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Published byDebra Dalton Modified over 8 years ago
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Face detection and recognition Many slides adapted from K. Grauman and D. Lowe
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Face detection and recognition DetectionRecognition “Sally”
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Consumer application: iPhoto 2009 http://www.apple.com/ilife/iphoto/
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Consumer application: iPhoto 2009 Can be trained to recognize pets! http://www.maclife.com/article/news/iphotos_faces_recognizes_cats
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Consumer application: iPhoto 2009 Things iPhoto thinks are faces
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Outline Face recognition Eigenfaces Face detection The Viola & Jones system
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The space of all face images When viewed as vectors of pixel values, face images are extremely high-dimensional 100x100 image = 10,000 dimensions However, relatively few 10,000-dimensional vectors correspond to valid face images We want to effectively model the subspace of face images
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The space of all face images We want to construct a low-dimensional linear subspace that best explains the variation in the set of face images
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Principal Component Analysis 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? Forsyth & Ponce, Sec. 22.3.1, 22.3.2
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Principal Component Analysis Direction that maximizes the variance of the projected data: Projection of data point Covariance matrix of data The direction that maximizes the variance is the eigenvector associated with the largest eigenvalue of Σ N N
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Principal component analysis The direction that captures the maximum covariance of the data is the eigenvector corresponding to the largest eigenvalue of the data covariance matrix Furthermore, the top k orthogonal directions that capture the most variance of the data are the k eigenvectors corresponding to the k largest eigenvalues
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Eigenfaces: Key idea Assume that most face images lie on a low-dimensional subspace determined by the first k (k<d) directions of maximum variance Use PCA to determine the vectors or “eigenfaces” u 1,…u k that span that subspace Represent all face images in the dataset as linear combinations of eigenfaces M. Turk and A. Pentland, Face Recognition using Eigenfaces, CVPR 1991Face Recognition using Eigenfaces
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Eigenfaces example Training images x 1,…,x N
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Eigenfaces example Top eigenvectors: u 1,…u k Mean: μ
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Eigenfaces example Principal component (eigenvector) u k μ + 3σ k u k μ – 3σ k u k
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Eigenfaces example Face x in “face space” coordinates: =
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Eigenfaces example Face x in “face space” coordinates: Reconstruction: =+ µ + w 1 u 1 +w 2 u 2 +w 3 u 3 +w 4 u 4 + … = ^ x=
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Reconstruction demo
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Recognition with eigenfaces Process labeled training images: Find mean µ and covariance matrix Σ Find k principal components (eigenvectors of Σ) u 1,…u k Project each training image x i onto subspace spanned by principal components: (w i1,…,w ik ) = (u 1 T (x i – µ), …, u k T (x i – µ)) Given novel image x: Project onto subspace: (w 1,…,w k ) = (u 1 T (x – µ), …, u k T (x – µ)) Optional: check reconstruction error x – x to determine whether image is really a face Classify as closest training face in k-dimensional subspace ^ M. Turk and A. Pentland, Face Recognition using Eigenfaces, CVPR 1991Face Recognition using Eigenfaces
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Recognition demo
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Limitations Global appearance method: not robust to misalignment, background variation
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Limitations PCA assumes that the data has a Gaussian distribution (mean µ, covariance matrix Σ) The shape of this dataset is not well described by its principal components
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Limitations The direction of maximum variance is not always good for classification
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