2540-2551, Aug. 2015

[Combrexelle16] A Bayesian framework for the multifractal analysis of images using data

augmentation and a Whittle approximation, Proc. ICASSP, Shanghai, China, March 2016

ICIP 2016 – Phoenix, Arizona, USA — September 25-28, 2016

.Bayesian model for image patches- Likelihood:

-{Xk=(kx,ky)} - partition of imageX into non-overlapping patches

-Y ={yk} - collection of Fourier coe�cients-M ={µk} - collection of latent variables

-V i={vi,k} - collection of MF parameters

+ assuming indepence between patches

�! p(Y ,M |V ) /Y

k

p(yk,µk|vk), V = {V 1,V 2}

- Gamma Markov random field (GMRF) prior:

- assuming smooth spatial evolution of MF parameters

- positive auxiliary variables Z = {Z1,Z2}, Z i = {zi,k}, i = 1, 2

�! induce positive correlation between neighbouring vi,k

- each vi,k is connected to 4 variables zi,k0

k0 2 Vv(k) = {(kx, ky) + (ix, iy)}ix,iy=0,1via edges with weights ai

- each zi,k is connected to 4 variables vi,k0

k0 2 Vz(k) = {(kx, ky) + (ix, iy)}ix,iy=�1,0

kx

ky

vi,k

zi,k

′

−→−→

−→−→

- associated distribution [Dikmen10]

p(V i,Z i|ai)=1

C(ai)

Yke�(4ai+1) log vi,k e(4ai�1) log zi,k ⇥e

� aivi,k

Pk02Vv(k) zi,k0

pconditionals for vi,k (zi,k) are inverse-gamma IG (gamma G)

- Posterior distribution & Bayesian estimator:

- Posterior distribution via Bayes’ theorem

p(V ,Z,M |Y , ai) / p(Y |V 2,M ) p(M |V 1)| {z }augmented likelihood

⇥Y

ip(V i,Z i|ai)| {z }

independent GMRF priors

-Marginal posterior mean (MMSE) estimator

V MMSEi = E[V i|Y , ai]| {z }

untractable

⇡ 1

Nmc �Nbi

XNmc

q=Nbi

V (q)i

with {V (q)i }Nmc

q=1 distributed according to p(V ,Z,M |Y , ai)

�! computation via Markov chain Monte Carlo algorithm

- Gibbs sampler: [Robert05]

�! iterative sampling according to conditional distributions

µk |Y ,V⇠CN⇣v1,kF��1

vkyk,

⇣(v1,kF )�1+(v2,kG)�1

⌘�1⌘

v1,k |M ,Z1⇠IG⇣NY+4ai,µ

Hk F

�1µk+�1,k

⌘

v2,k |Y ,M ,Z2⇠IG⇣NY+4ai, (yk � µk)

HG�1(yk � µk)+�2,k

⌘

zi,k |V i⇠G(4ai, �i,k)

with �i,k=aiP

k02Vv(k)zi,k0 and �i,k=(ai

Pk02Vz(k)

v�1i,k0)�1

pstandard distributions �! no acceptance/reject moves

.Numerical experiments-Scenario

- synthetic multifractal 2048⇥ 2048 image

�! 2D multifractal random walk (MRW)

-piece-wise constant values of c2 2 {�0.04,�0.02}- decomposition into non-overlapping 64⇥ 64 patches

- Illustration on a single realization

- estimates of c2

- k-means classification

-Estimation performance

evaluated on 100 realizations of heterogeneous 2D MRW

|bias| STD RMSE classification % time (s)

LF 0.0055 0.0406 0.0413 54.2 ⇠ 10IG 0.0018 0.0123 0.0125 76.5 ⇠ 50

GMRF 0.0027 0.0032 0.0044 94.6 ⇠ 50

Conclusion & Future workTake-away message- smooth joint estimation of c2 for image patches

-GMRF prior inducing positive correlation

- STD/RMSE reduced by ⇠ 4� 10 vs. state-of-the-art estimators

- e�cient MCMC algorithm (only ⇠ 5 times LF)

Future work- automatic estimation of ai- additional MF parameters (c1, c3, . . . )