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- 1. Natural Image Statistics Siwei Lyu SUNY Albany CIFAR NCAP Summer School 2009 August 7, 2009 Thanks to Eero Simoncelli for sharing some of the slidesMonday, August 10, 2009 1
- 2. big numbersMonday, August 10, 2009 2
- 3. big numbers • seconds since big bang: ~ 1017Monday, August 10, 2009 2
- 4. big numbers • seconds since big bang: ~ 1017 • atoms in the universe: ∼1080Monday, August 10, 2009 2
- 5. big numbers • seconds since big bang: ~ 1017 • atoms in the universe: ∼1080 • 65×65 8-bit gray-scale images: ∼1010000Monday, August 10, 2009 2
- 6. ... ...Monday, August 10, 2009 3
- 7. ... ... “The distribution of natural images is complicated. Perhaps it is something like beer foam, which is mostly empty but contains a thin mesh-work of ﬂuid which ﬁlls the space and occupies almost no volume. The ﬂuid region represents those images which are natural in character.” [Ruderman 1996]Monday, August 10, 2009 3
- 8. natural image statistics • natural images are rare in image space • they distinguish by nonrandom structures • common statistical properties of natural images is the focal element in the study of natural image statisticsMonday, August 10, 2009 4
- 9. why are we interested in natural image statistics? Visual! cortex LGN RetinaMonday, August 10, 2009 5
- 10. why are we interested in natural image statistics? Visual! cortex LGN RetinaMonday, August 10, 2009 5
- 11. why are we interested in natural image statistics? Visual! cortex LGN RetinaMonday, August 10, 2009 5
- 12. why are we interested in natural image statistics? Visual! cortex LGN RetinaMonday, August 10, 2009 5
- 13. (a) = ? ! + noise (b) (c)Monday, August 10, 2009 6
- 14. (a) = ? ! + noise (b) (c)Monday, August 10, 2009 6
- 15. computer vision applications • image restoration - de-noising, de-blurring and de-mosaicing, super-resolution and in-painting • image compression • texture synthesis • image segmentation • features for object detection and classiﬁcation (SIFT, gist, “primal sketch”, saliency, etc) • many othersMonday, August 10, 2009 7
- 16. optimization math perception machine! neuro-! learning science biology statistics natural! image! statistics computer! signal! science processing image! processing computer! visionMonday, August 10, 2009 8
- 17. scope of this tutorial • important developments following a general theme • focusing on concepts - light on math or speciﬁc applications • gray-scale intensity image, do not cover - color - time (video) - multi-image information (stereo)Monday, August 10, 2009 9
- 18. main components representationMonday, August 10, 2009 10
- 19. Neural characterization representation Transform Representationdients:• stimuli Monday, August 10, 2009 11
- 20. why representation matters? • example (from David Marr) • representation for numbers • Arabic: 123 • Roman: MCXXIII • binary: 1111011 • English: one hundred and twenty threeMonday, August 10, 2009 12
- 21. why representation matters? • example (from David Marr) • representation for numbers • Arabic: 123 × 10 • Roman: MCXXIII × X • binary: 1111011 × 110 • English: one hundred and twenty three × tenMonday, August 10, 2009 13
- 22. why representation matters? • example (from David Marr) • representation for numbers • Arabic: 123 × 4 • Roman: MCXXIII × IV • binary: 1111011 × 100 • English: one hundred and twenty three × fourMonday, August 10, 2009 14
- 23. linear representations • pixel • Fourier • wavelet - localized - orientedMonday, August 10, 2009 15
- 24. main components observations representationMonday, August 10, 2009 16
- 25. image data • calibrated - linearized response • relatively large number [van Hateren & van der Schaaf, 1998]Monday, August 10, 2009 17
