PCA and LDA based features

2.3 Gray-level image based methods

2.3.2 Image transform features

2.3.2.2 PCA and LDA based features

The PCA and the LDA are important multivariate data analysis methods used in pattern recognition. The LDA is also known as the Fisher’s discriminant analysis method. The PCA finds a set of the most represen- tative projection vectors such that the projected samples retain most information about original samples. The dimensionality of the projected samples is less than the dimensionality of the original samples yielding compact representations.

Consider a set of I images{f₁,f₂,· · ·,f_I}each of dimension (N+1)×(M+1). Assume that there are P classes with Q number of samples belonging to each class, such that I = PQ. The 2D images are represented as 1-dimensional vectors by concatenating the rows. Therefore, we get I vectors of size l = (N+1) (M+1) represented as{f1,f2,· · ·, fI}. The steps in PCA are as follows.

Step 1: Compute the mean centered image vectors f_iby f_i = fi− 1

I XI

i=1

fi (2.14)

Step 2: Find the eigen vectors and the eigen values of the covariance matrix R= 1

I

I

X

i=1

f_if^T_i (2.15)

The eigenvectors{e1, e2,· · ·, ek}corresponding to the k largest eigenvalues{λ₁, λ₂, · · ·, λ_k}of R form a trans- formation matrix

Wpca=[e1 e2 e3 . . . ek], (2.16)

for k << l. The matrix W_pca forms the orthonormal basis that projects each l−dimensional vector f_i in the original space to an k−dimensional vector gi defined as

g_i =W_pca^T f_i (2.17)

The k largest eigenvalues{λ₁, λ₂,· · ·, λ_k}are known as the principal components and the corresponding eigen- vectors{e₁,e₂,· · ·,e_k}are known as the eigenimages. The subspace spanned by the eigenvectors is known as the eigenspace. Each eigenimage forms the feature descriptor for image classification. The classification is performed by finding a match that minimises the Euclidean distance between the input image projected in to

the eigenspace and an image class represented in the eigenspace.

Unlike the PCA, the LDA uses the class information and finds a set of vectors that maximise the between- class scatter while minimizing the within-class scatter of the original samples. In the LDA technique, the projection vector is selected in such a way that the ratio of the between-class scatter and the within-class scatter is maximised.

Consider a set of I image vectors{f₁, f₂,· · ·,f_I}with I = PQ. For notational convenience, let us label the image vectors in terms of the image class and rewrite{f₁,· · ·,f_I}as

f₁₁,· · ·, f_1Q,· · ·, f_P1,· · · ,f_PQ . The steps involved in the LDA are summarised as follows.

Step 1: For p∈ {1,2,· · · ,P}and q∈ {1,2,· · · ,Q}, compute the mean image vectors for each image class by f_mean^p = 1

Q

Q

X

q=1

fpq (2.18)

Compute the global mean image vector f_meanthrough f_mean= 1

P

P

X

p=1

f_mean^p (2.19)

Step 2: Compute the between-class scatter matrix (S_b) and the within-class scatter matrix (S_w) as S_b=

P

X

p=1

pr (p)

f_mean^p − f_mean f_mean^p − f_meanT

(2.20)

Sw = XP

p=1

pr (p)

Q

X

q=1

fpq− f_mean^p fpq− f_mean^p T

(2.21) respectively. pr (p) denotes the probability of the image class p.

Step 3: Compute the transformation matrix W_ldasuch that it maximises the class separability with regard to a chosen separability criterion. One of the most widely used discriminant criteria is given by

W_lda=argmax

W

W^TSbW

W^TSwW

(2.22) If S_wis non-singular, W_ldais formed by the k eigenvectors{e₁,e₂, · · ·, e_k}of the matrix (S_w)⁻¹S_bcorrespond- ing to the k largest eigenvaluesn

λ₁, λ₂,· · ·, λ_ko

. However, if S_wis singular, W_ldais computed by first projecting the images in to a lower dimensional space through PCA so that S_w is non-singular. Then, the standard defi- nition in (2.21) is applied to the reduced data set. Accordingly, the transformation matrix is defined as [117]

W_lda^T =W^T_{f ld}W^T_pca (2.23)

2.3 Gray-level image based methods

where

W_{f ld} =argmax

W

W^TW^T_pcaSbWpcaW

W^TW^T_pcaSwWpcaW

(2.24) The projected image vectors gpqare obtained using the following linear transformation.

g_pq=W_lda^T f_pq (2.25)

In the case of LDA, the upper bound on k is P−1, where P is the number of classes. The classification is performed by computing the minimum distance between the projected image vectors.

The efficiency PCA and the LDA methods is widely studied in the field of face recognition. The comparative studies show that the PCA and the LDA features are robust under varying view-angle, illumination changes and other variations [118, 119].

A few works on hand posture recognition based on the PCA have been reported in [120–123]. The gray- values of the segmented hand posture regions constitute the desired input for PCA based classification. Birk et al [121] combined PCA based description with the Bayes classifier for classifying 25 hand signs in the American sign language. The segmented gray-level images were normalised for geometric transformations during the preprocessing stage. The database used for the experiment consisted of 2500 samples of 25 hand postures. The results demonstrated an over-all recognition rate of 99% on a test database containing 1500 images. Dardas and Petriu [123] developed a robust hand posture classification system using the PCA features derived from hand posture images with different scales, orientations and illumination variations. The features were used to classify 4000 samples of 4 gesture signs acquired at various scales, orientations and illumination conditions. The system achieved an average recognition rate of 93%.

The LDA based features were employed for hand posture analysis in [124–127]. The performance of the PCA and the LDA features in hand posture classification is studied in detail in [124, 125]. The hand postures were segmented from the background and the intensity maps in the posture segment are used as the input. The gesture signs used in their experiment were dynamic gestures in which the hand posture changes with time.

The system was trained with 504 samples of 28 gesture signs acquired at varying illumination. Similarly, the test dataset consisted of 540 samples of 28 gesture signs. The performance of the LDA features were shown to be superior to the PCA based classification method. Deng and Tsui [127] investigated the performance of the LDA features in classifying 100 hand gesture signs taken from the American sign language. The dimensionality of the dataset was initially reduced through the PCA and the discriminant features were derived from the

reduced dataset as defined in (2.25). They combined the LDA features with the HMM classifier and achieved a classification accuracy of 93.5%. In [126], LDA was employed to classify the hand posture signs in the Japanese sign language. The LDA features representing the hand postures were classified through the K-means clustering method. The system achieved an average recognition accuracy of 98% on the samples of 41 hand posture signs taken from 4 subjects.

From the above studies we can infer that the PCA and the LDA techniques provide potential features for classifying large classes of hand postures and results in dimensionality reduction offering efficient compact representations and better discriminations. However, more research is required to examine the performance of these multivariate data analysis methods in the presence of view-angle distortions and user-variations.