NIPS2008: tutorial: statistical models of visual images

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NIPS2008: tutorial: statistical models of visual images

  1. 1. Statistical Image Models Eero Simoncelli Howard Hughes Medical Institute, Center for Neural Science, andCourant Institute of Mathematical Sciences New York University
  2. 2. Photographic ImagesDiverse specialized structures:• edges/lines/contours• shadows/highlights• smooth regions• textured regions
  3. 3. Photographic Images Diverse specialized structures: • edges/lines/contours • shadows/highlights • smooth regions • textured regionsOccupy a small region of the full space
  4. 4. spa ce o f all ima ges typical imagesOne could describe this set as adeterministic manifold....
  5. 5. • Step edges are rare (lighting, junctions, texture, noise)
  6. 6. • Step edges are rare (lighting, junctions, texture, noise)• One scale’s texture is another scale’s edge
  7. 7. • Step edges are rare (lighting, junctions, texture, noise)• One scale’s texture is another scale’s edge• Need seamless transitions from isolated features to dense textures
  8. 8. spa ce o f all ima ges typical imagesOne could describe this set as adeterministic manifold....
  9. 9. spa ce o f all ima ges typical imagesOne could describe this set as adeterministic manifold....But seems more natural to use probability
  10. 10. spa ce o f all ima ges typical imagesOne could describe this set as a P(x)deterministic manifold....But seems more natural to use probability
  11. 11. “Applications”• Engineering: compression, denoising, restoration, enhancement/modification, synthesis, manipulation [Hubel ‘95]
  12. 12. “Applications”• Engineering: compression, denoising, restoration, enhancement/modification, synthesis, manipulation• Science: optimality principles for neurobiology (evolution, development, learning, adaptation) [Hubel ‘95]
  13. 13. Density modelsnonparametric parametric/ constrained
  14. 14. Density modelsnonparametric parametric/ constrainedbuild a histogramfrom lots ofobservations...
  15. 15. Density modelsnonparametric parametric/ constrainedbuild a histogram use “natural constraints”from lots of (geometry/photometryobservations... of image formation, computation, maxEnt)
  16. 16. Density modelsnonparametric parametric/ historical trend constrained (technology driven)build a histogram use “natural constraints”from lots of (geometry/photometryobservations... of image formation, computation, maxEnt)
  17. 17. histogramOriginal imageRange: [0, 237]Dims: [256, 256] 0 50 100 150 200 250
  18. 18. histogramOriginal image Range: [0, 237] Dims: [256, 256] 0 50 100 150 200 250 histogramEqualized imageRange: [1.99, 238] Dims: [256, 256] 0 50 100 150 200 250
  19. 19. histogramOriginal image Range: [0, 237] Dims: [256, 256] 0 50 100 150 200 250 histogramEqualized imageRange: [1.99, 238] Dims: [256, 256] 0 50 100 150 200 250
  20. 20. General methodologyObserve “interesting” Transform to Joint Statistics Optimal Representation
  21. 21. General methodologyObserve “interesting” Transform to Joint Statistics Optimal Representation
  22. 22. General methodologyObserve “interesting” Transform to Joint Statistics Optimal Representation“Onion peeling”
  23. 23. Evolution of image modelsI. (1950’s): Fourier + GaussianII. (mid 80’s - late 90’s): Wavelets + kurtotic marginalsIII. (mid 90’s - present): Wavelets + local context • local amplitude (contrast) • local orientationIV. (last 5 years): Hierarchical models
  24. 24. a. Pixel correlation b. 1 Correlation I(x+2,y) I(x+4,y)I(x+1,y) I(x,y) I(x,y) I(x,y) 0 10 Spatia
  25. 25. a. Pixel correlation b. 1 Correlation I(x+2,y) I(x+4,y) I(x+1,y) I(x,y) I(x,y) I(x,y) 0 10 b. Spatia 1 CorrelationI(x+4,y) I(x,y) 0 10 20 30 40 Spatial separation (pixels)
  26. 26. Translation invarianceAssuming translation invariance,
  27. 27. Translation invarianceAssuming translation invariance, => covariance matrix is Toeplitz (convolutional)
  28. 28. Translation invarianceAssuming translation invariance, => covariance matrix is Toeplitz (convolutional) => eigenvectors are sinusoids
  29. 29. Translation invarianceAssuming translation invariance, => covariance matrix is Toeplitz (convolutional) => eigenvectors are sinusoids => can diagonalize (decorrelate) with F.T.