- 26. observations • second-order pixel correlations • 1/f power law of frequency domain energy • importance of phases • heavy-tail non-Gaussian marginals in wavelet domain • near elliptical shape of joint densities in wavelet domain • decay of dependency in wavelet domain • ………….Monday, August 10, 2009 18
- 27. main components observations representation modelMonday, August 10, 2009 19
- 28. models • physical imaging process (e.g., occlusion) • nonlinear manifold of natural images • non-parametric implicit model based on large set of images Gaussian image signals weak • matching statistics of natural model is with density models <-- our focus ω −2 −1 1/f2 F natural F -1 images P(x) P(c) all possible images a. b.Monday, August 10, 2009 20
- 29. main components observations representation model Bayesian applications frameworkMonday, August 10, 2009 21
- 30. Bayesian framework image (x) corruption obs. (y) ?? est. (x’) min L(x, x (y))p(x|y)dx x x ∝ min L(x, x (y))p(y|x)p(x)dx x x • x’(y): estimator • L(x,x’(y)): loss functional • p(x): prior model for natural images • p(y|x): likelihood -- from corruption processMonday, August 10, 2009 22
- 31. application: Bayesian denoising • additive Gaussian noise y = x+w p(y|x) ∝ exp[−(y − x) /2σw ] 2 2 • maximum a posterior (MAP) xMAP = argmax p(x|y) = argmax p(y|x)p(x) x x • minimum mean squares error (MMSE) xMMES = argmin x−x 2 p(x|y)dx x x xp(y|x)p(x)dx = x = E(x|y) x p(y|x)p(x)dxMonday, August 10, 2009 23
- 32. main components observations efﬁcient coding representation model applicationsMonday, August 10, 2009 24
- 33. Neural characterization representation Transform Representation • unsupervised learning •dients: specify desired properties of the transform outputs• stimuli what are such properties?• response modelMonday, August 10, 2009 25
- 34. what makes a good representation? • intuitively, transformed signal should be +?.*-8,@A/-*BCD !"#$%&()*+&%$",-$%(%*.,/&01#,2%,3%(%1".&&45#)&2$#,2(+%5(1(%$&1%#21,%1-,%5#)&2$#,2$ “simpler” ! " !"#$#%&&()*+,*-$)./0)$1*)*&/"2%$#/")/.)$1*),&/34()$1*0*)#5)"/)%--%0*"$)5$03,$30*)#")$1*)5*$)/.)-/#"$5 " 61//5#"2)%")%--0/-0#%$*)0/$%$#/")%&&/75)35)$/)3"8*#&)$1*)3"4*0&#"2)5$03,$30*9):;/3),%")$1#"<)/.)$1#5)0/$%$#/") %5)=7%&<#"2)%0/3"4=)$1*)$10**>4#?*"5#/"%&)5*$()&//<#"2)./0)$1*)@*5$)8#*7-/#"$A - reduced dimensionality ! 678%0(2%"&+*%3#25%$90"%925&.+:#2;%$1.9019.&<%=1%$&+&01$%(%.,1(1#,2%$90"%1"(1%),$1%,3%1"&% >(.#(?#+#1:%-#1"#2%1"&%5(1(%$&1%#$%.&*.&$&21&5%#2%1"&%3#.$1%3&-%5#)&2$#,2$%,3%1"&%.,1(1&5%5(1( " !")/30)$10**>4#?*"5#/"%&),%5*()$1#5)?%)5**?)/.)&#$$&*)35* " B/7*8*0()71*")$1*)4%$%)#5)1#21&)?3&$#4#?*"5#/"%&):CDE5)/.)4#?*"5#/"5A()$1#5)%"%&5#5)#5)F3#$*)-/7*0.3& !"#$%&(#)%"*#%*+,##-$"*.",/01)1 => 2)(,$&%*3#)-$$-4561", 7-8,1*.9:*;")<-$1)#0Monday, August 10, 2009 26
- 35. what makes a good representation? • intuitively, transformed signal should be “simpler” - reduced dependency x r r has less dependency than x d - optimum: r is independent, p(r) = p(ri ) i=1 • reducing dependency is a general approach to relieve the curse of dimensionality • are there dependency in natural images?Monday, August 10, 2009 27
- 36. redundancy in natural images • structure = predictability = redundancy [Kersten, 1987]Monday, August 10, 2009 28
- 37. measure of statistical dependency multi-information (MI): I(x) = DKL p(x) p(xk ) k p(x) = p(x) log dx x k p(xk ) d = H(xk ) − H(x) i=1 [Studeny and Vejnarova, 1998]Monday, August 10, 2009 29
