22 October, 2012 MVA ENS Cachan. Lecture 5: Introduction to generative models Iasonas Kokkinos
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1 Machine Learning for Computer Vision 1 22 October, 2012 MVA ENS Cachan Lecture 5: Introduction to generative models Iasonas Kokkinos Iasonas.kokkinos@ecp.fr Center for Visual Computing Ecole Centrale Paris Galen Group INRIA-Saclay
2 Lecture outline 2 Bayes rule and generative models Density estimation Parametric deformable models
3 Decision Theory 3 What is an optimal decision rule? Consider loss matrix Consider underlying joint distribution of data-label pairs Find decision rule f that minimizes expected loss:
4 Optimal Classifier 4 Consider zero-one loss function: Form `Expected Prediction Error : Optimal decision at any : `Bayes-optimal classifier
5 Bayes theorem 5 P(X Y): likelihood of observations X, given class Y. P(Y): Prior probability of class Y P(Y X): Posterior probability of class Y, given observations Y. Why is this identity important?
6 Two approaches to pattern recognition Discriminative Generative 6 Y Observations, X Observations, X Y Posterior Model-based Likelihood Discriminative Vision: Function approximation Bayesian Vision: Inverse Graphics
7 Generative or disciminative? 7 Training Set Density Estimation (e.g. ML) Class Distributions Bayes Rule Class posteriors Discriminative Models (lectures 1-4) : Skip density estimation More robust to wrong distribution assumptions (e.g. outliers) V. Vapnik: `one should solve the classification problem directly and never solve a more general problem as an intermediate step Generative Models (Lectures 5-7) : Core task: density estimation If we know the distributions, requires smaller training sets Dealing with missing/corrupt data Explicit modelling of sources of variation (e.g. translation) Conceptual clarity, ability for `visual debugging
8 Lecture outline 8 Bayes rule and generative models Density estimation Gaussian distributions Mixture-of-Gaussian models Hidden Variables and Expectation-Maximization algorithm Factor Analysis & PCA Mixed Discrete/Continuous hidden variable models Parametric deformable models
9 Density Estimation 9 Training set: Examples corresponding to class k: Training data for class k: One density estimate per class: for short:
10 Parametric Distributions: Gaussian 10 1 D N D e.g. 2D:
11 Covariance matrix reminder 11 Covariance matrix: Height, Income Height, Weight Uncorrelated coordinates: diagonal covariance
12 12 Density Estimation for a Gaussian distribution Given: Notation: Maximum Likelihood Estimation for class k:
13 13 Classification task: Ferrari or Fiat? Consider placing a personalized ad. Which car will you try? New client: Classification problem: new client is likely to buy Fiat/Ferarri. Build class-specific probability distributions
14 Ferrari or Fiat, continued 14 Class-specific Gaussian Distributions Parameter estimation: Maximum Likelihood (ML) Can we proceed to classification? Bayes rule: Need to estimate: ML estimate:
15 Bayes rule, 1D 15
16 Classifier form for Gaussian Distributions 16 Choose class Decision boundary for the binary case: Quadratic Decision Boundaries Special case: Linear Decision Boundaries
17 Lecture outline 17 Bayes rule and generative models Density estimation Gaussian distributions Mixture-of-Gaussian models Expectation-Maximization algorithm and hidden variables Factor Analysis & PCA Mixed Discrete/Continuous hidden variable models Parametric deformable models
18 Mixture of Gaussians model 18 Main challenge: parameter estimation Which points go with which cluster? P(x) =.1 P(x) =.2 P(x) =.5
19 K-Means algorithm 19 Coordinate descent on distortion cost: Local minima (multiple initializations to find better solution)
20 Lecture outline 20 Bayes rule and generative models Density estimation Gaussian distributions Mixture-of-Gaussian models Expectation-Maximization algorithm and hidden variables Factor Analysis & PCA Mixed Discrete/Continuous hidden variable models Parametric deformable models
21 K-Means algorithm 21
22 Adaptation for Gaussian distributions 22
23 Expectation Maximization algorithm 23 E-step M-step
24 K-means vs. EM 24 k-means EM Closest center s index Soft assignment, R Isotropic Distance Anisotropic Likelihood (Euclidean) (Covariance-based,`Mahalanobis ) Fast (e.g. kd-trees) Accurate & more flexible More robust to initalization Prone to local minima Typical usage: initialize EM with k-means results Coordinate Descent on Coordinate descent on?
25 Mixture of Gaussians 25 Maximum Likelihood Estimation:
26 26
27 Hidden Variables: 27 Criterion: Problem: Summation inside logarithm We do not know which component generated each point What if we knew?
