Probabilistic Image Processing by Extended Gauss-Markov Random - PowerPoint PPT Presentation

Probabilistic Image Processing by Extended Gauss-Markov Random Fields Kazuyuki Tanaka, Kazuyuki Tanaka Muneki Yasuda Yasuda, Nicolas Morin Nicolas Morin Muneki Graduate School of Information Sciences, Tohoku University, Japan and D. M. Titterington Department of Statistics, University of Glasgow, UK Department of Statistics, University of Glasgow, UK SSP2009, Cardiff, UK 3 September, 2009 1

Image Restoration by Bayesian Statistics Noise Assumption 2: Degraded Transmission images are randomly Posterior generated from the original Degraded Original image by according to the Image Image Estimate conditional probability of Assumption 1: Original degradation process. images are randomly generated by according to a Bayes Formula prior probability. Posterior 6 4 4 4 4 4 4 4 7 4 4 4 4 4 4 4 8 Pr{ Original Image | Degraded Image } Degradatio n Process Prior 6 4 4 4 4 4 4 4 7 4 4 4 4 4 4 4 8 6 4 4 4 7 4 4 4 8 ∝ Pr{ Degraded Image | Original Image } Pr{ Original Image } SSP2009, Cardiff, UK 3 September, 2009 2

Bayesian Image Analysis Posterior 6 4 4 4 4 4 4 4 7 4 4 4 4 4 4 4 8 Assumption 1: Prior Probability Pr{ Original Image | Degraded Image } consists of a product of functions Likelihood 6 4 4 4 4 4 4 4 7 4 4 4 4 4 4 4 8 defined on the neighbouring pixels. ∝ Pr{ Degraded Image | Original Image } Prior 6 4 4 4 7 4 4 4 8 Prior Probability × Pr{ Original Image } ⎛ ⎞ ⎛ ⎞ r ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ r r r 1 1 1 ∏ ∏ = α ∝ − α − − αγ − = − α − α γ ⎜ 2 ⎟ ⎜ 2 ⎟ ⎜ T ⎟ X x x x x x x x Pr{ | } exp ( ) exp ( ) exp ( I C ( )) ⎜ ⎟ ⎜ ⎟ i j i j ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 2 2 2 ∈ ∈ ⎝ i j E ⎠ ⎝ i j E ⎠ { , } { , } 1 2 + γ = ∈ ⎧ i j V 4 4 , ⎪ α > − ∈ i j E ⎪ 1 , { , } 0 γ = 1 i j ⎨ C ( ) − γ ∈ i j E , { , } ⎪ 2 ⎪ ⎩ 0 , otherwise α = α = α = 0 . 0001 0 . 0005 0 . 0030 Gibbs Sampler γ = 0 γ = − 0 . 45 SSP2009, Cardiff, UK 3 September, 2009 3

Bayesian Image Analysis Posterior 6 4 4 4 4 4 4 4 7 4 4 4 4 4 4 4 8 Assumption 2: Degraded image is Pr{ Original Image | Degraded Image } generated from the original image by Likelihood 6 4 4 4 4 4 4 4 7 4 4 4 4 4 4 4 8 Additive White Gaussian Noise. ∝ Pr{ Degraded Image | Original Image } Prior 6 4 4 4 7 4 4 4 8 × Pr{ Original Image } r r ⎛ ⎞ ⎛ ⎞ r r r r 1 1 ∏ 2 = = σ ∝ − − = − − Y y X x ⎜ x y 2 ⎟ ⎜ x y ⎟ Pr{ | , } exp ( ) exp i i σ σ ⎝ 2 ⎠ ⎝ 2 ⎠ 2 2 ∈ i V V :Set of all the pixels σ > 0 3 September, 2009 SSP2009, Cardiff, UK 4

Bayesian Image Analysis Degraded Original r r Degradation Process Image Image r r r y r r r x = α γ = = σ X x Y y X x Pr{ | , } Pr{ | , } g Prior Probability Posterior Probability Estimate + γ = ∈ Bayesian Network ⎧ i j V 4 4 , ⎪ − ∈ ⎪ i j E r r 1 , { , } r r 1 γ = i j = = α γ σ ⎨ X x Y y C ( ) Pr{ | , , , } − γ ∈ i j E , { , } r r r ⎪ r r r 2 = = σ = α γ Y y X x X x Pr{ | , } Pr{ , , } ⎪ = r ⎩ 0 , otherwise r = α γ σ Y y Pr{ | , , } ⎛ ⎞ ⎜ ⎟ 1 1 1 ∑ ∑ ∑ ∝ − − − α − − αγ − x y 2 x x 2 x x 2 exp ( ) ( ) ( ) ⎜ ⎟ i i i j i j ⎜ σ ⎟ 2 2 2 2 ∈ ∈ ∈ ⎝ i V i j E i j E ⎠ { , } { , } 1 2 ⎛ ⎞ r r r r 1 1 2 = − − − α γ ⎜ x y x x T ⎟ exp C ( ) σ ⎝ 2 ⎠ 2 2 Data Dominant Smoothing r r r r r ∈ −∞ +∞ X ( , ) ∫ i = = = α γ σ x x X x Y y d x ˆ Pr{ | , , , } i i ⇒ Gauss Markov Random Field Model 3 September, 2009 SSP2009, Cardiff, UK 5

