1. 2D-LDA: A statistical linear discriminant analysis for image matrix
Dongchul Kim Presented by,
Suhasini Rajulapudi

2. Context: Feature Extraction Linear Discriminant Analysis
Two-Dimensional Linear Discriminant Analysis
Reconstruction of an image
Classification
Experiment and Analysis
Conclusion

3. Feature Extraction:
In pattern classification, a pattern is represented by a set of input variables. For better classification, feature extraction has been widely used to construct new features from input variables. This reduces the dimension of input space while preserving as much discriminative information as possible.
Over last decades, many algorithms such as Principle Component Analysis (PCA), Linear Discriminant Analysis (LDA), Independent Component Analysis (ICA), Non-negative matrix factorization (NMF), locality preserving projection and Bayesian probabilistic subspace, etc. have been proposed for feature extraction (also known as dimension reduction) in pattern recognition.

4. Most well-known feature extraction algorithms receive vector form input patterns, so the algorithm requires change matrix form images to vector form.
Recently some idea has been developed to extract features directly from image samples without any vectorization work on input image.
Two well-known developed algorithms are Two-Dimensional Principle Component Analysis (2DPCA) and Two-Dimensional Linear Discriminant Analysis (2DLDA).

5. Linear Discriminant Analysis
Linear Discriminant Analysis (LDA) is a well-known scheme for feature extraction and dimension reduction. It has been used widely in many applications involving high-dimensional data, such as face recognition and image retrieval.
An intrinsic limitation of classical LDA is the so-called singularity problem, that is, it fails when all scatter matrices are singular. A well-known approach to deal with the singularity problem is to apply an intermediate dimension reduction stage using Principle Component Analysis (PCA) before LDA.
The algorithm, called PCA+LDA, is used widely in face recognition. However, PCA+LDA has high costs in time and space, due to the need for an eigen-decomposition involving the scatter matrices.

6. Linear Discriminant Analysis (LDA) just reduces the number of dimensions of the input feature vector by preserving the inter-class separation as present in the original feature vector.

7. 2D-LDA, which stands for 2-Dimensional Linear Discriminant Analysis
2D-LDA, which stands for 2-Dimensional Linear Discriminant Analysis. 2D-LDA overcomes the singularity problem implicitly, while achieving efficiency.
The key difference between 2D-LDA and classical LDA lies in the model for data representation.
Classical LDA works with vectorized representations of data, while the 2DLDA works with data in matrix representation.
To further reduce the dimension by 2DLDA, the combination of 2DLDA and classical LDA, namely 2DLDA+LDA, where LDA is preceded by 2DLDA.

8. Two-Dimensional Linear Discriminant Analysis
2D-LDA deals with matrix form images instead of vector form and vectorization is not required and collection of data is represented as a collection of small matrices, instead of a single large matrix.
2D-LDA aims to find the optimal transformations (projections) such that the class structure of the original high-dimensional space is preserved in the low-dimensional space.
2D-LDA tries to maximize the Fisher’s criterion and finds an optimal projection using dimension-reduction.

9. Principle: The construction of Fisher projection axis
Let A is projected onto x by the following linear transformation y = Ax
The jth training image is denoted as Aj (j = 1, 2, …, M), and the mean image of all training sample is denoted by 𝐴 and Ai (i = 1, 2, …, L) denoted the mean image of class Ti and Ni is the number of samples in class Ti, the projected class is Pi.
After the projection of training image onto x, we get the projection feature vector yj = Aj x
The total scatter of the projected samples can be characterized by the trace of the covariance matrix of the projected feature vectors.
From this point of view, we introduced a criterion at first, J(x) = P𝐵 P𝑊
Two parameters, PB = tr(TSB),
PW = tr(TSW),

11. The criterion could be expressed by,
This criterion is called Fisher linear projection criterion.
When the criterion is maximized, i.e.,
If SW is nonsingular, the solution to above optimization problem is to solve the generalized eigen-value problem,
The traditional LDA must face to the singularity problem. However, 2D-LDA overcomes this problem successfully. This is because, for each training image, Aj , we have rank(Aj) = min(m, n).
From the previous equation, we have

