This preview shows pages 1–10. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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...
View
Full
Document
This note was uploaded on 09/27/2010 for the course CS 667 at Cornell University (Engineering School).
 '08
 MARSCHNER

Click to edit the document details