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Image Analysis

PCA and Eigenfaces

Christophoros [email protected]

PCA and Eigenfaces

University of Ioannina - Department of Computer Science

Images taken from:D. Forsyth and J. Ponce. Computer Vision: A Modern Approach, Prentice Hall, 2003.Computer Vision course by Svetlana Lazebnik, University of North Carolina at Chapel Hill.Computer Vision course by Michael Black, Brown University.Research page of Antonio Torralba, MIT.

2 Face detection and recognition

C. Nikou – Image Analysis (T-14)

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3 Face detection and recognition

Detection Recognition “Sally”


4 Consumer application: iPhoto 2009


http://www.apple.com/ilife/iphoto/

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It can be trained to recognize pets!


http://www.maclife.com/article/news/iphotos_faces_recognizes_cats


iPhoto decides that this is a face


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7 Outline

• Face recognition– EigenfacesEigenfaces

• Face detection– The Viola and Jones algorithm.


P. Viola and M. Jones. Robust real-time face detection. International Journal of Computer Vision, 57(2), 2004. Also in CVPR 2001.

M. Turk and A. Pentland. "Eigenfaces for recognition". Journal of Cognitive Neuroscience, 3(1):71–86, 1991. Also in CVPR 1991.

8 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


correspond to valid face images.

• Is there a compact and effective representation of the subspace of face images?

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9 The space of all face images

• Insight into principal t l icomponent analysis

(PCA).• We want to construct

a low-dimensional linear subspace that


linear subspace that best explains the variation in the set of face images.

10 Principal Component Analysis

• Given: N data points x1, … ,xN in Rd

• We want to find a new set of features that are linear combinations of the original ones:

w(xi) = uT(xi – µ)( : mean of data points)


(µ: mean of data points)

• What unit vector u in Rd captures the most variance of the data?

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11Principal Component Analysis

(cont.)• The variance of the projected data:

Projection of data point

( ) ( )1 1

1 1var ( ) ( ) ( ) ( ) ( )N N TT T T

i i i i ii i

w w wN N= =

= = − −∑ ∑x x x u x μ u x μ

1 1( )( ) ( )( )N N

T T Ti i i i

⎡ ⎤= − − = − −⎢ ⎥∑ ∑u x μ x μ u u x μ x μ u


Covariance matrix of data

1 1

( )( ) ( )( )i i i ii iN N= =

⎢ ⎥⎣ ⎦∑ ∑u x μ x μ u u x μ x μ u

T= u Σu


(cont.)• We now estimate vector u maximizing the variance:

Tu Σusubject to: 2|| || 1T = =u u ubecause any multiple of u maximizes the objective function.

• The Lagrangian is ( ; ) (1 )T TJ λ λ= + −u u Σu u u

u Σu


The Lagrangian isleading to the solution:

( ; ) (1 )J λ λ= +u u Σu u uλ=Σu u

which is an eigenvector of Σ. The one maximizing J corresponds to the largest eigenvalue of Σ.

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(cont.)• The direction that captures the maximum

covariance of the data is the eigenvector gcorresponding to the largest eigenvalue of the data covariance matrix.

• The top k orthogonal directions that


capture the most variance of the data are the k eigenvectors corresponding to the klargest eigenvalues.

14 Eigenfaces: Key idea

• Assume that most face images lie on a low-dimensional subspace determineda 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

b


subspace.• Represent all face images in the dataset

as linear combinations of eigenfaces.

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15 Eigenfaces example

Training images


16 Eigenfaces example (cont.)Top eigenvectors

Mean: μ


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17 Eigenfaces example (cont.)

uk

3μ - uk kλ


3μ uk kλ+


• Face x in “face space” coordinates:

( ) ( ) ( )TT T T⎡ ⎤

[ ]1 2

1 2

( ), ( ),..., ( )

, ,...,

x u x μ u x μ u x μT T Tk

Tkw w w

⎡ ⎤→ − − −⎣ ⎦

=• Reconstruction:


+=

+x̂ = μ 1 1 2 2 ...u u uk kw w w+ + +

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• Any face of the training set may be expressed as a linear combination of a pnumber of eigenfaces.

• In matrix-vector form:

[ ]1

21 2x μ u u uk

ww⎡ ⎤⎢ ⎥⎢ ⎥= +⎢ ⎥

1=K

ii kk KE

λ

λ= +∑

∑

Error:


+=

[ ]1 2μ k

kw⎢ ⎥⎢ ⎥⎣ ⎦

1

Kiiλ

=∑


Intra-personal subspace (variations in expression)


Extra-personal subspace (variations between people)

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21 Face Recognition with eigenfaces

Process the training images:• Find mean µ and covariance matrix Σ.• Find k principal components (eigenvectors of Σ) u u• Find k principal components (eigenvectors of Σ) u1,…uk.• Project each training image xi onto subspace spanned by

principal components:(wi1,…,wik) = (u1

T(xi – µ), … , ukT(xi – µ))

Given a new image x:• Project onto subspace:

T T


(w1,…,wk) = (u1T(x – µ), … , uk

T(x – µ)).• Optional: check the reconstruction error x – x to determine

whether the new image is really a face.• Classify as closest training face in k-dimensional subspace.

