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# Image compression

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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