1. Eigenfaces and simple face recognition [52 points; including 2 bonus points.] This question is a simplified illustration of using PCA for face recognition. We will use a subset of data from the famous Yale Face dataset. Remark: You will have to perform downsampling of the image by a factor of 4 to turn them into a lower resolution image as a preprocessing (e.g., reduce a picture of size 16-by-16 to 4-by-4). In this question, you can implement your own code or call packages. First, given a set of images for each person, we generate the eigenface using these images. You will treat one picture from the same person as one data point for that person. Note that you will first vectorize each image, which was originally a matrix. Thus, the data matrix (for each person) is a matrix; each row is a vectorized picture. You will find weight vectors to combine the pictures to extract different "eigenfaces" that correspond to that person's pictures' first few principal components. (a) (25 points) Perform analysis on the Yale face dataset for Subject 1 and Subject 2, respectively, using all the images EXCEPT for the two pictures named subject01-test.gif and subject02-test.gif. Plot the first 6 eigenfaces for each subject. When visualizing, please reshape the eigenvectors into proper images. Please explain can you see any patterns in the top 6 eigenfaces? (b) (25 points) Now we will perform a simple face recognition task. Face recognition through PCA is proceeded as follows. Given the test image subject01-test.gif and subject02-test.gif, first downsize by a factor of 4 (as before), and vectorize each image. Take the top eigenfaces of Subject 1 and Subject 2, respectively. Then we calculate the projection residual of the 2 vectorized test images with the vectorized eigenfaces: sij = (test image); - (eigenface, )(eigenface)? (test image); ll; Report all four scores: s;;, i = 1,2, j = 1, 2. Explain how to recognize the faces of the test images using these scores. (c) (Bonus: 2 points) Explain if face recognition can work well and discuss how we can improve it possibly