28 Plato s cave 28 Observations: B&W Images Models: 3D surfaces Hidden variables: positions
29 Hidden Variables: 29 Criterion: Problem: Summation inside logarithm We do not know which component generated each point What if we knew? Hidden variable Indicate which component is responsible for each point Multinomially distributed variable
30 Rewriting the MoG distribution 30 Marginalization Chain rule We have
31 Complete Log-Likelihood 31 Assume hidden variables are given Data+ hidden variables = complete observations Complete log-likelihood Summation falls outside the logarithm!
32 Full Observation Log Likelihood 32 Given: Hidden Variables Maximize w.r.t. parameters
33 Expected Complete Log-Likelihood 33 We do not know the hidden variables (`missing data ) Complete log-likelihood is a random quantity. Form its expectation, using a distribution q(h) on hidden variables: Expected complete log-likelihood
34 Full Observation Log-Likelihood 34 Given: Hidden Variables Maximize w.r.t. parameters
35 Expected Log-Likelihood 35 Given: Probability of assignment Maximize w.r.t. parameters M-step!
36 Lecture outline 36 Bayes rule and generative models Density estimation Gaussian distributions Mixture-of-Gaussian models Expectation-Maximization algorithm and hidden variables Factor Analysis & PCA Mixed Discrete/Continuous hidden variable models Parametric deformable models
37 P(Grades MVA) 10 students, 20 courses How can we model the distribution of the grades? Consider a Gaussian distribution.. 20X19/2 Parameters in covariance, 10 measurements Could we `summarize performance in a more compact way? 3 `hidden causes Math skills, CS skills, Effort Different skills per student Different effects of skills on grade per course 37 Observed grades Influence of skills on grade Skills per student
38 Generative Model: Factor Analysis 38 Hidden variables (skills) Observations `factor loading matrix Λ (course-specific effect of skills on grade) noise covariance matrix Ψ (performance on exam) Linear model Distribution of x (see end of slides) Density estimation: recover optimal µ, Λ, Ψ, for a set of data Χ
39 Continuous Hidden Variables: Factor Analysis 39 Find low-dimensional subspace (`skills ) explaining data Hidden variables: coordinates on subspace E-step: posterior on coordinates M-step: subspace
40 EM for Factor Analysis 40 E-step: distribution on h(skills), conditioned on x (grades) M-step: plug in distribution on h, and maximize w.r.t. parameters
41 Principal Component Analysis (PCA) 41 Find a low-dimensional subspace to reconstruct high-dimensional data Reconstruction on orthogonal basis Approximation with K terms
42 Relation with Factor Analysis? 42 PCA criterion: Regularize solution Equivalently: Difference from FA: What we gain: no EM, factorization-based estimate of Λ, h What we lose: proper probabilistic framework.
43 Principal component analysis 43 The k orthogonal directions that capture most of the data variance are the k leading (largest-eigenvalue) covariance eigenvectors Factor Analysis Λ matrix Hidden variables PCA Leading K eigenvectors of covariance Inner product of data with eigenvectors
44 PCA: decorrelation/dimensionality reduction 44 `Hidden variables : projection onto eigenvectors of covariance matrix Dimensionality reduction by using only leading eigenvectors Grades in 60 courses -> Good in math, computer science
45 Lecture outline 45 Bayes rule and generative models Density estimation Gaussian distributions Mixture-of-Gaussian models Expectation-Maximization algorithm and hidden variables Factor Analysis & PCA Mixed Discrete/Continuous hidden variable models Parametric deformable models
46 Continuous Hidden Variables: Factor Analysis 46 Also known as Dimensionality Reduction
47 Discrete hidden variables: Mixture of Gaussians 47 Also known as Clustering
48 Transformation-resilient image averaging 48 Consider shift as a hidden variable, l Estimate model with EM Shift Deformation-free image Observed Image Input Plain mean & std With transformation & EM
49 Transformed Components Analysis 49 Latent variables for synthesis (continuous) Latent variables for shift (discrete) Estimate mean basis using EM Plain mean & PCA Input With offset Samples of model
50 Transformed Mixture of Gaussians 50 Latent variables for cluster (discrete) Latent variables for shift (discrete) Input Plain Mixture-of-Gaussians With offset
51 Transformed Mixture of Gaussians 51 Input Plain Mixture-of-Gaussians With offset
52 Mixture of Transformed Components 52 Latent variables for cluster Latent variables for components Latent variables for shift
53 Lecture outline 53 Bayes rule and generative models Density estimation Parametric deformable models Statistical active shape models Eigenfaces Active appearance models 3D Morphable models
54 Example: bone contours 54 Task: localize anatomical structures 54
55 Task: Analyze a hand radiograph 55
56 Task: Analyze a hand radiograph 56 MC2 PP2 Assume: we are looking for proximal phalanx 2 MP2 MC3 PP3 MP3 MC4 MC5 PP5 PP4 MP5 MP4
57 Analyzing a hand radiograph 57 PP2 We have a priori knowledge about the typical appearance: e.g. bone shapes and texture How can we represent this knowledge? How can we exploit it?
