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Linear Subspace Transforms

Linear Subspace Transforms. PCA, Karhunen-Loeve, Hotelling C306, 2000. New idea: Image dependent basis. Previous transforms, (Fourier and Cosine) uses a fixed basis. Better compression can be achieved by tailoring the basis to the image (pixel) statistics. Sample data.

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Linear Subspace Transforms

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  1. Linear Subspace Transforms PCA, Karhunen-Loeve, Hotelling C306, 2000

  2. New idea: Image dependent basis • Previous transforms, (Fourier and Cosine) uses a fixed basis. • Better compression can be achieved by tailoring the basis to the image (pixel) statistics.

  3. Sample data • Let x =[x1, x2,…,xw] be a sequence of random variables (vectors) Exampe 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 I could be sampled from only one image.

  4. 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

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

  6. 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

  7. Dimensionality reduction • 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

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

  9. SummaryHotelling,PCA,KLT • Advantages: • Optimal coding in the least square sense • Theoretcially well founded • Disadvantages: • Computationally intensive • Image dependent basis

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