1. Eigen Value Analysis in Pattern RecognitionBy
Dr. M. Asmat Ullah Khan
COMSATS Institute of Information Technology,
Abbottabad

10. MULTI SPECTRAL IMAGE COMPRESSION

11. MULTI SPECTRAL IMAGE COMPRESSION

12. MULTI SPECTRAL IMAGE COMPRESSION

13. MULTI SPECTRAL IMAGE COMPRESSION

14. MULTI SPECTRAL IMAGE COMPRESSION

15. Principal Component Analysis (PCA)
Pattern recognition in high-dimensional spaces
Problems arise when performing recognition in a high-dimensional space (curse of dimensionality).
Significant improvements can be achieved by first mapping the data into a lower-dimensional sub-space.
The goal of PCA is to reduce the dimensionality of the data while retaining as much as possible of the variation present in the original dataset.

16. Principal Component Analysis (PCA)
Dimensionality reduction
PCA allows us to compute a linear transformation that maps data from a high dimensional space to a lower dimensional sub-space.

17. Principal Component Analysis (PCA)
Lower dimensionality basis
Approximate vectors by finding a basis in an appropriate lower dimensional space.
(1) Higher-dimensional space representation:
(2) Lower-dimensional space representation:

18. Principal Component Analysis (PCA)
Example

19. Principal Component Analysis (PCA)
Information loss
Dimensionality reduction implies information loss !!
Want to preserve as much information as possible, that is:
How to determine the best lower dimensional sub-space?

20. Principal Component Analysis (PCA)
Methodology
Suppose x1, x2, ..., xM are N x 1 vectors

21. Principal Component Analysis (PCA)
Methodology – cont.

22. Principal Component Analysis (PCA)
Linear transformation implied by PCA
The linear transformation RN  RK that performs the dimensionality reduction is:

23. Principal Component Analysis (PCA)
Geometric interpretation
PCA projects the data along the directions where the data varies the most.
These directions are determined by the eigenvectors of the covariance matrix corresponding to the largest eigenvalues.
The magnitude of the eigenvalues corresponds to the variance of the data along the eigenvector directions.

24. Principal Component Analysis (PCA)
How to choose the principal components?
To choose K, use the following criterion:

25. Principal Component Analysis (PCA)
What is the error due to dimensionality reduction?
We saw above that an original vector x can be reconstructed using its principal components:
It can be shown that the low-dimensional basis based on principal components minimizes the reconstruction error:
It can be shown that the error is equal to:

26. Principal Component Analysis (PCA)
Standardization
The principal components are dependent on the units used to measure the original variables as well as on the range of values they assume.
We should always standardize the data prior to using PCA.
A common standardization method is to transform all the data to have zero mean and unit standard deviation:

27. Principal Component Analysis (PCA)
PCA and classification
PCA is not always an optimal dimensionality-reduction procedure for classification purposes:

28. Principal Component Analysis (PCA)
Case Study: Eigenfaces for Face Detection/Recognition
M. Turk, A. Pentland, "Eigenfaces for Recognition", Journal of Cognitive Neuroscience, vol. 3, no. 1, pp , 1991.
Face Recognition
The simplest approach is to think of it as a template matching problem
Problems arise when performing recognition in a high-dimensional space.
Significant improvements can be achieved by first mapping the data into a lower dimensionality space.
How to find this lower-dimensional space?

29. Principal Component Analysis (PCA)
Main idea behind eigenfaces

30. Principal Component Analysis (PCA)
Computation of the eigenfaces

31. Principal Component Analysis (PCA)
Computation of the eigenfaces – cont.

32. Principal Component Analysis (PCA)
Computation of the eigenfaces – cont.

33. Principal Component Analysis (PCA)
Computation of the eigenfaces – cont.

34. Principal Component Analysis (PCA)
Representing faces onto this basis

35. Principal Component Analysis (PCA)
Representing faces onto this basis – cont.

36. Principal Component Analysis (PCA)
Face Recognition Using Eigenfaces

37. Principal Component Analysis (PCA)
Face Recognition Using Eigenfaces – cont.
The distance er is called distance within the face space (difs)
Comment: we can use the common Euclidean distance to compute er, however, it has been reported that the Mahalanobis distance performs better:

38. Principal Component Analysis (PCA)
Face Detection Using Eigenfaces

39. Principal Component Analysis (PCA)
Face Detection Using Eigenfaces – cont.

40. Principal Component Analysis (PCA)
Problems
Background (de-emphasize the outside of the face – e.g., by multiplying the input image by a 2D Gaussian window centered on the face)
Lighting conditions (performance degrades with light changes)
Scale (performance decreases quickly with changes to head size)
multi-scale eigenspaces
scale input image to multiple sizes
Orientation (performance decreases but not as fast as with scale changes)
plane rotations can be handled
out-of-plane rotations are more difficult to handle