- 38. efﬁcient coding [Attneave ’54; Barlow ’61; Laughlin ’81; Atick ’90; Bialek etal ‘91] x r I(r, x) = H(r) − H(r|x) • maximize mutual information of stimulus & response, subject to constraints (e.g. metabolic) • noiseless case => redundancy reduction: d H(r|x) = 0 ⇒ I(r, x) = H(r) = H(ri ) − I(r) - i=1 independent components - efﬁcient (maxEnt) marginalsMonday, August 10, 2009 30
- 39. main components observations representation model applicationsMonday, August 10, 2009 31
- 40. closed loop observations representation model applicationsMonday, August 10, 2009 32
- 41. pixel domainMonday, August 10, 2009 33
- 42. a. observation b. 1 Correlation I(x+2,y) I(x+4,y) I(x+1,y) b. 0 I(x,y) I(x,y) I(x,y) 1 Sp CorrelationI(x+4,y) I(x,y) 0 10 20 30 40 Spatial separation (pixels) Monday, August 10, 2009 34
- 43. model • maximum entropy density [Jaynes 54] - assume zero mean - Σ = E(xxT ) : consistent w/ second order statistics - ﬁnd p(x) with maximum entropy - solution: 1 T −1 p(x) ∝ exp − x Σ x 2Monday, August 10, 2009 35
- 44. Gaussian model for Bayesian denoising • additive Gaussian noise y = x+w p(y|x) ∝ exp[− y − x /2σw ] 2 2 • Gaussian model 1 T −1 p(x) ∝ exp − x Σ x 2 • posterior density (another Gaussian) 1 T −1 x−y 2 p(x|x) ∝ exp − x Σ x − 2 2σw 2 • inference (Wiener ﬁlter) xMAP = xMMSE = Σ(Σ + 2 −1 σw I) yMonday, August 10, 2009 36
- 45. efﬁcient coding transform • for Gaussian p(x) d I(x) ∝ log(Σ)ii − log det(Σ) i=1 • minimum (independent) when Σ is diagonal • a transform that diagonalizes Σ can eliminate all dependencies (second-order)Monday, August 10, 2009 37
- 46. PCA • eigen-decomposition of Σ: Σ = U ΛU T - U: orthonormal matrix (rotation) UTU = UUT = I - Λ: diagonal matrix, Λii ≥ 0 -- eigenvalue E{U x(U x) } = T T T T T U E{xx }U = U U ΛU U = Λ T T • s = UTx, or x = Us, s is independent Gaussian • principal component analysis (PCA) - Karhunen Loeve transformMonday, August 10, 2009 38
- 47. PCA xMonday, August 10, 2009 39
- 48. PCA x xPCA = U x TMonday, August 10, 2009 40
- 49. PCA bases learned from natural images (U)Monday, August 10, 2009 41
- 50. representation • PCA is for local patches - data dependent - expensive for large images • assume translation invariance cyclic boundary handling - image lattice on a torus - covariance matrix is block circulant - eigenvectors are complex exponential - diagonalized (decorrelated) with DFT - PCA => Fourier representationMonday, August 10, 2009 42
- 51. Spectral power observationsStructural:spectral power • Empirical: 6Assume scale-invariance: 5 F (sω) = s F (ω) p 4 Log power 3 10 then: 2 1 F (ω) ∝ p ω 1 0 0 1 2 3 Log spatialfrequency (cycles/image) 10 [Ritterman 52; DeRiugin 56; Field 87; Tolhurst 92; Ruderman/Bialek 94; ...] ﬁgure from [Simoncelli 05]Monday, August 10, 2009 43
- 52. model • power law A F (ω) = γ ω - scale invariance F (sω) = sp F (ω) • denoising (Wiener ﬁlter in frequency domain) ˆ A/ω γ X(ω) = · Y (ω) A/ω γ + σ 2Monday, August 10, 2009 44
- 53. further observations [Torralba and Oliva, 2003]Monday, August 10, 2009 45
- 54. PCA x Σ = U ΛU T T U x whitening −1 Λ 2 T U x 1 not unique! V Λ −2 T U xMonday, August 10, 2009 46
- 55. zero-phase (symmetric) whitening (ZCA) −1 −1 A = UΛ 2 U T minimum wiring length receptive ﬁelds of retina neurons [Atick & Redlich, 92]Monday, August 10, 2009 47
- 56. second-order constraints is weak Gaussian model are weak 1/ϖγ Gaussian sample ω −2 −1 1/f2 F F -1 P(x) P(c) a. b. whitened natural image F 2 −1 ω F ﬁgure courtesy of Eero SimoncelliMonday, August 10, 2009 48
- 57. summary pixelMonday, August 10, 2009 49
- 58. summary second order pixelMonday, August 10, 2009 49
- 59. summary second order pixel GaussianMonday, August 10, 2009 49
- 60. summary second order pixel Gaussian PCA/FourierMonday, August 10, 2009 49
- 61. summary second order power spectrum pixel Gaussian PCA/FourierMonday, August 10, 2009 49
