20130722

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20130722

  1. 1. PDPTA‘2013/07/24 @ Las Vegas Poisson Observed Image Restoration using a Latent Variational Approximation with Gaussian MRF Hayaru Shouno Univ. of Electro-Communications shouno@uec.ac.jp Masato Okada The Univ. of Tokyo RIKEN, BSI 13年7月22日月曜日
  2. 2. PDPTA‘2013/07/24 @ Las Vegas Contents Introduction Formulation Bayesian image restoration framework Observation Process Prior probability Image restoration algorithm Simulation Results Conclusion & Future works 13年7月22日月曜日
  3. 3. PDPTA‘2013/07/24 @ Las Vegas Introduction Low contrast Image restoration: night vision, radioactive image, ... small number of photon counts Observation with Poisson process. discrete photon counts Inference the parameter of the Poisson process. http://medical-checkup.info/image/spect01.jpg 13年7月22日月曜日
  4. 4. PDPTA‘2013/07/24 @ Las Vegas Introduction It’s just differ the conventional Gaussian image restoration framework! How can we extend the conventional image restoration framework? http://medical-checkup.info/image/spect01.jpg 13年7月22日月曜日
  5. 5. PDPTA‘2013/07/24 @ Las Vegas Contents Introduction Formulation Bayesian image restoration framework Observation Process Prior probability Image restoration algorithm Simulation Results Conclusion & Future works 13年7月22日月曜日
  6. 6. PDPTA‘2013/07/24 @ Las Vegas Conventional Framework: Bayesian image restoration (Geman&Geman 1988, Tanaka 2002) Obtaining the posterior p(x | z) Observation Process p(z | x): Obsevation value z: Gauss noise,Poisson noise,… Noisy Observation Observed zxOriginal p(x) Knowledge p(z| x) Bayes’ Theorem p(x | z) p(z | x) p(x) Prior probability p(x): Property of image MRF, Sparse prior,... 13年7月22日月曜日
  7. 7. PDPTA‘2013/07/24 @ Las Vegas Conventional Framework: Bayesian image restoration (Geman&Geman 1988, Tanaka 2002) Obtaining the posterior p(x | z) Observation Process p(z | x): Obsevation value z: Gauss noise,Poisson noise,… Noisy Observation Observed zxOriginal p(x) Knowledge p(z| x) Bayes’ Theorem p(x | z) p(z | x) p(x) Prior probability p(x): Property of image MRF, Sparse prior,... 13年7月22日月曜日
  8. 8. PDPTA‘2013/07/24 @ Las Vegas Formualation of Observation p(z | x): Poisson noise model Assuming Poisson parameter λm > 0 for each pixel Observed value zm obeys Poisson distribution Problem: Inference for non-negative parameter λm Solution strategy: Considering the corresponding Bernoulli process, and converting the parameter with logit transform (Watanabe & Okada 2011) Noisy Observation Observed zxOriginal p(z| x) λm λm zm Poisson dist. #photonsparameter 13年7月22日月曜日
  9. 9. PDPTA‘2013/07/24 @ Las Vegas Bernoulli process for each pixel Each pixel (indexed by m) contains K on/off events. On/Off events are labeled as ζmk = {+1, -1} with prob. ρm. Observation counts On events labels i.e. ζmk =+1 Observation value zm obeys binomial distribution, also obeys Poisson distribution under the K ρm = λm condition Noisy Observation Observed zxOriginal p(z| x) x Block m y mk K units Binominal dist. Poisson dist. Formulation of Observation p(z | x): Equivalent Bernoulli process λm 13年7月22日月曜日
  10. 10. PDPTA‘2013/07/24 @ Las Vegas Non-negative value ρm are intractable form (for us). Applying logit transform ρm → xm Resting problem: How can we get the evaluation of Non-linear function ln 2cosh xm ? → Local latent variational method (Palmer 2005) Noisy Observation Observed zxOriginal p(z| x) x Block m y mk K units λm Binominal dist. Logit transform p(zm | xm) = K zm exp(zmxm K ln 2 cosh xm) where zm = 2zm K Formulation of Observation p(z | x): Logit transform for the parameters 13年7月22日月曜日