41. Linear Discriminant Analysis (LDA)
Multiple classes and PCA
Suppose there are C classes in the training data.
PCA is based on the sample covariance which characterizes the scatter of the entire data set, irrespective of class-membership.
The projection axes chosen by PCA might not provide good discrimination power.
What is the goal of LDA?
Perform dimensionality reduction while preserving as much of the class discriminatory information as possible.
Seeks to find directions along which the classes are best separated.
Takes into consideration the scatter within-classes but also the scatter between-classes.
More capable of distinguishing image variation due to identity from variation due to other sources such as illumination and expression.

42. Linear Discriminant Analysis (LDA)
Methodology

43. Linear Discriminant Analysis (LDA)
Methodology – cont.
LDA computes a transformation that maximizes the between-class scatter while minimizing the within-class scatter:
Such a transformation should retain class separability while reducing the variation due to sources other than identity (e.g., illumination).

44. Linear Discriminant Analysis (LDA)
Linear transformation implied by LDA
The linear transformation is given by a matrix U whose columns are the eigenvectors of Sw-1 Sb (called Fisherfaces).
The eigenvectors are solutions of the generalized eigenvector problem:

45. Linear Discriminant Analysis (LDA)
Does Sw-1 always exist?
If Sw is non-singular, we can obtain a conventional eigenvalue problem by writing:
In practice, Sw is often singular since the data are image vectors with large dimensionality while the size of the data set is much smaller (M << N )

46. Linear Discriminant Analysis (LDA)
Does Sw-1 always exist? – cont.
To alleviate this problem, we can perform two projections:
PCA is first applied to the data set to reduce its dimensionality.
LDA is then applied to further reduce the dimensionality.

47. Linear Discriminant Analysis (LDA)
Case Study: Using Discriminant Eigenfeatures for Image Retrieval
D. Swets, J. Weng, "Using Discriminant Eigenfeatures for Image Retrieval", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 18, no. 8, pp , 1996.
Content-based image retrieval
The application being studied here is query-by-example image retrieval.
The paper deals with the problem of selecting a good set of image features for content-based image retrieval.

48. Linear Discriminant Analysis (LDA)
Assumptions
"Well-framed" images are required as input for training and query-by-example test probes.
Only a small variation in the size, position, and orientation of the objects in the images is allowed.

49. Linear Discriminant Analysis (LDA)
Some terminology
Most Expressive Features (MEF): the features (projections) obtained using PCA.
Most Discriminating Features (MDF): the features (projections) obtained using LDA.
Computational considerations
When computing the eigenvalues/eigenvectors of Sw-1SBuk = kuk numerically, the computations can be unstable since Sw-1SB is not always symmetric.
See paper for a way to find the eigenvalues/eigenvectors in a stable way.
Important: Dimensionality of LDA is bounded by C this is the rank of Sw-1SB

50. Linear Discriminant Analysis (LDA)
Case Study: PCA versus LDA
A. Martinez, A. Kak, "PCA versus LDA", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 23, no. 2, pp , 2001.
Is LDA always better than PCA?
There has been a tendency in the computer vision community to prefer LDA over PCA.
This is mainly because LDA deals directly with discrimination between classes while PCA does not pay attention to the underlying class structure.
This paper shows that when the training set is small, PCA can outperform LDA.
When the number of samples is large and representative for each class, LDA outperforms PCA.

51. Linear Discriminant Analysis (LDA)
Is LDA always better than PCA? – cont.

52. Linear Discriminant Analysis (LDA)
Is LDA always better than PCA? – cont.

53. Linear Discriminant Analysis (LDA)
Is LDA always better than PCA? – cont.

54. Linear Discriminant Analysis (LDA)
Critique of LDA
Only linearly separable classes will remain separable after applying LDA.
It does not seem to be superior to PCA when the training data set is small.

55. Appearance-based Recognition
Directly represent appearance (image brightness), not geometry.
Why?
Avoids modeling geometry, complex interactions
between geometry, lighting and reflectance.
Why not?
Too many possible appearances!
m “visual degrees of freedom” (eg., pose, lighting, etc)
R discrete samples for each DOF
How to discretely sample the DOFs?
How to PREDICT/SYNTHESIS/MATCH with novel views?