- 62. summary second order power spectrum pixel Gaussian PCA/Fourier power law modelMonday, August 10, 2009 49
- 63. summary second order power spectrum pixel Gaussian PCA/Fourier power law model Not enough!Monday, August 10, 2009 49
- 64. bandpass ﬁlter domain ⊗ =Monday, August 10, 2009 50
- 65. observation • sparseness log p(x) 0 [Burt&Adelson 82; Field 87; Mallat 89; Daugman 89, ...]Monday, August 10, 2009 51
- 66. model • if we only enforce consistency on 1D marginal densities, i.e., p(xi) = qi(xi) - maximum entropic density is the d factorial density p(x) = i=1 qi (xi ) - multi-information is non-negative, and achieves minimum (zero) when xis are independent H(x) = i H(xi ) − I(x) • there are second order dependencies, so derived model is a linearly transformed factorial (LTF) modelMonday, August 10, 2009 52
- 67. model • linearly transformed factorial (LTF) d - independent sources: p(s) = i=1 p(si ) - A: invertible linear transform (basis) s1 | ··· | x = As = a1 ··· . ad . . | ··· | sd = s 1 a1 + · · · + s d a d - A-1: ﬁlters for analysis s=A −1 xMonday, August 10, 2009 53
- 68. LTF model • SVD of matrix A: A = U Λ V 1/2 T - U,V: orthonormal matrices (rotation) UTU = UUT = I and VTV = VVT = I - Λ: diagonal matrix (Λii)1/2 ≥ 0 -- singular value rotation scale rotation s VTs Λ1/2VTs x = U Λ1/2VTsMonday, August 10, 2009 54
- 69. marginal model • well ﬁt with generalized Gaussian |s|p p(s) ∝ exp − σ Marginal densities [Mallat 89; Simoncelli&Adelson 96; Moulin&Liu 99; …] log(Probability) log(Probability) log(Probability) log(Probability) p = 0.46 p = 0.58 p = 0.48 !H/H = 0.0031 !H/H = 0.0011 !H/H = 0.0014 Wavelet coefficient value Wavelet coefficient value Wavelet coefficient value Fig. 4. Log histograms of a single wavelet subband of four example images (see Fig. 1 for image histogram, tails are truncated so as to show 99.8% of the distribution. Also shown (dashed lines)55Monday, August 10, 2009 ar
- 70. II. BLSBayesian denoising prior for non-Gaussian • Assume marginal distribution [Mallat ‘89]: P (x) ∝ exp −|x/s| p • Then Bayes estimator is generally nonlinear: p = 2.0 p = 1.0 p = 0.5 [Simoncelli & Adelson, ‘96]Monday, August 10, 2009 56
- 71. scale mixture of Gaussians (GSM) • = p(x) p(x) x x √ [Andrews & Mallows 74, Wainwright & Simoncelli, 99] x=u z - u: zero mean Gaussian with unit variance - z: positive random variable - special cases (different p(z)) generalized Gaussian, Student’s t, Bessel’s K, Cauchy, α-stable, etcMonday, August 10, 2009 57
- 72. efﬁcient coding transform • LTF model => independent component analysis (ICA) [Comon 94; Cardoso 96; Bell/Sejnowski 97; …] - many different implementations (JADE, InfoMax, FastICA, etc.) - interpretation using SVD s=A −1 x=VΛ −1/2 U x T - where to get U E{xx } = T AE{ss }A T T = U Λ V IV Λ U 1/2 T 1/2 T = U ΛU TMonday, August 10, 2009 58
- 73. ICA PCA xMonday, August 10, 2009 59
- 74. ICA PCA x xPCA = U x TMonday, August 10, 2009 59
- 75. ICA PCA x xPCA = U x T −1 xwht = Λ 2 U x TMonday, August 10, 2009 59
- 76. ICA PCA x xPCA = U x T −1 xwht = Λ 2 U x T 1 xICA = V Λ −2 T U xMonday, August 10, 2009 59
- 77. ﬁnding V 48/73 Higher-order redundancy reduction: • ﬁnd ﬁnal rotation that maximizes non- Independent Component Analysis (ICA) Gaussianity - linear mixing makes more Gaussian (CLT) Find the most non-Gaussian directions: - equivalent to maximize sparseness ﬁgure from [Bethge 2008]Monday, August 10, 2009 60
- 78. ICA bases (squared columns of A) learned from natural images - similar shape to receptive ﬁeld of V1 simple cells [Olshausen & Field 1996, Bell & Sejnowski 1997]Monday, August 10, 2009 61
- 79. breakMonday, August 10, 2009 62