58 Statistical Shape Models 58 Each example is represented by a vector containing the coordinates of the landmarks. Learning: Model Acquisition Inference: Model Fitting
59 The space of all bone shapes 59 Bone shapes: vectors in Goal: project data onto a low-dimensional linear subspace that best explains their variation.
60 New subspace: `better coordinate system 60 Mean New coordinates reflect the distribution of the data. Few coordinates suffice to represent a high dimensional vector 1. Active Shape Models They can be viewed as parameters of a model 60
61 Using PCA to model shape 61 = + + +
62 Machine Learning for Computer Vision Lecture 5 Active shape models (ASM) A set of training examples (images) A set of landmarks, that are present on all images Build a statistical model of shape variation (PCA) Build a statistical model of the local texture (PCA) Use the model for the search in a new image 62
63 ASM search 63 Initialize Adjust to texture Fit to shape model
64 ASM search 64 64
65 Lecture outline 65 Bayes rule and generative models Density estimation Parametric deformable models Statistical active shape models Eigenfaces Active appearance models 3D Morphable models = + ^ x = µ + w 1 u 1 +w 2 u 2 +w 3 u 3 +w 4 u 4 +
66 Appearance modelling for faces When viewed as vectors of pixel values, face images are extremely high-dimensiona 100x100 image = 10,000 dimensions Very few vectors correspond to valid face images 66 Original coordinates are not revealing about face properties We want to model the subspace (`manifold ) of face images
67 Continuous Hidden Variables: Appearance Manifolds 67 x 2 x n x 1 Lighting x Pose [Murase and Nayar 1993]
68 Eigenfaces (Murase & Nayar, 91) 68 Training images x 1,,x N
69 Eigenfaces 69 Top eigenvectors: u 1, u k Mean: µ
70 Eigenfaces 70 Principal component (eigenvector) u k µ + 3σ k u k µ 3σ k u k
71 Eigenfaces example 71 Face x in face space coordinates: Reconstruction: = = + ^ x = µ + w 1 u 1 +w 2 u 2 +w 3 u 3 +w 4 u 4 +
72 Limitations 72 Global appearance method: not robust to misalignment, background variation
73 Lecture outline 73 Bayes rule and generative models Density estimation Parametric deformable models Statistical active shape models Eigenfaces Active appearance models 3D Morphable models
74 Active Appearance Models (AAMs) 74 Shape: Appearance: Synthesis: I(S (x)) = T (x) X S(X) T emplate Ins tance
75 Playing with the AAM parameters 75 First two modes of shape variation First two modes of gray-level variation First four modes of appearance variation
76 Active Appearance Model Search (Results) 76
77 AAM Search 77
78 Lecture outline 78 Bayes rule and generative models Density estimation Parametric deformable models Statistical active shape models Eigenfaces Active appearance models 3D Morphable models
79 3-D surface acquisition 79 Laser Range Scanners Stereo Cameras Structured Light (Kinect) Photometric Stereo
80 What can we do with 3d shape models? 80 [Blanz and Vetter 1999, 2003]
81 Building a Morphable Face Model 81 [Blanz and Vetter 1999, 2003]
82 3-D Morphable Models 82 [Blanz and Vetter 1999, 2003]
83 3D Morphable models 83 Recover Shape Synthesize new views Synthesize new expressions
84 3-D Morphable Model fitting 84 Rough manual initialization Gradient descent to minimize reconstruction error functional And then
85 3D AAM for face tracking 85 CMU group: I. Matthews, S. Baker, R. Gross (230 Frames per second, 2004)
86 3D AAM for face tracking 86 86
87 Playing with Facial Attributes 87 Several classes of attributes are modeled: Facial expressions (smile, frown) Individual characteristics (double chin, hooked nose, maleness ) Distinctiveness
88 Manipulating Facial Attributes via Deformations 88 For each face in the database, two scans are recorded: S neutral, and S expression. The difference vector ΔS = S expression - S neutral is saved and later on simply added to the 3D reconstruction of the input image.
89 89
90 90 90
91 APPENDIX 91
92 Factor Analysis: Generative Model 92 Hidden variables Observations noise covariance matrix Linear model Distribution of x
93 Full observation distribution 93 Consider covariance of x, h: Full observations Distribution We will need to write Problem: non-diagonal matrix
94 Block matrix diagonalization 94 Schur Complement
95 Factorizing a Gaussian distribution 95
96 96 PCA criterion Minimize reconstruction error of training set
97 Spectral Decomposition of a matrix 97
98 Principal Component Analysis 98 Given: N data points x 1,,x N in R d We want to find a new set of features that are linear combinations of original ones: u(x i ) = u T (x i µ) (µ: mean of data points) What unit vector u in R d captures the most variance of the data?
99 Principal Component Analysis Variance of projection on u: 99 Projection of data point Covariance matrix of data Direction: Unit norm vector The direction that maximizes the variance: the eigenvector associated with the largest eigenvalue of Σ
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