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EigenFaces

EigenFaces. (squared) Variance. A measure of how "spread out" a sequence of numbers are. Covariance matrix. A measure of correlation between data elements. Example: Data set of size n Each data element has 3 fields: Height Weight Birth date. Covariance. [Collect data from class].

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EigenFaces

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  1. EigenFaces

  2. (squared) Variance • A measure of how "spread out" a sequence of numbers are.

  3. Covariance matrix • A measure of correlation between data elements. • Example: • Data set of size n • Each data element has 3 fields: • Height • Weight • Birth date

  4. Covariance • [Collect data from class]

  5. Covariance

  6. Covariance • The diagonals are the variance of that feature • Non-diagonals are a measure of correlation • High-positive == positive correlation • one goes up, other goes up • Low-negative == negative correlation • one goes up, other goes down • Near-zero == no correlation • unrelated • [How high depends on the range of values]

  7. Covariance • You can calculate it with a matrix: • Raw Matrix is a p x q matrix • p features • q samples • Convert to mean-deviation form • Calculate the average sample • Subtract this from all samples. • Multiply MeanDev (a p x q matrix) by its transpose (a q x p matrix) • Multiply by 1/n to get the covariance matrix.

  8. Covariance • [Calculate our covariance matrix]

  9. EigenSystems • An EigenSystem is: A vector (the eigenvector) A scalar λ(the eigenvalue) • Such that: (the zero vector isn't an eigenvector) • In general, not all matrices have eigenvectors.

  10. EigenSystems and PCA • When you calculate the eigen-system of an n x n Covariance matrix you get: • n eigenvectors (each of dimension n) • n matching eigenvalues • The biggest eigen-value "explains" the largest amount of variance in the data set.

  11. Example • Say we have a 2d data set • First eigen-pair (v1 = [0.8, 0.6], λ=800.0) • Second eigen-pair (v2 = [-0.6, 0.8], λ=100.0) • 8x as much variance is along v1 as v2. • v1 and v2 are perpendicular to each other • v1 and v2 define a new set of basis vectors for this data set. v2 v1

  12. Conversions between basis vectors • Let's take one data point… • Let's say it is [-1.5, 0.4] in "world units" • Project it onto v1 and v2 to get the coordinates relative to (v1, v2 unit-length basis vectors) v2 v1 To convert back to "world units":

  13. PCA and compression • Example: • n (the number of features) is high (~100) • Most of the variance is captured by 3 eigen-vectors. • You can throw out the other 97 eigen-vectors. • You can represent most of the data for each sample using just 3 numbers per sample (instead of 100) • For a large data set, this can be huge.

  14. EigenFaces • Collect database images • Subject looking straight ahead, no emotion, neutral lighting. • Crop: • on the top include all of the eyebrows • on the bottom include just to the chin • on the sides, include all of the face. • Size to 32x32, grayscale (a limit of the eigen-solver) • In code, include a way to convert to (and from) a VectorN.

  15. EigenFaces, cont. • Calculate the average image • Just pixel (Vector element) by element.

  16. EigenFaces, cont. • Calculate the Covariance matrix • Calculate the EigenSystem • Keep the eigen-pairs that preserve n% of the data variance (98% or so) • Your Eigen-database is the 32x32 average image and the (here) 8 32x32 eigen-face images.

  17. Eigenfaces, cont. • Represent each of your faces as a q-value vector (q = # of eigenfaces). • Subtract the average and project onto the q eigenfaces • The images I'm showing here are the original image and the 8-value "eigen-coordinates 9.08 187.4 -551.7 -114.4 -328.8 29.2 -371.9 -108.0 1277.0 150.9 -133.6 249.3 338.9 13.14 16.8 3.35

  18. EigenFaces, cont. • (for demonstration of compression) • You can reconstruct a compressed image by: • Start with a copy of the average image, X • Repeat for each eigenface: • Add the eigen-coord * eigenface to X • Here are the reconstructions of the 2 images on the last slide: Original Reconstruction

  19. EigenFaces, cont. • Facial Recognition • Take a novel image (same size as database images) • Using the eigenfaces computed earlier (this novel image is usually NOT part of this computation), compute eigen-coordinates. • Calculate the q-dimensional distance (pythagorean theorem in q-dimensions) between this image and each database image. • The database image with the smallest distance is your best match.

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