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Principal Component Analysis and Applications in Image Analysis

Principal Component Analysis and Applications in Image Analysis. Presented by Judy Jun Feng. Image Computing Group, Department of Computer Science . City University of Hong Kong. Outline. Introduction and History Bivariate or two variables case

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Principal Component Analysis and Applications in Image Analysis

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  1. Principal Component Analysis and Applications in Image Analysis Presented by Judy Jun Feng Image Computing Group, Department of Computer Science City University of Hong Kong

  2. Outline • Introduction and History • Bivariate or two variables case • Mathematical derivation in multivariate case • Graphical representation of PCA • Application in Face Recognition (FR) • Application in Active Shape Models (ASM) • A useful deriving for computation in practice • Conclusion

  3. Introduction and History • Basic idea • One of the multivariate techniques for data analysis • Main purpose is to reduce the dimension of the data set and retain information as much as possible, especially retain information which characterizes the most important variation of the data. • Transforming the original data to a new set of uncorrelated variables (PCs) • History • 1901: Karl Pearson -----First proposed • 1933: Harold Hotelling----General procedures • 1963: Anderson----Theoretical developments of variance of sample PCs • 1963: Rao ----New ideas about interpretation and extension • 1966: Gower ----The links between PCA and other statistic techniques and geometric insights • 1970’s ------Popularly used in the variety of areas • 1989: Esbensen ----Multivariate Image Analysis

  4. PCA in Bivariate Case • Data Set: Length and width of the similar shape objects in images: (XiLXiW ) (i=1..n) • Objective: Find 2 uncorrelated components which are the normalized linear combinations of XL, XW with maximum variance . • Procedures: • Transform the data set to the center: • Find two components: with the constraint: That is

  5. Statistic and Geometric meaning Transform each pair of (XL, XW) to a pair (U1, U2), by: Transform each pair of (U1, U2) to a pair (XL, XW), by:

  6. Principal Components in the multivariate case (1) • Data Set: ** X=(X1, X2, X3…Xp) T are vectors of p-dimensional random variables with covariance matrix C. ** X have been standardized to the same scales • Statistic background: The symmetric matrix C is positive definite with eigenvalues . Matrix formed by the eigenvectors is an orthogonal matrix From the relations the covariance matrix C can be diagonalized to And

  7. Principal Components in the multivariate case (2) • If principal components are chosen to be : Then That means the covariance matrix of U has been diagonalized to with the largest variance So Ui is known as the i-th principal component of the original variables and Ui can be calculated from And X can be reconstructed by the principle components:

  8. Principal Components in the multivariate case (3) • Total variance • Proportion variance • The cumulative proportion is often calculated to find if k components have reproduced the data sufficiently well Reduce the number of variables or dimension from p to k ( k<<p ).

  9. Graphical representation of PCA Euclidean Distance M=6.4 Mahalanobis Distance E=2.56 M=2.8 **The ith principal component lies on the ith longest axes of ellipsoids. **All of the points on the contour of one ellipse or on the surface of one ellipsoid with the orthogonal axes of the principal components, have the same Mahalanobis distance.

  10. Using PCA in the Face Recognition (FR) (1) Raw Faces Faces: complex natural objects with high dimension. Very difficult to develop a computational model Eigen Faces Feature images decomposed from face images Speed, simplicity, insensitivity to small or gradual changes of the faces.

  11. Using PCA in the Face Recognition (FR) (2) • Collect a number of face images as the training set, a few for each person.X --- NN, ----a row face vector of dimension N 2 1. • The eigenvectors and eigenvalues can be found from the covariance matrix C from the face vectors of those images . • Calculate the principal components from linear combinations: • Use principal components to reconstruct the raw image • We call these eigenvectors as "eigenfaces". • The reconstruction can also be viewed as the combinations of eigenfaces weighted by corresponding values of PCs. • In practice, each individual face is actually approximated by only a few k eigenfaces and PCs with the largest eigenvalues. 256256 8 bit 200 images 40 eigenfaces

  12. Using PCA in the Face Recognition (FR) (3) • Define a face class using a group of images of one person • Get a group of k-d PC vectors • Defined the face class centered at the mean of these PC vectors. • Choose a distance threshold to define the maximum allowable distance . • Define a face space using all of the images in the training set • Get the PC vectors of all the face images in the training set • Find the mean and allowable distance to determine if an input image is a face images • Face Recognition A new face is: (1) A known people : if it belongs to a face class. (2) An unknown people: add to training set. (3) Not a face at all: far away from the face space.

  13. Applying PCA in the Active Shape Models (1) Metacarpal shapes in X-ray images Flexible models, or deformable templates, can be used to allow for some degree of variability in the shape of the image objects PCA is often used to find some most important "mode of variation", and helps to obtain a more compressed statistical description of the shapes and their variation

  14. Applying PCA in the Active Shape Models (2) • Labeling and Aligning the shapes training set Each shape has been described by a position vector( A points in 2p-d space ) Mean shape: • Get the most important modes of variation • Find the eigenvectors from the 2p2p covariance matrix • The position vectors can be reproduced by the eigenvectors weighted by the values of PCs, and every eigenvectors accounts for a kind of mode of variation . • K largest eigenvalues and their eigenvectors, the most important modes of variations. • All of the PC vectors in the training set can consist of the "Allowable Shape Domain" . • A possible shape similar to the class of shapes can be expressed by those modes with different weights in the “Allowable Shape Domain” .

  15. A useful deriving for computation in practice • High computer load to find the eigenvectors in image analysis Face Recognition, 256256 N 2 =65,536 And C = 65,536  65,536 • There are only M meaningful eigenvectors ( M is the number of the images in the training data) Suppose V is the eigen vectors of is the eigen vectors of

  16. Conclusion PCA has been proved an effective approach for image recognition, segmentation and other kind of analysis. • Advantages: (1)Extract uncorrelated basic feature sets to describe the properties of a data set. (2)Reduce the dimensionality of the original image space. • Disadvantages: (1) Provide little quantitative information and visualization implications (2) No way to discriminate between variance due to object of interest and variance due to noise or background. • Researchers proposed different schemes and make many improvements for better applying PCA in image analysis.

  17. The End Thank you very much

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