56. Appearance-based Recognition
Example:
Visual DOFs: Object type P, Lighting Direction L, Pose R
Set of R * P * L possible images:
Image as a point in high dimensional space:
is an image of N pixels and
A point in N-dimensional space
Pixel 2 gray value
Pixel 1 gray value

57. 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
[Thanks to Chuck Dyer, Steve Seitz, Nishino]

58. Key Idea USE PCA! Images in the possible set are highly correlated.
So, compress them to a low-dimensional subspace that
captures key appearance characteristics of the visual DOFs.
EIGENFACES: [Turk and Pentland]
USE PCA!

59. Eigenfaces
Eigenfaces look somewhat like generic faces.

60. Linear Subspaces Classification can be expensive
convert x into v1, v2 coordinates
What does the v2 coordinate measure?
What does the v1 coordinate measure?
distance to line
use it for classification—near 0 for orange pts
position along line
use it to specify which orange point it is
Classification can be expensive
Must either search (e.g., nearest neighbors) or store large probability density functions.
Suppose the data points are arranged as above
Idea—fit a line, classifier measures distance to line

61. 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

62. 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

63. Higher Dimensions 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 by only 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

64. Problem: Size of Covariance Matrix A
Suppose each data point is N-dimensional (N pixels)
The size of covariance matrix A is N x N
The number of eigenfaces is N
Example: For N = 256 x 256 pixels,
Size of A will be x !
Number of eigenvectors will be !
Typically, only eigenvectors suffice. So, this
method is very inefficient! 2 2

65. Efficient Computation of Eigenvectors
If B is MxN and M<<N then A=BTB is NxN >> MxM
M  number of images, N  number of pixels
use BBT instead, eigenvector of BBT is easily
converted to that of BTB
(BBT) y = e y
=> BT(BBT) y = e (BTy)
=> (BTB)(BTy) = e (BTy)
=> BTy is the eigenvector of BTB

66. Eigenfaces – summary in words
Eigenfaces are
the eigenvectors of
the covariance matrix of
the probability distribution of
the vector space of
human faces
Eigenfaces are the ‘standardized face ingredients’ derived from the statistical analysis of many pictures of human faces
A human face may be considered to be a combination of these standardized faces

67. Generating Eigenfaces – in words
Large set of images of human faces is taken.
The images are normalized to line up the eyes, mouths and other features.
The eigenvectors of the covariance matrix of the face image vectors are then extracted.
These eigenvectors are called eigenfaces.

68. Eigenfaces for Face Recognition
When properly weighted, eigenfaces can be summed together to create an approximate gray-scale rendering of a human face.
Remarkably few eigenvector terms are needed to give a fair likeness of most people's faces.
Hence eigenfaces provide a means of applying data compression to faces for identification purposes.

69. Dimensionality Reduction
The set of faces is a “subspace” of the set
of images
Suppose it is K dimensional
We can find the best subspace using PCA
This is like fitting a “hyper-plane” to the set of faces
spanned by vectors v1, v2, ..., vK
Any face:

70. Eigenfaces PCA extracts the eigenvectors of A
Gives a set of vectors v1, v2, v3, ...
Each one of these vectors is a direction in face space
what do these look like?

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

72. Is this a face or not?

73. 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

74. Key Property of Eigenspace Representation
Given
2 images that are used to construct the Eigenspace
is the eigenspace projection of image
Then,
That is, distance in Eigenspace is approximately equal to the
correlation between two images.

75. Choosing the Dimension KNM
i =
eigenvalues
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

76. Papers

78. More Problems: Outliers
Sample Outliers
Intra-sample outliers
Need to explicitly reject outliers before or during computing PCA.
[De la Torre and Black]

79. Robustness to Intra-sample outliers
PCA
RPCA
RPCA: Robust PCA, [De la Torre and Black]

80. Robustness to Sample Outliers
Original
PCA
RPCA
Outliers
Finding outliers = Tracking moving objects

81. Research Questions
Does PCA encode information related to gender, ethnicity, age, and identity efficiently?
What information do PCA encode?
Are there components (features) of PCA that encode multiples properties?

82. PCA
The aim of the PCA is a linear reduction of D dimensional data to d dimensional data (d<D), while preserving as much information, in the data, as possible.
Linear functions
y1= w1 X
y2= w2 X *
yd= wd X
Y= W X
X – inputs; Y – outputs, components; W – eigenvectors, eigenfaces, basis vectors x1 w1 w2 x2

83. How many components?
Usual choice consider the first d PC’s which account for some percentage, usually above 90 %, of the cumulative variance of the data.
This is disadvantageous if the last components are interesting W2 W1 x1 x2