  30. 30. Translation invarianceAssuming translation invariance, => covariance matrix is Toeplitz (convolutional) => eigenvectors are sinusoids => can diagonalize (decorrelate) with F.T.Power spectrum captures full covariance structure
  31. 31. Spectral powerStructural:Assume scale-invariance: F (sω) = s F (ω) p then: 1 F (ω) ∝ p ω [Ritterman 52; DeRiugin 56; Field 87; Tolhurst 92; Ruderman/Bialek 94; ...]
  32. 32. Spectral powerStructural: 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; ...]
  33. 33. Principal Components Analysis (PCA) + whitening a. b. c.20 20 4-20 -20 -4 -20 20 -20 20 -4 4
  34. 34. PCA basis for image blocks
  35. 35. PCA basis for image blocks PCA is not unique
  36. 36. Maximum entropy (maxEnt) The density with maximal entropy satisfying E (f (x)) = c is of the form pME (x) ∝ exp (−λf (x)) where λ depends on cExamples: f (x) = x 2 f (x) = |x|
  37. 37. Model I (Fourier/Gaussian) Coefficient density: Basis set: Image: : : : :
  38. 38. Gaussian model is weak ω −2 −1 1/f2 F F -1P(x) P(c)
  39. 39. Gaussian model is weak ω −2 −1 1/f2 F F -1P(x) P(c) a. b. F 2 −1 ω F
  40. 40. Gaussian model is weak ω −2 −1 1/f2 F F -1 P(x) P(c) a. b. F 2 −1 ω F a. b. c.20 20 4-20 -20 -4 -20 20 -20 20 -4 4
  41. 41. Bandpass Filter Responses 0 10 Response histogram Gaussian density Probability -2 10 -4 10 500 0 500 Filter Response [Burt&Adelson 82; Field 87; Mallat 89; Daugman 89, ...]
  42. 42. “Independent” Components Analysis (ICA) a. b. c. d.20 20 4 4-20 -20 -4 -4 -20 20 -20 20 -4 4 -4 4 For Linearly Transformed Factorial (LTF) sources: guaranteed independence (with some minor caveats) [Comon 94; Cardoso 96; Bell/Sejnowski 97; ...]
