CSE 185 Introduction to Computer Vision

1. CSE 185 Introduction to Computer Vision Face Recognition

2. 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

3. We want to construct a low-dimensional linear subspace that best explains the variation in the set of face images The space of all face images

4. Covariance • Covariance is a measure of the extent to which corresponding elements from two sets of ordered data move in the samedirection

5. Variance-Covariance Matrix: Variance and covariance are displayed together in a variance-covariance matrix. The variances appear along the diagonal and covariancesappear in the off-diagonal elements Covariance (Variance-Covariance) matrix

6. Variance and covariance For any one pixel For any two pixels

7. Dimensionality reduction Distribution of data points

8. 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

9. Linear subspaces 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 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

10. 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

11. 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

12. 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

13. 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 = +

14. 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

15. Eigenfaces example • Training images x1,…,xN

16. Eigenfaces example Top eigenvectors: u1,…uk Mean: μ

17. Eigenfaces example Principal component (eigenvector) uk μ + 3σkuk μ – 3σkuk

18. Eigenfaces example • Face x in “face space” coordinates: =

19. Eigenfaces example • Face x in “face space” coordinates: • Reconstruction: = = + ^ x = µ + w1u1+w2u2+w3u3+w4u4+ …

20. Recognition with eigenfaces • Process labeled training images: • Find mean µ and covariance matrix Σ • Find k principal components (eigenvectors of Σ) u1,…uk • Project each training image xi onto subspace spanned by principal components:(wi1,…,wik) = (u1T(xi – µ), … , ukT(xi – µ)) • Given novel image x: • Project onto subspace:(w1,…,wk) = (u1T(x– µ), … , ukT(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

21. 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?

22. Projecting onto the eigenfaces The eigenfaces v1, ..., vK span the space of faces A face is converted to eigenface coordinates by

23. Recognition with eigenfaces Algorithm Process the image database (set of images with labels) Run PCA—compute eigenfaces Calculate the K coefficients for each image Given a new image (to be recognized) x, calculate K coefficients Detect if x is a face If it is a face, who is it? Find closest labeled face in database nearest-neighbor in K-dimensional space

24. Choosing the dimension K 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 i = K NM eigenvalues

25. Limitations • Global appearance method: not robust to misalignment, background variation