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CSE 185 Introduction to Computer Vision Face Recognition 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 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 Covariance • Covariance is a measure of the extent to which corresponding elements from two sets of ordered data move in the same direction x1 y1 X : ,Y : x n y n N covariance ( X ,Y ) i 1 x i x y i y N 1 Covariance (Variance-Covariance) matrix • Variance-Covariance Matrix: Variance and covariance are displayed together in a variance-covariance matrix. The variances appear along the diagonal and covariances appear in the offdiagonal elements æ v(x1 ) c(x1, x2 ) ........c(x1, x p ) ö ç ÷ ç c(x1, x2 ) v(x2 ) ........c(x2 , x p ) ÷ ç ÷ ç ÷ ç c(x , x ) c(x , x )..........v(x )÷ è 1 p 2 p p ø 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 Linear subspaces convert x into v1, v2 coordinates What does the v2 coordinate measure? - distance to line - use it for classification—near 0 for orange pts What does the v1 coordinate measure? - position along line - use it to specify which orange point it is • Classification can be expensive: – Big search prob (e.g., nearest neighbors) or store large PDF’s • Suppose the data points are arranged as above – Idea—fit a line, classifier measures distance to line Dimensionality reduction Dimensionality reduction • We can represent the orange points with only their v1 coordinates (since v2 coordinates are all essentially 0) • This makes it much cheaper to store and compare points • A bigger deal for higher dimensional problems Linear subspaces Consider the variation along direction v among all of the orange points: What unit vector v minimizes var? What unit vector v maximizes var? Solution: v1 is eigenvector of A with largest eigenvalue v2 is eigenvector of A with smallest eigenvalue Principal component analysis • Suppose each data point is N-dimensional – Same procedure applies: – The eigenvectors of A define a new coordinate system • eigenvector with largest eigenvalue captures the most variation among training vectors x • eigenvector with smallest eigenvalue has least variation – We can compress the data using the top few eigenvectors • corresponds to choosing a “linear subspace” – represent points on a line, plane, or “hyper-plane” • these eigenvectors are known as the principal components The space of faces = + • An image is a point in a high dimensional space – An N x M image is a point in RNM – We can define vectors in this space as we did in the 2D case 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” u1,…uk 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 1991 Eigenfaces example • Training images x1,…,xN Eigenfaces example Top eigenvectors: u1,…uk Mean: μ Eigenfaces example Principal component (eigenvector) uk μ + 3σkuk μ – 3σkuk Eigenfaces example • Face x in “face space” coordinates: = Eigenfaces example • Face x in “face space” coordinates: = • Reconstruction: = ^x = + µ + w1u1+w2u2+w3u3+w4u4+ … Recognition with eigenfaces • Process labeled training images: 1. Find mean µ and covariance matrix Σ 2. Find k principal components (eigenvectors of Σ) u1,…uk 3. Project each training image xi onto subspace spanned by principal components: (wi1,…,wik) = (u1T(xi – µ), … , ukT(xi – µ)) • Given novel image x: 1. Project onto subspace: (w1,…,wk) = (u1T(x – µ), … , ukT(x – µ)) 2. Optional: check reconstruction error x – x to determine whether image is really a face 3. Classify as closest training face in k-dimensional subspace Eigenfaces • PCA extracts the eigenvectors of A – Gives a set of vectors v1, v2, v3, ... – Each vector is a direction in face space • what do these look like? Projecting onto the eigenfaces • The eigenfaces v1, ..., vK span the space of faces – A face is converted to eigenface coordinates by Recognition with eigenfaces • Algorithm 1. Process the image database (set of images with labels) • • Run PCA—compute eigenfaces Calculate the K coefficients for each image 2. Given a new image (to be recognized) x, calculate K coefficients 3. Detect if x is a face 4. If it is a face, who is it? • Find closest labeled face in database – nearest-neighbor in K-dimensional space Choosing the dimension K eigenvalues i= K NM • How many eigenfaces to use? • Look at the decay of the eigenvalues – the eigenvalue tells you the amount of variance “in the direction” of that eigenface – ignore eigenfaces with low variance Limitations • Global appearance method: not robust to misalignment, background variation 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 Limitations • The direction of maximum variance is not always good for classification