PCA Method and Face Recognition

1. PCA Method and Face Recognition CPE488 && CPE631 [ 2012 – KMUTT ] Presented by Miss ChayanutPetpairote

2. Outline • What is PCA? • PCA Method • What are Eigenfaces? • Face Recognition • Training Steps • Testing Steps • Experimental Results • Conclusion • Demo

3. What is PCA? (1) • Principal Component Analysis • Eigen Vectors show the direction of axes of a fitted ellipsoid • Eigen Values show the significance of the corresponding axis • The larger the Eigen value, the more separation between mapped data • For high dimensional data, only few of Eigen values are significant

4. What is PCA? (2) • Finding Eigen Values and Eigen Vectors • Deciding on which are significant • Forming a new coordinate system defined by the significant Eigen vectors (lower dimensions for new coordinates) • Mapping data to the new space

5. PCA Method (1) • Step 1: Get some data

6. PCA Method (2) • Step 2: Subtract the mean • Mean x = 1.81 , Mean Y = 1.91

7. PCA Method (3) • Step 3: Calculate the covariance matrix • Covariance can analysis any relationship between the dimensions (for more 1D) • Ex: 3-dimensional data set (x,y,z), to measure the covariance between cov(x,y), cov(x,z), cov(y,z) , cov(x,x) = var(x), cov(y,y) = var(y), cov(z,z) = var(z) • Note!!!

8. PCA Method (4) • Step 3: Calculate the covariance matrix

9. PCA Method (5) • Step 4: Calculate the eigenvectors and eigenvalues of the covariance matrix • A is an n×n matrix, A is a linear operator on vectors in Cn • Eigenvector of A is a vector v ∈ Cn Av=λv • where λ = eigenvalue, v = eigenvector

10. PCA Method (6) • Step 4: Calculate the eigenvectors and eigenvalues of the covariance matrix • Find eigenvalues : det(A- λI)=0 and solve for λ

11. PCA Method (7) • Step 4: Calculate the eigenvectors and eigenvalues of the covariance matrix • Find eigenvectors • Found eigenvectors =

12. PCA Method (8) • Step 4: Calculate the eigenvectors and eigenvalues of the covariance matrix

13. PCA Method (9) • Step 5: Choosing components and forming a feature vector • Data compression • Reduce dimensionality

14. PCA Method (10) • Step 6: Deriving the new data set • Transformed data = DataAdjust * Eigenvectors

15. PCA Method (11) • Step 7: Getting the old data back • Original Data = (eigenvectorT x transformed data) + original mean

16. What are Eigenfaces? • Eigenfaces are the eigenvectors of the covariance matrix of the probability distribution of the vector space of human faces • Eigenfaces are the ‘standardized face components’ derived from the statistical analysis of many pictures of human faces • Eigenfaces is a set of the eigenvectors of face images by using PCA

17. Face Recognition • Training Steps • Training Image Set • Preprocessing • PCA / Eigenfaces • Dimensionality Reduction • Calculation Weight • Testing Steps • Testing Image • Preprocessing • Transformed into Eigenface Components • Finding minimizing the Euclidean distance

18. Training Steps (1) • Training Image Set • Each Image: is Row * Col size • Training Image: M = 15 Images

19. Training Steps (2) • Preprocessing • Normalized Training Image Set • Reduce nose • Reduce light

20. Training Steps (3) • PCA / Eigenfaces • Each Image is transformed into a vector of size N (N = Row * Col) • Obtain a set Row* Col N*M N*1

21. Training Steps (4) • PCA / Eigenfaces • Find the Mean Image

22. Training Steps (5) • PCA / Eigenfaces • Find the difference between input image and mean image

23. Training Steps (6) • PCA / Eigenfaces • Find the Covariance matrix C • Find AT • L = ATA  ATA vi = eigvali. vi • A ATA vi = A . eigvali. vi • CAvi= eigvali. Avi Eigenvectors uk = Avi

24. Training Steps (7) • PCA / Eigenfaces • Find Eigenvectors of Covariance matrix C • uk = Avi= size N * M (N = Row*Col)

25. Training Steps (8) • Dimensionality Reduction • Choose EigenVectors from large EigenValues most significant relationship between the data dimensions

26. Training Steps (9) • Calculation Weight • The weight describe the contribution of each eigenface in representing the training image set.

27. Testing Steps (1) • Testing Image: • Preprocessing • Transformed into Eigenface Components • The weight describe the contribution of each eigenface in representing the testing image.

28. Testing Steps (2) • Finding minimizing the Euclidean distance between the testing image and training image set weight vectors • If is minimum value that mean lower error between the testing image and training image so we can found the nearest image for face recognition.

29. Experimental Results (1)

30. Experimental Results (2)

31. Experimental Results (3)

32. Experimental Results (4)

33. Conclusion • Advantages • PCA can reduce the number of dimensions without much loss of information • PCA gives a high compression rate • Performance of recognition is good when noise is present • PCA can be used for face recognition and face reconstruction • Best low-dimensional space can be determined by the Best Eigenvectors of the covariance matrix • Limitations • Face Images must be the same image scale. If scale is changed, the performance of recognition is very bad • Face images are quite clear face and not occlusion

34. Demo

35. References • M. Turk and A. Penland, “Eigenfaces for recognition”, Journal of Congnitive Neuroscience, 1991. • M. Turk and A. Pentland, “Face recognition using eigenfaces”, In proc. Of Computer vision and Pattern Recognition, 1991 • “The ORL face database” , http://www.cl.cam.ac.uk/research/dtg/attarchive/facedatabase.html • http://www.pages.drexel.edu/~sis26/