  43. 43. ICA on image blocks [Olshausen/Field ’96; Bell/Sejnowski ’97] [example obtained with FastICA, Hyvarinen]
  44. 44. Marginal densities 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) are corresponding to equation (3). Text indicates the maximum-likelihood value of p used for the fitte Well-fit by a generalized Gaussian: the relative entropy (Kullback-Leibler divergence) of the model and histogram, as a fraction of th histogram. P (x) ∝ exp −|x/s| pnon-Gaussian than others. By the mid 1990s, a numberof authors had developed methods of optimizing a ba-sis of filters in order to to maximize the non-Gaussianityof the responses [e.g., 36, 4]. Often these methods oper- [Mallat 89; Simoncelli&Adelson 96; Moulin&Liu 99; ...]ate by optimizing a higher-order statistic such as kurto-
  45. 45. Kurtosis vs. bandwidth 16 14 12 Sample Kurtosis 10 8 6 4 0 0.5 1 1.5 2 2.5 3 Filter Bandwidth (octaves)Note: Bandwidth matters much more than orientation[see Bethge 06] [after Field 87]
  46. 46. Octave-bandwidth representations SpatialFrequencySelectivity: Filter:
  47. 47. Model II (LTF)Coefficient density: Basis set: Image: : : :
  48. 48. LTF also a weak model... Sample Gaussianized Sample ICA-transformed and Gaussianized
  49. 49. Trouble in paradise
  50. 50. Trouble in paradise• Biology: Visual system uses a cascade - Where’s the retina? The LGN? - What happens after V1? Why don’t responses get sparser? [Baddeley etal 97; Chechik etal 06]
  51. 51. Trouble in paradise• Biology: Visual system uses a cascade - Where’s the retina? The LGN? - What happens after V1? Why don’t responses get sparser? [Baddeley etal 97; Chechik etal 06]• Statistics: Images don’t obey ICA source model - Any bandpass filter gives sparse marginals [Baddeley 96] => Shallow optimum [Bethge 06; Lyu & Simoncelli 08] - The responses of ICA filters are highly dependent [Wegmann & Zetzsche 90, Simoncelli 97]
  52. 52. Conditional densities 1 1 0.6 0.6 0.2 0.2 -40 0 40 50 -40 0 40 40 0 -40 -40 0 40Linear responses are not independent, even for optimized filters! [Simoncelli 97; Schwartz&Simoncelli 01]CSH-02
  53. 53. [Schwartz&Simoncelli 01]
  54. 54. • Large-magnitude subband coefficients are found at neighboring positions, orientations, and scales.
  55. 55. Modeling heteroscedasticity (i.e., variable variance) Method 1: Conditional GaussianP (xn |{xk }) ∼ N 0; wnk |xk | + σ 2 2 k [Simoncelli 97; Buccigrossi&Simoncelli 99; see also ARCH models in econometrics!]
  56. 56. 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/ellipticalFig. 8. Empirical joint distributions of wavelet coefficients associated with different pairs of basis functions, for a singleimage of a New York City street scene (see Fig. 1 for image description). The top row shows joint distributions as contourplots, with lines drawn at equal intervals of log probability. The three leftmost examples correspond to pairs of basis func- • Distant: densities are approximately factorialtions at the same scale and orientation, but separated by different spatial offsets. The next corresponds to a pair at adjacentscales (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 fullrange of intensities. [Simoncelli, ‘97; Wainwright&Simoncelli, ‘99]
  57. 57. ICA-transformed joint densities d=2 d=16 d=32 12 12 12 10 10 10kurtosis 8 8 8 6 6 6 4 4 4 0 !/4 !/2 3!/4 ! 0 !/4 !/2 3!/4 ! 0 !/4 !/2 3!/4 ! orientation data (ICA’d): sphericalized: factorialized:
  58. 58. ICA-transformed joint densities d=2 d=16 d=32 12 12 12 10 10 10kurtosis • Local densities are elliptical (but non-Gaussian) 8 8 8 6 6 6 • Distant densities are factorial 4 4 4 0 !/4 !/2 3!/4 ! 0 !/4 !/2 3!/4 ! 0 !/4 !/2 3!/4 ! orientation data (ICA’d): [Wegmann&Zetzsche ‘90; Simoncelli ’97; + many recent models] sphericalized: factorialized:
  59. 59. 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]
  60. 60. non-Gaussian elliptical observationsand models of natural images: - Zetzsche & Krieger, 1999; - Huang & Mumford, 1999; - Wainwright & Simoncelli, 2000; - Hyvärinen and Hoyer, 2000; - Parra et al., 2001; - Srivastava et al., 2002; - Sendur & Selesnick, 2002; - Teh et al., 2003; - Gehler and Welling, 2006 - Lyu & Simoncelli, 2008 - etc.