^

M. Turk and A. Pentland, Face Recognition using Eigenfaces, CVPR 1991

22 Computation of eigenvectors

• Given the d-dimensional vectorized images xi(i=1,2…,N), we form matrix X=[x1, x2,…, xN], d >>N.

• Covariance matrix:

1 TXXN

Σ =

• Σ has dimensions dxd which, for example, for a


200x200 image is very large (40000x40000).• How do we compute its eigenvalues and

eigenvectors?

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23 Computation of eigenvectors (cont.)

• We create the NxN matrix:1 TT X X1 TT X XN

=

• T has N eigevalues (γi) and eigenvectors (ei):

i i iTe eγ= 1 Ti i iX Xe e

Nγ⇔ =

1 Ti i iXX Xe Xe

Nγ⇔ =


N N( ) ( )i i iXe Xeγ⇔ Σ =

• γi is an eigenvalue of Σ and Xei is the correspoding eigenvector.

24The importance of the mean image

and the variations


Average of 100 of the images from the Caltech-101 dataset. Reproduced from the web page of A. Torralba http://web.mit.edu/torralba/www/gallery/gallery11.html

What is this?

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and the variations (cont.)


The context plays an important role. Is it a car or a person? Both blobs correspond to the same shape after a 90 degrees rotation.

Reproduced from the web page of A. Torralba.http://web.mit.edu/torralba/www/gallery/gallery11.html


and the variations (cont.)


Average fashion model. Average of 60 aligned face images of fashion models.

Reproduced from the Perception Lab, University of St Andrews in Fife, UK. http://perception.st-and.ac.uk

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27Application: MRF features for texture

representation• Each pixel is described by its 7x7 neighborhood

in every Lab channel.• This results in a 7x7x3 = 147-dimensional vector

per pixel.• Applying PCA to these vectors leads to a

compact 6- to 8-dimensional representation of natural images.


• It captures more than 90% of the variation of the image pixels neighborhoods.

28Application: MRF features for texture

representation (cont.)


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29 Application: view-based modeling

• PCA on various views of a 3D object.


H. Murase and S. Nayar. Visual Learning and Recognition of 3D objects from appearance, International Journal of Computer Vision, Vol. 14, pp. 5-24,1995.

30Application: view-based modeling

(cont.)• Subspace of the first three PC (varying view angle).



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(cont.)• View angle recognition.




(cont.)• Extension to more 3D objects with varying pose and

illumination.



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33 Application: EigenheadsPrincipal components 1 and 2.


34 Application: Eigenheads (cont.)Principal components 3 and 4.


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35 Limitations

• Global appearance method– not robust to misalignment, backgroundnot robust to misalignment, background

variation.


36 Limitations (cont.)

• Projection may suppress important detail– The smallest variance directions may not be

unimportant. PCA assumes Gaussian data.


The shape of this dataset is not well described by its principal components

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37 Limitations (cont.)

• PCA does not take the discriminative task into account

– Typically, we wish to compute features that allow good discrimination.

• Projection onto the major axis can not separate the green from the red data.


• The second principal component captures what is required for classification.

38 Canonical Variates

• Also called “Linear Discriminant Analysis”• A labeled training set is necessaryA labeled training set is necessary.• We wish to choose linear functions of the

features that allow good discrimination.– Assume class-conditional covariances are

the same.


– We seek for linear feature maximizing the spread of class means for a fixed within-class variance.

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39 Canonical Variates (cont.)

• We have a set of vectorized images xi(i=1,2…,N).( , , )

• The images belong to C categories each having a mean μj, ( j=1,2,…,C ).

• The mean of the class means:

1 C


1

1j

jCμ μ

=

= ∑


• Between class covariance:C

T( )( )TB i i i

i 1

S N μ μ μ μ=

= − −∑with Ni being the number of images in class i.

• Within class covariance:


• Within class covariance:

( )( )iNC T

W k ,i i k ,i ii 1 k 1

S x xμ μ= =

= − −∑∑

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Using the same argument as in PCA (and graph-based segmentation) we seek vector u maximizing

which is equivalent to

maxT

BTu

W

u S uu S u

⎧ ⎫⎨ ⎬⎩ ⎭

max{ }TBu

u S u s.t. 1TWu S u =


B WS u S uλ=

The solution is the top eigenvector of the generalized eigenvalue problem:



• 71 views of 10 objects at a variety of poses at a black background.• 60 images were used to determine a set of canonical variates.• 11 images were used for testing.

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• The first two canonical variates. The clusters are tight and well separated (a different symbol per object is used).

• We could probably get a quite good classification.