### Document

```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
```