  11. 11. PDPTA‘2013/07/24 @ Las Vegas Approximation with Local latent variational method (1) Local latent variational method (Palmer 2005) Approximation method for super-Gaussian Derived from Legendre transform: convex duality (Rockafellar 1972, Jordan et al. 1999) Pair of Legendre transformation 13年7月22日月曜日
  12. 12. PDPTA‘2013/07/24 @ Las Vegas Assumption: g(u) is convex The lower bound for g(u) - ηu: the point with slope η y(u) = ηu - g*(η) is the tangent line. g(u) u ηu Approximation with Local latent variational method (2) ηu - g*(η) - g*(η) g(u) u slope η g⇤ (⌘) = inf u>0 g(u) ⌘u g⇤ (⌘) = sup u>0 ⌘u g(u) 13年7月22日月曜日
  13. 13. PDPTA‘2013/07/24 @ Las Vegas Assumption: g(u) is convex The collection of tangent lines of g(u): envelope g(u): the maximum in the tangent lines: ηu - g*(η) Approximation with Local latent variational method (2): contd. u g(u) g(u) u u g(u) = sup ⌘>0 ⌘u g⇤ (⌘) 13年7月22日月曜日
  14. 14. PDPTA‘2013/07/24 @ Las Vegas Formulation of Observation p(z | x) : Approximated as Gaussian Evaluation of ln 2cosh xm Quadratic form of xm with variational parameter ξm Upper bound can be treated as the Gaussian form Noisy Observation Observed zxOriginal p(z| x) Quadratic form can be regarded as Gaussian form p(zm | xm) = K zm exp(zmxm K ln 2 cosh xm) where zm = 2zm K -2 -1 0 1 2 -25-20-15-10-5 Comparison of Likelihoods x True( ln2cosh(x) type) Lat.( xi=-0.5 ) Lat.(xi=1.5) p(zm | xm) p m (zm | xm) = K zm exp zmxm K tanh m 2 m (x2 m 2 m) + ln 2 cosh m 13年7月22日月曜日
  15. 15. PDPTA‘2013/07/24 @ Las Vegas Formulation of Observation p(z | x) : Approximated as Gaussian (contd.) Evaluation of ln 2cosh xm Quadratic form of xm with variational parameter ξm Upper bound can be treated as the Gaussian form Noisy Observation Observed zxOriginal p(z| x) Quadratic form can be regarded as Gaussian form p(zm | xm) p m (zm | xm) = K zm exp zmxm K tanh m 2 m (x2 m 2 m) + ln 2 cosh m p⇠(z | x) = Y m p⇠m (zm | xm) / exp 1 2 xT ⌅x + zT x + 1 2 ⇠T ⌅⇠ K X ln 2 cosh ⇠m ! ⌅ = diag[K tanh(⇠m)/⇠m] 13年7月22日月曜日
  16. 16. PDPTA‘2013/07/24 @ Las Vegas Noisy Observation Observed zxOriginal Conventional Framework: Bayesian image restoration (Geman&Geman 1988, Tanaka 2002) Obtaining the posterior p(x | z) Observation Process p(z | x): Obsevation value z: Gauss noise,Poisson noise,… p(x) Knowledge p(z| x) Bayes’ Theorem p(x | z) p(z | x) p(x) Original x Prior probability p(x): Property of image MRF, Sparse prior,... 13年7月22日月曜日
  17. 17. PDPTA‘2013/07/24 @ Las Vegas Prior formulation: Gaussian MRF p(x) Knowledge Bayes’Theorem Noisy Observation Observed zxOriginal p(z| x) Reconstruct x xm zm Gaussian Markov Random Field (GMRF) Energy function: Quadrature of the difference of neighbor pixels Prior prob. can be described as Boltzmann distribution Hyper-parameter α should be controlled properly What is image?: GMRF assumes smoothness between neighbors xn HL2(x) = 1 2 X (m,n) (xm xn)2 pL2(x) = 1 Z(↵, h) exp ( ↵HL2(x)) exp( hkxk2 ) = 1 Z(↵, h) exp 1 2 xT (↵⇤ + hI)x ! 13年7月22日月曜日
