BAYESIAN JOINT ESTIMATION OF THE MULTIFRACTALITY PARAMETER OF IMAGE PATCHES USING GAMMA MARKOV RANDOM FIELD PRIORS S. Combrexelle 1 , H. Wendt 1 , Y. Altmann 2 , J.-Y. Tourneret 1 , S. McLaughlin 2 and P. Abry 3 1 IRIT, CNRS, Toulouse Univ., France [email protected] 2 School Engineering Physical Sci., Heriot-Watt Univ., Edinburgh, UK, [email protected] 3 Physics Dept., CNRS, ENS Lyon, France [email protected] Summary Multifractal (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 smoothing gamma 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 - H ¨ older exponent: Roughness of image X around t 0 -! comparison to a local power law behavior -! ||X (t) - X (t 0 )|| C ||t - t 0 || ↵ C> 0, ↵ > 0 -! largest such ↵: H¨ older exponent h(t 0 ) X (t 0 ) ⇢ rough h(t 0 ) ⇠ 0 smooth h(t 0 ) ⇠ 1 - Multifractal spectrum: D (h) = dim Hausdor↵ {t : h(t)= h} -! geometric structure of iso-H¨ older sets of points t i t i h(t i ) = 0.2 0.2 D(h) h 0 d - Wavelet-leaders: Local supremum of wavelet coefficients d (m) X (j, k ) – 2D DWT coefficients (m =1, 2, 3) 9λ j,k – dyadic cube centred at k 2 j and its 8 neighbors l (j, k )= sup m2(1,2,3),λ 0 ⇢9λ j,k |d (m) X (λ 0 )| - Multifractal formalism: - D (h) 1 - (h - c 1 ) 2 /(2|c 2 |)+ ··· C p (j )= c 0 p + c p ln 2 j x p-th cumulant of ln l (j, k ) - c 2 – multifractality parameter c 1 c 2 D(h) h 0 d C 2 (j ) = Var [ln l (j, k )] = c 0 2 + c 2 ln 2 j -! 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 (Δk ; ✓ ) – parametrized by ✓ =[✓ 1 , ✓ 2 ] T =[c 2 ,c 0 2 ] T – assuming independence between scales: p(`|✓ )= j 2 Y j =j 1 p(` j |✓ ) / j 2 Y j =j 1 |⌃(✓ )| - 1 2 e ⇣ - 1 2 ` T j ⌃(✓ ) -1 ` j ⌘ , ` =[` T j 1 , ..., ` T j 2 ] T ⌃(✓ ) - covariance matrix induced by % j (Δk ; ✓ ) – Whittle approximation of p(` j |✓ ) p(` j |✓ ) / exp ⇣ - X m log φ j (m; ✓ )+ y ⇤ j (m)y j (m) φ j (m; ✓ ) ⌘ y j = DFT (` j ) - Fourier transform of centered log-leaders φ j (m; ✓ ) - spectral density associated with % j (Δk ; ✓ ) - Data augmented model in the Fourier domain: 1 [Combrexelle16] – statistical interpretation of Whittle approximation -! -! y =[y T j 1 , ..., y T j 2 ] T complex Gaussian random variable – reparametrization v = (✓ ) 2 R +2 ? y ⇠ CN (v 1 ˜ F + v 2 ˜ G) independent positivity contraints on v i ˜ F and ˜ G - diagonal, positive-definite, known and fixed – data augmentation of the model ( y |μ,v 2 ⇠ CN (μ,v 2 ˜ G) observed data μ|v 1 ⇠ CN (0,v 1 ˜ F ) hidden mean -! associated with augmented likelihood p(y , μ|✓ ) / v 2 -N Y exp ⇣ - 1 v 2 (y - μ) H ˜ G -1 (y - μ) ⌘ ⇥ v 1 -N Y exp ⇣ - 1 v 1 μ H ˜ F -1 μ ⌘ p inverse-gamma IG priors on v i 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: - {X k=(k x ,k y ) } - partition of image X into non-overlapping patches - Y = {y k } - collection of Fourier coefficients - M = {μ k } - collection of latent variables - V i = {v i,k } - collection of MF parameters + assuming indepence between patches -! p(Y , M |V ) / Y k p(y k , μ k |v k ), V = {V 1 , V 2 } - Gamma Markov random field (GMRF) prior: - assuming smooth spatial evolution of MF parameters - positive auxiliary variables Z = {Z 1 , Z 2 }, Z i = {z i,k }, i =1, 2 -! induce positive correlation between neighbouring v i,k - each v i,k is connected to 4 variables z i,k 0 k 0 2 V v (k )= {(k x ,k y )+(i x ,i y )} i x ,i y =0,1 via edges with weights a i - each z i,k is connected to 4 variables v i,k 0 k 0 2 V z (k )= {(k x ,k y )+(i x ,i y )} i x ,i y =-1,0 k x k y v i,k z i,k ′ − → − → − → − → - associated distribution [Dikmen10] p(V i , Z i |a i )= 1 C (a i ) Y k e -(4a i +1) log v i,k e (4a i -1) log z i,k ⇥e - a i v i,k P k 0 2V v (k) z i,k 0 p conditionals for v i,k (z i,k ) are inverse-gamma IG (gamma G ) - Posterior distribution & Bayesian estimator: - Posterior distribution via Bayes’ theorem p(V , Z , M |Y ,a i ) / p(Y |V 2 , M ) p(M |V 1 ) | {z } augmented likelihood ⇥ Y i p(V i , Z i |a i ) | {z } independent GMRF priors - Marginal posterior mean (MMSE) estimator V MMSE i = E[V i |Y ,a i ] | {z } untractable ⇡ 1 N mc - N bi X N mc q =N bi V (q ) i with {V (q ) i } N mc q =1 distributed according to p(V , Z , M |Y ,a i ) -! computation via Markov chain Monte Carlo algorithm - Gibbs sampler: [Robert05] -! iterative sampling according to conditional distributions μ k | Y ,V ⇠ CN ⇣ v 1,k ˜ F Γ -1 v k y k , ⇣ (v 1,k ˜ F ) -1 +(v 2,k ˜ G) -1 ⌘ -1 ⌘ v 1,k | M ,Z 1 ⇠ IG ⇣ N Y +4a i , μ H k ˜ F -1 μ k + β 1,k ⌘ v 2,k | Y ,M ,Z 2 ⇠ IG ⇣ N Y +4a i , (y k - μ k ) H ˜ G -1 (y k - μ k )+ β 2,k ⌘ z i,k | V i ⇠ G (4a i , ˜ β i,k ) with β i,k = a i P k 0 2V v (k) z i,k 0 and ˜ β i,k =(a i P k 0 2V z (k) v -1 i,k 0 ) -1 p standard distributions -! no acceptance/reject moves . Numerical experiments - Scenario - synthetic multifractal 2048 ⇥ 2048 image -! 2D multifractal random walk (MRW) -piece-wise constant values of c 2 2 {-0.04, -0.02} - decomposition into non-overlapping 64 ⇥ 64 patches - Illustration on a single realization - estimates of c 2 - 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 ⇠ 10 IG 0.0018 0.0123 0.0125 76.5 ⇠ 50 GMRF 0.0027 0.0032 0.0044 94.6 ⇠ 50 Conclusion & Future work Take-away message - smooth joint estimation of c 2 for image patches - GMRF prior inducing positive correlation - STD/RMSE reduced by ⇠ 4 - 10 vs. state-of-the-art estimators -efficient MCMC algorithm (only ⇠ 5 times LF) Future work - automatic estimation of a i - additional MF parameters (c 1 , c 3 ,...)