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Linear Subspace Transforms

Linear Subspace Transforms. PCA, Karhunen-Loeve, Hotelling csc610, 2001. Next: Biological vision. Please look at eye book and pathways readings on www page. Image dependent basis. Other transforms, (Fourier and Cosine) uses a fixed basis.

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Linear Subspace Transforms

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  1. Linear Subspace Transforms PCA, Karhunen-Loeve, Hotelling csc610, 2001

  2. Next:Biological vision • Please look at eye book and pathways readings on www page.

  3. Image dependent basis • Other transforms, (Fourier and Cosine) uses a fixed basis. • Better compression can be achieved by tailoring the basis to the image (pixel) statistics.

  4. Sample data • Let x =[x1, x2,…,xw] be a sequence of random variables (vectors) Example 1: x = Column flattening of an image, so x is a sample over several images. Example 2: x = Column flattening of an image block. x could be sampled from only one image. Example 3: x = optic flow field.

  5. Natural images = stochastic variables • The samplings x =[x1, x2,…,xw] of natural image intensities or optic flows are far from uniform! • Draw x1, x2,…,xw: Localized clouds of points. • Particularly localized for e.g. • Groups of faces or similar objects • Lighting • Small motions represented as intensity changes or optic flow. (Note coupling through optic flow equation)

  6. Dimensionality reduction • Find an ON basis B which “compacts” x • Can write: x = y1B1 + y2B2 +…+ ywBw • But can drop some and only use a subset, i.e. x = y1B1 + y2B2 Result: Fewer y values encode almost all the data in x

  7. Estimated Covariance Matrix Note 1: Can more compactly write: C = X’*X and m = sum(x)/w Note 2: Covariance matrix is both symmetric and real, therefore the matrix can be diagonalized and its eigenvalues will be real

  8. Coordinate Transform • Diagonalize C, ie find B s.t. B’CB = D = diag( ) Note: • C symmetric => B orthogonal • B columns = C eigenvectors

  9. DefinitionKL, PCA, Hotelling transform • The forward coordinate transform y = B’*(x-mx) • Inverse transform x = B*y + mx Note: my = E[y] = 0 zero mean C = E[yy’] = D diagonal cov matrix

  10. Study eigenvalues • Typically quickly decreasing: • Write: x = y1B1 + y2B2 +…+ ym Bm +…+ ywBw • But can drop some and only use a subset, i.e. xm = y1B1 + y2B2 +…+ ym Bm Result: Fewer y values encode almost all the data in x m

  11. Remaining error • Mean square error of the approximation is • Proof: (on blackboard)

  12. Example eigen-images1: Lighting variation

  13. Example eigen-images 2Motions Motions with my arm sampled densely

  14. Human Arm Animation Training sequence Simulated Animation

  15. Modulating Basis Vectors Vectors: 5 and 6 1-6

  16. Practically: C=XX’ quite large, eig(C) impractical. Instead: • C=X’X, [d,V]=eig(C), eigenvectors U=XV • [U,s,V] = svd(X) (since C = XX’ = Us^2U’)

  17. SummaryHotelling,PCA,KLT • Advantages: • Optimal coding in the least square sense • Theoretcially well founded • Image dependent basis: We can interpret variation in our particular set of images. • Disadvantages: • Computationally intensive • Image dependent basis (we have to compute it)

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