  61. 61. Modeling heteroscedasticityMethod 2: Hidden scaling variable for each patchGaussian scale mixture (GSM)[Andrews & Mallows 74]: √ x= zu• u is Gaussian, z > 0• z and u are independent• x is elliptically symmetric, with covariance ∝ Cu• marginals of x are leptokurtotic [Wainwright&Simoncelli 99]
  62. 62. GSM - prior on z• Empirically, z is approximately lognormal [Portilla etal, icip-01] exp (−(log z − µl )2 /(2σl )) 2 pz (z) = 2 )1/2 z(2πσl• Alternatively, can use Jeffrey’s noninformative prior [Figueiredo&Nowak, ‘01; Portilla etal, ‘03] pz (z) ∝ 1/z
  63. 63. GSM simulation Image data GSM simulation ! !#" #" " "#" #" !!" " !" !!" " !" [Wainwright & Simoncelli, NIPS*99]
  64. 64. Model III (GSM)Coefficient density: Basis set: Image: X X X Xsqrt(z) X X u
  65. 65. √ Original coefficients Normalized by z !2 !4 !4 !5marginal Log probability Log probability !6 !6 !7 [Ruderman&Bialek 94] !8 !8 !9 !10 !500 0 500 !10 !5 0 5 100 8 50 6 joint 0 4 [Schwartz&Simoncelli 01] !50 2 !100 0 !100 !50 0 50 100 0 2 4 6 8subband
  66. 66. 6Model Encoding Cost (bits/coeff) Model Encoding cost (bits/coeff) 5.5 5 5 4.5 4 4 3 3.5 3 2 Gaussian Model First Order Ideal 2.5 Generalized Laplacian Conditional Model 1 3 4 5 1 2 3 4 5 6 Empirical First Order Entropy (bits/coeff) Empirical Conditional Entropy [Buccigrossi & Simoncelli 99]
  67. 67. Bayesian denoising• Additive Gaussian noise: y =x+w 2 2 P (y|x) ∝ exp[−(y − x) /2σw ]• Bayes’ least squares solution is conditional mean: x(y) = IE(x|y) ˆ = dxP(y|x)P(x)x/P(y)
  68. 68. I. ClassicalIf signal is Gaussian, BLS estimator is linear: denoised (ˆ) x 2 σxx(y) =ˆ 2 2 ·y σx + σn=> suppress fine scales, noisy (y) retain coarse scales
  69. 69. Non-Gaussian coefficients " #" -*./01.*,6.)07+48 94:..41,;*1.)5,, 2+0343()5 !% #" !$ #" !!"" " !"" &()*+,-*./01.* [Burt&Adelson ‘81; Field ‘87; Mallat ‘89; Daugman ‘89; etc]
  70. 70. II. BLS for non-Gaussian prior• 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]
  71. 71. MAP shrinkagep=2.0 p=1.0 p=0.5 [Simoncelli 99]
  72. 72. Denoising: Joint IE(x|y) = dz P(z|y) IE(x|y, z) −1y    = dz P(z|y)  zCu(zCu + Cw ) ctrwhere P(y|z) P(z) exp(−y T (zCu + Cw )−1y/2)P(z|y) = , P(y|z) = Py (2π)N |zCu + Cw |Numerical computation of solution is reasonably efficient ifone jointly diagonalizes Cu and Cw ...[Portilla, Strela, Wainwright, Simoncelli, ’03]IPAM, 9/04 20
  73. 73. ESTIMATED COEFF. Example estimators !" +1&2/1+3)*%+,,- " !w !!" #" !" " " NOISY COEFF. !#" $%&()./0+$1 !!" $%&()*%+,,-Estimators for the scalar and single-neighbor cases [Portilla etal 03]
  74. 74. Comparison to other methods "& "& " " ,456748+91:;<=/456,(74-.)/-0123 !"& !"& :>6965#8*>?6< !! !! !!& !!& !# !# !#& !#& )89!:1:;<=>? !$ .=9@A?-9?=@BC8D !$ !$& !$& !" #" $" %" &" !" #" $" %" &" ()*+,-*.)/-0123 ()*+,-*.)/-0123 Results averaged over 3 images [Portilla etal 03]
  75. 75. NoisyOriginal (22.1 dB)Matlab’s BLS-GSMwiener2 (30.5 dB)(28 dB)
  76. 76. Noisy Original (8.1 dB)UndWvlt BLS-GSM Thresh (21.2 dB)(19.0 dB)
  77. 77. Real sensor noise400 ISO denoised
  78. 78. GSM summary• GSM captures local variance• Underlying Gaussian leads to simple computation• Excellent denoising results• What’s missing? • Global model of z variables [Wainwright etal 99; Romberg etal ‘99; Hyvarinen/Hoyer ‘02; Karklin/ Lewicki ‘02; Lyu/Simoncelli 08] • Explicit geometry: phase and orientation
