Principal component analysis

1. Features extraction and representation / Image processing
Farah Al-Tufaili

2. The reader is probably familiar with the common saying that goes
something along the lines of ‘Why use a hundred words when ten
will do?’ This idea of expressing information in it’s most succinct
and compact form embodies very accurately the central idea behind
Principal Component Analysis (PCA) a very important and
powerful statistical technique.
Purpose: to reduce dimensionality of a vector image while maintaining
information as much as possible.
PCA- by Farah Al-Tufaili 2

3. Some of the more significant aspects of digital imaging in which it
has found useful application include the classification of
photographic film, remote sensing and data compression, and
automated facial recognition and facial synthesis, an Application of
PCA.
It is also commonly used as a low-level image processing tool for
certain tasks, such as determination of the orientation of basic
shapes.
PCA- by Farah Al-Tufaili 3

4. PCA steps: transform an 𝑁×𝑑 matrix 𝑋 into an 𝑁×𝑚 matrix
𝑌:
Centralized the data (subtract the mean).
Calculate the 𝑑×𝑑 covariance matrix: C=
1
𝑁−1
𝑋 𝑇 𝑋
 𝐶𝑖,𝑗=
1
𝑁−1 𝑞=1
𝑁
𝑋 𝑞,𝑖. 𝑋 𝑞,𝑗
 𝐶𝑖,𝑖 (diagonal) is the variance of variable i.
 𝐶𝑖,𝑗 (off-diagonal) is the covariance between variables i and j.
Calculate the eigenvectors of the covariance matrix (orthonormal).
PCA- by Farah Al-Tufaili 4

5. Let
Covariance matrix


K
k
T
xx
T
kk
T
xxx
K
E
1
1
}))({( mmxxmxmxC
T
nxxx ]...[ 21x
PCA- by Farah Al-Tufaili 5

6. A 2-D facial image can be represented as 1-D vector by concatenating each row (or column) into a
long thin vector. Let’s suppose we have M vectors of size N (= rows of image × columns of image)
representing a set of sampled images. pj’s represent the pixel values.
xi = [p1 ...pN]T,i = 1,...,M (1)
The images are mean centered by subtracting the mean image from each image vector. Let m
represent the mean image.
(2)
And let wi be defined as mean centered image
wi = xi − m (3)
Our goal is to find a set of ei’s which have the largest possible projection onto each of the wi’s. We
wish to find a set of M orthonormal vectors ei for which the quantity
(4)
is maximized with the orthonormality constraint
PCA- by Farah Al-Tufaili 6

7.  This example illustrates the primary and essential requirement for PCA to be
useful: there must be some degree of correlation between the data vectors. In
fact, the more strongly correlated the original data, the more effective PCA is
likely to be.
 To provide some concrete but simple data to work with, consider that we have
just M=2 measurements on the height h and the weight w of a sample group of
N=12 people as given in Table 9.2. Figure 9.7 plots the mean-subtracted data in
the 2-D feature space (w, h) and it is evident that the data shows considerable
variance over both
PCA- by Farah Al-Tufaili 8

8. variables and that they are correlated. The sample covariance matrix for these
two variables is calculated as:
Where x=(h w)T and x=(h w)T is the simple mean.
PCA- by Farah Al-Tufaili 9

9. Weight (X) Height (Y) X-mean Y-mean
65 170 -4.416666667 -3.25
75 176 5.583333333 2.75
53 154 -16.41666667 -19.25
54 167 -15.41666667 -6.25
61 171 -8.416666667 -2.25
88 184 18.58333333 10.75
70 182 0.583333333 8.75
78 190 8.583333333 16.75
52 166 -17.41666667 -7.25
95 168 25.58333333 -5.25
70 176 0.583333333 2.75
72 175 2.583333333 1.75
Mean: 69.41666667 173.25 Variance: 182.9924 90.5682
Applying the equation: 