84. Dataset A subset of FERET dataset 2670 grey scale frontal face images
Property
No.
Categories
No. Faces
Gender 2
Male
1603
Female
1067
Ethnicity 3
Caucasian
1758
African
320
East Asian
363
Age 5
20 – 29
665
30 – 39
1264
40 – 49
429
50 – 59
206
60+
106
Identity
358
Individuals with 3 or more examples
1161
A subset of FERET dataset
2670 grey scale frontal face images
Rich in variety: face images vary in pose, background lighting, presence or absence of glasses, slight change in expression

85. Dataset Each image is pre-processed to a 65 X 75 resolution.
Aligned based on eye locations
Cropped such that little or no hair information is available
Histogram equalisation is applied to reduce lighting effects

86. Does PCA efficiently represents information in face images?
Images of 65 × 75 resolution leads to a dimensionality of 4875.
The first 350 components accounted for 90% variance of the data.
Each face is thus represented using 350 components instead of 4875 dimensions
Classification employing 5-fold cross validation, with 80% of faces in each category for training and 20% of faces in each category for testing
for identity recognition leave-one-out method is used.
LDA is performed on the PCA data
Euclidean measure is used for classification
Property
Classification
Gender
86.4%
Ethnicity
81.6%
Age
91.5%
Identity
90%

87. What information does PCA encode? – Gender
Gender encoding power estimated using the LDA
3rd component carries highest gender encoding power followed by the 4th components
All important components are among the first 50 components

88. What information does PCA encode? – Gender
-6 SD
-4 SD
-2 SD
Mean
+2 SD
+4 SD
+6 SD
Reconstructed images from the altered components (a) third and (b) fourth components. The components are progressively added by quantities of -6 S.D (extreme left) to +6 S.D (extreme right) in steps of 2 S.D.
Third component encodes information related to the complexion, length of the nose, presence or absence of hair on the forehead, and texture around the mouth region.
Fourth component encodes information related to the eyebrow thickness, presence or absence of smiling expression

89. Gender
(a) Face examples with the first two being female and the next two being male faces. (b) Reconstructed faces of (a) using the top 20 gender important components. (c) Reconstructed faces of (a) using all components, except the top 20 gender important components.

90. What information does PCA encode? – Ethnicity
6th component carries highest ethnicity encoding power followed by the 15th components
All ethnicity important components are among the first 50 components

91. Ethnicity
-6 SD SD SD Mean SD SD SD
Reconstructed images from the altered components (a) 6th and (b) 4th components. The components are progressively added by quantities of -6 S.D (extreme left) to +6 S.D (extreme right) in steps of 2 S.D.
6th component encodes information related to complexion, broadness and length of the nose
15th component encodes information related to length of the nose, complexion, and presence or absence of smiling expression

92. What information does PCA encode? – Age
Age – and age groups termed as young and old)
10th component is found to be the most important for age
-6 SD SD SD Mean SD SD SD
Reconstructed images from the altered tenth component. The component is progressively added by quantities of -6 S.D (extreme left) to +6 S.D (extreme right) in steps of 2 S.D

93. What information does PCA encode? – Identity
Many components are found to be important for identity. However, their importance magnitude is small.
These components are widely distributed and not restricted to the first 50 components

94. Can a single component encode multiple properties?
A grey beard informs that the person is a male and also, most probably, old.
As all important components of gender, ethnicity, and age are among the first 50 components there are overlapping components.
One example is the 3rd component which is found to be the most important for gender and second most important for age

95. Can a single component encode multiple properties?
Normal distribution plots of the (a) third (b) and fourth components for male and female classes of young and old age groups.

96. Summary
PCA encodes face image properties such as gender, ethnicity, age, and identity efficiently.
Very few components are required to encode properties such as gender, ethnicity and age and these components are amongst the first few components which capture large part of the variance of the data. Large number of components are required to encode identity and these components are widely distributed.
There may be components which encode multiple properties.

97. Principal Component Analysis (PCA)
PCA and classification
PCA is not always an optimal dimensionality-reduction procedure for classification purposes.
Multiple classes and PCA
Suppose there are C classes in the training data.
PCA is based on the sample covariance which characterizes the scatter of the entire data set, irrespective of class-membership.
The projection axes chosen by PCA might not provide good discrimination power.

98. Linear Discriminant Analysis (LDA)
What is the goal of LDA?
Perform dimensionality reduction “while preserving as much of the class discriminatory information as possible”.
Seeks to find directions along which the classes are best separated.
Takes into consideration the scatter within-classes but also the scatter between-classes.
More capable of distinguishing image variation due to identity from variation due to other sources such as illumination and expression.

99. Linear Discriminant Analysis (LDA)

100. Angiograph Image Enhancement

104. Webcamera Calibration

108. THANKS
QUESTIONS