- 80. representation • ICA basis resemble wavelet and other multi-scale oriented linear representations - localized in spatial location, frequency band and local orientation • ICA basis are learned from data, while wavelet basis are ﬁxedGabor wavelet basisMonday, August 10, 2009 63
- 81. summary band-passMonday, August 10, 2009 64
- 82. summary sparsity band-passMonday, August 10, 2009 64
- 83. summary sparsity band-pass LTFMonday, August 10, 2009 64
- 84. summary sparsity band-pass LTF ICA/waveletMonday, August 10, 2009 64
- 85. summary sparsity higher-order dependency band-pass LTF ICA/waveletMonday, August 10, 2009 64
- 86. summary sparsity higher-order dependency band-pass LTF ICA/wavelet Not enough!Monday, August 10, 2009 64
- 87. problems with LTF • any band-pass or high-pass ﬁlter will lead to heavy tail marginals (even random ones) • if natural images are truly linear mixture of independent non-Gaussian sources, random projection (ﬁltering) should look like Gaussian - central limit theoremMonday, August 10, 2009 65
- 88. problems with LTF [Simoncelli ’97; Buccigrossi &Simoncelli ’99] • Large-magnitude subband coefﬁcients are found at neighboring positions, orientations, and scales.Monday, August 10, 2009 66
- 89. 0.5 raw pca/ica 0.4 MI (bits/coeff) 0.3 0.2 0.1 0 1 2 4 8 16 32 Separation ICA achieves very little improvement over PCA in terms of dependency reduction [Bethge 06, Lyu & Simoncelli 08]Monday, August 10, 2009 67
- 90. LTF also a weak model... natural images after sample from LTF Sample ICA ﬁltering Gaussianized Sample ICA-transformed and Gaussianized ﬁgure courtesy of Eero SimoncelliMonday, August 10, 2009 68
- 91. remedy • assumptions in LTF model and ICA - factorial marginals for ﬁlter outputs - linear combination - invertibleMonday, August 10, 2009 69
- 92. remedy • assumptions in LTF model and ICA - factorial marginals for ﬁlter outputs - linear combination - invertible • model => [Zhu, Wu & Mumford 1997; Portilla & Simoncelli 2000] MaxEnt joint density with constraints on ﬁlter output • representation => sparse coding [Olshausen & Field 1996] - ﬁnd ﬁlters giving optimum sparsity - compressed sensing [Candes & Donoho 2003]Monday, August 10, 2009 70
- 93. remedy • assumptions in LTF model and ICA - factorial marginals for ﬁlter outputs - linear combination nonlinear - invertibleMonday, August 10, 2009 71
- 94. joint density of natural image band-pass ﬁlter responses with separation of 2 pixelsMonday, August 10, 2009 72
- 95. elliptically symmetric density spherically symmetric density whitening 1 1 T −1 1 1 T pesd (x) = 1 f − x Σ x pssd (x) = f − x x α|Σ| 2 2 α 2 (Fang et.al. 1990)Monday, August 10, 2009 73
- 96. Monday, August 10, 2009 74
- 97. Monday, August 10, 2009 75
- 98. Spherical vs LTF 0.2 blk blk 0.4 blk blk size = 3x3 0.2 blk size = 7x7 blk size = 11x11 spherical spherical spherical factorial factorial 0.35 factorial 0.15 0.3 0.15 0.25 0.1 0.2 0.1 0.15 0.05 0.1 0.05 0.05 0 0 0 3 6 9 12 15 18 20 3 6 9 12 15 18 20 3 6 9 12 15 18 20 kurtosis kurtosis kurtosis 3x3 7x7 15x15 data (ICA’d): sphericalized: factorialized:• Histograms, kurtosis of projections of image blocks onto randomunit-norm basis functions.• These imply data are closer to spherical than factorial [Lyu & Simoncelli 08]Monday, August 10, 2009 76
- 99. Linearly transformed Elliptical factorial Factorial Gaussian SphericalMonday, August 10, 2009 77
- 100. Linearly transformed Elliptical factorial Factorial Gaussian SphericalMonday, August 10, 2009 78
- 101. Linearly transformed Elliptical factorial Factorial Gaussian SphericalMonday, August 10, 2009 79
- 102. Linearly transformed Elliptical factorial Factorial Gaussian SphericalMonday, August 10, 2009 80