K
k
T
xx
T
kk
T
xxx
K
E
1
1
}))({( mmxxmxmxC
PCA- by Farah Al-Tufaili 10

10. clc
x=[65 75 53 54 61 88 70 78 52 95 70 72]
y=[170 176 154 167 171 184 182 190 166 168 176 175]
x_mean=mean(x) , y_mean=mean(y)
x1=x-x_mean; y1=y-y_mean;
var_x=sum(x1.^2)/(length(x)-1)
var_y=sum(y1.^2)/(length(y)-1)
cv=(x1).*(y1);
cov=sum(cv)/(length(x)-1)
Output:
x_mean =
69.4167
y_mean =
173.2500
var_x =
182.9924
var_y =
90.5682
cov =
73.4318
PCA- by Farah Al-Tufaili 11

11. • The basic aim of PCA is to effect a rotation of the coordinate system and
thereby express the data in terms of a new set of variables or equivalently
axes which are uncorrelated.
• How are these found? The first principal axis is chosen to satisfy the following
criterion:
• The axis passing through the data points which maximizes the sum of the
squared lengths of the perpendicular projection of the data points onto that
axis is the principal axis.
PCA- by Farah Al-Tufaili 12

12. Figure 9.7 Distribution of weight and height values within a 2-D feature space. The mean
value of each variable has been subtracted from the data
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13. Thus, the principal axis is oriented so as to maximize the overall variance of the data with respect to it
(a variance-maximizing transform). We note in passing that an alternative but entirely equivalent
criterion for a principal axis is that it minimizes the sum of the squared errors (differences) between
the actual data points and their perpendicular projections onto the said straight line. This concept is
illustrated in Figure 9.8.
Let us suppose that the first principal axis has been found. To calculate the second principal axis we
proceed as follows:
 Calculate the projections of the data points onto the first principal axis.
 Subtract these projected values from the original data. The modified set of data points is termed the
residual data.
 The second principal axis is then calculated to satisfy an identical criterion to the first (i.e. the
variance of the residual data is maximized along this axis).
PCA- by Farah Al-Tufaili 14

14. Note also that this is distinct from fitting a straight line by regression. In regression, x is an
independent variable and y the dependent variable and we seek to minimize the sum of the
squared errors between the actual y values and their predicted values.
PCA- by Farah Al-Tufaili 15

15. PCA- by Farah Al-Tufaili 16

16. In fact, this alignment is precisely the mechanism that decorrelatcs the data. Furthermore, as the
eigenvalues appear along the main diagonal of Cy ,λ i is the variance of component yi along
eigenvector ei . The two eigenvectors are perpendicular. The y-axes sometimes are called the
eigen axes for obvious reasons.
PCA- by Farah Al-Tufaili 17

17.  let the covariance matrix is :
7 3
3 −1
 First Step: compute eigenvalues. To do this, we find the values of which satisfy
the characteristic equation of the matrix A, namely those values of for which
det(A − I) = 0,
1. λ I= λ
1 0
0 1
=
λ 0
0 λ
2. A- λI=
7 3
3 −1
-
λ 0
0 λ
=
7 − λ 3
3 −1 − λ
3. det
7 3
3 −1
= (7- λ)(-1- λ)-(3)(3)= -7-7λ+λ+ λ2-9= λ2-6 λ-16
(λ-8)(λ+2)=0 λ1=8 λ2=-2
EigenValues
Let this
equation equal
to zero and
solve it
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18.  Second step: find corresponding eigenvectors.
1. For each eigenvalue λ, we have (A − λ I)V = 0, where x is the eigenvector
associated with eigenvalue λ.
2. Find x by Gaussian elimination. That is, convert the augmented matrix
(A − λ I ⋮ 0)
to row echelon form, and solve the resulting linear system by back
substitution.
We find the eigenvectors associated with each of the eigenvalues
PCA- by Farah Al-Tufaili 19