  18. 18. PDPTA‘2013/07/24 @ Las Vegas Image restoration with Posterior p(x) Knowledge Bayes’Theorem Noisy Observation Observed zxOriginal p(z| x) Reconstruct x Prior Prob. Observation Process x z How can we get {α, h, Ξ} ? → EM method Posterior Mean m as the restored image p (z | x) exp 1 2 xT x + zT x + 1 2 T K ln 2 cosh m p(x | ↵, h) / exp 1 2 xT (↵⇤ + hI)x ! Posterior Product of Gaussians →Yes, it’s also Gaussian :-) p⇠(x |z, ↵, h) ⇠ N(x | m, (⌅ + ↵⇤ + hI) 1 ) m = (⌅ + ↵⇤ + hI) 1 z 13年7月22日月曜日
  19. 19. PDPTA‘2013/07/24 @ Las Vegas Hyper-parameter inferece by EM method: (Dempster 1977, Neal & Hinton 1999) x : Hidden variables θ = {α, h, Ξ} : inference parameters Maximizing of the Q function E-step: Calculation of expectation with posterior p(x|z,θ(t)) M-step: Maximizing the Q for inference parameters θ Expectation operator is evaluated by the posterior x z from prior from observation Q(✓ | ✓(t) ) = D ln p⇠(x, z | ✓) E(t) = ln |↵⇤ + hI| 2 1 2 D xT (↵⇤ + hI)x E(t) * 1 2 xT ⌅x zT x +(t) + 1 2 ⇠T ⌅⇠ K X m ln 2 cosh ⇠m 13年7月22日月曜日
  20. 20. PDPTA‘2013/07/24 @ Las Vegas Contents Introduction Formulation Bayesian image restoration framework Observation Process Prior probability Image restoration algorithm Simulation Results Conclusion & Future works 13年7月22日月曜日
  21. 21. PDPTA‘2013/07/24 @ Las Vegas Application to Artificial Image Simulation environment Original image: Fourier Basis Observation: Poisson process Control parameter: contrast Lmax/Lmin (Poisson parameter) Original image x Observed z λmax/λmin= 80/5 λmax/λmin= 20/5 x y Lambda λmax λmin contrast 13年7月22日月曜日
  22. 22. PDPTA‘2013/07/24 @ Las Vegas Restoration example λmax/λmin 20/2 λmax/λmin 80/2 Low contrast (High noise) Original x Observed z Restored x’ High contrast (Low noise) Restoration mechanism looks work well 13年7月22日月曜日
  23. 23. PDPTA‘2013/07/24 @ Las Vegas Quantitative Evaluation with PSNR value Evaluation with Peak Signal to Noise Ratio (PSNR) Improvement of the PSNR in any noise ratio High improvement is confirmed in the low contrast area. 20 40 80 160 202224262830 PSNR for Artificial Image Contrast(Lambda Max) PSNR[dB] Restored Observed High noise Low noise Good Poor 13年7月22日月曜日
  24. 24. PDPTA‘2013/07/24 @ Las Vegas Natural image application Original image: part of “Cameraman” Observation: Poisson noise controlled by contrast Original image x Observed image z 13年7月22日月曜日
  25. 25. PDPTA‘2013/07/24 @ Las Vegas Restoration result: L=20(High noise), 80(Low noise) cases High noise Low noise Original image x Observed image z Restored image x’ 13年7月22日月曜日
  26. 26. PDPTA‘2013/07/24 @ Las Vegas Contents Introduction Formulation Bayesian image restoration framework Observation Process Prior probability Image restoration algorithm Simulation Results Conclusion & Future works 13年7月22日月曜日
  27. 27. PDPTA‘2013/07/24 @ Las Vegas Summary We treat Poisson process observation image as following: Deriving the corresponding binomial distribution Applying logit transformation for the parameter Applying latent variational method We introduce the Gaussian MRF as the prior probability. Inferring the hyper-parameters as well as the restoring image, we introduce an EM algorithm. In the simple computer simulation, we confirm the restoration ability of our algorithm. 13年7月22日月曜日
  28. 28. PDPTA‘2013/07/24 @ Las Vegas Future works We should compare with the conventional Gaussian image restoration. Consideration of the Appropriateness of the prior? Is the Gaussian type prior appropriate one? Introducing of TotalVariation, Sparse prior. Finding application problems Medical imaging, such like SPECT, PET and so on. 13年7月22日月曜日

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