- 103. ICA PCAMonday, August 10, 2009 81
- 104. ICA PCAMonday, August 10, 2009 82
- 105. elliptical models of natural images - Simoncelli, 1997; - Zetzsche and Krieger, 1999; - Huang and Mumford, 1999; - Wainwright and Simoncelli, 2000; - Hyvärinen et al., 2000; - Parra et al., 2001; - Srivastava et al., 2002; - Sendur and Selesnick, 2002; - Teh et al., 2003; - Gehler and Welling, 2006 - etc. [Fang et.al. 1990]Monday, August 10, 2009 83
- 106. joint GSM model √ x=u z p(x) GSM simulation x Image data GSM simulation ! ! #" #" " " #" #" !!" " !" !!" " !"Monday, August 10, 2009 84
- 107. PCA/whitening Linearly transformed Elliptical factorial Factorial Gaussian SphericalMonday, August 10, 2009 85
- 108. PCA/whitening ICA Linearly transformed Elliptical factorial Factorial Gaussian SphericalMonday, August 10, 2009 86
- 109. PCA/whitening ICA ??? Linearly transformed Elliptical factorial Factorial Gaussian SphericalMonday, August 10, 2009 87
- 110. nonlinear representations • complex wavelet phase-based [Ates & Orchid, 2003] • orientation-based [Hammand & Simoncelli 2006] • nonlinear whitening [Gluckman 2005] • local divisive normalization [Malo et.al. 2004] • global divisive normalization [Lyu & Simoncelli 2007,2008]Monday, August 10, 2009 88
- 111. 1 xT x p(x) = exp − (2π)d 2 d 1 1 = exp − x2 i (2π)d 2 i=1 d 1 1 2 = √ exp − xi i=1 2π 2 d = p(xi ) i=1 Gaussian is the only density that can be both factorial and spherically symmetric [Nash and Klamkin 1976]Monday, August 10, 2009 89
- 112. ICA PCA ?Monday, August 10, 2009 90
- 113. radial Gaussianization (RG) [Lyu & Simoncelli, 2008,2009]Monday, August 10, 2009 91
- 114. radial Gaussianization (RG) [Lyu & Simoncelli, 2008,2009]Monday, August 10, 2009 92
- 115. radial Gaussianization (RG) pχ (r) ∝ r exp(−r2 /2) pr (r) ∝ rf (−r2 /2) [Lyu & Simoncelli, 2008,2009]Monday, August 10, 2009 93
- 116. radial Gaussianization (RG) pχ (r) ∝ r exp(−r2 /2) g(r) = Fχ Fr (r) −1 pr (r) ∝ rf (−r2 /2) [Lyu & Simoncelli, 2008,2009]Monday, August 10, 2009 94
- 117. radial Gaussianization (RG) pχ (r) ∝ r exp(−r2 /2) g(r) = Fχ Fr (r) −1 pr (r) ∝ rf (−r2 /2) g( xwht ) xrg = xwht xwht [Lyu & Simoncelli, 2008,2009]Monday, August 10, 2009 95
- 118. Monday, August 10, 2009 96
- 119. 0.5 raw pca/ica MI (bits/coeff) 0.4 rg 0.3 0.2 0.1 0! 1 2 4 8 16 32 SeparationMonday, August 10, 2009 97
- 120. 0.5 raw pca/ica MI (bits/coeff) 0.4 rg 0.3 0.2 0.1 0! 1 2 4 8 16 32 SeparationMonday, August 10, 2009 97
- 121. 0.5 raw pca/ica MI (bits/coeff) 0.4 rg 0.3 0.2 0.1 0! 1 2 4 8 16 32 SeparationMonday, August 10, 2009 97
- 122. 0.5 raw pca/ica MI (bits/coeff) 0.4 rg 0.3 0.2 0.1 0! 1 2 4 8 16 32 SeparationMonday, August 10, 2009 97
- 123. 0.5 raw pca/ica MI (bits/coeff) 0.4 rg 0.3 0.2 0.1 0! 1 2 4 8 16 32 SeparationMonday, August 10, 2009 97
- 124. 0.5 raw pca/ica MI (bits/coeff) 0.4 rg 0.3 0.2 0.1 0! 1 2 4 8 16 32 SeparationMonday, August 10, 2009 97
- 125. 0.5 raw pca/ica MI (bits/coeff) 0.4 rg 0.3 0.2 0.1 0! 1 2 4 8 16 32 SeparationMonday, August 10, 2009 97
- 126. 1.3 blk size = 15x15 blk size = 3x3 0.7 1.2 0.6 1.1 1 0.5 0.9 0.4 0.8 0.3 0.7 0.2 0.6 0.2 0.3 0.4 0.5 0.6 0.6 0.7 0.8 0.9 1 1.1 IPCA − IRAW IPCA − IRAW (+)IRG − IRAW (◦)IICA − IRAW blocks of local mean removed pixel blocks of natural images (Lyu & Simoncelli, Neural Computation, to appear)Monday, August 10, 2009 98
- 127. ICA PCA RG uniﬁcation as Gaussianization?Monday, August 10, 2009 99
- 128. marginal Gaussianization p(y) y p(x) xMonday, August 10, 2009 100
- 129. summary sparsity band-pass LTF ICA/waveletMonday, August 10, 2009 101
- 130. summary sparsity higher-order dependency band-pass LTF ICA/waveletMonday, August 10, 2009 101
