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Unformatted text preview: Lecture 15: Eigenfaces CS6670: Computer Vision Noah Snavely Announcements • Final project page up, at – http://www.cs.cornell.edu/courses/cs6670/2009fa/projects/p4/ – One person from each team should submit a proposal (to CMS) by tomorrow at 11:59pm • Project 3: Eigenfaces Skin detection results This same procedure applies in more general circumstances • More than two classes • More than one dimension General classification H. Schneiderman and T.Kanade Example: face detection • Here, X is an image region – dimension = # pixels – each face can be thought of as a point in a high dimensional space H. Schneiderman, T. Kanade. "A Statistical Method for 3D Object Detection Applied to Faces and Cars". IEEE Conference on Computer Vision and Pattern Recognition (CVPR 2000) http://www2.cs.cmu.edu/afs/cs.cmu.edu/user/hws/www/CVPR00.pdf Linear subspaces Classification can be expensive • Must either search (e.g., nearest neighbors) or store large PDF’s Suppose the data points are arranged as above • Idea—fit a line, classifier measures distance to line convert x into v 1 , v 2 coordinates What does the v 2 coordinate measure? What does the v 1 coordinate measure? distance to line use it for classification—near 0 for orange pts position along line use it to specify which orange point it is Dimensionality reduction How to find v 1 and v 2 ? Dimensionality reduction • We can represent the orange points with only their v 1 coordinates – since v 2 coordinates are all essentially 0 • This makes it much cheaper to store and compare points • A bigger deal for higher dimensional problems Linear subspaces Consider the variation along direction v among all of the orange points: What unit vector v minimizes var ? What unit vector v maximizes var ? Solution: v 1 is eigenvector of A with largest eigenvalue v 2 is eigenvector of A with smallest eigenvalue 2 Principal component analysis Suppose each data point is Ndimensional • Same procedure applies: • The eigenvectors of A define a new coordinate system – eigenvector with largest eigenvalue captures the most variation among training vectors x – eigenvector with smallest eigenvalue has least variation • We can compress the data by only using the top few eigenvectors – corresponds to choosing a ―linear subspace‖ » represent points on a line, plane, or ―hyperplane‖ – these eigenvectors are known as the principal components The space of faces An image is a point in a high dimensional space • An N x M intensity image is a point in R NM • We can define vectors in this space as we did in the 2D case + = Dimensionality reduction...
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 '08
 MARSCHNER
 Eigenvalue, eigenvector and eigenspace, eigenface, Visual Object Recognition, Sensory Augmented Visual, Augmented Visual Object

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