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  • 1. ImageCompression
  • 2.
  • 3. Basis
    • A set of linearly independent vectors whose linear combination can be used to express any vector in a given vector space (In our case, the vector space is the 8 x 8 matrix, X). There can be infinitely many bases for a given vector space. So, for representing our image, we are free to choose any basis that is convenient to us. The coefficient matrix will vary accordingly. ( Since X = BC and C = B-1X)
    • 4. B = [ b0 | b1 | b2 | b3 | b4 | b5 | b6 | b7 ] where b0, b1,.., b7 are 8 x 1 linearly independent vectors.
    • 5. A “Good” basis should have more of low frequency vectors or bis (ideal: all ones in the column; imply less variation of pixel values in space) and very few high frequency vectors (alternate +1s and -1s; imply maximum variation of pixel values in space) in order to account for the general smoothness of images.
  • Bases To Choose From
    ‘w’ in Fourier Basis is the nth root of unity for a basis of dimension n x n.
  • 6. Choice of Basis
  • 7. What makes a basis good?
  • 8.
  • 9.
  • 10. Reference
    MIT OCW: Linear Algebra (Gilbert Strang) Lecture 31
    MIT OCW: Linear Algebra (Gilbert Strang) Lecture 26
    http://www.amara.com/IEEEwave/IW_wave_vs_four.html
    http://www.vidyasagar.ac.in/journal/maths/vol13/Art11.pdf
    http://users.rowan.edu/~polikar/WAVELETS
    http://en.wikipedia.org