  79. 79. Global models for z• Non-overlapping neighborhoods, tree-structured z [Wainwright etal 99; Romberg etal ’99] z u Coarse scale Fine scale• Field of GSMs: z is an exponentiated GMRF, u is a GMRF, subband is the product [Lyu&Simoncelli 08]
  80. 80. D MACHINE INTELLIGENCE, VOL. X, NO. X, XX 200X 9 State-of-the-art denoising Lena Boats " " # # !" !" ∆()*+, ∆()*+, !$ !$ ! ! !& !&"## ! "#"! $! !# %! "## ! "#"! $! !# %! "## 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 [Lyu&Simoncelli, PAMI 08]
  81. 81. Measuring Orientation2-band steerable pyramid: Image decomposition interms of multi-scale gradient measurements [Simoncelli et.al., 1992; Simoncelli & Freeman 1995]
  82. 82. Multi-scale gradient basis
  83. 83. Multi-scale gradient basis• Multi-scale bases: efficient representation
  84. 84. Multi-scale gradient basis• Multi-scale bases: efficient representation• Derivatives: good for analysis • Local Taylor expansion of image structures • Explicit geometry (orientation)
  85. 85. Multi-scale gradient basis• Multi-scale bases: efficient representation• Derivatives: good for analysis • Local Taylor expansion of image structures • Explicit geometry (orientation)• Combination: • Explicit incorporation of geometry in basis • Bridge between PDE / harmonic analysis approaches
  86. 86. orientation magnitude orientation[Hammond&Simoncelli 06; cf. Oppenheim and Lim 81]
  87. 87. Importance of local orientation Randomized orientation Randomized magnitude [Hammond&Simoncelli 05]
  88. 88. Reconstruction from orientation Original Quantized to 2 bits • Reconstruction by projections onto convex sets • Resilient to quantization [Hammond&Simoncelli 06]
  89. 89. Image patches related by rotation two-band steerable[Hammond&Simoncelli 06] pyramid coefficients
  90. 90. raw rotatedpatches patches PCA of normalized gradient patches --- Raw Patches Rotated Patches [Hammond&Simoncelli 06]
  91. 91. Orientation-Adaptive GSM modelModel a vectorized patch of wavelet coefficients as: patch rotation operator hidden magnitude/orientation variables [Hammond&Simoncelli 06]
  92. 92. Orientation-Adaptive GSM modelModel a vectorized patch of wavelet coefficients as: patch rotation operator hidden magnitude/orientation variablesConditioned on ; is zero mean gaussian with covariance [Hammond&Simoncelli 06]
  93. 93. Estimation of C(θ) from noisy data noisy patch unknown, approximate by measured from noisy data.Assuming independent and noise rotationally invariant (assuming w.l.o.g. E[z] =1 ) [Hammond&Simoncelli 06]
  94. 94. Bayesian MMSE Estimator [Hammond&Simoncelli 06]
  95. 95. Bayesian MMSE Estimator condition on and integrate over hidden variables [Hammond&Simoncelli 06]
  96. 96. Bayesian MMSE Estimator condition on and integrate over hidden variables [Hammond&Simoncelli 06]
  97. 97. Bayesian MMSE Estimator condition on and integrate over hidden variables Wiener estimate [Hammond&Simoncelli 06]
  98. 98. Bayesian MMSE Estimator condition on and integrate over hidden variables Wiener estimate has covarianceseparable prior for hidden variables [Hammond&Simoncelli 06]
