Principal Component Analysis (PCA) is a technique used to simplify complex data sets by identifying patterns in the data and expressing it in such a way to highlight similarities and differences. It works by subtracting the mean from the data, calculating the covariance matrix, and determining the eigenvectors and eigenvalues to form a feature vector representing the data in a lower dimensional space. PCA can be used to represent image data as a one dimensional vector by stacking the pixel rows of an image and applying this analysis to multiple images.

1. Principal Component AnalysisDr. Nidhi Mathur

2. Principal Component Analysis or PCA is a way of identifying patterns in data and expressing data in such a way as to highlight their similarities and differences.PCA is a powerful tool for analyzing data.2PCA - Dr. Nidhi Mathur

3. MethodGet some data,Subtract the mean,Calculate the covariance matrix,Calculate the eigenvectors and eigenvalues of covariance matrix,Choose components and form a feature vector,Derive the new data set.3PCA - Dr. Nidhi Mathur

4. Image RepresentationRows of the pixels in an (NxN) image are placedone after the other to form a one dimensional vector.N x N Image1 x N2 vector4PCA - Dr. Nidhi Mathur

5. If there are M images, thenNow, this is the starting point of PCA Analysis.5PCA - Dr. Nidhi Mathur

6. Let  be an 1 x N 2 vector corresponding to an N x N image.Obtain images I1, I2, …..IM.Represent every Ii as vector i.Compute average vector Subtract the mean vectorCompute the covariance matrix , where is an N 2 x M matrixCompute the eigenvalues ui of ATA6PCA - Dr. Nidhi Mathur

7. Matrix ATA is very large and computation is impractical.Consider AAT matrix.

8. Compute eigenvalues vi of AAT. (AT A and AAT have the same eigenvalues and their eigenvectors are related as: ui = A x vi ) AT A has N2 eigenvectors/eigenvalues

9. AAT has M eigenvectors/eigenvalues7PCA - Dr. Nidhi Mathur