Average of Posterior Probability Average of the posterior probability can be calculated by using the multi-dimensional Gauss integral Formula r r r +∞ +∞ +∞ r r r r ∫ ∫ ∫ ˆ = α γ σ = = = α σ x X z X z Y y dz dz dz L L , , Pr{ | , , } V 1 2 | | − ∞ − ∞ − ∞ ⎛ ⎞ ⎛ ⎞ ( ) ⎛ ⎞ + ∞ + ∞ + ∞ r r r 1 r I r I ∫ ∫ ∫ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ − − + ασ − z z y 2 z T y T dz dz dz L exp I C L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ V 1 2 | | − ∞ − ∞ − ∞ σ + ασ γ + ασ γ 2 2 2 2 ⎝ I C ( ) ⎠ ⎝ I C ( ) ⎠ ⎝ ⎠ = ⎛ ⎞ ⎛ ⎞ ( ) ⎛ ⎞ + ∞ + ∞ + ∞ r r r r 1 I I ⎜ ⎟ ∫ ∫ ∫ ⎜ ⎟ ⎜ ⎟ − − + ασ − 2 T T z y z y dz dz dz L L exp I C ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ V 1 2 | | − ∞ − ∞ − ∞ σ + ασ γ + ασ γ 2 2 2 ⎝ ⎠ ⎝ ⎠ ⎝ 2 I C ( ) I C ( ) ⎠ r r I I = = T y y + ασ γ + ασ γ 2 2 I C ( ) I C ( ) Gaussian Integral formula V | | 6 4 7 4 8 ⎛ ⎞ + ∞ + ∞ + ∞ r r r r 1 r r ∫ ∫ ∫ − μ − − μ − ασ γ − μ = z ⎜ z 2 z T ⎟ dz dz dz , , L L L ( ) exp ( )( I C ( ))( ) ( 0 0 , 0 ) V 1 2 | | − ∞ − ∞ − ∞ ⎝ ⎠ 2 SSP2009, Cardiff, UK 3 September, 2009 6

Statistical Estimation of Hyperparameters Hyperparameters α, σ are determined so as to maximize the marginal likelihood Pr{ Y = y| α , γ , σ } with respect to α , σ. r r α σ = = α γ σ Y y ˆ ˆ ( , ) arg max Pr{ | , , } ( ) α σ , r r r r r r r r r ∫ = α γ σ = = = σ = α γ Y y Y y X x X z d z Pr{ | , , } Pr{ | , } Pr{ | , } Original Image Degraded Image r r r r r r r r y = α γ = = σ x X x Y y X x Pr{ | , } Pr{ | , } g Marginalized with respect to X r r = α γ σ Y y Pr{ | , , } Marginal Likelihood SSP2009, Cardiff, UK 3 September, 2009 7

Statistical Estimation of Hyperparameters r α γ σ r r r r Z y ( , , , ) r r r r r ∫ POS = α γ σ = = = σ = α γ = Y y Y y X z X z d z Pr{ | , , } Pr{ | , } Pr{ | , } V πσ σ 2 | | / 2 Z ( 2 ) ( ) PR π V ⎛− ⎞ | | ≡ ∫ + ∞ + ∞ + ∞ 1 r r ( 2 ) ∫ ∫ α α γ = ⎜ T ⎟ Z z z dz dz dz L L ( ) exp C ( ) PR V 1 2 | | − ∞ − ∞ − ∞ α V γ ⎝ ⎠ | | 2 det C ( ) r α γ σ Z y ( , , , ) POS ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ( ) γ + ∞ + ∞ + ∞ r r r r r r 1 I I 1 C ( ) ∫ ∫ ∫ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ = − − + ασ γ − − α z y 2 z T y T y y T dz dz dz L L exp I C ( ) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ V 1 2 | | − ∞ − ∞ − ∞ σ + ασ γ + ασ γ + ασ γ 2 2 2 2 2 2 ⎝ I C ( ) ⎠ ⎝ I C ( ) ⎠ I C ( ) ⎝ ⎠ ⎛ ⎞ γ 1 r ( ) r C ⎜ ⎟ = − α T y y exp ⎜ ⎟ + ασ γ 2 ⎝ 2 ⎠ I C ( ) ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ( ) + ∞ + ∞ + ∞ r r r r 1 I I ∫ ∫ ∫ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ × − − + ασ γ − z y 2 z T y T dz dz dz L exp I C ( ) L ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ V 1 2 | | − ∞ − ∞ − ∞ σ + ασ γ + ασ γ 2 2 2 2 ⎝ I C ( ) ⎠ ⎝ I C ( ) ⎠ ⎝ ⎠ V ⎛ ⎞ πσ γ 2 | | ( 2 ) 1 r C ( ) r ⎜ ⎟ = − α T y y exp ⎜ ⎟ + ασ γ + ασ γ 2 2 2 det( I C ( )) ⎝ I C ( ) ⎠ V Gaussian π ⎛ ⎞ | | + ∞ + ∞ + ∞ r r 1 r r ( 2 ) ∫ ∫ ∫ − − μ − μ = ⎜ z z T ⎟ dz dz dz L L exp ( ) A ( ) V Integral formula 1 2 | | − ∞ − ∞ − ∞ ⎝ ⎠ 2 det A SSP2009, Cardiff, UK 3 September, 2009 8