  99. 99. Bayesian MMSE Estimator condition on and integrate over hidden variables Wiener estimate has covarianceseparable prior for hidden variables [Hammond&Simoncelli 06]
  100. 100. σ = 40 noisy 2.81 dB gsm2 oagsm12.4 dB 13.1 dB
  101. 101. Locally adaptive covariance • Karklin & Lewicki 08: Each patch is Gaussian, with covariance constructed from a weighted outer- product of fixed vectors: p(x) = G (x; C(y)) log C(y) = yn Bn n Tp(y) = exp(−|yn |) Bn = wnk bk bk n k • Guerrero-Colon, Simoncelli & Portilla 08: Each patch is a mixture of GSMs (MGSMs): p(x) = Pk p(zk ) G(x; zk Ck ) dzk k
  102. 102. MGSMs generative model √ √ √Patch x chosen from { z1 u1 , z2 u2 , ... zK uK } with probabilities {P1 , P2 , ..., PK }Parameters: • Covariances Ck • Scale densities pk (zk ) • Component probabilities Pk • Number of components KParameters can be fit to data of one or more imagesby maximizing likelihood (EM-like) [Guerrero-Colon, Simoncelli, Portilla 08]
  103. 103. MGSM “segmentation” image 1 2 4 First sixeigenvectors of GSM covariance matrices [Guerrero-Colon, Simoncelli, Portilla 08]
  104. 104. MGSM“segmentation”Eigenvectors of GSMcomponents represent invariant subspaces:“generalized complex cells”
  105. 105. Potential of local homogeneous models?Consider an implicit model: maxEnt subject to constraints on subband coefficients: • marginal statistics [var,skew,kurtosis] • local raw correlations • local variance correlations • local phase correlations [Portilla & Simoncelli 00; cf. Zhu, Wu & Mumford 97]
  106. 106. Visual texture
  107. 107. Visual textureHomogeneous, with repeated structures
  108. 108. Visual textureHomogeneous, with repeated structures “You know it when you see it”
  109. 109. All Images Texture Images Equivalence class (visually indistinguishable)
  110. 110. Iterative synthesis algorithm AnalysisExample Transform Measure Texture Statistics Synthesis Measure StatisticsRandom Transform Inverse Synthesized Adjust Seed Transform Texture [Portilla&Simoncelli 00; cf. Heeger&Bergen ‘95]
  111. 111. Examples: Artificial
  112. 112. Photographic, quasi-periodic
  113. 113. Photographic, aperiodic
  114. 114. Photographic, structured
  115. 115. Photographic, color
  116. 116. Non-textures?
  117. 117. Texture mixtures
  118. 118. Texture mixturesConvex combinations in parameter space
  119. 119. Texture mixturesConvex combinations in parameter space=> Parameter space includes non-textures
  120. 120. Summary• Fusion of empirical data with structural principles• Statistical models have led to state-of-the-art image processing, and are relevant for biological vision• Local adaptation to {variance, orientation, phase, ...} gives improvement, but makes learning harder• Cascaded representations emerge naturally• There’s still much room for improvement!
  121. 121. Cast• Local GSM model: Martin Wainwright, Javier Portilla• GSM Denoising: Javier Portilla, Martin Wainwright, Vasily Strela• Variance-adaptive compression: Robert Buccigrossi• Local orientation and OAGSM: David Hammond• Field of GSMs: Siwei Lyu• Mixture of GSMs: Jose-Antonio Guerrero-Colón, Javier Portilla• Texture representation/synthesis: Javier Portilla

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