- 131. summary sparsity higher-order dependency band-pass LTF ICA/wavelet elliptical modelMonday, August 10, 2009 101
- 132. summary sparsity higher-order dependency band-pass LTF ICA/wavelet elliptical model RGMonday, August 10, 2009 101
- 133. summary sparsity higher-order dependency band-pass LTF ICA/wavelet elliptical model RG Not enough!Monday, August 10, 2009 101
- 134. Joint densities adjacent near far other scale other ori 150 150 150 150 150 100 100 100 100 100 50 50 50 50 50 0 0 0 0 0 !50 !50 !50 !50 !50 !100 !100 !100 !100 !100 !150 !150 !150 !150 !150 !100 0 100 !100 0 100 !100 0 100 !500 0 500 !100 0 100 150 150 150 150 150 100 100 100 100 100 50 50 50 50 50 0 0 0 0 0 !50 !50 !50 !50 !50 !100 !100 !100 !100 !100 !150 !150 !150 !150 !150 !100 0 100 !100 0 100 !100 0 100 !500 0 500 !100 0 100 • Nearby: densities are approximately circular/elliptical Fig. 8. Empirical joint distributions of wavelet coefﬁcients associated with different pairs of basis functions, for a single image of a New York City street scene (see Fig. 1 for image description). The top row shows joint distributions as contour plots, with lines drawn at equal intervals of log probability. The three leftmost examples correspond to pairs of basis func- • Distant: densities are approximately factorial tions at the same scale and orientation, but separated by different spatial offsets. The next corresponds to a pair at adjacent scales (but the same orientation, and nearly the same position), and the rightmost corresponds to a pair at orthogonal orien- tations (but the same scale and nearly the same position). The bottom row shows corresponding conditional distributions: brightness corresponds to frequency of occurance, except that each column has been independently rescaled to ﬁll the full range of intensities. [Simoncelli, ‘97; Wainwright&Simoncelli, ‘99] remain. First, although the normalized coefﬁcients are cations. There are several issues that seem to be of pri-Monday, August 10, 2009 102
- 135. extended models • independent subspace and topographical ICA [Hoyer & Hyvarinen, 2001,2003; Karklin & Lewicki 2005] • adaptive covariance structures [Hammond & Simoncelli, 2006; Guerrero-Colon et.al. 2008; Karklin & Lewicki 2009] • product of t experts [Osindero et.al. 2003] • ﬁelds of experts [Roth & Black, 2005] • tree and ﬁelds of GSMs [Wainwright & Simoncelli, 2003; Lyu & Simoncelli, 2008] • implicit MRF model [Lyu 2009]Monday, August 10, 2009 103
- 136. z field x field d √ FoGSM: x=u⊗ z • u : zero mean homogeneous Gauss MRF • z : exponentiated homogeneous Gauss MRF • x|z : inhomogeneous Gauss MRF √ • x z : homogeneous Gauss MRF • marginal distribution is GSM • generative model: eﬃcient samplingMonday, August 10, 2009 104
- 137. x u log zMonday, August 10, 2009 105
- 138. Barbara boat housemarginal subband, sample, · · · · · · Gaussian joint simulation: marginal densities ∆=1 ∆=8 ∆ = 32 orientation scale subband FoGSMMonday, August 10, 2009 106
- 139. Lena Boats " " # # !" !" ∆()*+, ∆()*+, !$ !$ ! ! !& !&"## ! "#"! $! !# %! "## ! "#"! $! !# %! "## σ σ FoGSM BM3D kSVDthods for three diﬀerent images. Plotted are diﬀerences in PSNR for diﬀerent input noise levels (σ) between BLS-GSM [17], kSVD [39] and FoE [27]). The PSNR values for these methods were taken from GSM FoE peak-signal-to-noise-ratio (PSNR) [Lyu&Simoncelli, PAMI 08] 255 20 ∗ log10 2 i,j (Ioriginal (i, j) − Idenoised (i, j))Monday, August 10, 2009 107
- 140. original image noisy image (σ = 25) matlab wiener2 FoGSM (14.15dB) (27.19dB) (30.02dB)Monday, August 10, 2009 108
- 141. original image noisy image (σ = 100) (8.13dB) matlab wiener2(29.32dB) (18.38dB) FoGSM (23.01dB)Monday, August 10, 2009 109
