Notes 20110215 CIS 6320 Image Processing -- Tao - K-LT Transform

From SnOwy - Ed's Wiki Notebook

Karhunen-Loeve Transform

• image compression, object recognition, object alignment, image retrieval
• how variables relate to one another
• noise and dimensionality reduction
• covariance matrix
• covariance -- statistical measure of two random variables
• Cov(x, y) = E[(x-μ_x)(y-μ_y)] = E(xy) - E(x)E(y)
• covariance matrix of random vector (see definition)
• covariance matrix compares all variables with one another
• derived Σ_y is the diagonal of the original matrix
• the Eigen decomposition of a real symmetric matrix
• algorithm
  • covariance
  • Gram-Schmid orthogonal decomposition
    • must know direction of maximum variance
  • Lanczos algorithm
• K-LT -- why this is very useful for image processing ...
• dimensionality reduction -- example
  • requires projecting all points onto a one-dimensional line
• K-LT face recognition
  • make a few assumptions of these images ...
  • more restrictive = better recognition
  • faces must be aligned and greyscale
  • same lighting condition
• each image is a point vector in a high dimensional space
• each column of the matrix contains all the pixels of a single image (concatenate)
• each row is a different pixel location
• eigenface generation
• choose a small set of the eigen vectors to be all the images
• this is based on the principal component analysis -- choose the pixels that matter the most (vary the most)
• choose a small number of significant eigenvalues (2%)
• image compression ...
  • instead of dimensionality reduction, we have to map the transform data back to the input space
  • must save a mean image
• image recognition
  • dimensionality reduction
• general notes
  • finds a very compact representation of an image
  • K-LT -- rotates an object in the eigen space
  • is optimal with respect to projection
  • significant difference between K-LT transform vs orthogonal transforms (i.e. cosime, fourier, wavelet)
    • K-LT is based on the input data
    • sensitive to scales
    • principal components lack intuition (human cannot decide this by inspection)
    • coefficient of objects have no meaning -- combination of many bases