12. ≤ (M - L) . min(m, n)
So, in 2D-LDA, SW is nonsingular when M ≥ L + 𝑛 min 𝑚, 𝑛
Using these projection axes, x1 , … , xd we could form a new Fisher projection matrix X, which is n × d matrix,
X = [x1 x2 … xd]
For feature extraction, the optimal projection vectors of 2D-LDA is x1 , … , xd. For a given image A, we have yk = A xk , here k = 1, 2, … , d.
Then, family of Fisher feature vectors are y1 , … , yd and here Y is the Fisher feature matrix, Y = [y1 , … , yd ]

13. Reconstruction:
For a given image A, the Fisher feature matrix Y and the Fisher optimal projection axes X, then we have Y = AX
Reconstructed image A is, 𝐴 =𝑌 𝑋 𝑇 = 𝑘=1 𝑑 𝑦 𝑘 𝑥 𝑘 𝑇
𝐴 𝑘 = 𝑦 𝑘 𝑥 𝑘 𝑇 , reconstructed sub image of A.
If we select d = n, then we can completely reconstruct the images in the training set 𝐴 = 𝐴.
if d < n, the reconstructed image 𝐴 is an approximation for 𝐴.

14. Classification:
Given two images 𝐴 1 , 𝐴 2 represented by 2D-LDA feature matrix 𝑌 1 = [𝑌 1 1 ,…, 𝑌 𝑑 ] and 𝑌 2 = [𝑌 1 2 ,…, 𝑌 𝑑 ]. So, the similarity d( 𝑌 1 , 𝑌 2 ) is defined as,
Fisher feature matrix of training images are 𝑌 1 , 𝑌 2 ,…, 𝑌 𝑀 and each image is assigned to class 𝑇 𝑖 . Then, given a test image Y, if d(Y, 𝑌 𝑙 ) = 𝑚𝑖𝑛 𝑗 d( 𝑌 1 , 𝑌 𝑙 ), and 𝑌 𝑙 𝜖 𝑇 𝑖 . The resulting decision is Y 𝜖 𝑇 𝑖 .

15. Experiment and Analysis:
We evaluated our 2D-LDA algorithm on the ORL face image database.
The images were taken in dark homogeneous background with the subjects in an upright, frontal positions.
Here, five samples of one person in ORL database are shown in Fig. 1.

16. Using 2D-LDA, we could project the test face image onto the Fisher optimal projection axis , then we could use the Fisher feature vectors set to reconstruct the image. Fig. 2.
Here, observed these images, we could find that the reconstructed images are very like obtained by sample the original image on the spacing vertical scanning lines.
An experiment on the ORL database to evaluate performance of 2D-LDA , 2D-PCA, Eigenfaces, Fisherfaces.

17. In our experiment, select first five images samples per person for training, and the left five images samples for testing.
The size of training set and testing set were both 200.
So, in the 2D-LDA, the size of between class scatter matrix 𝑆 𝐵 and within-class scatter matrix 𝑆 𝑊 are both 92 × 92.
It shows the classification result in Fig. 3. We could find the two 2D feature extraction methods have outstanding performance in the low-dimension condition, but the conventional ones’ ability is very poor.

19. Table 1 shows comparison of the training time of the four algorithms.
In 2D condition, we only need to handle a 92 × 92 matrix. But using the Eigenfaces and Fisherfaces, we must face to a × matrix.
When used the Fisherfaces, we must reduced the dimension of the image data to avoid that 𝑆 𝑊 is singular.

20. Table 2 shows that the memory cost of 2D feature extraction is much larger than the 1D ones.
2D methods used a n × d matrix to present a face image. At the same time, the 1D techniques reconstructed face images by a d-dimension vector.

21. Conclusion:
This method uses the Fisher Linear Discriminant Analysis to enhance the effect of variation caused by different individuals, other than by illumination, expression, orientation, etc.
2D-LDA uses the image matrix instead of the image vector to compute the between-class scatter matrix and the within-class scatter matrix.
2D-LDA have many advantages over other methods. It achieves the best recognition accuracy in the four algorithms and computing cost is very low compared with Eigenfaces, and Fisherfaces, and close to 2D-PCA.
This method shows powerful performance in the low dimension.