- 142. pairwise conditional density E(x2|x1)+std (x2|x1) x2 E(x2|x1) E(x2|x1)-std (x2|x1) x1 p(x2|x1) “bow-tie” [Buccigrossi & Simoncelli, 97]Monday, August 10, 2009 110
- 143. pairwise conditional density E(x2|x1)+std (x2|x1) x2 E(x2|x1) E(x2|x1)-std (x2|x1) x1 E(x2 |x1 ) ≈ ax1 var(x2 |x1 ) ≈ b + 2 cx1Monday, August 10, 2009 111
- 144. conditional density µi = E(xi |xj,j∈N (i) ) = aj xj j∈N (i) 2 σi = var(xi |xj,j∈N (i) ) = b + cj x2 j j∈N (i) • maxEnt conditional density 1 (xi − µi )2 p(xi |xj,j∈N (i) ) = exp − 2πσi 2 2σi2 - singleton conditionals - joint MRF density can be determined by all singletons (Brook’s lemma)Monday, August 10, 2009 112
- 145. implicit MRF • deﬁned by all singletons • joint density (and clique potential) is implicit • learning: maximum pseudo-likelihoodMonday, August 10, 2009 113
- 146. ICM-MAP denoising argmax p(x|y) = argmax p(y|x)p(x) = argmax log p(y|x) + log p(x) x x x - set initial value for x(0) , and t = 1 - repeat until convergence - repeat for all i - compute the current estimation for xi , as (t) (t) (t) xi = argmax log p(x1 , · · · , xi−1 , xi (t−1) (t−1) xi , xi+1 , · · · , xd |y). - t ←t+1Monday, August 10, 2009 114
- 147. ICM-MAP denoising argmax log p(y|x) + log p(x) xi = argmax log p(y|x) + log p(x1 , · · · , xi−1 , xi , xi+1 , · · · , xn ) xi = argmax log p(y|x) + log p(xi |xj,j∈N (i) ) xi can be further simpliﬁed singleton conditional ( ((( ) . + (((j,j∈N (i) log p(x constant w.r.t xi local adaptive and iterative Wiener ﬁltering 2 2 σw σi yi µi xi = 2 + 2− wij (xj − yj ) σw + σi σw 2 2 σi i=j .Monday, August 10, 2009 115
- 148. summary observations representation model applicationsMonday, August 10, 2009 116
- 149. what need to be done • inhomogeneous structures - structural (edge, contour, etc.) - textual (grass, leaves, etc.) - smooth (fog, sky, etc.) • local orientations and relative phases holy grail: comprehensive model & representations to capture all these variationsMonday, August 10, 2009 117
- 150. big question marks • what are natural images, anyway? • ironically, white noises are “natural” as they are the result of cosmic radiations • naturalness is subjectiveMonday, August 10, 2009 118
- 151. peeling the onion natural images all possible imagesMonday, August 10, 2009 119
- 152. peeling the onion natural images images of same marginal stats all possible imagesMonday, August 10, 2009 119
- 153. peeling the onion images of same second order stats natural images images of same marginal stats all possible imagesMonday, August 10, 2009 119
- 154. peeling the onion images of same second order stats natural images images of same marginal stats images of same higher order stats all possible imagesMonday, August 10, 2009 119
- 155. resources • D. L. Ruderman. The statistics of natural images. Network: Computation in Neural Systems, 5:517–548, 1996. • E. P. Simoncelli and B. Olshausen. Natural image statistics and neural representation. Annual Review of Neuroscience, 24:1193– 1216, 2001. • S.-C. Zhu. Statistical modeling and conceptualization of visual patterns. IEEE Trans PAMI, 25(6), 2003 • A. Srivastava, A. B. Lee, E. P. Simoncelli, and S.-C. Zhu. On advances in statistical modeling of natural images. J. Math. Imaging and Vision, 18(1):17–33, 2003. • E. P. Simoncelli. Statistical modeling of photographic images. In Handbook of Image and Video Processing, 431–441. Academic Press, 2005. • A. Hyvärinen, J. Hurri, and P. O. Hoyer. Natural Image Statistics: A probabilistic approach to early computational vision. Springer, 2009.Monday, August 10, 2009 120
- 156. thank youMonday, August 10, 2009 121

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