PCA Based Face Recognition System
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This document describes a principal component analysis (PCA) based face recognition system. It discusses two main steps: initialization operations on the training faces and recognizing new faces. For training, faces are converted to vectors and normalized. Eigenvectors are calculated from the covariance matrix and used to reduce dimensionality. Each training face can then be represented as a linear combination of eigenvectors. For recognition, a new face is converted to a vector, normalized, projected onto the eigenspace to get its weight vector, and compared to stored weight vectors using Euclidean distance to identify the face.
![Calculate the Covariance Matrix (𝑪)
C = 𝑛=1
16
Ф 𝑛 Ф 𝑛
𝑇
= 𝐴𝐴 𝑇
= {(𝑁2× 𝑀). (𝑀 × 𝑁2)}
= 𝑁2× 𝑁2
= (10304 × 10304)
Where 𝐴 = {Ф1, Ф2, Ф3, … … … ., Ф16}
[𝐀 = 𝐍 𝟐
× 𝐌]……….. 𝜳
Ф𝒊
Face vector space
Converted
𝑀= 16 images in the training set
Converted](https://image.slidesharecdn.com/pcabasedfacerecognitionsystem-150426010219-conversion-gate01/85/PCA-Based-Face-Recognition-System-12-320.jpg)
PCA Based Face Recognition System
- 1. PCA Based Face Recognition System MD. ATIQUR RAHMAN
- 2. Face Recognition using PCA Algorithm PCA- Principal Component Analysis Goal- Reduce the dimensionality of the data by retaining as much as variation possible in our original data set. The best low-dimensional space can be determined by best principal- components.
- 3. Eigenface Approach Pioneered by Kirby and Sirivich in 1988 There are two steps of Eigenface Approach Initialization Operations in Face Recognition Recognizing New Face Images
- 4. Steps Initialization Operations in Face Recognition Prepare the Training Set to Face Vector Normalize the Face Vectors Calculate the Eigen Vectors Reduce Dimensionality Back to original dimensionality Represent Each Face Image a Linear Combination of all K Eigenvectors Recognizing An Unknown Face
- 5. Prepare the Training Set to Face Vector
- 6. ……….. 112 × 92 10304 × 1𝜞𝒊 Face vector space Images converted to vector Each Image size column vector 𝑀= 16 images in the training set Convert each of face images in Training set to face vectors
- 8. Average face vector/Mean image (𝜳) 𝑀= 16 images in the training set ……….. 𝜳 Converted Face vector space Mean Image 𝜳 𝜞𝒊 Calculate Average face vector Save it into face vector space
- 9. Subtract the Mean from each Face Vector ……….. Ф𝒊 𝜳 Converted Face vector space 𝑀= 16 images in the training set − = 𝜞 𝟏 𝚿 Ф 𝟏 Normalized Face vector
- 10. Result of Normalization Figure: Normalized Data set
- 12. Calculate the Covariance Matrix (𝑪) C = 𝑛=1 16 Ф 𝑛 Ф 𝑛 𝑇 = 𝐴𝐴 𝑇 = {(𝑁2× 𝑀). (𝑀 × 𝑁2)} = 𝑁2× 𝑁2 = (10304 × 10304) Where 𝐴 = {Ф1, Ф2, Ф3, … … … ., Ф16} [𝐀 = 𝐍 𝟐 × 𝐌]……….. 𝜳 Ф𝒊 Face vector space Converted 𝑀= 16 images in the training set Converted
- 13. C = 10304 × 10304 10304 eigenvectors ……… Each 10304×1 dimensional ……….. 𝜳 Ф𝒊 Face vector space 𝒖𝒊 Converted 𝑀= 16 images in the training set In 𝑪, 𝑵 𝟐 is creating 𝟏𝟎𝟑𝟎𝟒 eigenvectors Each of eigenvector size is 𝟏𝟎𝟑𝟎𝟒 × 𝟏 dimensional Calculate Eigenvector (𝒖𝒊)