19. −1𝑣1+ 3𝑣2 = 0 ………..(1
3𝑣1- 9𝑣2 = 0 ………..(2
2𝑣1- 6𝑣2 = 0 ………..(3
Lets 𝑠 =
(2)2+(−6)2=6.3246
v1 = 2/6.3246= 0.3162
v2 = -6/6.3246= -0.9487
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When λ1=8
A- λI=
7 − 8 3
3 −1 − 8
=
−1 3
3 −9
(A − λ I)V = 0
−1 3
3 −9
⋮
𝑣1
𝑣2
=0
−𝟏 × 𝐯 𝟏 + 𝟑 × 𝐯 𝟐 = 𝟎
𝟑 × 𝐯 𝟏 + −𝟗 × 𝐯 𝟐 = 𝟎

20. PCA- by Farah Al-Tufaili 21
9𝑣1+ 3𝑣2 = 0 ………..(1
3𝑣1+1𝑣2 = 0 ………..(2
6𝑣1+2𝑣2 = 0 ………..(3
Lets 𝑠 =
(2)2+(6)2=6.3246
v1 = -6/6.3246= -0.9487
v2 = -2/6.3246= -0.3162
When λ1=-2
A- λI=
7 + 2 3
3 −1 + 2
=
9 3
3 1
(A − λ I)V = 0
9 3
3 1
⋮
𝑣1
𝑣2
=0
𝟗 × 𝐯 𝟏 + 𝟑 × 𝐯 𝟐 = 𝟎
𝟑 × 𝐯 𝟏 + 𝟏 × 𝐯 𝟐 = 𝟎

21. check our solution in Matlab:
>> x=[7 3;3 -1]
x =
7 3
3 -1
>>[a b]=eig(x)
a =
0.3162 -0.9487
-0.9487 -0.3162
b =
-2 0
0 8
PCA- by Farah Al-Tufaili 22
EigenValues
EigenVictor

22.  There is a subtle difference between the two alternative derivations we have
offered. In the first case we considered our variables to define the feature space
and derived a set of principal axes. The dimensionality of our feature space
was determined by the number of variables and the diagonalizing matrix of
eigenvectors R to produce a new set of axes in that 2-D space. Our second
formulation leads to essentially the same procedure (i.e. to diagonalize the
covariance matrix), but it nonetheless admits an alternative viewpoint. In this
instance, we view the observations on each of the variables to define our
(higher dimensional) feature vectors. The role of the diagonalizing matrix of
eigenvectors R here is thus to multiply and transform these higher dimensional
vectors to produce a new orthogonal (principal) set. Thus, the dimensionality
of the space here is determined by the number of observations made on each of
the variables.
PCA- by Farah Al-Tufaili 23

23. W=[65 75 53 54 61 88 70 78 52 95 70 72]' %1. Form data vector on weight
H=[170 176 154 167 171 184 182 190 166 168 176 175]' %Form data vector on height
XM=[mean(W).*ones(length(W),1) mean(H).*ones(length(H),1)] %matrix with mean values replicated
X=[W H]-XM %Form mean-subtracted data matrix
Cx=cov(X) %calculate covariance on data
[R,LAMBDA,Q]=svd(Cx) %Get eigenvalues LAMBDA and eigenvectors R
V=X*R %Calculate principal components
subplot(1,2,1), plot(X(:,1),X(:,2),'ko'); grid on; %2. display data on original axes
subplot(1,2,2), plot(V(:,1),V(:,2),'ro'); grid on; %display PCs as data in rotated space
XR=XM+V*R' %3. Reconstruct data in terms of PCs
XR-[W H] %Confirm reconstruction (diff = 0)
V'*V./(length(W)-1) %4. Confirm covariance terms
%MATLAB FUNCTIONS cov, mean, svd,
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24. The weight–height data referred to the original axes x1 and x2 and to the principal axes v1 and v2
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25. • It is self-evident that the basic
placement, size and shape of human
facial features are similar. Therefore,
we can expect that an ensemble of
suitably scaled and registered images
of the human face will exhibit fairly
strong correlation. Each image
consisted of 21 054 grey-scale pixel
values.
Figure 9.13 A sample of human faces scaled and
registered such that each can be described by the
same number of pixels (21 054).
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26. Figure 9.15 The first six facial principal components from a sample of 290 faces.
The first principal is the average face. Note the strong male appearance coded by
principal component no. 3
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27. Any Questions?
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