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Unformatted text preview: EE263 S. Lall 2009.12.07.01 Final exam 09-10 Solutions This is a 24-hour take-home exam. You may use any books, lecture-notes or computer programs that you wish, but you may not discuss this exam with anyone until Wednesday Dec 9, after everyone has taken it. The only exception is that you can ask the TAs or Sanjay Lall for clarification. Web page for typos and datafiles. The web page http://junction.stanford.edu/~lall/ee263/final contains any needed files, and it should always contain all known typos; check this before contacting one of us for clarification. Please note that we have tried pretty hard to make the exam unambiguous and clear, so we are unlikely to say much. If you choose to send out an email for clarification, please use the staff email address email@example.com so that you can get reply as soon as possible. Since you have 24 hours, we expect your solutions to be legible, neat, and clear. Do not hand in your rough notes, and please try to simplify your solutions as much as possible. Good Luck! 1. Eigenfaces 9921 Eigenfaces are a set of eigenvectors used in the computer vision problem of human face recognition. A typical eigenface might look like 50 100 150 200 250 300 50 100 150 200 Given a set of independent vectors v i , any other vector can be projected onto the R ( V ), using least squares. Similarly, given an image of a face, we can project it onto a set of independent face images. If this set of face images is a good set, then it is reasonable to expect that any human face image can be reconstructed as a linear combination of these images, with reasonable accuracy. Elements of such a good set of images are called eigenfaces. In other words, eigenfaces serve as a representative sample of human face images i.e. they capture the main characteristics of a human face image. The linear combination of these eigenfaces is called the facespace . Eigenfaces can be used for the purposes of facial recognition. Given a new image, we can project it onto the facespace, and if the error between the actual image and the projection is small, the face is recognized. This method can then be implemented in security systems. To generate a set of eigenfaces, a large set of digitized images of human faces, taken under the same lighting conditions, are normalized to line up the eyes and mouth. They are then all re-sampled at the same pixel resolution. Eigenfaces can be extracted out of the image data by means of a mathematical tool called principal component analysis (PCA). In this problem you will get to do all this. 1 EE263 S. Lall 2009.12.07.01 We provide you 165 images of 15 different people in 11 different lighting conditions and emotions. For example, yalefaces \ subject01.happy is the image of person 1 in happy mood. Each of these image can be represented by a matrix of size 243 by 320 (or a column vector of size 77760 by 1)....
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