- 14. C = 10304 × 10304 10304 eigenvectors Each 10304×1 dimensional ……….. 𝜳 Ф𝒊 Face vector space Converted ……… 𝒖𝒊 𝑀= 16 images in the training set Find the Significant 𝑲 𝒕𝒉 eigenfaces Where, 𝑲 < 𝑴
- 15. C = 10304 × 10304 10304 eigenvectors Each 10304×1 dimensional ……….. 𝜳 Ф𝒊 Face vector space Converted ……… 𝒖𝒊 𝑀= 16 images in the training set Make system slow Required huge calculation
- 19. Consider lower dimensional subspace ……….. 𝜳 Ф𝒊 Lower dimensional Sub-space Face vector space Converted 𝑀= 16 images in the training set
- 20. 𝑳 = 𝑨 𝑻 𝑨 = 𝑴 × 𝑵2 𝑵2 × 𝑴 = 𝑴 × 𝑴 = 16 × 16 … … . . 16 eigenvectors Each 16 ×1 dimensional Calculate eigenvectors 𝒗𝒊 𝒗𝒊 ……….. 𝜳 Ф𝒊 Lower dimensional Sub-space Face vector space Converted 𝑀= 16 images in the training set Calculate Co-variance matrix(𝑳) of lower dimensional
- 21. 𝑳 = 𝑨 𝑻 𝑨 = 𝑴 × 𝑵2 𝑵2 × 𝑴 = 𝑴 × 𝑴 = 16 × 16 … … . . 16 eigenvectors Each 16 ×1 dimensional 𝒖𝒊 V/S 𝒗𝒊 𝒗𝒊 ……….. 𝜳 Ф𝒊 Lower dimensional Sub-space Face vector space Converted 10304 eigenvectors ……… Each 10304×1 dimensional 𝒖𝒊 C = 10304 × 10304 v/s 𝑀 images in the training set
- 22. 𝑳 = 𝑨 𝑻 𝑨 = 𝑴 × 𝑵2 𝑵2 × 𝑴 = 𝑴 × 𝑴 = 16 × 16 Select K best eigenvectors ……….. 𝜳 Ф𝒊 Lower dimensional Sub-space Face vector space Converted … … . . 16 eigenvectors Each 16 ×1 dimensional 𝒗𝒊 Selected K eigenfaces MUST be in The ORIGINAL dimensionality of the Face vector space
- 24. 𝑳 = 𝑨 𝑻 𝑨 = 𝑴 × 𝑵2 𝑵2 × 𝑴 = 𝑴 × 𝑴 = 16 × 16 ……….. 𝜳 Ф𝒊 Lower dimensional Sub-space Face vector space Converted … … . . 16 eigenvectors Each 16 ×1 dimensional 𝒗𝒊 A= 𝒖𝒊 = 𝑨𝒗𝒊 10304 eigenvectors ……… Each 10304×1 dimensional 𝒖𝒊 𝑀= 16 images in the training set
- 25. 𝑳 = 𝑨 𝑻 𝑨 = 𝑴 × 𝑵2 𝑵2 × 𝑴 = 𝑴 × 𝑴 = 16 × 16 ……….. 𝜳 Ф𝒊 Lower dimensional Sub-space Face vector space Converted … … . . 16 eigenvectors Each 16 ×1 dimensional 𝒗𝒊 A= 𝒖𝒊 = 𝑨𝒗𝒊 10304 eigenvectors ……… Each 10304×1 dimensional 𝒖𝒊 𝑀 images in the training set
- 26. C = 𝐴𝐴 𝑇 10304 eigenvectors ……… Each 10304×1 dimensional 𝒖𝒊 The K selected eigenface ……….. 𝜳 Ф𝒊 Face vector space Converted 𝑀= 16 images in the training set
- 27. Result of Eigenfaces Calculation Figure: The selected K eigenfaces of our set of original images
- 28. Represent Each Face Image a Linear Combination of all K Eigenvectors
- 29. 𝛚 𝟏 𝛚 𝟐 𝛚 𝟑 𝛚 𝟒 𝛚 𝟓 𝛚 𝐊⋯ ⋯ ⋯ + 𝜳 (Mean Image) Each face from Training set can be represented a weighted sum of the K Eigenfaces + the Mean face
- 30. 𝛚 𝟏 𝛚 𝟐 𝛚 𝟑 𝛚 𝟒 𝛚 𝟓 𝛚 𝐊⋯ + 𝜳 (Mean Image) The K selected eigenface Each face from Training set can be represented a weighted sum of the K Eigenfaces + the Mean face ……….. Ф𝒊 𝜳 Converted Face vector space 𝑀= 16 images in the training set
- 31. Weight Vector (𝜴𝒊) 𝛚 𝟏 𝛚 𝟐 𝛚 𝟑 𝛚 𝟒 𝛚 𝟓 𝛚 𝐊⋯ + 𝜳 (Mean Image) = 𝜴𝒊 = 𝝎1 𝒊 𝝎2 𝒊 𝝎3 𝒊 . . . 𝝎 𝑲 𝒊 Each face from Training set can be represented a weighted sum of the K Eigenfaces + the Mean face A weight vector 𝛀𝐢 which is the eigenfaces representation of the 𝒊𝒕𝒉 face. We calculated each faces weight vector.
- 32. Recognizing An Unknown Face
- 33. Convert the Input to Face Vector Normalize the Face Vector Project Normalize Face Vector onto the Eigenspace Get the Weight Vector 𝜴 𝒏𝒆𝒘 = 𝝎 𝟏 𝝎 𝟐 𝝎 𝟑 . . . 𝝎 𝑲 Euclidian Distance (E) = (𝛀 𝒏𝒆𝒘 − 𝛀𝒊) If 𝑬 < 𝜽 𝒕 No Unknown Yes Input of a unknown Image Recognized as
- 34. Thank You