Modular PCA Face Recognition Based on Weighted Average

This paper presents an improved modular PCA approach, that is, modular PCA algorithm based on weighted average. This algorithm extracts weighted average for every sub-block of every training sample in each type of training sample, and normally operates the corresponding sub-block in training sample using weighted average, then all standardized sub-blocks constitute the overall scatter matrix, and thus the optimal projective matrix is obtained; From the middle value of sub-blocks in training set, and normally projecting sub-blocks of training samples and test samples to the projective matrix, then we can get identified characteristics; At last, use the recent distance classifier to class. The test results in the ORL face database show that the proposed method in identifying performance is superior to ordinary modular PCA approach

every training sample in each type of training sample, and normally operates the corresponding sub-block in training sample using weighted average.The test results in the ORL face database show that the proposed method in identifying performance is superior to ordinary modular PCA approach.

PCA Algorithm
PCA method is a statistical analysis method based on Karhunen-Loeve (KL) transformation, whose principle is that high-dimensional vector projects to low-dimensional vector space by a specific feature vector matrix.Through the vector of low-dimensional representation and the feature vector matrix, we can reconstruct the corresponding original high-dimensional vector.In the face recognition process, after the KL transformation, we can get a set of feature vectors to form a lower dimensional subspace.Any human face image can project it and get a set of coordinates factors.This group of coefficients shows that the image location in the sub-space can be used as a basis for face recognition.
In this method, generating matrix is the total scatter matrix of training samples, i.e:

(
)( ) Where i X is the image vector of the non-zero eigenvalues as the orthogonal basis, in the new orthogonal sub-space U , the face sample i X can be expressed as: (2)

Adaptive weighted average [Yin Hongtao, 2006]
When using PCA algorithm, we first spread the image matrix by row (column) as a vector.Suppose the vector spread by all the image matrix be: ( ) , , L , is the number of the i -th training sample; m is the vector dimension.Because the mean vector of several vectors is taken averagely from the scalar vector of the corresponding dimension, we explain the determined method of weighted value by taking the first dimension as example.
First of all, calculate the distance sum ( ) ( ) ( ) of every sample in the i -th sample and other sample, then find the minimum of them ( ) We believe that the sample whose distance sum with the same type sample is larger.It deviates greater from the class center.In calculating class average, it should be given a smaller weight and the weighted value in the first dimension of the j -th sample in the i -th class is where β is a constant greater than or equal to zero.For regulating the weight extent, when 0 β = , the algorithm becomes the traditional averaging method.The average of the first dimension in the i -th sample can be modified to Similarly, we can find the means of other dimensions of training samples.

Modular PCA algorithm based on the weighted average
The basic idea of modular PCA algorithm based on the weighted average is as follows: block m n × image matrix I into p q × blocked image matrix, namely,  PCA algorithm is that we derive all scatter matrix not using the sub-block average of all training samples, but using weighted average of sub-blocks.This can reduce the impact of deriving the optimal projective matrix from the mean deviation in training samples, thus improving the recognition rate.
Algorithm steps are as follows: For convenience, we first introduce the concept of quantization matrix.
Definition: Suppose ( ) , , , Where the vector is arranged in turn by column vector of the matrix A , which is called the quantification of matrix A .
1) Suppose the model category is C, the image matrix in the i -th class training sample is ( ) is 2) Require overall scatter matrix of sub-image matrix in all the training sample images.
Let ( ) ( ) . So the overall scatter matrix of all training sample sub-blocks is: Where ( ) , ( ) n i is the number of each class of training samples; ( ) is the total number of training sample sub-images matrix. .( ) kl i η is the weighted average image between the i -th sample image and the kl -th block.The specific calculation method is to spread all sub-blocks by row to column vector, then calculates its weighted mean vector by equations ( 4) and ( 5), and reverts the middle measures to a matrix.
Easily, we can prove S is a 1 1 1 1 m n m n × non-negative definite matrix.
3) Seek optimal projective matrix Take corresponding orthonormal eigenvectors (discriminant vectors) is quantified by equation ( 7) and normalized, then obtain the characteristics matrix of ij A after projecting to ( ) is quantified by equation ( 7) and normalized, then obtain the characteristics matrix of test samples after projecting to where ( ) , , , , carrying the most recent method to sort: ; x is identified the x -th sample under test.

Experiment and result analysis
Test the method of this paper in ORL (olivetti research laboratory) face database.This face database contains 40 individuals, and each person has 10 images.The image is a positive image of single dark background that contains a certain amount of illumination changes, facial changes (open eyes and closed eyes , laughing or not laughing), facial details changes(wearing glasses or not wearing glasses), and the depth rotation within a certain range.The sizes of these images are 112 × 92 pixels.Other part faces are showed in Figure 1.For each person, randomly selecte five images as training samples and the rest five images are used to test the identification method performance.
The experimental results are shown in Figure 2 and Figure 3. Figure 2 shows experimental result of traditional PCA method, 2 × 2 modular PCA method and 2 × 2 blocked PCA method based on weighted average.From the figure, we can see that recognition rate of traditional PCA method is lower, that is up to 77 %.Module PCA method improves the recognition rate, while the blocked PCA method based on weighted average is superior to ordinary blocked PCA method.Figure 3 respectively shows the test results of 4 × 2 sub-blocks and 4 × 4 sub-blocks conditions.From the figures we can see that in the 4 × 2 block case, modular PCA method based on weighted average has a higher recognition rate and a more robust than ordinary blocked PCA method; In addition, test result also shows that 4 × 2 blocked approach is superior to 2 × 2 blocked approach.In the blocked mode, the correct recognition rate is greatly decreased.The cause is that the more blocks number of each image is, the more reduced the distinguished information contained in each sub-block.So there will be more similar sub-blocks and it is not conducive to classification, thus correct identification rate has dropped.In this case, modular PCA method based on weighted average is still better than ordinary blocked PCA method.At the same time, we find in experiments the recognition performance of 4 × 2 sub-blocks is far better than that of 2 × 4 sub-blocks, which is shown in Table 1.The cause is that the difference between different people faces focus on eyes, nose, mouth, chin and other parts, so the vertical multi-block is not conducive to identification.

Conclusion
The prominent advantage of face recognition method based on modular PCA is the ability to extract the local features of image, which better reflects the difference between images.We can easily use discriminant analysis method in the smaller image for the process is simple.To further improve the recognition rate, this paper improves face recognition method based on modular PCA and proposes modular PCA algorithm based on weighted average.The experiment on ORL face database shows that this method is superior to the traditional PCA method and ordinary PCA method.For the same database, if the original image has different sub-blocks, the obtained highest recognition rate is generally different.How to find the best sub-blocks acquired highest recognition rate and how to simplify the sub-blocks PCA algorithm have yet to be further studied.
Table 1.recognition rate of 2× 4 blocks and 4 × 2 blocks of method in this paper (%)

=
, then taking the sub-image matrix of all training samples as the image vector of training sample to purpose PCA method.The difference from the traditional the total number of training samples, and each sample image is m , the sample x I belongs to the i -th category.

Figure 2 .
Figure 1.image in ORL face database i -th training sample; Vector dimension is n ; X is the average figure vector for training sample; M is the total number of training samples.