HW2 due on Thursday. Face Recognition: Dimensionality Reduction. Biometrics CSE 190 Lecture 11. Perceptron Revisited: Linear Separators

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1 HW due on Thursday Face Recognition: Dimensionality Reduction Biometrics CSE 190 Lecture 11 CSE190, Winter 010 CSE190, Winter 010 Perceptron Revisited: Linear Separators Binary classification can be viewed as the task of separating classes in feature space: Linear Separators Which of the linear separators is optimal? w T x + b > 0 w T x + b = 0 w T x + b < 0 f(x) = sign(w T x + b) 3 4 Classification Margin Distance from example x i to the separator is r = wt x i + b w Training examples closest to the hyperplane are support vectors. Margin ρ of the separator is the distance from the separator to support vectors. ρ Maximum Margin Classification Maximizing the margin is good according to intuition and PAC (Probably Approximately Correct) theory. Implies that only support vectors matter; other training examples are ignorable. r 5 Solved as a Quadratic Programming Problem 6 1

2 Soft Margin Classification What if the training set is not linearly separable? Slack variables ξ i can be added to allow misclassification of difficult or noisy examples, resulting margin called soft. Non-linear SVMs Datasets that are linearly separable with some noise work out great: 0 x But what are we going to do if the dataset is just too hard? ξ i ξ i 0 x How about mapping data to a higher-dimensional space: x 7 0 x 8 Non-linear SVMs: Feature spaces The Kernel Trick General idea: the original feature space can always be mapped to some higher-dimensional feature space where the training set is separable: Φ: x φ(x) 9 The linear classifier relies on inner product between vectors K(x i,x j )=x it x j If every datapoint is mapped into high-dimensional space via some transformation Φ: x φ(x), the inner product becomes: K(x i,x j )= φ(x i ) T φ(x j ) A kernel function is a function that is eqiuvalent to an inner product in some feature space. Example: -dimensional vectors x=[x 1 x ]; let K(x i,x j )=(1 + x it x j ), Need to show that K(x i,x j )= φ(x i ) T φ(x j ): K(x i,x j )=(1 + x it x j ),= 1+ x i1 x j1 + x i1 x j1 x i x j + x i x j + x i1 x j1 + x i x j = = [1 x i1 x i1 x i x i x i1 x i ] T [1 x j1 x j1 x j x j x j1 x j ] = = φ(x i ) T φ(x j ), where φ(x) = [1 x 1 x 1 x x x 1 x ] Thus, a kernel function implicitly maps data to a high-dimensional space (without the need to compute each φ(x) explicitly). 10 What Functions are Kernels? Examples of Kernel Functions For some functions K(x i,x j ) checking that K(x i,x j )= φ(x i ) T φ(x j ) can be cumbersome. Mercer s theorem: Every semi-positive definite symmetric function is a kernel Semi-positive definite symmetric functions correspond to a semi-positive definite symmetric Gram matrix: Linear: K(x i,x j )= x it x j Mapping Φ: x φ(x), where φ(x) is x itself Polynomial of power p: K(x i,x j )= (1+ x it x j ) p Mapping Φ: x φ(x), where φ(x) has dimensions K= K(x 1,x 1 ) K(x 1,x ) K(x 1,x 3 ) K(x 1,x n ) K(x,x 1 ) K(x,x ) K(x,x 3 ) K(x,x n ) K(x n,x 1 ) K(x n,x ) K(x n,x 3 ) K(x n,x n ) Gaussian (radial-basis function): K(x i,x j ) = Mapping Φ: x φ(x), where φ(x) is infinite-dimensional: every point is mapped to a function (a Gaussian); combination of functions for support vectors is the separator. Higher-dimensional space still has intrinsic dimensionality d (the mapping is not onto), but linear separators in it correspond to non-linear separators in original space. 11 1

3 Dual problem formulation: The solution is: Non-linear SVMs Mathematically Find α 1 α n such that Q(α) =Σα i - ½ΣΣα i α j y i y j K(x i, x j ) is maximized and (1) Σα i y i = 0 () α i 0 for all α i f(x) = Σα i y i K(x i, x j )+ b SVM applications SVMs were originally proposed by Boser, Guyon and Vapnik in 199 and gained increasing popularity in late 1990s. SVMs are currently among the best performers for a number of classification tasks ranging from text to genomic data. SVMs can be applied to complex data types beyond feature vectors (e.g. graphs, sequences, relational data) by designing kernel functions for such data. SVM techniques have been extended to a number of tasks such as regression [Vapnik et al. 97], principal component analysis [Schölkopf et al. 99], etc. Tuning SVMs remains a black art: selecting a specific kernel and parameters is usually done in a try-and-see manner. Optimization techniques for finding α i s remain the same! Face Face Recognition Face is the most common biometric used by humans Applications range from static, mug-shot verification to a dynamic, uncontrolled face identification in a cluttered background Challenges: automatically locate the face recognize the face from a general view point under different illumination conditions, facial expressions, and aging effects CSE190, Winter 010 Authentication vs Identification Face Authentication/Verification (1:1 matching) Applications Face Identification/recognition (1:N matching) 3

4 Applications Applications Face Scan at Airports Why is Face Recognition Hard? Who are these people? Many faces of Madonna [Sinha and Poggio 1996] Why is Face Recognition Hard? Face Recognition Difficulties Identify similar faces (inter-class similarity) Accommodate intra-class variability due to: head pose illumination conditions expressions facial accessories aging effects Cartoon faces 4

5 Inter-class Similarity Intra-class Variability Faces with intra-subject variations in pose, illumination, expression, accessories, color, occlusions, and brightness Different persons may have very similar appearance news.bbc.co.uk/hi/english/in_depth/ americas/000/us_elections Twins Father and son Sketch of a Pattern Recognition Architecture Example: Face Detection Scan window over image. Classify window as either: Face Non-face Image (window) Feature Extraction Classification Feature Vector CS5A, Winter 005 Detection Test Sets Object Identity Face Window Classifier CS5A, Winter 005 Non-face Profile views Schneiderman s Test set 5

6 Face Detection: Experimental Results Test sets: two CMU benchmark data sets Test set 1: 15 images with 483 faces Test set : 0 images with 136 faces Example: Finding skin Non-parametric Representation of CCD Skin has a very small range of (intensity independent) colors, and little texture Compute an intensity-independent color measure, check if color is in this range, check if there is little texture (median filter) See this as a classifier - we can set up the tests by hand, or learn them. get class conditional densities (histograms), priors from data (counting) Classifier is [See also work by Viola & Jones, Rehg, more recent by Schneiderman] CS5A, Winter 005 Face Detection Algorithm Lighting Compensation Color Space Transformation Skin Color Detection Input Image Variance-based Segmentation Connected Component & Grouping Face Localization Eye/ Mouth Detection! Face Boundary Detection" Verifying/ Weighting! Eyes-Mouth Triangles! Figure from Statistical color models with application to skin detection, M.J. Jones and J. Rehg, Proc. Computer Vision and Pattern Recognition, 1999 copyright 1999, IEEE Facial Feature Detection! Output Image CS5A, Winter 005 Face Detection Face Recognition: -D and 3-D Time (video) -D 3-D CS5A, Winter 005 Recognition Comparison -D 3-D Face Prior knowledge Recognition Database of face class Data (Probe) (Gallery) 6